Multiple Round Random Ball Placement: Power of Second Chance

Abstract

The traditional coupon collector’s problem studies the number of balls required to fill n bins if the balls are placed into bins uniformly at random. It is folklore that Θ(nln n) balls are required to fill the bins with high probability (w.h.p.). In this paper, we study a variation of the random ball placement process. In each round, we assume the ability to acquire the set of empty bins after previous rounds and exclusively place balls into them uniformly at random. For such a k-round random ball placement process (k -RBP), we derive a sharp threshold of nln ^[k] n balls for filling n bins.

We apply the bounds of k -RBP to the wireless sensor network deployment problem. Assume the communication range for the sensors is r and the deployment region is a 2D unit square. Let n = (1/r)². We show that the number of random nodes needed to achieve connectivity is Θ(nln ln n) if we are given a “second chance” to deploy nodes, improving the previous Θ(nln n) bounds [8] in the one round case. More generally, under certain deployment assumption, if the random deployment in i-th round can utilize the information from the previous i − 1 rounds, the asymptotic number of nodes to satisfy connectivity is Θ(n ln ^[k] n) for k rounds. Similar results also hold if the sensing regions of the deployed nodes are required to cover the region of interest.