Sorting Under 1-$\infty$ Cost Model

TL;DR

This paper investigates sorting with non-uniform comparison costs, specifically when comparisons are either 1 or infinite, and introduces algorithms that efficiently handle forbidden comparison pairs represented by a graph.

Contribution

It presents the first deterministic and a simple randomized algorithm for sorting under the 1-∞ cost model with forbidden pairs, with complexity bounds depending on the graph structure.

Findings

Deterministic algorithm with $O((q + n)\log n)$ comparisons.

Randomized algorithm with $ ilde{O}(n^2/\sqrt{q + n} + n\sqrt{q})$ probes.

Efficient sorting for random graphs with $ ilde{O}( ext{min}(n^{3/2}, pn^2))$ probes.

Abstract

In this paper we study the problem of sorting under non-uniform comparison costs, where costs are either 1 or $\infty$. If comparing a pair has an associated cost of $\infty$ then we say that such a pair cannot be compared (forbidden pairs). Along with the set of elements $V$ the input to our problem is a graph $G(V, E)$, whose edges represents the pairs that we can compare incurring an unit of cost. Given a graph with $n$ vertices and $q$ forbidden edges we propose the first non-trivial deterministic algorithm which makes $O((q + n)\log{n})$ comparisons with a total complexity of $O(n^2 + q^{\omega/2})$, where $\omega$ is the exponent in the complexity of matrix multiplication. We also propose a simple randomized algorithm for the problem which makes $\widetilde{O}(n^2/\sqrt{q + n} + n\sqrt{q})$ probes with high probability. When the input graph is random we show that $\widetilde{O}(\min{(n^{3/2}, pn^2)})$ probes suffice, where $p$ is the edge probability.