Geographic routing in Ad-Hoc and Sensor Networks
188
Geographic Routing in Ad-Hoc and Sensor Networks An algorithmic approach From ”An algorithmic approach to Geographic Routing in Ad Hoc and Sensor Networks”, by Kuhn et al. (2008) Alessandro Lenzi ALP Seminars December 4, 2013 Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 1 / 37
-
Upload
alessandro-lenzi -
Category
Engineering
-
view
149 -
download
1
Transcript of Geographic routing in Ad-Hoc and Sensor Networks
Geographic Routing in Ad-Hoc and Sensor NetworksAn algorithmic approach
From ”An algorithmic approach to Geographic Routing in Ad Hoc andSensor Networks”, by Kuhn et al. (2008)
Alessandro Lenzi
ALP Seminars
December 4, 2013
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 1 / 37
1 Introduction
2 Model
3 Greedy Routing
4 Graph Planarization
5 Routing with faces
6 GOAFR+
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 2 / 37
Introduction
Layout
1 Introduction
2 Model
3 Greedy Routing
4 Graph Planarization
5 Routing with faces
6 GOAFR+
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 3 / 37
Introduction
Scenario
An ad-hoc network is a wireless network created in a scenario where noprevious infrastructure is available prior to deployment of the networkitself.
Typical scenarios are emergency situations and disaster recovery.
Their aim is to provide a survivable, efficient and dynamiccommunication network.
Every node in the network also performs routing operations.
A sensor network is a particular kind of ad-hoc network in which devicesare also equipped with sensing capabilities, used to monitor and to collectdata.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 4 / 37
Introduction
Scenario
An ad-hoc network is a wireless network created in a scenario where noprevious infrastructure is available prior to deployment of the networkitself.
Typical scenarios are emergency situations and disaster recovery.
Their aim is to provide a survivable, efficient and dynamiccommunication network.
Every node in the network also performs routing operations.
A sensor network is a particular kind of ad-hoc network in which devicesare also equipped with sensing capabilities, used to monitor and to collectdata.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 4 / 37
Introduction
Scenario
An ad-hoc network is a wireless network created in a scenario where noprevious infrastructure is available prior to deployment of the networkitself.
Typical scenarios are emergency situations and disaster recovery.
Their aim is to provide a survivable, efficient and dynamiccommunication network.
Every node in the network also performs routing operations.
A sensor network is a particular kind of ad-hoc network in which devicesare also equipped with sensing capabilities, used to monitor and to collectdata.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 4 / 37
Introduction
Scenario
An ad-hoc network is a wireless network created in a scenario where noprevious infrastructure is available prior to deployment of the networkitself.
Typical scenarios are emergency situations and disaster recovery.
Their aim is to provide a survivable, efficient and dynamiccommunication network.
Every node in the network also performs routing operations.
A sensor network is a particular kind of ad-hoc network in which devicesare also equipped with sensing capabilities, used to monitor and to collectdata.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 4 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.
Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymore
Nodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Problems
Devices are battery powered, meaning thatenergy is an issue!
Radio transmission is one of the biggestpower-consumption sources.Routing overhead should be avoided asmuch as possible.
Nodes are typically cheap and lightweight.
Possibly, routing tables are too big to be maintained in memory
The network is not stable
Several transmission impairment are present. Nodes might move andthus be not reachable anymoreNodes often fail.
Traditional routing protocols generally fail to satisfy the requirements ofad-hoc networks!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 5 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Introduction
Geographic Ad Hoc Routing
Idea!
Exploit ”physical” network topology of the network to establish routes.
GPS devices nowadays are very small.
Position can also be inferred without using satellites, as shown in[RRP+03].
Possibly minimize power consumption by using a ”real” distancemeasure and reducing topology information.
Assumptions
1 Every node in the system knows its geographic position and theposition of all its neighbors.
2 Position of the destination node is known.
3 Nodes move slowly compared to topology changes
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 6 / 37
Model
Layout
1 Introduction
2 Model
3 Greedy Routing
4 Graph Planarization
5 Routing with faces
6 GOAFR+
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 7 / 37
Model
Geographic Ad Hoc Routing Algorithm
Definition: Let G = (V, E) be a Euclidean Graph. A geographic ad hocrouting algorithm A transmits a message from a source s ∈ V to adestination d ∈ V sending packets over the edges of G while respectingthe following conditions:
1 All nodes v ∈ V know their position in a coordinate system and alsotheir neighbor’s location.
2 A packet can store control information of size O(log n) bit, thus for aconstant number of nodes.
3 No node is allowed to maintain other routing information thantemporary storage of packets to be forwarded.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 8 / 37
Model
Geographic Ad Hoc Routing Algorithm
Definition: Let G = (V, E) be a Euclidean Graph. A geographic ad hocrouting algorithm A transmits a message from a source s ∈ V to adestination d ∈ V sending packets over the edges of G while respectingthe following conditions:
1 All nodes v ∈ V know their position in a coordinate system and alsotheir neighbor’s location.
2 A packet can store control information of size O(log n) bit, thus for aconstant number of nodes.
3 No node is allowed to maintain other routing information thantemporary storage of packets to be forwarded.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 8 / 37
Model
Geographic Ad Hoc Routing Algorithm
Definition: Let G = (V, E) be a Euclidean Graph. A geographic ad hocrouting algorithm A transmits a message from a source s ∈ V to adestination d ∈ V sending packets over the edges of G while respectingthe following conditions:
1 All nodes v ∈ V know their position in a coordinate system and alsotheir neighbor’s location.
2 A packet can store control information of size O(log n) bit, thus for aconstant number of nodes.
3 No node is allowed to maintain other routing information thantemporary storage of packets to be forwarded.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 8 / 37
Model
Geographic Ad Hoc Routing Algorithm
Definition: Let G = (V, E) be a Euclidean Graph. A geographic ad hocrouting algorithm A transmits a message from a source s ∈ V to adestination d ∈ V sending packets over the edges of G while respectingthe following conditions:
1 All nodes v ∈ V know their position in a coordinate system and alsotheir neighbor’s location.
2 A packet can store control information of size O(log n) bit, thus for aconstant number of nodes.
3 No node is allowed to maintain other routing information thantemporary storage of packets to be forwarded.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 8 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange. All edges are bidirectional!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange. All edges are bidirectional!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange. All edges are bidirectional!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange. All edges are bidirectional!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange. All edges are bidirectional!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange. All edges are bidirectional!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange.
All edges are bidirectional!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (1)
As usual, the communication network is abstracted as a Euclidean graphwhere nodes are devices and edges are network connections.
Def.: Euclidean graph
A graph G = (V ,E ) is called Euclidean whenever:
1 ∀v ∈ V , v is a point in the Euclidean plane
2 ∀e = (u, v) ∈ E ,weight(e) = |uv |
The graph considered is forced to be a unit disk graph.
Def.: Unit Disk Graph (UDG)
Let V ⊂ R2 be a set of points in the two dimensional plane. The graphwith edges between all nodes with distance at most 1 is called unit diskgraph of V.
