TY - JOUR

T1 - Ramsey Spanning Trees and Their Applications

AU - Abraham, Ittai

AU - Chechik, Shiri

AU - Elkin, Michael

AU - Filtser, Arnold

AU - Neiman, Ofer

N1 - Funding Information:
S. Chechik supported by the ISF grant No. 1528/15 and the Blavatnik Fund. M. Elkin supported by the ISF grant No. 724/15. A. Filtser and O. Neiman supported in part by ISF grant 1817/17, and by BSF Grant 2015813. Authors’ addresses: I. Abraham, Ampa Building, 5 Sapir St., Ground Floor, POB 12093, Herzliya 4685209, Israel; email: iabraham@vmware.com; S. Chechik, Department of Computer Science, Tel-Aviv University, Tel Aviv 69978, Israel; email: schechik@post.tau.ac.il; M. Elkin, A. Filtser, and O. Neiman, Department of Computer Science, Ben-Gurion University, Beer Sheva, Israel; emails: {elkinm, arnoldf, neimano}@cs.bgu.ac.il. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2020 Association for Computing Machinery. 1549-6325/2020/03-ART19 $15.00 https://doi.org/10.1145/3371039
Publisher Copyright:
© 2020 ACM.

PY - 2020/4/1

Y1 - 2020/4/1

N2 - The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor [29] devised the so-called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this article, we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees. We ask for the largest subset S ⊆ V of a given graph G=(V, E), such that there exists a spanning tree of G that has small stretch for S. Applied iteratively, this provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices. The union of these trees serves as a special type of spanner, a tree-padding spanner. We use this spanner to devise the first compact stateless routing scheme with O(1) routing decision time, and labels that are much shorter than in all currently existing schemes. We first revisit the metric Ramsey problem and provide a new deterministic construction. We prove that for every k, any n-point metric space has a subset S of size at least n1-1/k that embeds into an ultrametric with distortion 8k. We use this result to obtain the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every k, any n-vertex graph G=(V, E) has a subset S of size at least n1-1/k, and a spanning tree of G, that has stretch O(k log log n) between any point in S and any point in V.

AB - The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor [29] devised the so-called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this article, we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees. We ask for the largest subset S ⊆ V of a given graph G=(V, E), such that there exists a spanning tree of G that has small stretch for S. Applied iteratively, this provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices. The union of these trees serves as a special type of spanner, a tree-padding spanner. We use this spanner to devise the first compact stateless routing scheme with O(1) routing decision time, and labels that are much shorter than in all currently existing schemes. We first revisit the metric Ramsey problem and provide a new deterministic construction. We prove that for every k, any n-point metric space has a subset S of size at least n1-1/k that embeds into an ultrametric with distortion 8k. We use this result to obtain the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every k, any n-vertex graph G=(V, E) has a subset S of size at least n1-1/k, and a spanning tree of G, that has stretch O(k log log n) between any point in S and any point in V.

KW - Distortion

KW - compact routing

KW - distance oracles

KW - metric embedding

KW - spanning trees

UR - http://www.scopus.com/inward/record.url?scp=85084754046&partnerID=8YFLogxK

U2 - 10.1145/3371039

DO - 10.1145/3371039

M3 - Article

AN - SCOPUS:85084754046

VL - 16

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 2

M1 - 19

ER -