标题： 合并树的内在瓶颈距离 显示英文标题

标题： Intrinsic Bottleneck Distance for Merge Trees

摘要： 合并树是过滤空间的拓扑描述符，它通过合并结构丰富了零度条形码。 合并树的空间配备了交错距离$d_I$，这引发了一个简单的问题：两个合并树之间的交错距离是否等于它们对应条形码之间的瓶颈距离？ 由于从合并树到条形码的映射不是单射的，按照这种说法答案是否定的，但（如 Gasparovic 等人所猜想的）我们证明对于由合并树空间中的无限小路径长度实现的\emph{内在的}度量$\widehat{d}_I$和$\widehat{d}_B$来说这是正确的。 这个结果表明，在某些特殊情况下，可以将可以快速计算的瓶颈距离代替交错距离（一般来说，是 NP 难的）。 显示英文摘要

摘要： Merge trees are a topological descriptor of a filtered space that enriches the degree zero barcode with its merge structure. The space of merge trees comes equipped with an interleaving distance $d_I$, which prompts a naive question: is the interleaving distance between two merge trees equal to the bottleneck distance between their corresponding barcodes? As the map from merge trees to barcodes is not injective, the answer as posed is no, but (as conjectured in Gasparovic et al.) we prove that it is true for the \emph{intrinsic} metrics $\widehat{d}_I$ and $\widehat{d}_B$ realized by infinitesimal path length in merge tree space. This result suggests that in some special cases the bottleneck distance (which can be computed quickly) can be substituted for the interleaving distance (in general, NP-hard).