49 research outputs found
Distributed Strong Diameter Network Decomposition
For a pair of positive parameters , a partition of the
vertex set of an -vertex graph into disjoint clusters of
diameter at most each is called a network decomposition, if the
supergraph , obtained by contracting each of the clusters
of , can be properly -colored. The decomposition is
said to be strong (resp., weak) if each of the clusters has strong (resp.,
weak) diameter at most , i.e., if for every cluster and
every two vertices , the distance between them in the induced graph
of (resp., in ) is at most .
Network decomposition is a powerful construct, very useful in distributed
computing and beyond. It was shown by Awerbuch \etal \cite{AGLP89} and
Panconesi and Srinivasan \cite{PS92}, that strong network decompositions can be computed in
distributed time. Linial and Saks \cite{LS93} devised an
ingenious randomized algorithm that constructs {\em weak} network decompositions in time. It was however open till now
if {\em strong} network decompositions with both parameters can be constructed in distributed time.
In this paper we answer this long-standing open question in the affirmative,
and show that strong network decompositions can be
computed in time. We also present a tradeoff between parameters
of our network decomposition. Our work is inspired by and relies on the
"shifted shortest path approach", due to Blelloch \etal \cite{BGKMPT11}, and
Miller \etal \cite{MPX13}. These authors developed this approach for PRAM
algorithms for padded partitions. We adapt their approach to network
decompositions in the distributed model of computation
On Efficient Distributed Construction of Near Optimal Routing Schemes
Given a distributed network represented by a weighted undirected graph
on vertices, and a parameter , we devise a distributed
algorithm that computes a routing scheme in
rounds, where is the hop-diameter of the network. The running time matches
the lower bound of rounds (which holds for any
scheme with polynomial stretch), up to lower order terms. The routing tables
are of size , the labels are of size , and
every packet is routed on a path suffering stretch at most . Our
construction nearly matches the state-of-the-art for routing schemes built in a
centralized sequential manner. The previous best algorithms for building
routing tables in a distributed small messages model were by \cite[STOC
2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but
suffers from substantially larger routing tables of size ,
while the latter has sub-optimal running time of
Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion
This paper addresses the basic question of how well can a tree approximate
distances of a metric space or a graph. Given a graph, the problem of
constructing a spanning tree in a graph which strongly preserves distances in
the graph is a fundamental problem in network design. We present scaling
distortion embeddings where the distortion scales as a function of ,
with the guarantee that for each the distortion of a fraction
of all pairs is bounded accordingly. Such a bound implies, in
particular, that the \emph{average distortion} and -distortions are
small. Specifically, our embeddings have \emph{constant} average distortion and
-distortion. This follows from the following
results: we prove that any metric space embeds into an ultrametric with scaling
distortion . For the graph setting we prove that any
weighted graph contains a spanning tree with scaling distortion
. These bounds are tight even for embedding in arbitrary
trees.
For probabilistic embedding into spanning trees we prove a scaling distortion
of , which implies \emph{constant}
-distortion for every fixed .Comment: Extended abstrat apears in SODA 200