Memoryless Routing in Convex Subdivisions: Random Walks are Optimal

A memoryless routing algorithm is one in which the decision about the next edge on the route to a vertex t for a packet currently located at vertex v is made based only on the coordinates of v, t, and the neighbourhood, N(v), of v. The current paper explores the limitations of such algorithms by showing that, for any (randomized) memoryless routing algorithm A, there exists a convex subdivision on which A takes Omega(n^2) expected time to route a message between some pair of vertices. Since this lower bound is matched by a random walk, this result implies that the geometric information available in convex subdivisions is not helpful for this class of routing algorithms. The current paper also shows the existence of triangulations for which the Random-Compass algorithm proposed by Bose etal (2002,2004) requires 2^{\Omega(n)} time to route between some pair of vertices.Comment: 11 pages, 6 figure

Efficiently Navigating a Random Delaunay Triangulation

International audiencePlanar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. Whilst many algorithms have been proposed, very little theoretical analysis is available for the properties of the paths generated or the computational resources required to generate them. In this work, we propose and analyse a new planar navigation algorithm for the Delaunay triangulation. We then demonstrate a number of strong theoretical guarantees for the algorithm when it is applied to a random set of points in a convex region

Angle constrained paths in sensor networks

Short-length paths in geometric graphs are not necessarily feasible in sensor networks and robotics. Paths with sharp-turn angles cannot be used by robotic vehicles and tend to consume more energy in sensor networks; In this thesis, we investigate the development of short-length paths without sharp-turn angles. We present a critical review of existing algorithms for generating angle constrained paths. We then consider the construction of routes having directional properties---d-monotone routes which are special cases of angle constrained paths. We develop a centralized algorithm for computing shortest d-monotone paths in triangulated networks. Since local computations are highly desired in sensor networks, we also consider localized online algorithms for computing length-reduced d-monotone paths in Delaunay Triangulation networks; The proposed algorithms are implemented in the Java programming language. Performances of the proposed algorithms are evaluated by examining the routes constructed by them on several randomly-generated Delaunay networks

Competitive online routing in geometric graphs

AbstractWe consider online routing algorithms for finding paths between the vertices of plane graphs. Although it has been shown in Bose et al. (Internat. J. Comput. Geom. 12(4) (2002) 283) that there exists no competitive routing scheme that works on all triangulations, we show that there exists a simple online O(1)-memory c-competitive routing strategy that approximates the shortest path in triangulations possessing the diamond property, i.e., the total distance travelled by the algorithm to route a message between two vertices is at most a constant c times the shortest path. Our results imply a competitive routing strategy for certain classical triangulations such as the Delaunay, greedy, or minimum-weight triangulation, since they all possess the diamond property. We then generalize our results to show that the O(1)-memory c-competitive routing strategy works for all plane graphs possessing both the diamond property and the good convex polygon property

Expected Complexity of Routing in Θ6\Theta_6 and Half-Θ6\Theta_6 Graphs

We study online routing algorithms on the Θ6-graph and the half-Θ6-graph (which is equivalent to a variant of the Delaunay triangulation). Given a source vertex s and a target vertex t in the Θ6-graph (resp. half-Θ6-graph), there exists a deterministic online routing algorithm that finds a path from s to t whose length is at most 2 st (resp. 2.89 st) which is optimal in the worst case [Bose et al., siam J. on Computing, 44(6)]. We propose alternative, slightly simpler routing algorithms that are optimal in the worst case and for which we provide an analysis of the average routing ratio for the Θ6-graph and half-Θ6-graph defined on a Poisson point process. For the Θ6-graph, our online routing algorithm has an expected routing ratio of 1.161 (when s and t random) and a maximum expected routing ratio of 1.22 (maximum for fixed s and t where all other points are random), much better than the worst-case routing ratio of 2. For the half-Θ6-graph, our memoryless online routing algorithm has an expected routing ratio of 1.43 and a maximum expected routing ratio of 1.58. Our online routing algorithm that uses a constant amount of additional memory has an expected routing ratio of 1.34 and a maximum expected routing ratio of 1.40. The additional memory is only used to remember the coordinates of the starting point of the route. Both of these algorithms have an expected routing ratio that is much better than their worst-case routing ratio of 2.89

Celestial Walk: A Terminating, Memoryless Walk for Convex Subdivisions

International audienceA common solution for routing messages or performing point location in planar subdivisions consists in walking from one face to another using neighboring relationships. If the next face does not depend on the previously visited faces, the walk is called memoryless. We present a new memoryless strategy for convex subdivisions. The known alternatives are straight walk, which is a bit slower and not memoryless, and visibility walk, which is guaranteed to work properly only for Delaunay triangulations. We prove termination of our walk using a novel distance measure that, for our proposed walking strategy, is strictly monotonically decreasing

Navigation on a Poisson point process

On a locally finite point set, a navigation defines a path through the point set from one point to another. The set of paths leading to a given point defines a tree known as the navigation tree. In this article, we analyze the properties of the navigation tree when the point set is a Poisson point process on $\mathbb{R}^d$. We examine the local weak convergence of the navigation tree, the asymptotic average of a functional along a path, the shape of the navigation tree and its topological ends. We illustrate our work in the small-world graphs where new results are established.Comment: Published in at http://dx.doi.org/10.1214/07-AAP472 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

La pire marche par visibilité dans une triangulation de Delaunay de points aléatoires est en O(n)O(\sqrt{n})

We show that the memoryless routing algorithms Greedy Walk, Compass Walk,and all variants of visibility walk based on orientation predicates are asymptotically optimalin the average case on the Delaunay triangulation. Morespecifically, we consider the Delaunay triangulation of an unbounded Poisson pointprocess of unit rate and demonstrate that the worst-case path between any twovertices inside a domain of area $n$ has a number of steps that is not asymptotically morethan the shortest path which exists between those two vertices with probabilityconverging to one (as long as the vertices are sufficiently far apart.) As acorollary, it follows that the worst-case path has $O(\sqrt{n}\,)$ steps in thelimiting case, under the same conditions. Our results have applications inrouting in mobile networks and also settle a long-standing conjecture in pointlocation using walking algorithms. Our proofs use techniques frompercolation theory and stochastic geometry.Nous montrons que les algorithmes de routage sans mémoire de marchegloutonne, de marche au compas et toutes les variantes de marche parvisibilité sont asymptotiquement optimale en moyenne pour latriangulation de Delaunay.Plus précisément, nous considérons la triangulation de Delaunay d'unprocessus de Poisson non borné d'intensité un et démontrons que lerapport entre les nombre d'étapesdu pire et du meilleur chemin entre deux sommets suffisamment loin dans un domaine d'aire $n$est borné par une constante avec une probabilité convergeant vers 1.On en déduit comme corollaire que le pire chemin a au plus$O(\sqrt{n}\,)$ étapes.Ce résultat a des applications au routage dans les réseaux mobiles etréponds à une conjecture sur les algorithmes de localisation parmarche dans les triangulations.Nos démonstrations utilisent des résultats de percolation et degéométrie stochastique