Hop-Constrained Metric Embeddings and their Applications

Abstract

Given a metric space (X, d X ), a (β, s, ∆)-sparse cover is a collection of clusters C ⊆ P (X) with diameter at most ∆, such that for every point x ∈ X, the ball B X (x, ∆ β ) is fully contained in some cluster C ∈ C, and x belongs to at most s clusters in C. Our main contribution is to show that the shortest path metric of every K r -minor free graphs admits (O(r), O(r 2 ), ∆)sparse cover, and for every ϵ > 0, (4 + ϵ, O( 1 ϵ ) r , ∆)-sparse cover (for arbitrary ∆ > 0). We then use this sparse cover to show that every K r -minor free graph embeds into ℓ)⋅log n ∞ with distortion O(r)). Further, we provide applications of these sparse covers into padded decompositions, sparse partitions, universal TSP / Steiner tree, oblivious buy at bulk, name independent routing, and path reporting distance oracles.

Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded Hop-Diameter Spanners

“…There are other routing schemes [HP00, AM04, Tal04, Sli05, AGGM06] on metrics that do not rely on the underlying spanner, but still have a hop diameter of Ω(log n). Concurrently and independently of our work, a recent paper by Filtser [Fil21] presents a routing scheme for general graphs with O(n 1/ℓ • ℓ • log n) bits of memory per vertex, which routes, for any pair u, v of vertices in the graph, using a path of hop-length O( ℓ 2 •log log n ǫ ) • h and whose weight is at most a factor 3.68(1 + ǫ) greater than the minimum weight of any h-hop path between u and v, where h, ℓ ≥ 1 are arbitrary integers. The techniques in the two works are disjoint.…”

Low Treewidth Embeddings of Planar and Minor-Free Metrics

“…See also [HHZ21a]. Finally, clan and Ramsey type embeddings were also studied in the context of hop-constrained metric embeddings [HHZ21b,Fil21].…”

Linear constraints for ensuring k -hop connectivity using Mixed-Integer Programming for Multi-Agent Systems

“…Networks with a limited diameter are also desirable since the communication between agents connected through many relays (hops) requires more point-to-point transmissions. Each of these transmissions is subjected to potential failures and delays, decreasing the reliability and efficiency of the network 15 . This could be an issue when technologies such as Visual Light Communication (VLC) are employed 5 .…”