Space-Efficient Hierholzer: Eulerian Cycles in $\mathrm{O}(m)$ Time and $\mathrm{O}(n)$ Space
Ziad Ismaili Alaoui, Detlef Plump, Sebastian Wild
TL;DR
This paper introduces Space-Efficient Hierholzer, a variant that computes an Eulerian cycle in $O(n \lg m)$ bits of working memory while maintaining linear time $O(n+m)$. The key idea is to avoid storing the entire tour by maintaining only one incoming edge per vertex and per-vertex iterators, and to use a small set of edge marks (Black, Red, Green, Dashed) to guide reverse traversal and backtracking. The algorithm outputs the cycle incrementally to a write-only stream and is applicable to both directed and undirected Eulerian graphs, with particular efficiency on dense or multi-graph inputs. The results include a formal correctness proof and precise complexity analyses, making the approach practical and easy to implement in memory-constrained settings, and highlighting potential extensions to broader graph models.
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.