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

In the field of topological data analysis, persistence modules are used to express geometrical features of data sets. The matching distance d_ℳ measures the difference between 2-parameter persistence modules by taking the maximum bottleneck distance between 1-parameter slices of the modules. The previous fastest algorithm to compute d_ℳ exactly runs in O(n^8+ω), where ω is the matrix multiplication constant. We improve significantly on this by describing an algorithm with expected running time O(n^5 log^3 n). We first solve the decision problem d_ℳ≤λ for a constant λ in O(n^5log n) 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 ℝ^3 whose vertices represent possible values for d_ℳ, and use a randomized incremental method to search through the vertices and find d_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_ℳ≤λ.

• 6 publications
• 19 publications
research
03/07/2018

### Computing Bottleneck Distance for 2-D Interval Decomposable Modules

Computation of the interleaving distance between persistence modules is ...
research
08/17/2021

### Rectangular Approximation and Stability of 2-parameter Persistence Modules

One of the main reasons for topological persistence being useful in data...
research
12/21/2018

### Exact computation of the matching distance on 2-parameter persistence modules

The matching distance is a pseudometric on multi-parameter persistence m...
research
10/23/2022

### Computing the Matching Distance of 2-Parameter Persistence Modules from Critical Values

The exact computation of the matching distance for multi-parameter persi...
research
05/12/2022

### Bottleneck Matching in the Plane

We present an algorithm for computing a bottleneck matching in a set of ...
research
05/28/2018

### The reflection distance between zigzag persistence modules

By invoking the reflection functors introduced by Bernstein, Gelfand, an...
research
10/29/2020

### Fast Minimal Presentations of Bi-graded Persistence Modules

Multi-parameter persistent homology is a recent branch of topological da...