Terminal embeddings

Abstract

In this paper we study terminal embeddings, in which one is given a finite metric (X, d X ) (or a graph G = (V, E)) and a subset K ⊆ X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve ≈ |K| · |X| pairs, the distortion depends only on |K|, rather than on |X|.We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X × X and with respect to K × X.Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [ALN08] devised anÕ( √ log r)-approximation algorithm for sparsestcut instances with r demands. Building on their framework, we provide anÕ( log |K|)approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K| ≤ r, our bound generalizes that of [ALN08]. * A preliminary version of this paper appeared in APPROX'15. †

Optimal terminal dimensionality reduction in Euclidean space

“…In [EFN17,MMMR18] and the current work, a terminal embedding is obtained by, for each u ∈ R d \X, defining an outer extension f Ext act identically on X for any u, u ′ ∈ R d , we can then define our final terminal embedding bỹ…”

“…They called such an embedding a terminal embedding 1 . Though rather than achieving terminal distortion 1 + ε, their work only showed how to achieve constant terminal distortion with m = O(log n) for a constant that could be made arbitrarily close to √ 10 (see [EFN17,Theorem 1]). Terminal embeddings can be useful in static high-dimensional computational geometry data structural problems.…”

“…Remark. Unlike in the JL lemma for which the embedding f may be linear, the terminal embedding we analyze here (as was also in the case in [EFN17,MMMR18]) is nonlinear, as it must be. To see that it must be nonlinear, consider that any linear embedding with constant terminal distortion, x → Πx for some matrix Π ∈ R m×d , must have Π(x − y) = 0 for any x ∈ X and any y ∈ R d on the unit sphere centered at x.…”

Linear Size Distance Preservers

“…Perhaps surprisingly, Coppersmith and Elkin showed that nontrivial linear size distance preservers do indeed exist: in any undirected graph, one can preserve any p = O(n 1/2 ) pairwise distances using a subgraph on just O(n) edges. This unexpected fact has found several interesting applications: for example, Elkin and Pettie [26] used it to build the first linear-size log n stretch path reporting distance oracle, Bodwin and Vassilevska W. [13] have used it to build the current most accurate additive spanner of linear size, Pettie [38] used them as an ingredient in state-of-the-art constructions for mixed spanners, and they were employed by Elkin, Filtser, and Neiman [24] to build terminal subgraph spanners.…”

Lossless Prioritized Embeddings

“…The constant loss in the first part of the scheme has to do with an outer extension, implicitly developed in [EFN17], and explicated in [MMMR18,NN19]. Bi-Lipschitz outer extensions have been a focus of recent research [MMMR18,NN19,EN18], where they were studied in the context of Johnson-Lindenstrauss dimension reduction [MMMR18,NN19], and in the context of doubling metrics [EN18].…”