A UDG is a flat environment in which all devices have equal transmissionrange. All edges are bidirectional!Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 9 / 37
Model
The model (2)
Def.: Civilized Graph
A graph is civilized if, given any pair of nodes, the distance between thetwo is no less than a certain constant d0 > 0
Def.: Cost Function
We define as cost function a generic c : (0, 1] 7→ R+ such that∀d1 < d2, c(d1) ≤ c(d2)
Notice that this definition is generic enough to model several metrics:
1 Hop distance: ∀d , c(d) := 1
2 Euclidean distance: ∀d , c(d) := d
3 Energy distance: ∀d , c(d) := dα with α ≥ 2
4 Linear combinations of the above
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 10 / 37
Model
The model (2)
Def.: Civilized Graph
A graph is civilized if, given any pair of nodes, the distance between thetwo is no less than a certain constant d0 > 0
Def.: Cost Function
We define as cost function a generic c : (0, 1] 7→ R+ such that∀d1 < d2, c(d1) ≤ c(d2)
Notice that this definition is generic enough to model several metrics:
1 Hop distance: ∀d , c(d) := 1
2 Euclidean distance: ∀d , c(d) := d
3 Energy distance: ∀d , c(d) := dα with α ≥ 2
4 Linear combinations of the above
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 10 / 37
Model
The model (2)
Def.: Civilized Graph
A graph is civilized if, given any pair of nodes, the distance between thetwo is no less than a certain constant d0 > 0
Def.: Cost Function
We define as cost function a generic c : (0, 1] 7→ R+ such that∀d1 < d2, c(d1) ≤ c(d2)
Notice that this definition is generic enough to model several metrics:
1 Hop distance: ∀d , c(d) := 1
2 Euclidean distance: ∀d , c(d) := d
3 Energy distance: ∀d , c(d) := dα with α ≥ 2
4 Linear combinations of the above
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 10 / 37
Model
The model (2)
Def.: Civilized Graph
A graph is civilized if, given any pair of nodes, the distance between thetwo is no less than a certain constant d0 > 0
Def.: Cost Function
We define as cost function a generic c : (0, 1] 7→ R+ such that∀d1 < d2, c(d1) ≤ c(d2)
Notice that this definition is generic enough to model several metrics:
1 Hop distance: ∀d , c(d) := 1
2 Euclidean distance: ∀d , c(d) := d
3 Energy distance: ∀d , c(d) := dα with α ≥ 2
4 Linear combinations of the above
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 10 / 37
Model
The model (2)
Def.: Civilized Graph
A graph is civilized if, given any pair of nodes, the distance between thetwo is no less than a certain constant d0 > 0
Def.: Cost Function
We define as cost function a generic c : (0, 1] 7→ R+ such that∀d1 < d2, c(d1) ≤ c(d2)
Notice that this definition is generic enough to model several metrics:
1 Hop distance: ∀d , c(d) := 1
2 Euclidean distance: ∀d , c(d) := d
3 Energy distance: ∀d , c(d) := dα with α ≥ 2
4 Linear combinations of the above
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 10 / 37
Model
The model (2)
Def.: Civilized Graph
A graph is civilized if, given any pair of nodes, the distance between thetwo is no less than a certain constant d0 > 0
Def.: Cost Function
We define as cost function a generic c : (0, 1] 7→ R+ such that∀d1 < d2, c(d1) ≤ c(d2)
Notice that this definition is generic enough to model several metrics:
1 Hop distance: ∀d , c(d) := 1
2 Euclidean distance: ∀d , c(d) := d
3 Energy distance: ∀d , c(d) := dα with α ≥ 2
4 Linear combinations of the above
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 10 / 37
Greedy Routing
Layout
1 Introduction
2 Model
3 Greedy Routing
4 Graph Planarization
5 Routing with faces
6 GOAFR+
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 11 / 37
Greedy Routing
A straightforward approach...
Greedy routing is a very straightforward approach for geographic routingalgorithms.
while TRUE domsg ← receive(shared channel)if msg .dest 6= my pos then
n←MinDistN(Neighbors,msg .dest)if Distance(n,msg .dest) > Distance(my pos,msg .dest) then
StopGreedy()end ifsend(n,msg)
elseStoreMessage(msg)
end ifend while
StopGreedy is used when no neighbor nearer to the destination is found
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 12 / 37
Greedy Routing
Greedy forwarding failures
In some topologies, reaching the destination might require movingtemporarily farther from the destination in order to reach it.
Local Minimum
The forwarding procedureworks until a localminimum is found
Then the lack of globalknowledge makes thealgorithm stop.
This is caused by a void, around which we shall route to find thedestination
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 13 / 37
Greedy Routing
Greedy forwarding failures
In some topologies, reaching the destination might require movingtemporarily farther from the destination in order to reach it.
Local Minimum
The forwarding procedureworks until a localminimum is found
Then the lack of globalknowledge makes thealgorithm stop.
This is caused by a void, around which we shall route to find thedestination
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 13 / 37
Greedy Routing
Greedy forwarding failures
In some topologies, reaching the destination might require movingtemporarily farther from the destination in order to reach it.
Local Minimum
The forwarding procedureworks until a localminimum is found
Then the lack of globalknowledge makes thealgorithm stop.
This is caused by a void, around which we shall route to find thedestination
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 13 / 37
Greedy Routing
Greedy forwarding failures
In some topologies, reaching the destination might require movingtemporarily farther from the destination in order to reach it.
Local Minimum
The forwarding procedureworks until a localminimum is found
Then the lack of globalknowledge makes thealgorithm stop.
This is caused by a void, around which we shall route to find thedestination
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 13 / 37
Greedy Routing
Greedy forwarding failures
In some topologies, reaching the destination might require movingtemporarily farther from the destination in order to reach it.
Local Minimum
The forwarding procedureworks until a localminimum is found
Then the lack of globalknowledge makes thealgorithm stop.
This is caused by a void, around which we shall route to find thedestination
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 13 / 37
Greedy Routing
Greedy forwarding failures
In some topologies, reaching the destination might require movingtemporarily farther from the destination in order to reach it.
Local Minimum
The forwarding procedureworks until a localminimum is found
Then the lack of globalknowledge makes thealgorithm stop.
This is caused by a void, around which we shall route to find thedestination
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 13 / 37
Greedy Routing
Greedy forwarding failures
In some topologies, reaching the destination might require movingtemporarily farther from the destination in order to reach it.
Local Minimum
The forwarding procedureworks until a localminimum is found
Then the lack of globalknowledge makes thealgorithm stop.
This is caused by a void, around which we shall route to find thedestination
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 13 / 37
Greedy Routing
Dead Ends
Similar to the localminimum case, exceptthere’s no alternative: thepacket has to follow thereverse route.
Going back and avoidingthe dead end could solvethe problem
However the strict amount of information that can be stored on packetsdo not allow to perform backtracking to recover from this situation
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 14 / 37
Greedy Routing
Dead Ends
Similar to the localminimum case, exceptthere’s no alternative: thepacket has to follow thereverse route.
Going back and avoidingthe dead end could solvethe problem
However the strict amount of information that can be stored on packetsdo not allow to perform backtracking to recover from this situation
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 14 / 37
Greedy Routing
Dead Ends
Similar to the localminimum case, exceptthere’s no alternative: thepacket has to follow thereverse route.
Going back and avoidingthe dead end could solvethe problem
However the strict amount of information that can be stored on packetsdo not allow to perform backtracking to recover from this situation
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 14 / 37
Greedy Routing
Efficiency of Greedy Routing
If Greedy Routing reaches the destination t starting from s - i.e. it doesn’toccur in any local minimum - it does it with a cost O(d2) where d = |st|is the euclidean distance between source s and destination t.
Sketch of proof1
Let p = v1, ..., vk be the path followed during Greedy Routing. Noticethat as the index of v increases, the distance between vi and tdecreases.All vi belong to a circle centered in t and with radius d .
Because of the definition of the Greedy Algorithm,∀vi ∈ p,∀h ≥ 2, vi+h /∈ Neighbors(vi ). This means that givenpodd = v1, v3, ..., v2dk/2e−1 their distance is at least 1.
On the circle of radius d at most d2 points like those in podd mayexist.
1The complete proof can be found on [GGH+01]Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 15 / 37
Greedy Routing
Efficiency of Greedy Routing
If Greedy Routing reaches the destination t starting from s - i.e. it doesn’toccur in any local minimum - it does it with a cost O(d2) where d = |st|is the euclidean distance between source s and destination t.Sketch of proof1
Let p = v1, ..., vk be the path followed during Greedy Routing. Noticethat as the index of v increases, the distance between vi and tdecreases.
All vi belong to a circle centered in t and with radius d .
Because of the definition of the Greedy Algorithm,∀vi ∈ p,∀h ≥ 2, vi+h /∈ Neighbors(vi ). This means that givenpodd = v1, v3, ..., v2dk/2e−1 their distance is at least 1.
On the circle of radius d at most d2 points like those in podd mayexist.
1The complete proof can be found on [GGH+01]Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 15 / 37
Greedy Routing
Efficiency of Greedy Routing
If Greedy Routing reaches the destination t starting from s - i.e. it doesn’toccur in any local minimum - it does it with a cost O(d2) where d = |st|is the euclidean distance between source s and destination t.Sketch of proof1
Let p = v1, ..., vk be the path followed during Greedy Routing. Noticethat as the index of v increases, the distance between vi and tdecreases.
All vi belong to a circle centered in t and with radius d .
Because of the definition of the Greedy Algorithm,∀vi ∈ p,∀h ≥ 2, vi+h /∈ Neighbors(vi ). This means that givenpodd = v1, v3, ..., v2dk/2e−1 their distance is at least 1.
On the circle of radius d at most d2 points like those in podd mayexist.
