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 dM 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 dM exactly runs in O(n ), where ω is the matrix multiplication constant. We improve significantly on this by describing an algorithm with expected running time O(n log n). We first solve the decision problem dM ≤ λ for a constant λ in O(n logn) 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 whose vertices represent possible values for dM, and use a randomized incremental method to search through the vertices and find dM. The expected running time of this algorithm is O((n +T (n)) log n), where T (n) is an upper bound for the complexity of deciding if dM ≤ λ.

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