TY - JOUR

T1 - Randomized pursuit-evasion in graphs

AU - Adler, Micah

AU - Räcke, Harald

AU - Sivadasan, Naveen

AU - Sohler, Christian

AU - Vöcking, Berthold

PY - 2003/5

Y1 - 2003/5

N2 - We analyse a randomized pursuit-evasion game played by two players on a graph, a hunter and a rabbit. Let G be any connected, undirected graph with n nodes. The game is played in rounds and in each round both the hunter and the rabbit are located at a node of the graph. Between rounds both the hunter and the rabbit can stay at the current node or move to another node. The hunter is assumed to be restricted to the graph G: in every round, the hunter can move using at most one edge. For the rabbit we investigate two models: in one model the rabbit is restricted to the same graph as the hunter, and in the other model the rabbit is unrestricted, i.e., it can jump to an arbitrary node in every round. We say that the rabbit is caught as soon as hunter and rabbit are located at the same node in a round. The goal of the hunter is to catch the rabbit in as few rounds as possible, whereas the rabbit aims to maximize the number of rounds until it is caught. Given a randomized hunter strategy for G, the escape length for that strategy is the worst case expected number of rounds it takes the hunter to catch the rabbit, where the worst case is with regard to all (possibly randomized) rabbit strategies. Our main result is a hunter strategy for general graphs with an escape length of only Script O Sign(n log(diam(G))) against restricted as well as unrestricted rabbits. This bound is close to optimal since Ω(n) is a trivial lower bound on the escape length in both models. Furthermore, we prove that our upper bound is optimal up to constant factors against unrestricted rabbits.

AB - We analyse a randomized pursuit-evasion game played by two players on a graph, a hunter and a rabbit. Let G be any connected, undirected graph with n nodes. The game is played in rounds and in each round both the hunter and the rabbit are located at a node of the graph. Between rounds both the hunter and the rabbit can stay at the current node or move to another node. The hunter is assumed to be restricted to the graph G: in every round, the hunter can move using at most one edge. For the rabbit we investigate two models: in one model the rabbit is restricted to the same graph as the hunter, and in the other model the rabbit is unrestricted, i.e., it can jump to an arbitrary node in every round. We say that the rabbit is caught as soon as hunter and rabbit are located at the same node in a round. The goal of the hunter is to catch the rabbit in as few rounds as possible, whereas the rabbit aims to maximize the number of rounds until it is caught. Given a randomized hunter strategy for G, the escape length for that strategy is the worst case expected number of rounds it takes the hunter to catch the rabbit, where the worst case is with regard to all (possibly randomized) rabbit strategies. Our main result is a hunter strategy for general graphs with an escape length of only Script O Sign(n log(diam(G))) against restricted as well as unrestricted rabbits. This bound is close to optimal since Ω(n) is a trivial lower bound on the escape length in both models. Furthermore, we prove that our upper bound is optimal up to constant factors against unrestricted rabbits.

UR - http://www.scopus.com/inward/record.url?scp=0038334104&partnerID=8YFLogxK

U2 - 10.1017/S0963548303005625

DO - 10.1017/S0963548303005625

M3 - Article

AN - SCOPUS:0038334104

SN - 0963-5483

VL - 12

SP - 225

EP - 244

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 3

ER -