1The complete proof can be found on [GGH+01]Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 15 / 37
Greedy Routing
Efficiency of Greedy Routing
If Greedy Routing reaches the destination t starting from s - i.e. it doesn’toccur in any local minimum - it does it with a cost O(d2) where d = |st|is the euclidean distance between source s and destination t.Sketch of proof1
Let p = v1, ..., vk be the path followed during Greedy Routing. Noticethat as the index of v increases, the distance between vi and tdecreases.All vi belong to a circle centered in t and with radius d .
Because of the definition of the Greedy Algorithm,∀vi ∈ p,∀h ≥ 2, vi+h /∈ Neighbors(vi ). This means that givenpodd = v1, v3, ..., v2dk/2e−1 their distance is at least 1.
On the circle of radius d at most d2 points like those in podd mayexist.
1The complete proof can be found on [GGH+01]Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 15 / 37
Greedy Routing
Efficiency of Greedy Routing
If Greedy Routing reaches the destination t starting from s - i.e. it doesn’toccur in any local minimum - it does it with a cost O(d2) where d = |st|is the euclidean distance between source s and destination t.Sketch of proof1
Let p = v1, ..., vk be the path followed during Greedy Routing. Noticethat as the index of v increases, the distance between vi and tdecreases.All vi belong to a circle centered in t and with radius d .
Because of the definition of the Greedy Algorithm,∀vi ∈ p,∀h ≥ 2, vi+h /∈ Neighbors(vi ). This means that givenpodd = v1, v3, ..., v2dk/2e−1 their distance is at least 1.
On the circle of radius d at most d2 points like those in podd mayexist.
1The complete proof can be found on [GGH+01]Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 15 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.
Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.
Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.
Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].
This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
How to solve greedy problems?
Flooding guarantees that, if the graph is connected, destination isreached.
Waste of bandwidth and of power.Definitely not a good choice!
Follow the perimeter: when greedy routing cannot go on, there’sprobably a void.
Use the right hand rule, like in GPSR [KK00].This leads to degenerate tours in case of crossing edges, but..
Degenerate Tour
The first message is sent on greedymode
Then the message is forwardedexploiting right hand rule
We won’t explore the perimeter!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 16 / 37
Greedy Routing
Planar Graphs
We want to avoid edge crossing, i.e. operate on a planar graph.
Def. Planar Graph
A graph G = (V, E) is planar if can be drawn on a plain so that no edgescross, except in the vertices.
A planar graph divides the plane in non overlapping regions, calledfaces.
The edges dividing these regions are called boundary of a region.
We could force the routing algorithm to follow the boundaries toescape a void!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 17 / 37
Greedy Routing
Planar Graphs
We want to avoid edge crossing, i.e. operate on a planar graph.
Def. Planar Graph
A graph G = (V, E) is planar if can be drawn on a plain so that no edgescross, except in the vertices.
A planar graph divides the plane in non overlapping regions, calledfaces.
The edges dividing these regions are called boundary of a region.
We could force the routing algorithm to follow the boundaries toescape a void!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 17 / 37
Greedy Routing
Planar Graphs
We want to avoid edge crossing, i.e. operate on a planar graph.
Def. Planar Graph
A graph G = (V, E) is planar if can be drawn on a plain so that no edgescross, except in the vertices.
A planar graph divides the plane in non overlapping regions, calledfaces.
The edges dividing these regions are called boundary of a region.
We could force the routing algorithm to follow the boundaries toescape a void!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 17 / 37
Greedy Routing
Planar Graphs
We want to avoid edge crossing, i.e. operate on a planar graph.
Def. Planar Graph
A graph G = (V, E) is planar if can be drawn on a plain so that no edgescross, except in the vertices.
A planar graph divides the plane in non overlapping regions, calledfaces.
The edges dividing these regions are called boundary of a region.
We could force the routing algorithm to follow the boundaries toescape a void!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 17 / 37
Greedy Routing
Planar Graphs
We want to avoid edge crossing, i.e. operate on a planar graph.
Def. Planar Graph
A graph G = (V, E) is planar if can be drawn on a plain so that no edgescross, except in the vertices.
A planar graph divides the plane in non overlapping regions, calledfaces.
The edges dividing these regions are called boundary of a region.
We could force the routing algorithm to follow the boundaries toescape a void!
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 17 / 37
Graph Planarization
Layout
1 Introduction
2 Model
3 Greedy Routing
4 Graph Planarization
5 Routing with faces
6 GOAFR+
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 18 / 37
Graph Planarization
Graph Planarization with Gabriel Graph
Def. Gabriel Graph (for R2)
A graph G = (V ,E ) is called Gabriel graph iffor any u, v ∈ V , the edge (u, v) ∈ E if and onlyif the circle with uv as diameter doesn’t containany w ∈ V except u and v.
The Gabriel graph GGG of a UDG G can be computed locally byinspection of neighbors.The Gabriel graph GGG obtained from the UDG G is alwaysconnected if G is connected.The construction of GGG preserves an energy-minimal path betweenany pair of network nodes, given that the graph is civilized.
GGG can be either pre-calculated or calculated on the fly when there’sneed.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 19 / 37
Graph Planarization
Graph Planarization with Gabriel Graph
Def. Gabriel Graph (for R2)
A graph G = (V ,E ) is called Gabriel graph iffor any u, v ∈ V , the edge (u, v) ∈ E if and onlyif the circle with uv as diameter doesn’t containany w ∈ V except u and v.
The Gabriel graph GGG of a UDG G can be computed locally byinspection of neighbors.The Gabriel graph GGG obtained from the UDG G is alwaysconnected if G is connected.The construction of GGG preserves an energy-minimal path betweenany pair of network nodes, given that the graph is civilized.
GGG can be either pre-calculated or calculated on the fly when there’sneed.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 19 / 37
Graph Planarization
Graph Planarization with Gabriel Graph
Def. Gabriel Graph (for R2)
A graph G = (V ,E ) is called Gabriel graph iffor any u, v ∈ V , the edge (u, v) ∈ E if and onlyif the circle with uv as diameter doesn’t containany w ∈ V except u and v.
The Gabriel graph GGG of a UDG G can be computed locally byinspection of neighbors.The Gabriel graph GGG obtained from the UDG G is alwaysconnected if G is connected.The construction of GGG preserves an energy-minimal path betweenany pair of network nodes, given that the graph is civilized.
GGG can be either pre-calculated or calculated on the fly when there’sneed.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 19 / 37
Graph Planarization
Graph Planarization with Gabriel Graph
Def. Gabriel Graph (for R2)
A graph G = (V ,E ) is called Gabriel graph iffor any u, v ∈ V , the edge (u, v) ∈ E if and onlyif the circle with uv as diameter doesn’t containany w ∈ V except u and v.
The Gabriel graph GGG of a UDG G can be computed locally byinspection of neighbors.
The Gabriel graph GGG obtained from the UDG G is alwaysconnected if G is connected.The construction of GGG preserves an energy-minimal path betweenany pair of network nodes, given that the graph is civilized.
GGG can be either pre-calculated or calculated on the fly when there’sneed.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 19 / 37
Graph Planarization
Graph Planarization with Gabriel Graph
Def. Gabriel Graph (for R2)
A graph G = (V ,E ) is called Gabriel graph iffor any u, v ∈ V , the edge (u, v) ∈ E if and onlyif the circle with uv as diameter doesn’t containany w ∈ V except u and v.
The Gabriel graph GGG of a UDG G can be computed locally byinspection of neighbors.The Gabriel graph GGG obtained from the UDG G is alwaysconnected if G is connected.
The construction of GGG preserves an energy-minimal path betweenany pair of network nodes, given that the graph is civilized.
GGG can be either pre-calculated or calculated on the fly when there’sneed.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 19 / 37
Graph Planarization
Graph Planarization with Gabriel Graph
Def. Gabriel Graph (for R2)
A graph G = (V ,E ) is called Gabriel graph iffor any u, v ∈ V , the edge (u, v) ∈ E if and onlyif the circle with uv as diameter doesn’t containany w ∈ V except u and v.
The Gabriel graph GGG of a UDG G can be computed locally byinspection of neighbors.The Gabriel graph GGG obtained from the UDG G is alwaysconnected if G is connected.The construction of GGG preserves an energy-minimal path betweenany pair of network nodes, given that the graph is civilized.
