The Gromov--Hausdorff Distance between Simplexes and Two-Distance Spaces

Abstract

In the present paper we calculate the Gromov-Hausdorff distance between an arbitrary simplex (a metric space all whose non-zero distances are the same) and a finite metric space whose non-zero distances take two distinct values (so-called 2-distance spaces). As a corollary, a complete solution to generalized Borsuk problem for the 2-distance spaces is obtained. In addition, we derive formulas for the clique covering number and for the chromatic number of an arbitrary graph G in terms of the Gromov-Hausdorff distance between a simplex and an appropriate 2-distance space constructed by the graph G.

Lipschitz (non-)equivalence of the Gromov--Hausdorff distances, including on ultrametric spaces

“…Notice that the cardinality of ∆ must be finite as long as the Gromov-Hausdorff distances are defined between the compact metric spaces only. However, allowing d GH and d GH to take the value +∞ immediately extends their definitions to also include non-compact spaces, which was the approach taken by Ivanov and Tuzhilin in [IT16] and [IT19b]. We note that the proofs in this section are compatible with the extended definition, and in particular they remain valid if the cardinality of ∆ is allowed to be infinite.…”

“…Several bijection-based algorithms for approximating the distance have been proposed and applied for comparing point cloud [BBK + 10, VBBW16] and graph data [CLS + 17, LKC + 12, FVFM18]. Grigor'yev et al studied the GH distance between an arbitrary metric space and a regular simplex, and showed that various optimization problems in discrete geometry can be formulated in terms of this notion [IT16,IT19b]. Schmiedl proved that d GH (X, Y ) cannot be approximated up to a factor of max{|X|, |Y |} 1 2 (or up to a factor of 3 when restricted to ultrametric spaces) in polynomial time [Sch17].…”

“…It has been recently shown that certain problems in graph theory and metric geometry can be formulated in terms of the GH distance to a regular simplex. In particular, this notion can be used to calculate the weights of minimum spanning tree edges on a complete graph [IT16], solve the generalized Borsuk problem, or find clique cover and chromatic numbers of a graph [IT19b].…”