2 research outputs found
Fast Generation of Random Spanning Trees and the Effective Resistance Metric
We present a new algorithm for generating a uniformly random spanning tree in
an undirected graph. Our algorithm samples such a tree in expected
time. This improves over the best previously known bound
of -- that follows from the work of
Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] --
whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of
random spanning trees, random walks, and the notion of effective resistance, as
well as from devising a way to algorithmically relate these concepts to the
combinatorial structure of the graph. This involves, in particular,
establishing a new connection between the effective resistance metric and the
cut structure of the underlying graph
Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler
In the decremental -approximate Single-Source Shortest Path
(SSSP) problem, we are given a graph with ,
undergoing edge deletions, and a distinguished source , and we are
asked to process edge deletions efficiently and answer queries for distance
estimates for each , at any stage,
such that . In the decremental -approximate
All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for
distance estimates for every . In
this article, we consider the problems for undirected, unweighted graphs.
We present a new \emph{deterministic} algorithm for the decremental
-approximate SSSP problem that takes total update time . Our algorithm improves on the currently best algorithm for dense
graphs by Chechik and Bernstein [STOC 2016] with total update time
and the best existing algorithm for sparse graphs with running
time [SODA 2017] whenever .
In order to obtain this new algorithm, we develop several new techniques
including improved decremental cover data structures for graphs, a more
efficient notion of the heavy/light decomposition framework introduced by
Chechik and Bernstein and the first clustering technique to maintain a dynamic
\emph{sparse} emulator in the deterministic setting.
As a by-product, we also obtain a new simple deterministic algorithm for the
decremental -approximate APSP problem with near-optimal total
running time matching the time complexity of the
sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai
[FOCS 2013].Comment: Appeared in SODA'2