GGG can be either pre-calculated or calculated on the fly when there’sneed.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 19 / 37
Graph Planarization
Graph Planarization with Gabriel Graph
Def. Gabriel Graph (for R2)
A graph G = (V ,E ) is called Gabriel graph iffor any u, v ∈ V , the edge (u, v) ∈ E if and onlyif the circle with uv as diameter doesn’t containany w ∈ V except u and v.
The Gabriel graph GGG of a UDG G can be computed locally byinspection of neighbors.The Gabriel graph GGG obtained from the UDG G is alwaysconnected if G is connected.The construction of GGG preserves an energy-minimal path betweenany pair of network nodes, given that the graph is civilized.
GGG can be either pre-calculated or calculated on the fly when there’sneed.Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 19 / 37
Routing with faces
Layout
1 Introduction
2 Model
3 Greedy Routing
4 Graph Planarization
5 Routing with faces
6 GOAFR+
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 20 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.
Keep track of all points in which the boundaries cross the line stbetween source s and destination t.
Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.
Face Routing always finds the destination in O(n) steps!
The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.
Keep track of all points in which the boundaries cross the line stbetween source s and destination t.
Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.
Face Routing always finds the destination in O(n) steps!
The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.
Keep track of all points in which the boundaries cross the line stbetween source s and destination t.
Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.
Face Routing always finds the destination in O(n) steps!
The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.
Keep track of all points in which the boundaries cross the line stbetween source s and destination t.
Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.
Face Routing always finds the destination in O(n) steps!
The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.
Keep track of all points in which the boundaries cross the line stbetween source s and destination t.
Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.
Face Routing always finds the destination in O(n) steps!
The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.Keep track of all points in which the boundaries cross the line stbetween source s and destination t.Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.Face Routing always finds the destination in O(n) steps!The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.Keep track of all points in which the boundaries cross the line stbetween source s and destination t.Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.Face Routing always finds the destination in O(n) steps!The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.Keep track of all points in which the boundaries cross the line stbetween source s and destination t.Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.Face Routing always finds the destination in O(n) steps!The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.Keep track of all points in which the boundaries cross the line stbetween source s and destination t.Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.Face Routing always finds the destination in O(n) steps!The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.Keep track of all points in which the boundaries cross the line stbetween source s and destination t.Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.Face Routing always finds the destination in O(n) steps!The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Face Routing grants to always reach the destination
Explore face boundaries employing some criteria - e.g.: right handrule.
Keep track of all points in which the boundaries cross the line stbetween source s and destination t.
Select the point p of the face intersection which is nearer to thedestination, then switch to next face closer to t.
Face Routing always finds the destination in O(n) steps!
The problem is that the complete boundary of a face has to beexplored.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 21 / 37
Routing with faces
Other Face Routing
Other Face Routing (OFR)
Idea: similar to Face Routing, but instead of changing face to the pointnearest to the destination in which st crosses the boundary, switch in theexplored point nearer to the destination!
1. Start at s, selecting the face F containing st2. Explore F with the right hand rule and find p, the point on F ’s edge whosedistance from t is minimal3. Move back to p4. From p switch face to the one incident to pt. Go to 2 until t is reached.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 22 / 37
Routing with faces
Other Face Routing
Other Face Routing (OFR)
Idea: similar to Face Routing, but instead of changing face to the pointnearest to the destination in which st crosses the boundary, switch in theexplored point nearer to the destination!
1. Start at s, selecting the face F containing st2. Explore F with the right hand rule and find p, the point on F ’s edge whosedistance from t is minimal3. Move back to p4. From p switch face to the one incident to pt. Go to 2 until t is reached.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 22 / 37
Routing with faces
Other Face Routing
Other Face Routing (OFR)
Idea: similar to Face Routing, but instead of changing face to the pointnearest to the destination in which st crosses the boundary, switch in theexplored point nearer to the destination!
1. Start at s, selecting the face F containing st2. Explore F with the right hand rule and find p, the point on F ’s edge whosedistance from t is minimal3. Move back to p4. From p switch face to the one incident to pt. Go to 2 until t is reached.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 22 / 37
Routing with faces
Other Face Routing
Other Face Routing (OFR)
Idea: similar to Face Routing, but instead of changing face to the pointnearest to the destination in which st crosses the boundary, switch in theexplored point nearer to the destination!
1. Start at s, selecting the face F containing st2. Explore F with the right hand rule and find p, the point on F ’s edge whosedistance from t is minimal3. Move back to p4. From p switch face to the one incident to pt. Go to 2 until t is reached.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 22 / 37
Routing with faces
Other Face Routing
Other Face Routing (OFR)
Idea: similar to Face Routing, but instead of changing face to the pointnearest to the destination in which st crosses the boundary, switch in theexplored point nearer to the destination!
1. Start at s, selecting the face F containing st2. Explore F with the right hand rule and find p, the point on F ’s edge whosedistance from t is minimal3. Move back to p4. From p switch face to the one incident to pt. Go to 2 until t is reached.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 22 / 37
Routing with faces
Other Face Routing
Other Face Routing (OFR)
Idea: similar to Face Routing, but instead of changing face to the pointnearest to the destination in which st crosses the boundary, switch in theexplored point nearer to the destination!
1. Start at s, selecting the face F containing st2. Explore F with the right hand rule and find p, the point on F ’s edge whosedistance from t is minimal3. Move back to p4. From p switch face to the one incident to pt. Go to 2 until t is reached.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 22 / 37
Routing with faces
Other Face Routing
Other Face Routing (OFR)
Idea: similar to Face Routing, but instead of changing face to the pointnearest to the destination in which st crosses the boundary, switch in theexplored point nearer to the destination!
1. Start at s, selecting the face F containing st2. Explore F with the right hand rule and find p, the point on F ’s edge whosedistance from t is minimal3. Move back to p4. From p switch face to the one incident to pt. Go to 2 until t is reached.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 22 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.
(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
Lemma: OFR requires O(n) messages, where n = |V |. OFR always finds tif it is connected to s, otherwise it recognizes a disconnection.(Sketch of) Proof:
1 The switch between two faces always happens at the point of the face whichis closest to t.
2 The face selected next will always contains points which are nearer to t, sono face is visited twice.
3 Let p0, p1, ..., pt be the trace of OFR’s execution. By definition, pi is thepoint on the i-th face Fi which is nearer to t.
4 Since no face is visited more than once, ∀i > j : |pi t| < |pj t| - i.e. distancefrom t is decreasing on the trace.
5 OFR will arrive at t or stop whenever there’s i such that pi = pi+1
6 Moreover, each face can be visited at most once, meaning that each edgecan be used at most four times. (cont..)
OFR takes O(m) steps to find t or terminate
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 23 / 37
Routing with faces
It is possible to bound the number of edges to the number of nodes in thegraph.
Being the graph planar and civilized, it is possible to prove thatm ≤ 3n − 6.Thus OFR terminates in O(n) steps.Be aware that when applying OFR on a Gabriel Graph, OFR can besimplified so that the switching happens in the node nearer to t. Thisdoesn’t affect complexity.
Is O(n) a good complexity?
O(n) can be huge with respect to the optimal path!The problem is that in worst case we can find huge faces and explore themeven if there’s no need.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 24 / 37
Routing with faces
It is possible to bound the number of edges to the number of nodes in thegraph.Being the graph planar and civilized, it is possible to prove thatm ≤ 3n − 6.
Thus OFR terminates in O(n) steps.Be aware that when applying OFR on a Gabriel Graph, OFR can besimplified so that the switching happens in the node nearer to t. Thisdoesn’t affect complexity.
Is O(n) a good complexity?
O(n) can be huge with respect to the optimal path!The problem is that in worst case we can find huge faces and explore themeven if there’s no need.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 24 / 37
Routing with faces
It is possible to bound the number of edges to the number of nodes in thegraph.Being the graph planar and civilized, it is possible to prove thatm ≤ 3n − 6.Thus OFR terminates in O(n) steps.
Be aware that when applying OFR on a Gabriel Graph, OFR can besimplified so that the switching happens in the node nearer to t. Thisdoesn’t affect complexity.
Is O(n) a good complexity?
O(n) can be huge with respect to the optimal path!The problem is that in worst case we can find huge faces and explore themeven if there’s no need.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 24 / 37
Routing with faces
It is possible to bound the number of edges to the number of nodes in thegraph.Being the graph planar and civilized, it is possible to prove thatm ≤ 3n − 6.Thus OFR terminates in O(n) steps.Be aware that when applying OFR on a Gabriel Graph, OFR can besimplified so that the switching happens in the node nearer to t.
