3 research outputs found
A Simple Deterministic Distributed MST Algorithm, with Near-Optimal Time and Message Complexities
Distributed minimum spanning tree (MST) problem is one of the most central
and fundamental problems in distributed graph algorithms. Garay et al.
\cite{GKP98,KP98} devised an algorithm with running time , where is the hop-diameter of the input -vertex -edge
graph, and with message complexity . Peleg and Rubinovich
\cite{PR99} showed that the running time of the algorithm of \cite{KP98} is
essentially tight, and asked if one can achieve near-optimal running time
**together with near-optimal message complexity**.
In a recent breakthrough, Pandurangan et al. \cite{PRS16} answered this
question in the affirmative, and devised a **randomized** algorithm with time
and message complexity . They asked if
such a simultaneous time- and message-optimality can be achieved by a
**deterministic** algorithm.
In this paper, building upon the work of \cite{PRS16}, we answer this
question in the affirmative, and devise a **deterministic** algorithm that
computes MST in time , using messages. The polylogarithmic factors in the time
and message complexities of our algorithm are significantly smaller than the
respective factors in the result of \cite{PRS16}. Also, our algorithm and its
analysis are very **simple** and self-contained, as opposed to rather
complicated previous sublinear-time algorithms \cite{GKP98,KP98,E04b,PRS16}
Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time
Given parameters , a subgraph of an
-vertex unweighted undirected graph is called an
-spanner if for every pair of vertices,
. If the spanner is called a
multiplicative -spanner, and if , for an
arbitrarily small , the spanner is said to be a near-additive one.
Graph spanners are a fundamental and extremely well-studied combinatorial
construct, with a multitude of applications in distributed computing and in
other areas. Near-additive spanners, introduced in [EP01], preserve large
distances much more faithfully than multiplicative spanners. Also, recent lower
bounds [AB15] ruled out the existence of arbitrarily sparse purely additive
spanners (i.e., spanners with ), and therefore near-additive spanners
provide the best approximation of distances that one can hope for. Numerous
distributed algorithms for constructing sparse near-additive spanners exist. In
particular, there are now known efficient randomized algorithms in the CONGEST
model that construct such spanners [EN17], and also there are efficient
deterministic algorithms in the LOCAL model [DGPV09]. The only known
deterministic CONGEST-model algorithm for the problem [Elk01] requires
superlinear time in . We remedy the situation and devise an efficient
deterministic CONGEST-model algorithm for constructing arbitrarily sparse
near-additive spanners. The running time of our algorithm is low polynomial,
i.e., roughly , where is an arbitrarily small
positive constant that affects the additive term . In general, the
parameters of our algorithm and of the resulting spanner are at the same
ballpark as the respective parameters of the state-of-the-art randomized
algorithm for the problem due to [EN17]
Ramsey Spanning Trees and their Applications
The metric Ramsey problem asks for the largest subset 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 2007 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 of a given graph , such that there
exists a spanning tree of that has small stretch for . 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
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 , any -point metric space has a
subset of size at least which embeds into an ultrametric with
distortion . This results improves the best previous result of Mendel and
Naor that obtained distortion and required randomization. In addition,
it provides the state-of-the-art deterministic construction of a distance
oracle. Building on this result, we prove that for every , any -vertex
graph has a subset of size at least , and a spanning
tree of , that has stretch between any point in and
any point in