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 1parameter slices of the modules. The previous best algorithm to compute d(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(M) <= lambda for a constant lambda in O(n5log 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 R3 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) <= lambda. Moreover, we show how to compute the matching distance using only linear space, to the price of a much worse time complexity.