Thisdoesn’t affect complexity.
Is O(n) a good complexity?
O(n) can be huge with respect to the optimal path!The problem is that in worst case we can find huge faces and explore themeven if there’s no need.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 24 / 37
Routing with faces
It is possible to bound the number of edges to the number of nodes in thegraph.Being the graph planar and civilized, it is possible to prove thatm ≤ 3n − 6.Thus OFR terminates in O(n) steps.Be aware that when applying OFR on a Gabriel Graph, OFR can besimplified so that the switching happens in the node nearer to t. Thisdoesn’t affect complexity.
Is O(n) a good complexity?
O(n) can be huge with respect to the optimal path!The problem is that in worst case we can find huge faces and explore themeven if there’s no need.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 24 / 37
Routing with faces
It is possible to bound the number of edges to the number of nodes in thegraph.Being the graph planar and civilized, it is possible to prove thatm ≤ 3n − 6.Thus OFR terminates in O(n) steps.Be aware that when applying OFR on a Gabriel Graph, OFR can besimplified so that the switching happens in the node nearer to t. Thisdoesn’t affect complexity.
Is O(n) a good complexity?
O(n) can be huge with respect to the optimal path!
The problem is that in worst case we can find huge faces and explore themeven if there’s no need.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 24 / 37
Routing with faces
It is possible to bound the number of edges to the number of nodes in thegraph.Being the graph planar and civilized, it is possible to prove thatm ≤ 3n − 6.Thus OFR terminates in O(n) steps.Be aware that when applying OFR on a Gabriel Graph, OFR can besimplified so that the switching happens in the node nearer to t. Thisdoesn’t affect complexity.
Is O(n) a good complexity?
O(n) can be huge with respect to the optimal path!The problem is that in worst case we can find huge faces and explore themeven if there’s no need.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 24 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.
- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.
- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.
- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.
- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.
- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.
E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing
Other Bounded Face Routing
Suppose we know the Euclidean length of an optimal path connecting thesource s and the destination t.
Define an ellipse E whose size contains a complete optimal path.
Restrict the search to the ellipse, searching for p node of minimumdistance wrt. t.
1 When the packet ”hits” the ellipse for the first time, it continues toexplore the face in the opposite direction
2 The second time it ”hits” the ellipse, or has completed the tour theface exploration is finished.- If p is equal to the previous one and the whole face was explored, thegraph is disconnected. Terminate.
3 Move to the face incident to pt, and explore it (go back to step 1).
The size of the major axis of E will be c , Euclidean length of the optimalpath p∗ connecting s and t.E contains all paths of size ≤ c .
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 25 / 37
Routing with faces
Other Bounded Face Routing Example
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 26 / 37
Routing with faces
Other Bounded Face Routing Example
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 26 / 37
Routing with faces
Other Bounded Face Routing Example
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 26 / 37
Routing with faces
Other Bounded Face Routing Example
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 26 / 37
Routing with faces
Other Bounded Face Routing Example
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 26 / 37
Routing with faces
Other Bounded Face Routing Example
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 26 / 37
Routing with faces
Correctness and Cost of OBFR (1)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
(Sketch of) Proof:
1 Assume that the shortest Euclidean path p∗ is contained in E .
2 For contradiction, assume OBFR to report a failure: this can happen only ifthe ellipse cuts the region in two parts, one containing s and one containingt. So p∗ isn’t in E .
Contradiction!
3 If no path connects s and t in E this is detected as there will be a node vwhich is on the part of the graph where s stands and which is the nearestone to t. At second iteration, no progress is done thus this is reportedstarting OBFR in E with destination s.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 27 / 37
Routing with faces
Correctness and Cost of OBFR (1)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
(Sketch of) Proof:
1 Assume that the shortest Euclidean path p∗ is contained in E .
2 For contradiction, assume OBFR to report a failure: this can happen only ifthe ellipse cuts the region in two parts, one containing s and one containingt. So p∗ isn’t in E .
Contradiction!
3 If no path connects s and t in E this is detected as there will be a node vwhich is on the part of the graph where s stands and which is the nearestone to t. At second iteration, no progress is done thus this is reportedstarting OBFR in E with destination s.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 27 / 37
Routing with faces
Correctness and Cost of OBFR (1)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
(Sketch of) Proof:
1 Assume that the shortest Euclidean path p∗ is contained in E .
2 For contradiction, assume OBFR to report a failure: this can happen only ifthe ellipse cuts the region in two parts, one containing s and one containingt. So p∗ isn’t in E .
Contradiction!
3 If no path connects s and t in E this is detected as there will be a node vwhich is on the part of the graph where s stands and which is the nearestone to t. At second iteration, no progress is done thus this is reportedstarting OBFR in E with destination s.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 27 / 37
Routing with faces
Correctness and Cost of OBFR (1)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
(Sketch of) Proof:
1 Assume that the shortest Euclidean path p∗ is contained in E .
2 For contradiction, assume OBFR to report a failure: this can happen only ifthe ellipse cuts the region in two parts, one containing s and one containingt. So p∗ isn’t in E . Contradiction!
3 If no path connects s and t in E this is detected as there will be a node vwhich is on the part of the graph where s stands and which is the nearestone to t. At second iteration, no progress is done thus this is reportedstarting OBFR in E with destination s.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 27 / 37
Routing with faces
Correctness and Cost of OBFR (2)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
4 Every face is traversed once or twice, thus edges are used at most fourtimes.
OBFR is O(m) = O(n) for planarity of graph!
5 Since the graph is civilized, there’s d0 for which circles of radius d0/2 aroundall nodes do not intersect.
6 Upper bound E ’s area with π(c
2)2
7 Let n′ be the number of nodes inside E , we can state: n′ ≤π( c
2 )2
π( d02 )2
=c2
d20
8 This proves that OBFR visits at most O(c2) nodes.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 28 / 37
Routing with faces
Correctness and Cost of OBFR (2)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
4 Every face is traversed once or twice, thus edges are used at most fourtimes.
OBFR is O(m) = O(n) for planarity of graph!
5 Since the graph is civilized, there’s d0 for which circles of radius d0/2 aroundall nodes do not intersect.
6 Upper bound E ’s area with π(c
2)2
7 Let n′ be the number of nodes inside E , we can state: n′ ≤π( c
2 )2
π( d02 )2
=c2
d20
8 This proves that OBFR visits at most O(c2) nodes.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 28 / 37
Routing with faces
Correctness and Cost of OBFR (2)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
4 Every face is traversed once or twice, thus edges are used at most fourtimes. OBFR is O(m) = O(n) for planarity of graph!
5 Since the graph is civilized, there’s d0 for which circles of radius d0/2 aroundall nodes do not intersect.
6 Upper bound E ’s area with π(c
2)2
7 Let n′ be the number of nodes inside E , we can state: n′ ≤π( c
2 )2
π( d02 )2
=c2
d20
8 This proves that OBFR visits at most O(c2) nodes.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 28 / 37
Routing with faces
Correctness and Cost of OBFR (2)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
4 Every face is traversed once or twice, thus edges are used at most fourtimes. OBFR is O(m) = O(n) for planarity of graph!
5 Since the graph is civilized, there’s d0 for which circles of radius d0/2 aroundall nodes do not intersect.
6 Upper bound E ’s area with π(c
2)2
7 Let n′ be the number of nodes inside E , we can state: n′ ≤π( c
2 )2
π( d02 )2
=c2
d20
8 This proves that OBFR visits at most O(c2) nodes.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 28 / 37
Routing with faces
Correctness and Cost of OBFR (2)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
4 Every face is traversed once or twice, thus edges are used at most fourtimes. OBFR is O(m) = O(n) for planarity of graph!
5 Since the graph is civilized, there’s d0 for which circles of radius d0/2 aroundall nodes do not intersect.
6 Upper bound E ’s area with π(c
2)2
7 Let n′ be the number of nodes inside E , we can state: n′ ≤π( c
2 )2
π( d02 )2
=c2
d20
8 This proves that OBFR visits at most O(c2) nodes.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 28 / 37
Routing with faces
Correctness and Cost of OBFR (2)
Lemma: If c length of the major axis of E is at least the Euclidean lengthof the shortest path between s and t, OBFR reaches the destination (if thegraph is connected) or reports an error expending at most O(c2)
4 Every face is traversed once or twice, thus edges are used at most fourtimes. OBFR is O(m) = O(n) for planarity of graph!
