Ad-Hoc Networks Beyond Unit Disk Graphs

Fabian KuhnRoger Wattenhofer

Aaron Zollinger

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Overview

• Introduction– Graph Models for Mobile Ad-Hoc Networks– Quasi Unit Disk Graphs– Related Work– Volatile Memory Routing

• Flooding– Lower Bound for Message Complexity– Topology for Optimal Flooding

• Greedy + Flooding– Volatile Memory Routing Algorithms– Combining Greedy and Flooding

• Geometric Routing for– How to obtain a planar graph– Optimality of AFR/GOAFR

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Mobile Ad-Hoc Networks

• Mobile Devices communicating via radio• Network without centralized control (base station)

• We consider the abstraction level of graphs

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Graph Models for Ad-Hoc Networks

Simple Models• Unit Disk Graph is most widely

applied model:

• Underlying assumption:All nodes are in R2, have exactly the same transmission range (normalized to one), and there are no obstacles.

• Far from realityBUT: There are numerous theoretical results.

Realistic Models• We need more general graph

models

• However, arbitrary graphs are too general to obtain strong results for routing, etc.

• We need something between UDG and arbitrary graphs:– general enough to model

reality as close as possible

– restrictive enough to allow useful theoretical results

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Quasi Unit Disk Graph

Definition Unit Disk Graph:• Edge between u and v if |u-v|·1• No edge between u and v if |u-v|>1

Definition Quasi Unit Disk Graph:• Edge between u and v if |u-v|·d• No edge between u and v if |u-v|>1• May have an edge if d<|u-v|<1

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Related Work

• The Quasi Unit Disk Graph model is not new:Barrière, Fraigniaud, and Narayanan have shown that correct geometric routing is possible if .Dial-M 2001 and Wireless Networks Journal Vol. 3(2) 2003

• Other generalizations of the unit disk graph have been proposed, e.g. (r,s)-civilized graphs by Krumke, Marathe, and RaviDial-M 1998 and Wireless Networks Journal Vol. 7(6) 2001

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Volatile Memory Routing

• We want to consider routing without routing tables

• We need to allow nodes to temporarily store some information

• Volatile Memory Routing Algorithm:

For each message, each node is allowed to temporarily store O(log n) bits.(temporary = while the message has not reached the destination)

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Overview

• Introduction– Graph Models for Mobile Ad-Hoc Networks– Quasi Unit Disk Graphs– Related Work– Volatile Memory Routing

• Flooding– Lower Bound for Message Complexity– Topology for Optimal Flooding

• Greedy + Flooding– Volatile Memory Routing Algorithms– Combining Greedy and Flooding

• Geometric Routing for– How to obtain a planar graph– Optimality of AFR/GOAFR

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Message Complexity Lower Bound

• Lower Bound Graph is a Quasi Unit Disk Graph (parameter d)

• To find destination t, all vertical chains (all nodes) have to be visited

• Length of one chain: c

• Optimal path has length O(c)

• There are O(c2/d2) nodes! O(c2/d2) messages

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Flooding

• Flooding on the Quasi UDG gives unbounded message complexity

• We need a subgraph on which flooding is efficient (a kind of topology control)

• Desired properties:– Nodes form a dominating set– O(A/d2) nodes per area A– O(A/d2) edges per area A– Optional: spanner

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Topology Control I

• Construct a Minimal Independent Set (MIS)

! dominating set and O(A/d2) nodes per area A

• If we make all 2- and 3-hop connections, we have a spanner, but too many nodes and edges

• Solution: Choose only a subset of the 2- and 3-hop connections (“virtual edges” of length · 3)

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Topology Control II

Place grid over the nodes

(cell size = 6)

Add another grid shifted

by (3,3)

Add another grid shifted

by (3,0)

Add another grid shifted

by (0,3)

Each “virtual edge” is

completely covered

by a cell of at least

one of the grids

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Topology Control III

• In each cell, we calculate a spanner of the nodes (MIS) and “virtual edges” lying completely inside the cell

! Applying a randomized construction of Linial and Saks (SODA 91) yields a O(log(1/d2))-spanner with O(1/d2) virtual edges.

• Combining all local spanners gives a O(log(1/d2))-spanner with O(A/d2) “virtual edges” per area A.

! Backbone Graph

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Flooding on the Backbone Graph

• Flooding/Echo with exponentially growing TTL on the Backbone Graph gives:– O(log(1/d2)c) time and O(c2/d2) message complexity in the

synchronous model– O(log(1/d2)log3(c/d)c) time and O(log3(c/d)c2/d2) message

complexity in the asynchronous model (using a synchronizer described by Awerbuch and Peleg, FOCS 90)

• Geometric Flooding/Echo uses disks with exponentially growing radius instead of TTL:– O(c2/d2) time and message complexity (synchronous and

asynchronous)

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Overview

• Introduction– Graph Models for Mobile Ad-Hoc Networks– Quasi Unit Disk Graphs– Related Work– Volatile Memory Routing

• Flooding– Lower Bound for Message Complexity– Topology for Optimal Flooding

• Greedy + Flooding– Geometric (Volatile Memory) Routing Algorithms– Combining Greedy and Flooding

• Geometric Routing for– How to obtain a planar graph– Optimality of AFR/GOAFR

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Geometric Routing

• A.k.a. location-based, position-based, geographic, etc.

• Each node knows its own position and position of neighbors

• Source knows the position of the destination• No routing tables in the nodes, all routing information is

in the message!

• Volatile Geometric Routing:– Geometric Routing + O(log n) bits per message in each node

(while message is on the way from s to t)

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Geometric Routing

t

s

s

???

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Greedy Routing

• Each node forwards message to “best” neighbor

t

s

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Greedy Routing

• Each node forwards message to “best” neighbor

• But greedy routing may fail: message may get stuck in a “dead end”

• Needed: Correct geometric routing algorithm

t

?s

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Greedy + Flooding

• Straight-forward idea to make greedy routing correct(i.e. always find the destination)

! combine greedy and flooding

• We want to keep worst-case optimality

• Apply geometric flooding with exponentially increasing radius and the right criterion to fall back from flooding to greedy

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Greedy + Flooding, Fall Back Criterion

• Flooding phase with radii r0, r1, … where ri=r02i

• Flooding starts at node u, node vi is best (closest to destination t) node for radius ri

• Go back to greedy if |u-t| - |vi-t| ¸ q¢ri

(q is a predefined constant)

• Message and time complexity: O(c2/d2)

• Simulations on UDG suggest that the algorithm is efficient in the average case

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Overview

• Introduction– Graph Models for Mobile Ad-Hoc Networks– Quasi Unit Disk Graphs– Related Work– Volatile Memory Routing

• Flooding– Lower Bound for Message Complexity– Topology for Optimal Flooding

• Greedy + Flooding– Geometric (Volatile Memory) Routing Algorithms– Combining Greedy and Flooding

• Geometric Routing for– How to obtain a planar graph– Optimality of AFR/GOAFR

DIALM-POMC 2003

Questions?Comments?