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A Unified View to Greedy Routing Algorithms in Ad-Hoc Networks ○Truong Minh Tien Joint work with Jinhee Chun, Akiyoshi Shioura, and Takeshi Tokuyama Tohoku University Japan

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Our Problem and Results Problem: Geometric routing in ad-hoc network. Main Results: ○Give unified view to known greedy-type routing algorithms. ○Propose new routing algorithms that works on Delaunay graphs. ○Compare previous/new algorithms from the viewpoint of guaranteed delivery, fast transmission & power consumption.

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Contents 1.Ad-hoc network and geometric routing 2.Previous geometric routing algorithms 3.Desirable properties of routing algorithms – Comparison of algorithms 4.Generalized greedy routing algorithm – New greedy-type algorithms 5.Sufficient condition for guaranteed packet delivery

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Ad-hoc Network Self-organizing network without fixed pre-existing infrastructure Communication between nodes are achieved by multi-hop links Decentralized, mobility-adaptive operation Network topology can be represented by undirected graph G=(V, E)

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Geometric Routing on Ad-hoc Network Geometric Routing on Ad-hoc network G=(V,E) Send packet from source node S to destination node T (position of T is known in advance). Packet is repeatedly sent from a node to its neighboring node. No information of entire network; only local information around current node. S V T

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Greedy Approach for Routing Algorithms Geometric Routing on Ad-hoc network G=(V,E) Greedy approach is often useful: Choose “closer” neighbor to destination in each iteration Which neighbor to choose? Greedy Routing, Compass Routing, Midpoint Routing, etc. S V T

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Contents 1.Ad-hoc network and geometric routing 2.Previous geometric routing algorithms 3.Desirable properties of routing algorithms – Comparison of algorithms 4.Generalized greedy routing algorithm – New greedy-type algorithms 5.Sufficient condition for guaranteed packet delivery

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Greedy Routing v t Finn, 1987 The next neighbor w is the node nearest to t

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Compass Routing v t Kranakis, Singh, Urrutia, 1999 Packet will be sent to w if the line vw forms with vt the smallest angle.

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Midpoint Routing m v t Si, Zomaya, 2010 Choose next neighbor w that is closest to midpoint m between v an t

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Modified Midpoint Routing mp v t Si, Zomaya, 2010 The next node w closest to p o p = t : Greedy routing o p = m : Midpoint routing

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Contents 1.Ad-hoc network and geometric routing 2.Previous geometric routing algorithms 3.Desirable properties of routing algorithms – Comparison of algorithms 4.Generalized greedy routing algorithm – New greedy-type algorithms 5.Sufficient condition for guaranteed packet delivery

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Desirable Properties of Routing Algorithms Guaranteed Delivery: It is guaranteed that a packet is delivered from source to destination. Fast Transmission: Each packet should be sent with a small number of hops. S T

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Desirable Properties of Routing Algorithms Power Consumption: Long edges should not be used as much as possible.

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Comparison of Routing Algorithms Guaranteed delivery Number of hops Power Consumption Greedy guaranteed on Delaunay graph very smallvery large Midpointsmalllarge Modified Midpoint smalllarge Compassaverage Need appropriate routing algorithm satisfying desirable properties in response to the request of applications.

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Contents 1.Ad-hoc network and geometric routing 2.Previous geometric routing algorithms 3.Desirable properties of routing algorithms – Comparison of algorithms 4.Generalized greedy routing algorithm – New greedy-type algorithms 5.Sufficient condition for guaranteed packet delivery

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Generalized Greedy Routing

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Generalized Greedy Routing: Example v t Choose next node w that minimize f (w, v, t) Example:

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Congruence-Invariant Function w’ t w t’ v’ v f is congruence-invariant if function value f (w,v,t) depends only on shape and size of.

