Abstract
We analyze the Hunter vs. Rabbit game on a graph, which is a model of communication in adhoc mobile networks. Let G be a cycle graph with N nodes. The hunter can move from a vertex to a vertex along an edge. The rabbit can jump from any vertex to any vertex on the graph. We formalize the game using the random walk framework. The strategy of the rabbit is formalized using a one dimensional random walk over . We classify strategies using the order O(k −β−1) of their Fourier transformation. We investigate lower bounds and upper bounds of the probability that the hunter catches the rabbit. We found a constant lower bound if β∈(0,1) which does not depend on the size N of the graph. We show the order is equivalent to O(1/logN) if β=1 and a lower bound is 1/N (β−1)/β if β∈(1,2]. These results help us to choose the parameter β of a rabbit strategy according to the size N of the given graph. We introduce a formalization of strategies using a random walk, theoretical estimation of bounds of a probability that the hunter catches the rabbit, and also show computing simulation results.
Highlights
We consider a game played by two players: the hunter and the rabbit
The presented hunter strategy is based on random walk on a graph and it is shown that the hunter catches an unrestricted rabbit within O(nm2) rounds, where n and m denote the number of nodes and edges, respectively
If β ∈ (1, 2], the lower bound of the probability that the hunter catches the rabbit is c4N−(β−1)/β where c4 > 0 is are constant defined by the given strategy
Summary
We consider a game played by two players: the hunter and the rabbit. This game is described using a graph G(V , E) where V is a set of vertices and E is a set of edges. The presented hunter strategy is based on random walk on a graph and it is shown that the hunter catches an unrestricted rabbit within O(nm2) rounds, where n and m denote the number of nodes and edges, respectively. We have the general lower bound formula for the probability that the hunter catches the rabbit. If β = 1, the lower bound of a probability that the hunter catches the rabbit is ((c∗π )−1 log N + c2)−1 where c2 and c∗ are constants defined by the given strategy. If β ∈ (1, 2], the lower bound of the probability that the hunter catches the rabbit is c4N−(β−1)/β where c4 > 0 is are constant defined by the given strategy. If X1 takes three values −1, 0, 1 with equal probability, there exists a constant c7 > 0 such that for N ∈ N, c7 ≤ P(RN).
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