A Practical and Worst-Case Efficient Algorithm for Divisor Methods of Apportionment

TL;DR

This paper introduces a new algorithm for divisor methods of apportionment that combines worst-case efficiency with practical speed, improving over existing methods.

Contribution

The paper presents a novel algorithm that is both practically fast and theoretically worst-case efficient for divisor-based apportionment.

Findings

The new algorithm outperforms existing methods in practical running time.

It maintains worst-case optimality while being simpler and faster in practice.

Experimental analysis compares the three algorithms' performance.

Abstract

Proportional apportionment is the problem of assigning seats to parties according to their relative share of votes. Divisor methods are the de-facto standard solution, used in many countries. In recent literature, there are two algorithms that implement divisor methods: one by Cheng and Eppstein (ISAAC, 2014) has worst-case optimal running time but is complex, while the other (Pukelsheim, 2014) is relatively simple and fast in practice but does not offer worst-case guarantees. We demonstrate that the former algorithm is much slower than the other in practice and propose a novel algorithm that avoids the shortcomings of both. We investigate the running-time behavior of the three contenders in order to determine which is most useful in practice.