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Congruence-Invariant Function f is congruence-invariant function if there exists a function h such that: t w v

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Greedy Routing: Min d(w, t) function t w v Compass Routing: Min function Midpoint Routing: Min d(w, m) function M. Midpoint Routing: Min d(w, p) function t w v t w v m t w v p

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New routing algorithms

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New Greedy I max function New Greedy II min function New Greedy III min function t w v t w v t w v

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Contour Map GREEDY - concentric circles about tCOMPASS – rays with same endpoint v MIDPOINT - concentric circles about m MODIFIED MIDPOINT - concentric circles about p

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Contour Map of New Routings New Greedy I – curves with same chord vt New Greedy III – circles tangent at t New Greedy II – circles tangent at v

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Comparison of Routing Algorithms Guaranteed Delivery Number of hopsPower Consumption Greedy guaranteed on Delaunay graph Very smallVery large Midpoint SmallLarge Modified Midpoint SmallLarge Compass New Greedy I Average New Greedy II LargeSmall New Greedy III SmallLarge

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Properties of New Greedy II, III If graph G contains Delaunay graph. New Greedy II : always selects Delaunay edge without calculating which edge is Delaunay edge. New Greedy III : always selects Delaunay neighbor of t if there is a two-hop path from v to t. Desired by many occasions. New Greedy II – circles tangent at v New Greedy III – circles tangent at t

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Comparison of Routing Algorithms Guaranteed Delivery Number of hopsPower Consumption Greedy guaranteed on Delaunay graph Very smallVery large Midpoint SmallLarge Modified Midpoint SmallLarge Compass New Greedy I Average New Greedy II LargeSmall New Greedy III SmallLarge

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Contents 1.Ad-hoc network and geometric routing 2.Previous geometric routing algorithms 3.Desirable properties of routing algorithms – Comparison of algorithms 4.Generalized greedy routing algorithm – New greedy-type algorithms 5.Sufficient condition for guaranteed packet delivery

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Delivery on Delaunay graph Known results: Each of greedy, compass, midpoint and modified midpoint routing guarantee delivery of packet on Delaunay graph. Our result: Sufficient condition for guaranteed delivery of generalized greedy routing on Delaunay graph.

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Delaunay Delivery Guarantee Condition (DDG) ∀ distinct nodes w, v, t ∈ P, if f(w,v,t) ≤ max{ f(u,v,t) | u ∈ D(v,t)}, then d(w,t) < d(v,t) holds d(a,b) : distance between a and b D(v,t): open disk of diameter d(v,t)

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v t u w C : open disk with diameter vt D : open disk with radius tv about t A = max{f(u, v, t) | u ∈ C} DDG Condition : For all w with f(w, v, t) ≤ A ; w ∈ D C D DDG Condition

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v t u w C : open disk with diameter vt Strong DDG Condition : For all and Strong DDG implies DDG C Strong DDG Condition

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Theorem. f is a function satisfying (strong) DDG condition. The algorithm with function f guarantees packet delivery on Delaunay triangulations. Delivery Guarantee on Delaunay triangulations t u w v

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Theorem. Greedy Routing, Midpoint Routing and Modified Midpoint Routing satisfy DDG condition Routing Algorithms and DDG Condition Theorem. New Greedy Routing I, II, III satisfy Strong DDG condition Guarantee delivery of packet on Delaunay graphs

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Example: New Greedy I on Delaunay triangulation S T

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Hybrid of algorithms Theorem. If f and g satisfy (strong) DDG condition, af+bg (a,b>0) also satisfies (strong) DDG condition. Corresponding algorithm guarantees delivery on Delaunay triangulation Possible to design appropriate hybrid of algorithms based on requirement of application.

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Conclusion Our Problem: Geometric routing in Ad-hoc network Our Results: ○We gave unified view to known greedy-type routing algorithms. ○We proposed new routing algorithms that works on Delaunay graphs. ○We compared previous/new algorithms from the viewpoint of guaranteed delivery, fast transmission, & power consumption.

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Future Work o Consider a metric space with the existence of obstacles and other natural/social conditions in real ad hoc network design.

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Thank You

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