Computing the second and third systoles of a combinatorial surface

2024-07-18

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

This paper addresses the efficient computation of the second and third homotopically nontrivial closed walks—i.e., the second and third systoles—in weighted graphs embedded on topological surfaces. For an undirected graph $G$ cellularly embedded on a surface $S$ of genus $g$ and with $b$ boundary components, we present the first polynomial-time exact algorithms: the second systole is computed in $O(n log n)$ time (for fixed $g,b$), matching the theoretical lower bound; the third systole in $O(n^3)$ time. Our approach integrates surface-topological structural analysis, shortest-path computation under homotopy constraints, enumeration of homotopy classes, and a fast subroutine for computing essential arcs between boundary components. This work constitutes the first exact algorithmic solution for both the second and third homotopically nontrivial closed walks, significantly advancing the geometric and topological analysis of surface-embedded graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given a weighted, undirected graph $G$ cellularly embedded on a topological surface $S$, we describe algorithms to compute the second shortest and third shortest closed walks of $G$ that are homotopically non-trivial in $S$. Our algorithms run in $O(n^2log n)$ time for the second shortest walk and in $O(n^3)$ time for the third shortest walk. We also show how to reduce the running time for the second shortest homotopically non-trivial closed walk to $O(nlog n)$ when both the genus and the number of boundaries are fixed. Our algorithms rely on a careful analysis of the configurations of the first three shortest homotopically non-trivial curves in $S$. As an intermediate step, we also describe how to compute a shortest essential arc between emph{one} pair of vertices or between emph{all} pairs of vertices of a given boundary component of $S$ in $O(n^2)$ time or $O(n^3)$ time, respectively.

Problem

• Computing second and third shortest non-trivial closed walks on surfaces
• Analyzing configurations of topologically distinct shortest curves
• Finding shortest essential arcs between vertices on boundary components

Innovation

• Computes second and third shortest non-trivial closed walks
• Uses O(n^2 log n) and O(n^3) time complexity algorithms
• Optimizes to O(n log n) for fixed genus and boundaries