Parallel Equivalence Class Sorting: Algorithms, Lower Bounds, and Distribution-Based Analysis

TL;DR

This paper introduces new parallel algorithms and lower bounds for equivalence class sorting, a problem where only pairwise same/different comparisons are available, with applications in distributed security and privacy.

Contribution

It presents novel parallel algorithms, establishes lower bounds, and offers a distribution-based analysis for equivalence class sorting under comparison constraints.

Findings

New parallel algorithms for equivalence class sorting.

Lower bounds on the complexity of the problem.

Distribution-based analysis of algorithm performance.

Abstract

We study parallel comparison-based algorithms for finding all equivalence classes of a set of $n$ elements, where sorting according to some total order is not possible. Such scenarios arise, for example, in applications, such as in distributed computer security, where each of $n$ agents are working to identify the private group to which they belong, with the only operation available to them being a zero-knowledge pairwise-comparison (which is sometimes called a "secret handshake") that reveals only whether two agents are in the same group or in different groups. We provide new parallel algorithms for this problem, as well as new lower bounds and distribution-based analysis.