Fast Geometric Routing with Concurrent Face Traversal

Tom ClouserMark MiyashitaMikhail Nesterenko

Kent State University

OPODISDecember 17, 2008

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Geometric Routing: Routing without Overhead

• static nodes, each node knows its global coordinates, source knows coords of destination• little overhead

no routing tables – each node only knows coords of neighors no memory – no info kept at node after message is routed no message overhead – messages of constant size no global knowledge

• simple approaches flooding – expensive greedy routing

message carries coords of dest. each node forwards to

neighbor closer todestination

problem: local minimum what if no closer neighbor?

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Outline

• terms & assumptions• planar graph routing

sequential face routing concurrent face routing (CFR)

motivation & description example operation correctness & optimality performance evaluation

• non-planar routing motivation & description performance evaluation

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• face – area where any two points can be connected by a line non-intersecting graph edges

single infinite external face

Graph Terms

• unit disk graph – two vertices are adjacent iff they are within unit distance approximates radio model

• planar (embedding) graph – can be effectively constructed from a unit disk graph

• source (s), destination (d) vertices, sd-line coords assumed carried by message

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Routing Terms and Assumptions

• right-hand-rule – if node receives left message, node forwards message to next clockwise from sender node traverses internal face clockwise similar left-hand-rule

• juncture – vertex adjacent to edge intersecting sd-line• adjacent faces – intersecting sd-line and sharing a juncture• message cost – # of messages sent• path stretch – ratio to shortest path

• assumptions reliable transmission asynchronous communication zero channel capacity single send queue per vertex atomic message sent/receipt

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Outline

• terms & assumptions• planar graph routing

sequential face routing concurrent face routing (CFR)

motivation & description example operation correctness & optimality performance evaluation

• non-planar routing motivation & description performance evaluation

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COMPASS [KSU99]

traverse entire face, find furthest adjacent face, switch to it

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• message cost O(|E|)• path stretch ?

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FACE [BMSU01,DSW02]

traverse face, switch as soon as juncture is found (cross sd-line and change traversal direction)

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• message cost O(|V|2)• path stretch ?

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GPSR [KK00]

variant: traverse face, switch as soon as juncture is found (do not cross sd-line keep traversal direction)

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• message cost O(|V|2)• path stretch ?

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OAFR [KWZ03a]

traverse in one direction, if found bounding ellipse, change direction, if bounded in both directions, double ellipse size

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• message cost: outperforms others on average• path stretch: O(ρ2) – optimal

- where ρ is shortest path

bounding ellipse E

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Combination of Greedy and Face Routing

• guarantees delivery while shortening the path• go greedy until local minimum is encountered• in local minimum – switch to face routing• switch back to greedy if closer to destination than this local minimum• adding greedy generates combined geometric algorithms

GFG [BMSU01,DSW02] GPSR [KK00] GOAFR+ [KWZZ03]

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What Is Wrong with Sequential Face Routing?

• traversing a face in one direction may be significantly longer than the other esp. external face

• idea: send messages in all directions concurrently• issues

have to handle multiple messages in the same face what to do when one of the messages reaches junction vertex?

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CFR Description & Operation

• source injects left and right msgs in each face intersecting sd-line

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CFR Description & Operation

• source injects left and right msgs in each face intersecting sd-line

• when juncture receives msg, it forwards the msg and injects left/right pair in each adjacent face

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CFR Description & Operation

• source injects left and right msgs in each face intersecting sd-line

• when juncture receives msg, it forwards the msg and injects left/right pair in each adjacent face

• matching – left/right messages at a vertex and at least one is not originated at this vertex

• matching messages are destroyed

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CFR Description & Operation

• source injects left and right msgs in each face intersecting sd-line

• when juncture receives msg, it forwards the msg and injects left/right pair in each adjacent face

• matching – left/right messages at a vertex and at least one is not originated at this vertex

• matching messages are destroyed

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CFR Description & Operation

• source injects left and right msgs in each face intersecting sd-line

• when juncture receives msg, it forwards the msg and injects left/right pair in each adjacent face

• matching – left/right messages at a vertex and at least one is not originated at this vertex

• matching messages are destroyed

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CFR Description & Operation

• source injects left and right msgs in each face intersecting sd-line

• when juncture receives msg, it forwards the msg and injects left/right pair in each adjacent face

• matching – left/right messages at a vertex and at least one is not originated at this vertex

• matching messages are destroyed

• destination delivers and forwards the message, but still forwards it

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CFR Description & Operation

• source injects left and right msgs in each face intersecting sd-line

• when juncturedestination delivers the message, but still forwards it

• receives msg, it forwards the msg and injects left/right pair in each adjacent face

• matching – left/right messages at a vertex and at least one is not originated at this vertex

• matching messages are destroyed

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CVR

GCFR: Greedy + CFR

• can start in greedy mode and then switch to CFR in local minimum

• cannot switch back due to multiple concurrent messages

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CFR Correctness & Optimality

• correctness proof: every edge of the face is visited exactly onceCorollary: The message cost

of CFR is in O(|E|).