5 Since the graph is civilized, there’s d0 for which circles of radius d0/2 aroundall nodes do not intersect.
6 Upper bound E ’s area with π(c
2)2
7 Let n′ be the number of nodes inside E , we can state: n′ ≤π( c
2 )2
π( d02 )2
=c2
d20
8 This proves that OBFR visits at most O(c2) nodes.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 28 / 37
Routing with faces
Other Adaptive Face Routing
However the estimation of the size of an optimal path p∗ is not easy.
Other Adaptive Face Routing faces this problem as follows:1. Initialize E with s and t in the foci and c0 = 2|st|. i = 0.
2. Run OBFRE3.a. If t is not reached, increment i and initialize E to the ellipse with major axis
ci = 2ic0. Go back to 2.
3.b. If the graph is disconnected, terminate.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 29 / 37
Routing with faces
Other Adaptive Face Routing
However the estimation of the size of an optimal path p∗ is not easy.Other Adaptive Face Routing faces this problem as follows:1. Initialize E with s and t in the foci and c0 = 2|st|. i = 0.
2. Run OBFRE3.a. If t is not reached, increment i and initialize E to the ellipse with major axis
ci = 2ic0. Go back to 2.
3.b. If the graph is disconnected, terminate.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 29 / 37
Routing with faces
Other Adaptive Face Routing
However the estimation of the size of an optimal path p∗ is not easy.Other Adaptive Face Routing faces this problem as follows:1. Initialize E with s and t in the foci and c0 = 2|st|. i = 0.
2. Run OBFRE
3.a. If t is not reached, increment i and initialize E to the ellipse with major axis
ci = 2ic0. Go back to 2.
3.b. If the graph is disconnected, terminate.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 29 / 37
Routing with faces
Other Adaptive Face Routing
However the estimation of the size of an optimal path p∗ is not easy.Other Adaptive Face Routing faces this problem as follows:1. Initialize E with s and t in the foci and c0 = 2|st|. i = 0.
2. Run OBFRE3.a. If t is not reached, increment i and initialize E to the ellipse with major axis
ci = 2ic0. Go back to 2.
3.b. If the graph is disconnected, terminate.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 29 / 37
Routing with faces
Other Adaptive Face Routing
However the estimation of the size of an optimal path p∗ is not easy.Other Adaptive Face Routing faces this problem as follows:1. Initialize E with s and t in the foci and c0 = 2|st|. i = 0.
2. Run OBFRE3.a. If t is not reached, increment i and initialize E to the ellipse with major axis
ci = 2ic0. Go back to 2.
3.b. If the graph is disconnected, terminate.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 29 / 37
Routing with faces
Other Adaptive Face Routing
However the estimation of the size of an optimal path p∗ is not easy.Other Adaptive Face Routing faces this problem as follows:1. Initialize E with s and t in the foci and c0 = 2|st|. i = 0.
2. Run OBFRE3.a. If t is not reached, increment i and initialize E to the ellipse with major axis
ci = 2ic0. Go back to 2.
3.b. If the graph is disconnected, terminate.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 29 / 37
Routing with faces
OAFR Features
Theorem: If s and t are connected, OAFR reaches the destination with acost of O(c2(p∗)) where p∗ is an optimal path between the two. If theyare disconnected, OAFR detects the situation.
1 Let k be the index such that ck−1 = 2k−1c0 < cd(p∗) ≤ ck where cd is theEuclidean cost metric of a path.
2 Since the hop cost cl(OBFRc) is O(c2), then cl(OBFRc) ≤ λc2 for a certainλ constant.
3 So cl(OAFR) ≤k∑
i=0
cl(OBFRci ≤k∑
i=0
λ(2i c0)2 = λc20
4k+1 − 1
3<
16
3λ(2k−1c0)2 <
16
3λc2
d (p∗) ∈ O(c2d (p∗))
4 In few words, the cost of the last iteration dominates the complexity of thealgorithm, which is ∈ O(c2
d (p∗)).
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 30 / 37
Routing with faces
OAFR Features
Theorem: If s and t are connected, OAFR reaches the destination with acost of O(c2(p∗)) where p∗ is an optimal path between the two. If theyare disconnected, OAFR detects the situation.
1 Let k be the index such that ck−1 = 2k−1c0 < cd(p∗) ≤ ck where cd is theEuclidean cost metric of a path.
2 Since the hop cost cl(OBFRc) is O(c2), then cl(OBFRc) ≤ λc2 for a certainλ constant.
3 So cl(OAFR) ≤k∑
i=0
cl(OBFRci ≤k∑
i=0
λ(2i c0)2 = λc20
4k+1 − 1
3<
16
3λ(2k−1c0)2 <
16
3λc2
d (p∗) ∈ O(c2d (p∗))
4 In few words, the cost of the last iteration dominates the complexity of thealgorithm, which is ∈ O(c2
d (p∗)).
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 30 / 37
Routing with faces
OAFR Features
Theorem: If s and t are connected, OAFR reaches the destination with acost of O(c2(p∗)) where p∗ is an optimal path between the two. If theyare disconnected, OAFR detects the situation.
1 Let k be the index such that ck−1 = 2k−1c0 < cd(p∗) ≤ ck where cd is theEuclidean cost metric of a path.
2 Since the hop cost cl(OBFRc) is O(c2), then cl(OBFRc) ≤ λc2 for a certainλ constant.
3 So cl(OAFR) ≤k∑
i=0
cl(OBFRci ≤k∑
i=0
λ(2i c0)2 = λc20
4k+1 − 1
3<
16
3λ(2k−1c0)2 <
16
3λc2
d (p∗) ∈ O(c2d (p∗))
4 In few words, the cost of the last iteration dominates the complexity of thealgorithm, which is ∈ O(c2
d (p∗)).
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 30 / 37
Routing with faces
OAFR Features
Theorem: If s and t are connected, OAFR reaches the destination with acost of O(c2(p∗)) where p∗ is an optimal path between the two. If theyare disconnected, OAFR detects the situation.
1 Let k be the index such that ck−1 = 2k−1c0 < cd(p∗) ≤ ck where cd is theEuclidean cost metric of a path.
2 Since the hop cost cl(OBFRc) is O(c2), then cl(OBFRc) ≤ λc2 for a certainλ constant.
3 So cl(OAFR) ≤k∑
i=0
cl(OBFRci ≤k∑
i=0
λ(2i c0)2 = λc20
4k+1 − 1
3<
16
3λ(2k−1c0)2 <
16
3λc2
d (p∗) ∈ O(c2d (p∗))
4 In few words, the cost of the last iteration dominates the complexity of thealgorithm, which is ∈ O(c2
d (p∗)).
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 30 / 37
Routing with faces
OAFR Features
Theorem: If s and t are connected, OAFR reaches the destination with acost of O(c2(p∗)) where p∗ is an optimal path between the two. If theyare disconnected, OAFR detects the situation.
1 Let k be the index such that ck−1 = 2k−1c0 < cd(p∗) ≤ ck where cd is theEuclidean cost metric of a path.
2 Since the hop cost cl(OBFRc) is O(c2), then cl(OBFRc) ≤ λc2 for a certainλ constant.
3 So cl(OAFR) ≤k∑
i=0
cl(OBFRci ≤k∑
i=0
λ(2i c0)2 = λc20
4k+1 − 1
3<
16
3λ(2k−1c0)2 <
16
3λc2
d (p∗) ∈ O(c2d (p∗))
4 In few words, the cost of the last iteration dominates the complexity of thealgorithm, which is ∈ O(c2
d (p∗)).
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 30 / 37
Routing with faces
OAFR’s Asymptotic Optimality
Theorem: Being the cost of a best route for a pair ofnodes c , there exists graphs where any deterministicGeographic Ad Hoc Routing Algorithms has expectedcost of Ω(c2).
No proof, just intuition!The graph has 2k nodes in the circumference with adistance of one, and contains O(k2) nodes.
Think at what will happen when routing from a node in the border to onein the center of the circle, without having calculated shortest path.At worst case all nodes have to be checked before arriving at w and thenfollowing the line to the center. O(k2)The optimal path, instead has at most k/2 nodes in the border to reach wand k/π to reach the destination. O(k). Thus OAFR on civilized graphs isasymptotically optimal.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 31 / 37
Routing with faces
OAFR’s Asymptotic Optimality
Theorem: Being the cost of a best route for a pair ofnodes c , there exists graphs where any deterministicGeographic Ad Hoc Routing Algorithms has expectedcost of Ω(c2).No proof, just intuition!The graph has 2k nodes in the circumference with adistance of one, and contains O(k2) nodes.
