Statistical Properties of a Class of Randomized Binary Search Algorithms

In this paper, we analyze the statistical properties of a randomized binary search algorithm and its variants. The basic discrete version of the problem is as follows. Suppose there are total m items, numbered 1, 2, ..., m, out of which the first k items are marked, where k is unknown. The objective is to choose one of the marked items uniformly at random. In each step of the basic algorithm, a number y is chosen uniformly at random from 1 to x, where x is the number chosen in the previous step and is equal to m in the first step. A query is made about y. If y is marked, the algorithm returns. We will also consider batch versions of this algorithm in which multiple numbers are chosen in each step and multiple queries are made in parallel. We give the mean and variance (exact or asymptotic) for the number of search steps in each version of the algorithm, and when possible, we give its distribution. We also analyze the access or hit pattern to the entire search space. The basic algorithm is fairly efficient in terms of the number of search steps, and also has small variance. The two batch versions of the algorithm can be used separately or combined to further reduce the number search steps and its variance. Email: [email protected]. Email:[email protected]