Analysis of Heuristics for the Freeze-Tag Problem

Abstract

In the Freeze Tag Problem (FTP) we are given a swarm of n asleep (frozen or inactive) robots and a single awake (active) robot, and we want to awaken all robots in the shortest possible time. A robot is awakened when an active robot “touches” it. The goal is to compute an optimal awakening schedule such that all robots are awake by time t*, for the smallest possible value of t*. We devise and test heuristic strategies on geometric and network datasets. Our experiments show that all of the strategies perform well, with the simple greedy strategy performing particularly well. A theoretical analysis of the greedy strategy gives a tight approximation bound of Θ(√log n) for points in the plane. We show more generally that the (tight) performance bound is Θ((log n)^1-1/d) in d dimensions. This is in contrast to general metric spaces, where greedy is known to have a Θ(log n) approximation factor, and no method is known to achieve an approximation bound of o(log n).