Ohio State is in the process of revising websites and program materials to accurately reflect compliance with the law. While this work occurs, language referencing protected class status or other activities prohibited by Ohio Senate Bill 1 may still appear in some places. However, all programs and activities are being administered in compliance with federal and state law.

The Gromov-Hausdorff distance between ultrametric spaces

Zhengchao Wan
October 12, 2021
4:00 pm - 5:00 pm
Zoom

Title:  The Gromov-Hausdorff distance between ultrametric spaces

Speaker:  Zhengchao Wan (OSU)

Speaker's URL:  https://zhengchaow.github.io

Abstract:  The Gromov-Hausdorff distance ($d_{\mathrm{GH}}$) is a natural distance between metric spaces. However, computing $d_{\mathrm{GH}}$ is NP-hard, even in the case of finite ultrametric spaces. We identify a one parameter family $\{d_{\mathrm{GH}, p}\}_{p\in[1,\infty]}$ of Gromov-Hausdorff type distances on the collection of ultrametric spaces such that $d_{\mathrm{GH}, 1} =d_{\mathrm{GH}}$. The extreme case when $p=\infty$, which we also denote by $u_{\mathrm{GH}}$, turns out to be an ultrametric on the collection of ultrametric spaces. We discuss various geometric and topological properties of $d_{\mathrm{GH}, p}$ as well as some of its structural results. These structural results in turn allow us to study the computational aspects of the distance. In particular, we establish that (1) $u_{\mathrm{GH}}$ is computationally tractable and (2) when $p<\infty$, although $d_{\mathrm{GH}, p}$ is NP-hard to compute, we identify a fixed-parameter tractable algorithm for computing the exact value of $d_{\mathrm{GH}, p}$ between finite ultrametric spaces.

Events Filters: