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arXiv · 2508.05251

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

Ziad Ismaili Alaoui·Detlef Plump·Sebastian Wild
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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.

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Ziad Ismaili Alaoui, Detlef Plump, Sebastian Wild. 2025-10-28. Space-Efficient Hierholzer: Eulerian Cycles in $\mathrm{O}(m)$ Time and $\mathrm{O}(n)$ Space. https://arxiv.org/abs/2508.05251

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