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According to the software Hera, bottleneck distance is computed by finding an perfect matching with minimal cost using the Hopcroft-Karp algorithm.

furthermore, the wasserstein distance between persistence diagrams is computed by the auction algorithm.

Why do we need two separate algorithms to compute the perfect matching needed for both bottleneck and wasserstein distances?

What is the difference between the computation of wasserstein distance and bottle neck distance?

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