Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft-Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft-Karp algorithm for bottleneck matching (based on previous work by Efrat el al.), and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.

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In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We introduce a different model, where we assume that the computation of a single distance is an expensive operation and consequently, the goal is to minimize the number of such distance queries. This model is motivated by metric spaces that appear in the context of topological data analysis. We consider two standard operations on metric spaces, namely the construction of a [Formula: see text]-spanner and the computation of an approximate nearest neighbor for a given query point. In both cases, we partially explore the metric space through distance queries and infer lower and upper bounds for yet unexplored distances through triangle inequality. For spanners, we evaluate several exploration strategies through extensive experimental evaluation. For approximate nearest neighbors, we prove that our strategy returns an approximate nearest neighbor after a logarithmic number of distance queries.