Computational topology and normal surfaces: Theoretical and experimental complexity bounds

TL;DR

This paper investigates the complexity of normal surfaces in 3D computational topology, providing new theoretical exponential lower bounds and experimental analysis over billions of cases, revealing insights into practical algorithm performance.

Contribution

It introduces the first exponential lower bounds for normal surface complexity and offers extensive experimental data on typical and worst-case scenarios.

Findings

New exponential lower bounds on worst-case complexities

Experimental analysis over three billion input cases

Evidence that theoretical bounds are asymptotically tight

Abstract

In three-dimensional computational topology, the theory of normal surfaces is a tool of great theoretical and practical significance. Although this theory typically leads to exponential time algorithms, very little is known about how these algorithms perform in "typical" scenarios, or how far the best known theoretical bounds are from the real worst-case scenarios. Here we study the combinatorial and algebraic complexity of normal surfaces from both the theoretical and experimental viewpoints. Theoretically, we obtain new exponential lower bounds on the worst-case complexities in a variety of settings that are important for practical computation. Experimentally, we study the worst-case and average-case complexities over a comprehensive body of roughly three billion input triangulations. Many of our lower bounds are the first known exponential lower bounds in these settings, and experimental evidence suggests that many of our theoretical lower bounds on worst-case growth rates may indeed be asymptotically tight.