# 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.

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