TY - GEN

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:
†SchoolofComputerScience, Tel-Aviv University. Email: shiri.chechik@gmail.com. Supported by the ISF grant No. 1528/15andtheBlavatnik Fund.
Publisher Copyright:
© Copyright 2018 by SIAM.

PY - 2018/1/1

Y1 - 2018/1/1

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 [MN07] 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 paper 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 which 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 which 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 [MN07] 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 paper 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 which 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 which 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 .

UR - http://www.scopus.com/inward/record.url?scp=85045546836&partnerID=8YFLogxK

U2 - 10.1137/1.9781611975031.108

DO - 10.1137/1.9781611975031.108

M3 - Conference contribution

AN - SCOPUS:85045546836

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1650

EP - 1664

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

A2 - Czumaj, Artur

PB - Association for Computing Machinery

T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

Y2 - 7 January 2018 through 10 January 2018

ER -