Lower Bounds for Coin-Weighing Problems

Abstract

Among a set of n coins of two weights (good and bad), and using a balance, we wish to determine the number of bad coins using as few measurements as possible. There is a known adaptive decision tree that answers this question in O ((log( n )) 2 ) measurements, and a slight modification of this decision tree determines the parity of the number of bad coins in O (log n ). In this article, we prove an Ω(\sqrt{ n }) lower bound on the depth of any oblivious decision tree which solves either the counting or the parity problem. Our lower bound can also be applied to any function of high average sensitivity, which includes most random functions and most random symmetric functions. With a slight generalization of this result, we derive lower bounds for the size of threshold circuits for a wide class of Boolean functions. We demonstrate an exponential gap between the nonadaptive and adaptive coin-weighing complexities of the counting and parity problems. We prove a tight Θ (log n ) bound on the adaptive coin-weighing complexity of the parity problem.