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. These algorithms have applications in caching and load balancing in distributed environments such as peer-to-peer networks, cloud storage, data centers, and content distribution networks. The basic discrete version of the problem is as follows. Suppose there are $m$ servers, numbered 1, 2, …, $m$, out of which the first $k$ servers are marked as special, where $k$ is unknown. These $k$ servers may contain a particular file or service that clients want. The objective is to select one of the marked servers uniformly at random. Considering the intended applications, we impose the constraint that there is no central controller to facilitate the selection process. We start with a basic algorithm: In each step, the client requesting the service chooses a number $y$ uniformly at random from $1,2,\dots ,x$, where $x$ is the number chosen in the previous step, initially set to $m$ in the first step. A query is then sent to server $y$ asking whether $y$ is marked. If the answer is yes, the algorithm returns $y$; otherwise, the process is repeated with $x\leftarrow y$. In this paper, we primarily consider two batch versions of this algorithm in which multiple numbers are chosen in each step and multiple queries are made in parallel. We derive the mean and variance (exact and/or asymptotic) for the number of search steps in each version of the algorithm, and when possible, we give its distribution. Additionally, we analyze the access pattern of queries across the entire search space.