Memoryless Routing in Convex Subdivisions: Random Walks are Optimal

Details Memoryless Routing in Convex Subdivisions: Random Walks are Optimal Memoryless Routing in Convex Subdivisions: Random Walks are Optimal

2009-11-12

arxiv.org/abs/0911.2484v1

Computational Geometry: Theory and Applications, Volume 45, Issue 4, May 2012, Pages 178-185

Computer Science

Computational Geometry

11 pages, 6 figures

Scientific paper

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.

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