Affiliation:
1. VMWare, Herzliya, Israel
2. Tel-Aviv University, Tel Aviv, Israel
3. Ben-Gurion University of the Negev, Beer Sheva, Israel
Abstract
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
n
1−1/
k
that embeds into an ultrametric with distortion 8
k
. 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
n
1−1/
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
.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
8 articles.
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1. One Tree to Rule Them All: Poly-Logarithmic Universal Steiner Tree;2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS);2023-11-06
2. Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n log log n);2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS);2023-11-06
3. Low Treewidth Embeddings of Planar and Minor-Free Metrics;2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS);2022-10
4. Can't See the Forest for the Trees;Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing;2022-07-20
5. Light Spanners for High Dimensional Norms via Stochastic Decompositions;Algorithmica;2022-06-28