An Optimal Competitive Strategy for Walking in Streets

Abstract

A simple polygon P with two distinguished vertices, s and t, is called a street if the two boundary chains from s to t are mutually weakly visible. We present an on-line strategy that walks from s to t, in any unknown street, on a path at most $\sqrt{2}$ times longer than the shortest path. This matches the best lower bound previously known and settles an open problem in the area of competitive path planning. (The result was simultaneously and independently obtained by the first three authors and by the last two authors. Both papers, [C. Icking, R. Klein, and E. Langetepe, Proceedings of the 16th Symposium on Theoretical Aspects in Computer Science, Lecture Notes in Comput. Sci. 1563, Springer-Verlag, Berlin, 1999, pp. 110--120] and [S. Schuierer and I. Semrau, Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science, pp. 121--131], were presented at STACS'99. The present paper contains a joint full version.)