Constant Amortized Time Enumeration of Eulerian trails

TL;DR

This paper introduces optimal algorithms for enumerating edge-distinct and vertex-distinct Eulerian trails with constant amortized time per solution, using reverse search and push out amortization techniques.

Contribution

It presents the first algorithms achieving constant amortized time for enumerating all Eulerian trails, improving efficiency over previous methods.

Findings

Algorithms run in O(N) total time for N solutions.

Uses reverse search and push out amortization techniques.

Achieves optimal enumeration performance.

Abstract

In this paper, we consider enumeration problems for edge-distinct and vertex-distinct Eulerian trails. Here, two Eulerian trails are \emph{edge-distinct} if the edge sequences are not identical, and they are \emph{vertex-distinct} if the vertex sequences are not identical. As the main result, we propose optimal enumeration algorithms for both problems, that is, these algorithm runs in $\mathcal{O}(N)$ total time, where $N$ is the number of solutions. Our algorithms are based on the reverse search technique introduced by [Avis and Fukuda, DAM 1996], and the push out amortization technique introduced by [Uno, WADS 2015].