Improved Routing on the Delaunay Triangulation

A geometric graph G=(P,E) is a set of points in the plane and edges between pairs of points, where the weight of an edge is equal to the Euclidean distance between its two endpoints. In local routing we find a path through G from a source vertex s to a destination vertex t, using only knowledge of the current vertex, its incident edges, and the locations of s and t. We present an algorithm for local routing on the Delaunay triangulation, and show that it finds a path between a source vertex s and a target vertex t that is not longer than 3.56|st|, improving the previous bound of 5.9|st|

The Stretch Factor of L1L_1- and L∞L_\infty-Delaunay Triangulations

In this paper we determine the stretch factor of the $L_1$-Delaunay and $L_\infty$-Delaunay triangulations, and we show that this stretch is $\sqrt{4+2\sqrt{2}} \approx 2.61$. Between any two points $x,y$ of such triangulations, we construct a path whose length is no more than $\sqrt{4+2\sqrt{2}}$ times the Euclidean distance between $x$ and $y$, and this bound is best possible. This definitively improves the 25-year old bound of $\sqrt{10}$ by Chew (SoCG '86). To the best of our knowledge, this is the first time the stretch factor of the well-studied $L_p$-Delaunay triangulations, for any real $p\ge 1$, is determined exactly

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

Improvement of the robustness on geographical networks by adding shortcuts

In a topological structure affected by geographical constraints on liking, the connectivity is weakened by constructing local stubs with small cycles, a something of randomness to bridge them is crucial for the robust network design. In this paper, we numerically investigate the effects of adding shortcuts on the robustness in geographical scale-free network models under a similar degree distribution to the original one. We show that a small fraction of shortcuts is highly contribute to improve the tolerance of connectivity especially for the intentional attacks on hubs. The improvement is equivalent to the effect by fully rewirings without geographical constraints on linking. Even in the realistic Internet topologies, these effects are virtually examined.Comment: 14 pages, 10 figures, 1 tabl

There are Plane Spanners of Maximum Degree 4

Let E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a t-spanner, or simply a spanner, if for any pair of vertices u,v in E the distance between u and v in G is at most t times their distance in E. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper we show that the complete Euclidean graph always contains a plane spanner of maximum degree at most 4 and make a big step toward closing the question. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's L1-Delaunay triangulation

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