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

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.

Contents Ad-hoc network and geometric routing Previous geometric routing algorithms Desirable properties of routing algorithms Comparison of algorithms Generalized greedy routing algorithm New greedy-type algorithms Sufficient condition for guaranteed packet delivery

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) First, I will explain briefly about wireless ad hoc network.

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. The routing algorithm used in the ad hoc network is geometric routing. More specifically, geometric routing is a routing algorithm from the source S to the destination T that satisfies: T S V

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. The routing algorithm used in the ad hoc network is geometric routing. More specifically, geometric routing is a routing algorithm from the source S to the destination T that satisfies: T S V

Contents Ad-hoc network and geometric routing Previous geometric routing algorithms Desirable properties of routing algorithms Comparison of algorithms Generalized greedy routing algorithm New greedy-type algorithms Sufficient condition for guaranteed packet delivery

Greedy Routing v t Well, the first and most popular routing algorithm is greedy routing , that was proposed by Finn in 1987. The greedy routing algorithm is based on the idea that we want to get as closer to the destination as possible. Therefore when the packet arrives at V, the next neighbor that we want to send the packet to is the node nearest to the destination. In the figure, the next neighbor is w4 Finn, 1987 The next neighbor w is the node nearest to t smallest 𝑑𝑖𝑠𝑡(𝑤,𝑡)

Compass Routing v t Kranakis, Singh, Urrutia, 1999 smallest ∠𝑤𝑣𝑡 v t The next routing algorithm is compass routing proposed in 1999. In the compass routing, among all the neighbors of V, the packet will be sent to w if the line vw form with the line vt the smallest angle. Kranakis, Singh, Urrutia, 1999 Packet will be sent to w if the line vw forms with vt the smallest angle.

Midpoint Routing v m t Si, Zomaya, 2010 smallest 𝑑𝑖𝑠𝑡(𝑤,𝑚) v m t Next is midpoint routing, proposed recently by Si and Zomaya in 2010. Here we will choose the next neighbor w that is closest to the midpoint m, the midpoint between v and t. Si, Zomaya, 2010 Choose next neighbor w that is closest to midpoint m between v an t

Modified Midpoint Routing v m p t smallest 𝑑𝑖𝑠𝑡(𝑤,𝑝) Finally, we consider the Modified midpoint routing. Let p be any node between m and t and the algorithms will choose the next neighbor that is closest to p. We notice that if p coincide t, the algorithm will become greedy routing. If p coincide m, the algorithm will become midpoint routing. Si, Zomaya, 2010 The next node w closest to p p = t : Greedy routing p = m : Midpoint routing

Contents Ad-hoc network and geometric routing Previous geometric routing algorithms Desirable properties of routing algorithms Comparison of algorithms Generalized greedy routing algorithm New greedy-type algorithms Sufficient condition for guaranteed packet delivery

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

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

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

Contents Ad-hoc network and geometric routing Previous geometric routing algorithms Desirable properties of routing algorithms Comparison of algorithms Generalized greedy routing algorithm New greedy-type algorithms Sufficient condition for guaranteed packet delivery

Generalized Greedy Routing Unify greedy-type routing algorithms using general objective function. Obtain better understanding of previous algorithms. Propose new algorithms. T = {(w ,v ,t) | w ,v ,t: distinct nodes} (w: next node, v: current node, t: terminal node) General objective function Generalized greedy routing: Choose a neighbor w of v that minimizes f (w, v, t) in each iteration Let T be the set of triplets of w, v and t, where w, v, and t are distinct nodes. We define the general objective function f to be function from T to the union of R and infinity. Here we can unify greedy routing algorithms by generalized greedy routing with objective functions.

Generalized Greedy Routing: Example Choose next node w that minimize f (w, v, t) Example: 7 3 v t 2 The algorithm is that: we choose the next node w that minimize the function value f(w,v,t). In the example here, the function values are 7, 3, 2, and infinity, so we choose w3 to be next node on the route to the destination. +∞

Congruence-Invariant Function w t v w’ v’ In particular, we consider a restricted class of objective functions, that is congruence-invariant function. We say that f is congruence-invariant if the function value f(w ,v ,t) depends only on the shape and size of the triangle wvt. In other words, the function value does not change if we transform the triangle wvt by a combination of translation, rotation and reflection. f is congruence-invariant if function value f (w ,v ,t) depends only on shape and size of .

Congruence-Invariant Function f is congruence-invariant function if there exists a function h such that: w More precisely, f is said to be congruence-invariant function if there exists a function h such that … Now we show that many existing greedy routing algorithms that I explained earlier are special cases of generalized greedy routing. t v

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

New routing algorithms

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

Contour Map GREEDY - concentric circles about t COMPASS – rays with same endpoint v Figures here show the contour maps of the corresponding functions of these four routings. For the greedy routing, it’s a set of concentric circles about t. For compass routing it’s a set of rays with same endpoint v. For the midpoint routing, it’s a set of concentric circles about m. For the modified midpoint routing, it’s a set of concentric circles about p. Next we will propose 3 new routing algorithms that are also special cases of generalized greedy routing. MIDPOINT - concentric circles about m MODIFIED MIDPOINT - concentric circles about p

Contour Map of New Routings New Greedy I – curves with same chord vt Figures here show the contour maps of the corresponding functions of these four routings. For the greedy routing, it’s a set of concentric circles about t. For compass routing it’s a set of rays with same endpoint v. For the midpoint routing, it’s a set of concentric circles about m. For the modified midpoint routing, it’s a set of concentric circles about p. Next we will propose 3 new routing algorithms that are also special cases of generalized greedy routing. New Greedy II – circles tangent at v New Greedy III – circles tangent at t

Comparison of Routing Algorithms Guaranteed Delivery Number of hops Power Consumption Greedy guaranteed on Delaunay graph Very small Very large Midpoint Small Large Modified Midpoint Compass New Greedy I Average New Greedy II New Greedy III

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 Let T be the set of triplets of w, v and t, where w, v, and t are distinct nodes. We define the general objective function f to be function from T to the union of R and infinity. Here we can unify greedy routing algorithms by generalized greedy routing with objective functions. New Greedy III – circles tangent at t

Comparison of Routing Algorithms Guaranteed Delivery Number of hops Power Consumption Greedy guaranteed on Delaunay graph Very small Very large Midpoint Small Large Modified Midpoint Compass New Greedy I Average New Greedy II New Greedy III

Contents Ad-hoc network and geometric routing Previous geometric routing algorithms Desirable properties of routing algorithms Comparison of algorithms Generalized greedy routing algorithm New greedy-type algorithms Sufficient condition for guaranteed packet delivery

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.

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) The DGS condition is that : For every distinct nodes w, v and t of P such that w and v are adjacent and t and v are not adjacent, if … , then the distance between w and t is less than the distance between v and t. Here c is the midpoint between v and t.

DDG Condition D 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 w C v t u We can see how the DGS condition works in this figure.

Strong DDG Condition w C : open disk with diameter vt For all 　　　　　and Strong DDG implies DDG C v t u We can see how the DGS condition works in this figure.

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

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

Example: New Greedy I on Delaunay triangulation S Or by using midpoint routing And it can also be delivered by using modified midpoint routing, and New greedy I, II and III T

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. Now we move on to the second part of finding the sufficient condition for delivery guarantee on the Delaunay triangulation.

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.

Future Work Consider a metric space with the existence of obstacles and other natural/social conditions in real ad hoc network design. That concludes my talk. Thank you for your attention

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