Memoryless Routing in Convex Subdivisions: Random Walks are Optimal

TL;DR

This paper demonstrates that memoryless routing algorithms in convex subdivisions are fundamentally limited, with random walks being optimal, and shows that certain triangulations cause exponential routing times for specific algorithms.

Contribution

It establishes lower bounds on the efficiency of memoryless routing algorithms and identifies cases where these algorithms perform exponentially poorly.

Findings

Memoryless algorithms have an Omega(n^2) lower bound in convex subdivisions.

Random walks are asymptotically optimal among memoryless algorithms.

Some triangulations cause the Random-Compass algorithm to require exponential time.

Abstract

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.