Space-Efficient Hierholzer: Eulerian Cycles in $\mathrm{O}(m)$ Time and $\mathrm{O}(n)$ Space

TL;DR

This paper presents a space-efficient variant of Hierholzer's algorithm that finds Eulerian cycles in linear time using significantly less memory, especially beneficial for dense graphs.

Contribution

It introduces a simple, linear-time algorithm for Eulerian cycles that uses only O(n) bits of space, improving over traditional methods.

Findings

Uses O(n log m) bits of memory, reducing space compared to O(m log n).

Runs in linear time, matching classical algorithms.

First known linear-time, space-efficient Eulerian cycle algorithm.

Abstract

We describe a simple variant of Hierholzer's algorithm that finds an Eulerian cycle in a (multi)graph with $n$ vertices and $m$ edges using $\mathrm{O}(n \lg m)$ bits of working memory. This substantially improves the working space compared to standard implementations of Hierholzer's algorithm, which use $\mathrm{O}(m \lg n)$ bits of space. Our algorithm runs in linear time, like the classical versions, but avoids an $\mathrm{O}(m)$-size stack of vertices or storing information for each edge. To our knowledge, this is the first linear-time algorithm to achieve this space bound, and the method is very easy to implement. The correctness argument, by contrast, is surprisingly subtle; we give a detailed formal proof. The space savings are particularly relevant for dense graphs or multigraphs with large edge multiplicities.