## 1 Introduction
In 2017, [PERSON] et al. [13] provided an improved implementation of the bottleneck distance in the library Hera by exploiting geometry. Their approach follows closely the work of [PERSON] et al. [14]. Figure 1: A Rips filtration over a point cloud captures the merging dynamics of clusters evolving from points across multiple scales. The dimension zero persistence diagram produced by this filtration is a set of points positioned along an extended vertical line at the merging heights in the corresponding dendrogram, except for the last point positioned at \(\infty\) representing the eventual single component. The neighborhoods around points are colored by the persistent cluster determined by the elder rule. For the sets \(X_{0}\) and \(Y_{0}\) of orthogonal projections on the diagonal \(\Delta\) of points respectively from \(X\) and \(Y\), and the sets \(U=X\cup Y_{0}\) and \(V=X_{0}\cup Y\), they consider the weighted complete bipartite graph \(G=(U\sqcup V,U\times V,w)\) where \(w:U\times V\rightarrow\mathbb{R}_{\geq 0}\) is given by  
\[w(u,v)= \begin{cases}||u-v||_{\infty}&\text{ if }u\in X\text{ or }v\in Y\\ 0&\text{ otherwise.}\end{cases} \]  
With this, the bottleneck computation problem can be recast in the following manner: if \(G[r]\) is the subgraph of \(G\) with all edges \(e\) of weight \(w(e)\leq r\), then the bottleneck distance of \(G\) is the minimal value \(r\) such that \(G[r]\) contains a perfect matching. Hence the bottleneck distance can be recovered by combining a binary search on the edge weights of \(G\) with a test for a perfect matching. For the matching step, they augment the Hopcroft-Karp algorithm [15] by appealing to a near-neighbor data structure (a k-d tree) to search for the best candidate pair for a query point, pruning from the search the subtrees (and hence all other candidates within them) whose enclosing box is further away from the query than the current best candidate. This circumvents the overwhelming matching problem by significantly shrinking down the combination pool to retrieve the best matching. To approximate complexity, they fit curves of the form \(cn^{\alpha}\) and found a best fit with \(\alpha=1.4\). This translates to speed-up from Dionysus already by a factor of 400 on diagrams with 2800 points, and opened opportunities for several works that examine larger [16] or more complex [17; 18] data sets. We take inspiration from this idea of exploiting the geometry of persistence diagrams to extract computational speed-up. By considering dimension 0 persistence diagrams induced from the Rips filtration, we can approach the problem via a different framework, birthing a new efficient algorithm for computing the bottleneck distance. The key idea is to begin with a specific initial bijection that one can methodically modify to optimize the norm between matched points. This process allows us to identify all possible instances where the bottleneck matching is achieved, and the exact value for the bottleneck distance, significantly bypassing the overwhelming matching step in previous implementations. We remark that while this strategy only works for persistence diagrams of a specific kind--those whose detected signatures all begin at the same time--this class is in no way less significant than diagrams induced from other filtrations. Moreover, in addition to diagrams induced from the above setting, this class also includes diagrams obtained from the output of any hierarchical clustering algorithm applied to point cloud data. Hence, the computational speed-up for the bottleneck distance we obtain benefits the comparison of these diagrams as well. Furthermore, we note that there are other metrics used in the literature to compare persistence diagrams, and we make no preference claim in favor of the bottleneck distance. In fact, it is a good question to ask whether the above strategy can be followed to generate computational speed-up for these metrics as well (We credit [PERSON] for raising this question first in relation to the Wasserstein distance.). We name this algorithm Lumawig.