Asymptotic Improvements on the Exact Matching Distance for 2-parameter Persistence

TL;DR

This paper presents a significantly faster algorithm for computing the exact matching distance between 2-parameter persistence modules in topological data analysis, reducing time complexity from polynomial to near-quadratic with high probability.

Contribution

The authors develop an expected-time algorithm with improved efficiency for calculating the matching distance, including a decision procedure and space optimization techniques.

Findings

Expected running time of $O(n^5 \, \log^3 n)$ for the new algorithm.

Efficient decision procedure for $d_\mathcal{M} \leq \lambda$ using line arrangement traversal.

Linear space computation method with higher time complexity.

Abstract

In the field of topological data analysis, persistence modules are used to express geometrical features of data sets. The matching distance $d_\mathcal{M}$ measures the difference between $2$-parameter persistence modules by taking the maximum bottleneck distance between $1$-parameter slices of the modules. The previous best algorithm to compute $d_\mathcal{M}$ exactly runs in $O(n^{8+\omega})$ time using $O(n^4)$ space, where $n$ is the number of generators and relations of the modules and $\omega$ is the matrix multiplication constant. We improve significantly on this by describing an algorithm with expected running time $O(n^5 \log^3 n)$ and using $O(n^2)$ space. We first solve the decision problem $d_\mathcal{M}\leq \lambda$ for a constant $\lambda$ in $O(n^5\log n)$ time by traversing a line arrangement in the dual plane, where each point represents a slice. Then we lift the line arrangement to a plane arrangement in $\mathbb{R}^3$ whose vertices represent possible values for $d_\mathcal{M}$, and use a randomized incremental method to search through the vertices and find $d_\mathcal{M}$. The expected running time of this algorithm is $O((n^4+T(n))\log^2 n)$, where $T(n)$ is an upper bound for the complexity of deciding if $d_\mathcal{M}\leq \lambda$. Moreover, we show how to compute the matching distance using only linear space, to the price of a much worse time complexity.