Online Multi-Path Routing in a Maze

1

University of FreiburgComputer Networks and Telematics

Prof. Christian Schindelhauer

Online Multi-Path Routing in a Maze

Christian Schindelhauer

joint work with

Stefan Rührup

Workshop of Flexible Network Design

Bertinoro, 1.-6.10.2006

to appear at ISAAC 2006

[email protected]

University of FreiburgInstitute of Computer Science

Computer Networks and TelematicsProf. Christian Schindelhauer

Worksho of Flexible Network DesignBertinoro, 1-6.10.2006

Online Multi-Path Routing in a Maze- 2

Target: geographic position instead of network address

Idea: Iteratively choose neighbor closest to the target(Greedy-Strategie)

Position based Routing

(2,5)

(13,5)(5,7)

(4,2)

(3,9)

13,513,5

(0,8)

st

Advantages:– local decisisions– no routing tables– scalable





Prerequisits:

– All nodes know their positions (e.g. GPS)

– Position of all neighbors are known (Beacon Messages)

– Target position is known (Location Service)


(2,5)

(13,5)(5,7)

(4,2)

(3,9)

13,513,5

(0,8)

st





First Works (1)

Routing in Packet Radio Networks Greedy-Strategies:

– MFR: Most Forwarding within Radius [Takagi, Kleinrock 1984]

– NFP: Nearest with Forwarding Progress [Hou, Li 1986]

s t

MFR

NFP

Transmission radius





First Works (2)

Cartesian Routing [Finn 1987]

Routing with geographic Coordinates

n-hop Cartesian regular: Every node has a node in its n-hop-neighborhood which is closer to an arbitrary target

Greedy-Routing and Limited Flooding (restricted to n Hops)

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X

Problems: Greedy routing may end in local minima

No neighbors closer to the target available

Recovery-strategy necessary (e.g. GPSR [Karp, Kung 2000])

Example:


st

?

Advance circleAdvance circle

Right hand ruleRight hand rule





Lower bounds und alternatives

Lower bound for position based routing [Kuhn et al. 2002]:

s t

Alternative strategyy: flooding

–Time: O(d)–Traffic: O(d2)

Position based single-path routingstrategies

–Time and traffic: O(d2)

Is Flooding more efficient?

Worst case analysis not useful

Time: Ω(d2)Time: Ω(d2)

Time = #Hops, Traffic = #MessagesTime = #Hops, Traffic = #Messagesd = length of shortest path (distance)d = length of shortest path (distance)





Grid networks and Unit-Disk Networks

Online routing in grid network with faulty nodes is equivalent to position based routing in wireless ad-hoc networks

Implicit geographic clustering

Partitioning of the plane into cells, empty regions = barriers

Distributed protocol for construction and routing





Finding Cells for a Unit-Disk Graph

Cell size ≤ 1/3– transmission-distance=1

Cell is NOT a barrier if– it is inside of a circle

around a node with radius 1/2

– if an edge (u,v) with |u,v| ≤ 1 touches this cell

Cell clustering– Gateways (and leader)

Two-hop communication gives a complete local view of the cell network

v

x

w

u





Lower bounds and comparative analysiss

Ω(d + p) lower bound for traffic

(online)

Instead of worst-case-analysis:Compare the algorithm with the best online-algorithm for the class of problems

Characterize the class of problems by the perimeter p and the distance d

p

Path length: Ω(d + p)Path length: Ω(d + p)

Lower bound for Online Navigation [Lumelsky, Stepanov 1987]:

d = length of shortest pathp = Perimeter of the barriersd = length of shortest pathp = Perimeter of the barriers

p

st





The Network Model

Grid network with faulty nodesFaulty blocks = barriers

Barriers are unkown (a priori),decisions need to be madeonline

Comparative analysis– Competitive time-ratio

– Comparatives traffic-ratio

StartStart

TargetTarget

PerimeterPerimeter

BarrierBarrier

Time (# Hops)

Messages





Time: O(d + p) Rt = O(d)

Time: O(d) Traffic: O(d2) RTr= O(d)

Traffic: O(d)

Single-Path versus Flooding

Single-Path (sequential)

Flooding (parallel)

StartStartStartStart

TargetTarget

d = length ofthe shortest path

AA BBNo Barriers (p<d) Maze (p=d2)

AA BB

TargetTarget

PerimeterPerimeter

O(d)

maxRt,RTr

O(d)

Is there a strategy,as fast as flooding

and with as low traffic as single-path... for all scenarios ?