Think at what will happen when routing from a node in the border to onein the center of the circle, without having calculated shortest path.At worst case all nodes have to be checked before arriving at w and thenfollowing the line to the center. O(k2)The optimal path, instead has at most k/2 nodes in the border to reach wand k/π to reach the destination. O(k). Thus OAFR on civilized graphs isasymptotically optimal.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 31 / 37
Routing with faces
OAFR’s Asymptotic Optimality
Theorem: Being the cost of a best route for a pair ofnodes c , there exists graphs where any deterministicGeographic Ad Hoc Routing Algorithms has expectedcost of Ω(c2).No proof, just intuition!The graph has 2k nodes in the circumference with adistance of one, and contains O(k2) nodes.
Think at what will happen when routing from a node in the border to onein the center of the circle, without having calculated shortest path.
At worst case all nodes have to be checked before arriving at w and thenfollowing the line to the center. O(k2)The optimal path, instead has at most k/2 nodes in the border to reach wand k/π to reach the destination. O(k). Thus OAFR on civilized graphs isasymptotically optimal.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 31 / 37
Routing with faces
OAFR’s Asymptotic Optimality
Theorem: Being the cost of a best route for a pair ofnodes c , there exists graphs where any deterministicGeographic Ad Hoc Routing Algorithms has expectedcost of Ω(c2).No proof, just intuition!The graph has 2k nodes in the circumference with adistance of one, and contains O(k2) nodes.
Think at what will happen when routing from a node in the border to onein the center of the circle, without having calculated shortest path.At worst case all nodes have to be checked before arriving at w and thenfollowing the line to the center.
O(k2)The optimal path, instead has at most k/2 nodes in the border to reach wand k/π to reach the destination. O(k). Thus OAFR on civilized graphs isasymptotically optimal.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 31 / 37
Routing with faces
OAFR’s Asymptotic Optimality
Theorem: Being the cost of a best route for a pair ofnodes c , there exists graphs where any deterministicGeographic Ad Hoc Routing Algorithms has expectedcost of Ω(c2).No proof, just intuition!The graph has 2k nodes in the circumference with adistance of one, and contains O(k2) nodes.
Think at what will happen when routing from a node in the border to onein the center of the circle, without having calculated shortest path.At worst case all nodes have to be checked before arriving at w and thenfollowing the line to the center. O(k2)The optimal path, instead has at most k/2 nodes in the border to reach wand k/π to reach the destination.
O(k). Thus OAFR on civilized graphs isasymptotically optimal.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 31 / 37
Routing with faces
OAFR’s Asymptotic Optimality
Theorem: Being the cost of a best route for a pair ofnodes c , there exists graphs where any deterministicGeographic Ad Hoc Routing Algorithms has expectedcost of Ω(c2).No proof, just intuition!The graph has 2k nodes in the circumference with adistance of one, and contains O(k2) nodes.
Think at what will happen when routing from a node in the border to onein the center of the circle, without having calculated shortest path.At worst case all nodes have to be checked before arriving at w and thenfollowing the line to the center. O(k2)The optimal path, instead has at most k/2 nodes in the border to reach wand k/π to reach the destination. O(k). Thus OAFR on civilized graphs isasymptotically optimal.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 31 / 37
GOAFR+
Layout
1 Introduction
2 Model
3 Greedy Routing
4 Graph Planarization
5 Routing with faces
6 GOAFR+
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 32 / 37
GOAFR+
GOAFR+
GOAFR+
GOAFR+ is a combination of greedy and face routing, which tries toexploit the advantages of both.
Intuitively:
Greedy is particularly good in dense networks, but doesn’t guaranteedelivery.
OAFR grants delivery, but could be very expensive.
Use OAFR to escape from greedy’s local minimum!
Don’t get back to Greedy mode as soon as the local minimum is escaped- i.e. the current node explored by OAFR is nearer to the destination wrt.the point in which it started. This will impair asymptotic optimality ofOAFR.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 33 / 37
GOAFR+
GOAFR+
GOAFR+
GOAFR+ is a combination of greedy and face routing, which tries toexploit the advantages of both.Intuitively:
Greedy is particularly good in dense networks, but doesn’t guaranteedelivery.
OAFR grants delivery, but could be very expensive.
Use OAFR to escape from greedy’s local minimum!
Don’t get back to Greedy mode as soon as the local minimum is escaped- i.e. the current node explored by OAFR is nearer to the destination wrt.the point in which it started. This will impair asymptotic optimality ofOAFR.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 33 / 37
GOAFR+
GOAFR+
GOAFR+
GOAFR+ is a combination of greedy and face routing, which tries toexploit the advantages of both.Intuitively:
Greedy is particularly good in dense networks, but doesn’t guaranteedelivery.
OAFR grants delivery, but could be very expensive.
Use OAFR to escape from greedy’s local minimum!
Don’t get back to Greedy mode as soon as the local minimum is escaped- i.e. the current node explored by OAFR is nearer to the destination wrt.the point in which it started. This will impair asymptotic optimality ofOAFR.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 33 / 37
GOAFR+
GOAFR+
GOAFR+
GOAFR+ is a combination of greedy and face routing, which tries toexploit the advantages of both.Intuitively:
Greedy is particularly good in dense networks, but doesn’t guaranteedelivery.
OAFR grants delivery, but could be very expensive.
Use OAFR to escape from greedy’s local minimum!
Don’t get back to Greedy mode as soon as the local minimum is escaped- i.e. the current node explored by OAFR is nearer to the destination wrt.the point in which it started. This will impair asymptotic optimality ofOAFR.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 33 / 37
GOAFR+
GOAFR+
GOAFR+
GOAFR+ is a combination of greedy and face routing, which tries toexploit the advantages of both.Intuitively:
Greedy is particularly good in dense networks, but doesn’t guaranteedelivery.
OAFR grants delivery, but could be very expensive.
Use OAFR to escape from greedy’s local minimum!
Don’t get back to Greedy mode as soon as the local minimum is escaped- i.e. the current node explored by OAFR is nearer to the destination wrt.the point in which it started. This will impair asymptotic optimality ofOAFR.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 33 / 37
GOAFR+
GOAFR+
GOAFR+
GOAFR+ is a combination of greedy and face routing, which tries toexploit the advantages of both.Intuitively:
Greedy is particularly good in dense networks, but doesn’t guaranteedelivery.
OAFR grants delivery, but could be very expensive.
Use OAFR to escape from greedy’s local minimum!
Don’t get back to Greedy mode as soon as the local minimum is escaped- i.e. the current node explored by OAFR is nearer to the destination wrt.the point in which it started.
This will impair asymptotic optimality ofOAFR.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 33 / 37
GOAFR+
GOAFR+
GOAFR+
GOAFR+ is a combination of greedy and face routing, which tries toexploit the advantages of both.Intuitively:
Greedy is particularly good in dense networks, but doesn’t guaranteedelivery.
OAFR grants delivery, but could be very expensive.
Use OAFR to escape from greedy’s local minimum!
Don’t get back to Greedy mode as soon as the local minimum is escaped- i.e. the current node explored by OAFR is nearer to the destination wrt.the point in which it started. This will impair asymptotic optimality ofOAFR.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 33 / 37
GOAFR+
Keeping OAFR Asymptotic Optimality
When visiting face F with OAFR, take two counters, p and q
Let d be the distance from the local minimums and the destination t.
p counts the number of nodes v ∈ F such that |vt| < d
q counts the number of nodes u ∈ F such that |ut| ≥ d
When some condition cond(p, q) is true, get back to Greedy mode. 2.
According to the authors, it can be proved that GOAFR+ maintainsOAFR’s asymptotic optimality and thus guarantee’s delivery in O(c2(p∗))However on the average this algorithm clearly outperforms all the others inthe Geographic Ad Hoc Routing class.
2The condition was not disclosed by the authors in the paperAlessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 34 / 37
GOAFR+
Keeping OAFR Asymptotic Optimality
When visiting face F with OAFR, take two counters, p and q
Let d be the distance from the local minimums and the destination t.
p counts the number of nodes v ∈ F such that |vt| < d
q counts the number of nodes u ∈ F such that |ut| ≥ d
When some condition cond(p, q) is true, get back to Greedy mode. 2.