• latency path is in O(ρ2) • latency for any such algorithm

is in O(ρ2)

CFR latency path is optimal

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Model Description

• replicated [KWZ03a]• implemented CFR and others

20x20 units square field uniformly distributed nodes

according to density edges according to unit-disk

graph 21 density levels each experiment – a new graph

and a source destination pair 2,000 experiments per level

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example graph, latency path of CFR,density is 5,

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Model Observations

[KWZ03a] observed

too sparse s and d are either disconnected or adjacent

critical density region

too densepath is straightgreedy succeeds

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Pure Face Routing, Path Stretch

path stretch = latency path/shortest path

~ 5 timesbetter thanbest serial

alg

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Face+Greedy, Path Stretch

path stretch = latency path/shortest path

~ 2.5 timesbetter

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Pure Face, Message Cost

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Face+Greedy, Message Cost

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Pure Face, Message Cost, Normalized to Flooding

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Face+Greedy, Message Cost, Normalized to Flooding

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Outline

• terms & assumptions• planar graph routing

sequential face routing concurrent face routing (CFR)

motivation & description example operation correctness & optimality performance evaluation

• non-planar routing motivation & description performance evaluation

3112/17/2008 OPODIS

Radio Networks are Not Unit-Disk

• realistic radio message propagation

patterns are highly irregular

• unit-disk graph model is inadequate

[Culler, D., UCB]

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Traversing Non-Planar Graphs

idea: follow the segment of the edge that borders the void [VN05]

• nodes have to store info on edges intersecting each adjacent edge

two parts• edge_change message sent to

node adjacent to next segment edge, node selects beginning of next segment (next intersecting edge)the selection minimizes the currentedge segment

• sends edge_selection message to the other adjacent node to confirm selection and forward message to node adjacent tonext segment edge

• algorithms VOID GVG CVR GCVR

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Non-Planar Model

quasi-unit-disk graph [BFNO03, KWZ03b]

u and v are• adjacent if |uv| ≤ 0.75• adjacent with probability 0.5 if

0.75 <|uv| ≤ 1• not adjacent if |uv| > 1

21 density levels3000 experiments at each

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Non-Planar, Path Stretch

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Related Literature

[BFNO03] Barrière, L., Fraignaud, P., Narayannan, L., Opatrny, J., “Robust Position-Based Routing in Wireless Ad Hoc Networks with Irregular Transmission Ranges”, Wireless Communication and Mobile Computing 3(2), pp. 141-153, 2003

[BMSU01] Bose, P., Mortin, P., Stojmenovic, I., Urrutia,. “Routing with guaranteed Delivery in Ad Hoc Wireless Networks”, The Journal of Mobile Communication, Computation and Information 7(6), pp. 48-55, 2001

[DSW02] Datta, S., Stojmenovic, I., Wu, J., “Internal Node and Shortcut Based Routing with Guaranteed Delivery in Wireless Networks, Cluster Computing, 5(2), pp. 169-178, 2002

[KK00] Karp, B., Kung, H., “GPSR: Greey Perimeter Stateless Routing for Wireless Networks”, MobiCom, pp. 243-254, August 2000

[KSU99] Kranakis, E., Singh, H., Urrutia, J. “Compass Routing on Geometric Networks”, Canadian Conference on Computational Geometry, pp. 51-54, August 1999, Vancouver, Canada

[KWZ02] Kuhn, F., Wattenhofer, R., Zollinger, A., “Asymptotically Optimal Geometric Mobil Ad-Hoc Routing, Dial-M, September 2002, Atlanta, GA

[KWZ03a] Kuhn, F., Wattenhofer, R., Zollinger, A., “Worst-Case Optimal and Average-Case Efficient Geometric Ad-Hoc Routing”, MobiHoc, pp. 267-278, Annapoils, MD, July 2003

[KWZ03b] Kuhn, F., Wattenhofer, R., Zollinger, A., “Ad-Hoc Networks Beyond Unit Disk Graphs”, DialM-POMC, pp. 69-78, September 2003, San Diego, CA

[KWZZ03] Kuhn, F., Wattenhofer, R., Zhang,Y., Zollinger, A., “Geometric Ad-Hoc Routing: Of Theory and Practice”, PODC, July 2003

[VN05] Vora, A., Nesterenko, M., “Void Traversal for Guaranteed Delivery in Geometric Routing”, MASS, pp. 63-67, Washington, DC, November 2005.

Fast Geometric Routing withConcurrent Face Traversal

Thomas Clouser

Mark MiyashitaMikhail Nesterenko

thank you

summary• presented CFR• matches theoretical performance bounds• significantly speeds up delivery in both planar and non-planar face traversal• modest message cost increase