Linear-time Algorithms for Proportional Apportionment

TL;DR

This paper introduces an efficient linear-time algorithm for proportional apportionment using highest averages methods, significantly improving computational speed for distributing resources fairly among entities.

Contribution

It presents the first $O(n)$-time algorithm applicable to all widely used highest averages apportionment methods, enhancing efficiency in resource allocation.

Findings

Algorithm runs in linear time, $O(n)$

Applicable to all common highest averages methods

Improves computational efficiency for apportionment tasks

Abstract

The apportionment problem deals with the fair distribution of a discrete set of $k$ indivisible resources (such as legislative seats) to $n$ entities (such as parties or geographic subdivisions). Highest averages methods are a frequently used class of methods for solving this problem. We present an $O(n)$-time algorithm for performing apportionment under a large class of highest averages methods. Our algorithm works for all highest averages methods used in practice.