Lucas Algorithm[Lucas 88]

1: repeat 2: Follow the straight line

connecting source andtarget.

3: if a barrier is hit then 4: Start a complete right-hand

traversal around the barrier and remember allpoints where the straight line is crossed.

5: Go to the crossing point that is nearest to the target.

6: end if 7: until target is reachedTime: d + 3/2 pTraffic: d + 3/2 p





Expanding Ring Search

[Johnson, Maltz 96]Start flooding with restricted

search depthRepeat flooding while doubling the

search depth until the destination is reached

Time: O(d)Traffic: O(d2)





Continuous Ring Search

Modification of Expanding Ring Search:Source starts flooding

– but with a delay of σ time steps for each hopIf the target is reached, a notification message is sent

back to the sourceThen the source starts flooding without slow-down a

second time

– Second wave is sent out to stop the first wave

Time: O(d)Traffic: O(d2)





The JITE Algorithmus

Message efficient parallel BFS (breadth first search) – using Continuous Ring Search

Just-In-Time Exploration (JITE) and Construktion of search path insteadflooding

Search paths surround barriers

Slow Search:slow BFS on a sparse grid

Fast Exploration:Construction of the sparsegrid near to the shoreline

TargetTarget

StartStart

Barrier

ShorelineShoreline





EE

E

EE

E

EE

E

E

E

E E

E

E

E

Slow Search & Fast Exploration

Slow Search visits only explored paths

Fast Exploration is started in the vicinity of the BFS-shoreline

Exploration must be terminated before a frame is reached by the BFS-shoreline

ExplorationExploration

ShorelineShoreline





• Construction of a path network for the BFS

• Partition into „Frames“

• Frame borders provide an approximation of the shortest path tree

Fast Exploration (1)

DetourDetour

entrypointentrypoint

• Frame traversal(Right hand rule)

• Time limit: If the traversal takes too long then the fram is divided into smaller frames





Fast Exploration (2)

Problems:• Exploration causes traffic

explore only frames in the vicinity of the shoreline

• Small barriers cause further subdivision (traffic!) Allow small detours

• Exploration needs time Slow down BFS-Shoreline by a

constant factor Size limit for new neighbor frames

• Multiple entry points Coordinate exploration

gallowed detour: g/(t)

allowed detour: g/(t)

E

E

E

E

E

E

E

E

E





Frame Exploration

A frame can be explored in parallel from different sides (entry points)– All messages stop after at most 2g+g/(t) rounds– If a message is stopped then no messages of type 3 or 4 occured after a

specific time• further subdivision is triggered when the messages of type 3 do not

occur in time1. Wake up:

Tell all frame border nodes about the exploration in progressFind a coordinator

2. Count: Coordinator sends counting messages

3. Stop:Frame has been explored (in time)

4. Close: Stop exploration within frame

Ø NotifyShoreline enters frame: Start exploration in neighbor frames





Slow Search

Path network/frame network gives a constant factor approximation of the shortest path tree

Constant factor slow down of the BFS-Shoreline

Allowed detours of g/(t) per gg-frame. Choose (t)= log t.For a portion of 1-1/log d of all frames we observe g/(t) = O(g/log d) (log g = 1..log d)

Target is reached in time O(d) (constant competitive ratio)

Traffic O(d + p log2 d)– O(p log d) is the size of the path network/frame network– further logarithmic factor for allowed detours

Time Traffic maxRt, RTr

Greedy (Single-Path) O(d+p) O(d+p) O(d)

Flooding O(d) O(d2) O(d)

JITE O(d) O(d + p log2 d) O(log2 d)

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Summary

New efficient strategy for position based routing

Comparative analysis for time and traffic

Lower bounds, linear trade-off

Single-Path versus Flooding

JITE Algorithm

– asymptotical as fast as flooding

– small polylogarithmic overhead for traffic

Results applicable for wireless ad-hoc-networks

23

University of FreiburgComputer Networks and Telematics

Prof. Christian Schindelhauer

Thank you

Position based Routing StrategiesChristian Schindelhauer

joint work with

Stefan RührupWorkshop of Flexible Network Design

Bertinoro, 1.-6.10.2006

to appear at ISAAC 2006

[email protected]

Online Multi-Path Routing in a Maze

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