Sorting and Selection with Random Costs

TL;DR

This paper explores sorting and selection algorithms in models where comparison costs are random variables, providing bounds and algorithms tailored to different stochastic cost distributions.

Contribution

It introduces the first analysis of sorting and selection with stochastic comparison costs, establishing bounds and designing algorithms for various random cost models.

Findings

Established lower and upper bounds for stochastic cost models

Designed algorithms ensuring cost independence during execution

Analyzed properties of random partial orders in the context

Abstract

There is a growing body of work on sorting and selection in models other than the unit-cost comparison model. This work is the first treatment of a natural stochastic variant of the problem where the cost of comparing two elements is a random variable. Each cost is chosen independently and is known to the algorithm. In particular we consider the following three models: each cost is chosen uniformly in the range $[0,1]$, each cost is 0 with some probability $p$ and 1 otherwise, or each cost is 1 with probability $p$ and infinite otherwise. We present lower and upper bounds (optimal in most cases) for these problems. We obtain our upper bounds by carefully designing algorithms to ensure that the costs incurred at various stages are independent and using properties of random partial orders when appropriate.