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

Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a function $x : V(G) \to X$ such that in the embedding for every pair of non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent to $x(s)$ which is closer to $x(t)$ than $x(s)$. This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph $G = (V,E)$, an embedding x: V \to \bbbr^2 of $G$ is a planar convex greedy embedding if and only if, in the embedding $x$, weight of the maximum weight spanning tree ($T$) and weight of the minimum weight spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq (\card{V}-1)^{1 - \delta}, for some $0 < \delta \leq 1$.Comment: 19 pages, A short version of this paper has been accepted for presentation in FCT 2009 - 17th International Symposium on Fundamentals of Computation Theor

Upper and Lower Bounds for Competitive Online Routing on Delaunay Triangulations

Consider a weighted graph G where vertices are points in the plane and edges are line segments. The weight of each edge is the Euclidean distance between its two endpoints. A routing algorithm on G has a competitive ratio of c if the length of the path produced by the algorithm from any vertex s to any vertex t is at most c times the length of the shortest path from s to t in G. If the length of the path is at most c times the Euclidean distance from s to t, we say that the routing algorithm on G has a routing ratio of c.We present an online routing algorithm on the Delaunay triangulation with competitive and routing ratios of 5.90. This improves upon the best known algorithm that has competitive and routing ratio 15.48. The algorithm is a generalization of the deterministic 1-local routing algorithm by Chew on the L1-Delaunay triangulation. When a message follows the routing path produced by our algorithm, its header need only contain the coordinates of s and t. This is an improvement over the currently known competitive routing algorithms on the Delaunay triangulation, for which the header of a message must additionally contain partial sums of distances along the routing path.We also show that the routing ratio of any deterministic k-local algorithm is at least 1.70 for the Delaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case of the L1-Delaunay triangulation, this implies that even though there exists a path between two points x and y whose length is at most 2.61|[xy]| (where |[xy]| denotes the length of the line segment [xy]), it is not always possible to route a message along a path of length less than 2.70|[xy]|. From these bounds on the routing ratio, we derive lower bounds on the competitive ratio of 1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations

Routing with Guaranteed Delivery on Virtual Coordinates

We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for non-triangular graphs, and their virtual coordinates are easy to construct.Engineering and Applied Science

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

Constant memory routing in quasi-planar and quasi-polyhedral graphs

AbstractWe address the problem of online route discovery for a class of graphs that can be embedded either in two or in three-dimensional space. In two dimensions we propose the class of quasi-planar graphs and in three dimensions the class of quasi-polyhedral graphs. In the former case such graphs are geometrically embedded in R2 and have an underlying backbone that is planar with convex faces; however within each face arbitrary edges (with arbitrary crossings) are allowed. In the latter case, these graphs are geometrically embedded in R3 and consist of a backbone of convex polyhedra and arbitrary edges within each polyhedron. In both cases we provide a routing algorithm that guarantees delivery. Our algorithms need only “remember” the source and destination nodes and one (respectively, two) reference nodes used to store information about the underlying face (respectively, polyhedron) currently being traversed. The existence of the backbone is used only in proofs of correctness of the routing algorithm; the particular choice is irrelevant and does not affect the behaviour of the algorithm

BLR: Beacon-Less Routing Algorithm for Mobile Ad-Hoc Networks

Routing of packets in a mobile ad-hoc network with a large number... this paper is a routing protocol that makes use of location information to reduce routing overhead. However, unlike other position-based routing protocols, BLR does not require nodes to periodically broadcast Hello-messages (called beaconing), and thus avoids drawbacks such as extensive use of scarce battery-power, interferences with regular data transmission, and performance degradation. BLR selects a forwarding node in a distributed manner among all its neighboring nodes with having information neither about their positions nor even about their existence. Data packets are broadcasted and the protocol takes care that just one of the receiving nodes forwards the packet. Optimized forwarding is achieved by applying a concept of Dynamic Forwarding Delay (DFD). Consequently, the node which computes the shortest forwarding delay relays the packet first. This forwarding is detected by the other nodes and suppresses them to relay the same packet any further. Analytical results and simulation experiments indicate that BLR provides efficient and robust routing in highly dynamic mobile ad-hoc networks

La marche céleste: une marche oublieuse dans les subdivisions convexes à terminaison garantie

We present a new oblivious walking strategy for convex subdivisions. Our walk isfaster than the straight walk and more generally applicable than the visiblity walk. To provetermination of our walk we use a novel monotonically decreasing distance measure.Nous présentons une nouvelle stratégie de marche pour les subdivisions convexes.Cette stratégie est oublieuse, c’est à dire que la prochaine cellule visitée ne dépends pas des cellulesvisitées précédemment. Notre marche est plus rapide que la marche rectiligne et s’applique à dessubdivisions plus générales que la marche par visibilité. La démonstration de terminaison reposesur la décroissance monotone d’une nouvelle distance mesurant le progrès de la march