A Note on the k -Canadian Traveller Problem Stephan Westphal University of Kaiserslautern, Department of Mathematics, P.O.Box 3049, Paul-Ehrlich-Str. 14, 67653 Kaiserslautern, Germany. westphal@mathematik.uni-kl.de Abstract We consider the online problem k-CTP, which is the problem to guide a vehicle from some site s to some site t on a road map given by a graph G =(V,E) in which up to k (unknown) edges are blocked by avalanches. An online algorithm learns from a blocked edge when reaching one of its endpoints. Thus, it might have to change its route to the target t up to k times. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 2k + 1 and give an easy algorithm which matches this lower bound. Furthermore, we show that randomization can not improve the competitive ratio substantially, by establishing a lower bound of k + 1 for the competitivity of randomized online algorithms against an oblivious adversary. Key words: routing, on-line algorithms, competitiveness 1 Introduction The shortest path problem is a well-studied problem in combinatorial opti- mization. Given an undirected graph G =(V,E) with two nodes s and t and a cost function d : E → R + representing the time it takes to traverse the edges, one seeks to determine a shortest path from s to t (with respect to d). We consider the following online variant of the problem, in which some of the edges of G are blocked, and an online algorithm only learns from the block- ing of an edge when reaching one of its endpoints. Whenever a blocked edge which is part of the planned route is reached, it cannot be passed, and for this reason the remaining path has to be changed. This problem is called the Cana- dian Traveller Problem (CTP) and has been introduced by Papadimitriou and Yannakakis [1]. A deterministic online algorithm alg for k-CTP is c-competitive [2, 3], if for any input σ the total length alg(σ) of the the s-t-path produced by alg on Preprint submitted to Elsevier 12 March 2007