According to the authors, it can be proved that GOAFR+ maintainsOAFR’s asymptotic optimality and thus guarantee’s delivery in O(c2(p∗))However on the average this algorithm clearly outperforms all the others inthe Geographic Ad Hoc Routing class.
2The condition was not disclosed by the authors in the paperAlessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 34 / 37
GOAFR+
Keeping OAFR Asymptotic Optimality
When visiting face F with OAFR, take two counters, p and q
Let d be the distance from the local minimums and the destination t.
p counts the number of nodes v ∈ F such that |vt| < d
q counts the number of nodes u ∈ F such that |ut| ≥ d
When some condition cond(p, q) is true, get back to Greedy mode. 2.
According to the authors, it can be proved that GOAFR+ maintainsOAFR’s asymptotic optimality and thus guarantee’s delivery in O(c2(p∗))However on the average this algorithm clearly outperforms all the others inthe Geographic Ad Hoc Routing class.
2The condition was not disclosed by the authors in the paperAlessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 34 / 37
GOAFR+
Keeping OAFR Asymptotic Optimality
When visiting face F with OAFR, take two counters, p and q
Let d be the distance from the local minimums and the destination t.
p counts the number of nodes v ∈ F such that |vt| < d
q counts the number of nodes u ∈ F such that |ut| ≥ d
When some condition cond(p, q) is true, get back to Greedy mode. 2.
According to the authors, it can be proved that GOAFR+ maintainsOAFR’s asymptotic optimality and thus guarantee’s delivery in O(c2(p∗))However on the average this algorithm clearly outperforms all the others inthe Geographic Ad Hoc Routing class.
2The condition was not disclosed by the authors in the paperAlessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 34 / 37
GOAFR+
Keeping OAFR Asymptotic Optimality
When visiting face F with OAFR, take two counters, p and q
Let d be the distance from the local minimums and the destination t.
p counts the number of nodes v ∈ F such that |vt| < d
q counts the number of nodes u ∈ F such that |ut| ≥ d
When some condition cond(p, q) is true, get back to Greedy mode. 2.
According to the authors, it can be proved that GOAFR+ maintainsOAFR’s asymptotic optimality and thus guarantee’s delivery in O(c2(p∗))However on the average this algorithm clearly outperforms all the others inthe Geographic Ad Hoc Routing class.
2The condition was not disclosed by the authors in the paperAlessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 34 / 37
GOAFR+
Keeping OAFR Asymptotic Optimality
When visiting face F with OAFR, take two counters, p and q
Let d be the distance from the local minimums and the destination t.
p counts the number of nodes v ∈ F such that |vt| < d
q counts the number of nodes u ∈ F such that |ut| ≥ d
When some condition cond(p, q) is true, get back to Greedy mode. 2.
According to the authors, it can be proved that GOAFR+ maintainsOAFR’s asymptotic optimality and thus guarantee’s delivery in O(c2(p∗))
However on the average this algorithm clearly outperforms all the others inthe Geographic Ad Hoc Routing class.
2The condition was not disclosed by the authors in the paperAlessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 34 / 37
GOAFR+
Keeping OAFR Asymptotic Optimality
When visiting face F with OAFR, take two counters, p and q
Let d be the distance from the local minimums and the destination t.
p counts the number of nodes v ∈ F such that |vt| < d
q counts the number of nodes u ∈ F such that |ut| ≥ d
When some condition cond(p, q) is true, get back to Greedy mode. 2.
According to the authors, it can be proved that GOAFR+ maintainsOAFR’s asymptotic optimality and thus guarantee’s delivery in O(c2(p∗))However on the average this algorithm clearly outperforms all the others inthe Geographic Ad Hoc Routing class.
2The condition was not disclosed by the authors in the paperAlessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 34 / 37
GOAFR+
Mean Algorithm Cost
We shall consider as cost of an algorithmthe following normalized function:
costA(N, s, t) =sa(N, s, t)
cl(p∗l (N, s, t))
Where:
s and t are a source and a destination
N is a randomly generated network
sA(N, s, t) counts the number of stepsused by the algorithm to reach tstarting from s in N.
cl(p∗l (N, s, t)) is the cost in hops ofthe shortest path in N connecting s tot.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 35 / 37
GOAFR+
Mean Algorithm Cost
We shall consider as cost of an algorithmthe following normalized function:
costA(N, s, t) =sa(N, s, t)
cl(p∗l (N, s, t))Where:
s and t are a source and a destination
N is a randomly generated network
sA(N, s, t) counts the number of stepsused by the algorithm to reach tstarting from s in N.
cl(p∗l (N, s, t)) is the cost in hops ofthe shortest path in N connecting s tot.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 35 / 37
GOAFR+
Mean Algorithm Cost
We shall consider as cost of an algorithmthe following normalized function:
costA(N, s, t) =sa(N, s, t)
cl(p∗l (N, s, t))Where:
s and t are a source and a destination
N is a randomly generated network
sA(N, s, t) counts the number of stepsused by the algorithm to reach tstarting from s in N.
cl(p∗l (N, s, t)) is the cost in hops ofthe shortest path in N connecting s tot.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 35 / 37
GOAFR+
Mean Algorithm Cost
We shall consider as cost of an algorithmthe following normalized function:
costA(N, s, t) =sa(N, s, t)
cl(p∗l (N, s, t))Where:
s and t are a source and a destination
N is a randomly generated network
sA(N, s, t) counts the number of stepsused by the algorithm to reach tstarting from s in N.
cl(p∗l (N, s, t)) is the cost in hops ofthe shortest path in N connecting s tot.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 35 / 37
GOAFR+
Mean Algorithm Cost
We shall consider as cost of an algorithmthe following normalized function:
costA(N, s, t) =sa(N, s, t)
cl(p∗l (N, s, t))Where:
s and t are a source and a destination
N is a randomly generated network
sA(N, s, t) counts the number of stepsused by the algorithm to reach tstarting from s in N.
cl(p∗l (N, s, t)) is the cost in hops ofthe shortest path in N connecting s tot.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 35 / 37
GOAFR+
Mean Algorithm Cost
We shall consider as cost of an algorithmthe following normalized function:
costA(N, s, t) =sa(N, s, t)
cl(p∗l (N, s, t))Where:
s and t are a source and a destination
N is a randomly generated network
sA(N, s, t) counts the number of stepsused by the algorithm to reach tstarting from s in N.
cl(p∗l (N, s, t)) is the cost in hops ofthe shortest path in N connecting s tot.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 35 / 37
GOAFR+
Conclusions
We have seen some algorithms to solve the problem of Geographic AdHoc Routing
Actually, to the best of our knowledge, GOAFR+ is not justasymptotically optimal, but also one of the best algorithms of its classon average case simulations.
The idea of exploiting geographic coordinates to route can beadopted for a variety of scenarios: redundancy with geographic hashtables, geocasting ...
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 36 / 37
GOAFR+
Conclusions
We have seen some algorithms to solve the problem of Geographic AdHoc Routing
Actually, to the best of our knowledge, GOAFR+ is not justasymptotically optimal, but also one of the best algorithms of its classon average case simulations.
The idea of exploiting geographic coordinates to route can beadopted for a variety of scenarios: redundancy with geographic hashtables, geocasting ...
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 36 / 37
GOAFR+
Conclusions
We have seen some algorithms to solve the problem of Geographic AdHoc Routing
Actually, to the best of our knowledge, GOAFR+ is not justasymptotically optimal, but also one of the best algorithms of its classon average case simulations.
The idea of exploiting geographic coordinates to route can beadopted for a variety of scenarios: redundancy with geographic hashtables, geocasting ...
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 36 / 37
GOAFR+
Jie Gao, Leonidas J. Guibas, John Hershberger, Li Zhang 0001, andAn Zhu.Geometric spanner for routing in mobile networks.In MobiHoc, pages 45–55. ACM, 2001.
Brad Karp and H. T. Kung.Gpsr: Greedy perimeter stateless routing for wireless networks.In Proceedings of the 6th Annual International Conference on MobileComputing and Networking, MobiCom ’00, pages 243–254, New York,NY, USA, 2000. ACM.
Ananth Rao, Sylvia Ratnasamy, Christos Papadimitriou, ScottShenker, and Ion Stoica.Geographic routing without location information.In Proceedings of the 9th Annual International Conference on MobileComputing and Networking, MobiCom ’03, pages 96–108, New York,NY, USA, 2003. ACM.
Alessandro Lenzi (ALP Seminars) Geographic Routing December 4, 2013 37 / 37
