Fast Schulze Voting Using Quickselect

TL;DR

This paper introduces a faster algorithm for the Schulze voting method, reducing the expected runtime from O(m^2 log^4 m) to O(m^2 log m) by employing a modified quickselect technique.

Contribution

It presents an improved randomized algorithm for computing the Schulze winner, optimizing the previous method's efficiency with a novel application of quickselect.

Findings

Reduced expected runtime from O(m^2 log^4 m) to O(m^2 log m)

Achieved faster computation of Schulze winners in large elections

Demonstrated the effectiveness of quickselect in voting algorithms

Abstract

The Schulze voting method aggregates voter preference data using maxmin-weight graph paths, achieving the Condorcet property that a candidate who would win every head-to-head contest will also win the overall election. Once the voter preferences among $m$ candidates have been arranged into an $m\times m$ matrix of pairwise election outcomes, a previous algorithm of Sornat, Vassilevska Williams and Xu (EC '21) determines the Schulze winner in randomized expected time $O(m^2\log^4 m)$. We improve this to randomized expected time $O(m^2\log m)$ using a modified version of quickselect.