A Fast Algorithm for Computing Zigzag Representatives

TL;DR

This paper introduces a new algorithm that efficiently computes compatible homological representatives in zigzag filtrations, improving upon previous methods in terms of computational complexity.

Contribution

The authors develop a novel $O(m^2n)$ time algorithm for computing compatible representatives in zigzag filtrations, surpassing existing approaches.

Findings

The algorithm computes compatible representatives in $O(m^2n)$ time.

It extends classical methods to the zigzag setting with improved efficiency.

The approach handles large complexes effectively within the given complexity bounds.

Abstract

Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features. Although one can locate a particular feature at any index in the filtration using existing algorithms, the resulting representatives may not be compatible with the zigzag: a representative cycle at one index may not map into a representative cycle at its neighbor. For this, one needs to compute compatible representative cycles along each bar in the barcode. It is known that the barcode for a zigzag filtration with $m$ insertions and deletions can be computed in $O(m^\omega)$ time, where $\omega< 2.373$ is the matrix multiplication exponent. However, it is not known how to compute the compatible representatives so efficiently. For a non-zigzag filtration, the classical matrix-based algorithm provides representatives in $O(m^3)$ time, which can be improved to $O(m^\omega)$. However, no known algorithm for zigzag filtrations computes the representatives with the $O(m^3)$ time bound. We present an $O(m^2n)$ time algorithm for this problem, where $n\leq m$ is the size of the largest complex in the filtration.