27 research outputs found
Almost Shortest Paths with Near-Additive Error in Weighted Graphs
Let be a weighted undirected graph with vertices and
edges, and fix a set of sources . We study the problem of
computing {\em almost shortest paths} (ASP) for all pairs in in
both classical centralized and parallel (PRAM) models of computation. Consider
the regime of multiplicative approximation of , for an arbitrarily
small constant . In this regime existing centralized algorithms
require time, where is the
matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic
depth (aka time) require work .
Our centralized algorithm has running time , and its PRAM
counterpart has polylogarithmic depth and work , for an
arbitrarily small constant . For a pair , it
provides a path of length that satisfies , where is the weight of the
heaviest edge on some shortest path. Hence our additive term depends
linearly on a {\em local} maximum edge weight, as opposed to the global maximum
edge weight in previous works. Finally, our .
We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a
parameter , this algorithm provides for {\em unweighted}
graphs a purely additive approximation of for {\em all pairs
shortest paths} (APASP) in time . Within the same
running time, our algorithm for {\em weighted} graphs provides a purely
additive error of , for every vertex pair , with defined as above.
On the way to these results we devise a suit of novel constructions of
spanners, emulators and hopsets
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
Path-Reporting Distance Oracles with Near-Logarithmic Stretch and Linear Size
Given an -vertex undirected graph , and a parameter , a
path-reporting distance oracle (or PRDO) is a data structure of size ,
that given a query , returns an -approximate shortest
path in within time . Here , and are
arbitrary functions.
A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen,
has , and . The
size of this oracle is for all . Elkin and Pettie and
Neiman and Shabat devised much sparser PRDOs, but their stretch was
polynomially larger than the optimal . On the other hand, for
non-path-reporting distance oracles, Chechik devised a result with
, and .
In this paper we make a dramatic progress in bridging the gap between
path-reporting and non-path-reporting distance oracles. We devise a PRDO with
size ,
stretch and query time . We can also have size , stretch
and query time
.
Our results on PRDOs are based on novel constructions of approximate distance
preservers, that we devise in this paper. Specifically, we show that for any
, any , and any graph and a collection
of vertex pairs, there exists a -approximate preserver with
edges, where
. These new
preservers are significantly sparser than the previous state-of-the-art
approximate preservers due to Kogan and Parter.Comment: 61 pages, 3 figure
Sparse Hopsets in Congested Clique
We give the first Congested Clique algorithm that computes a sparse hopset
with polylogarithmic hopbound in polylogarithmic time. Given a graph ,
a -hopset with "hopbound" , is a set of edges
added to such that for any pair of nodes and in there is a path
with at most hops in with length within of
the shortest path between and in .
Our hopsets are significantly sparser than the recent construction of
Censor-Hillel et al. [6], that constructs a hopset of size
, but with a smaller polylogarithmic hopbound. On the other
hand, the previously known constructions of sparse hopsets with polylogarithmic
hopbound in the Congested Clique model, proposed by Elkin and Neiman
[10],[11],[12], all require polynomial rounds.
One tool that we use is an efficient algorithm that constructs an
-limited neighborhood cover, that may be of independent interest.
Finally, as a side result, we also give a hopset construction in a variant of
the low-memory Massively Parallel Computation model, with improved running time
over existing algorithms
A Unified Framework for Hopsets
Given an undirected graph G = (V,E), an (?,?)-hopset is a graph H = (V,E\u27), so that adding its edges to G guarantees every pair has an ?-approximate shortest path that has at most ? edges (hops), that is, d_G(u,v) ? d_{G?H}^(?)(u,v) ? ?? d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter ?.
In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm.
In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^?),O(k^{1-?}))-hopset of size O?(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ? < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (?,?)-hopset of size O(n^{1+1/k}) must have ??? ? ?(k)
DISTRIBUTED, PARALLEL AND DYNAMIC DISTANCE STRUCTURES
Many fundamental computational tasks can be modeled by distances on a graph. This has inspired studying various structures that preserve approximate distances, but trade off this approximation factor with size, running time, or the number of hops on the approximate shortest paths.
Our focus is on three important objects involving preservation of graph distances: hopsets, in which our goal is to ensure that small-hop paths also provide approximate shortest paths; distance oracles, in which we build a small data structure that supports efficient distance queries; and spanners, in which we find a sparse subgraph that approximately preserves all distances.
We study efficient constructions and applications of these structures in various models of computation that capture different aspects of computational systems. Specifically, we propose new algorithms for constructing hopsets and distance oracles in two modern distributed models: the Massively Parallel Computation (MPC) and the Congested Clique model. These models have received significant attention recently due to their close connection to present-day big data platforms.
In a different direction, we consider a centralized dynamic model in which the input changes over time. We propose new dynamic algorithms for constructing hopsets and distance oracles that lead to state-of-the-art approximate single-source, multi-source and all-pairs shortest path algorithms with respect to update-time.
Finally, we study the problem of finding optimal spanners in a different distributed model, the LOCAL model. Unlike our other results, for this problem our goal is to find the best solution for a specific input graph rather than giving a general guarantee that holds for all inputs.
One contribution of this work is to emphasize the significance of the tools and the techniques used for these distance problems rather than heavily focusing on a specific model.
In other words, we show that our techniques are broad enough that they can be extended to different models
Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time
In the decremental single-source shortest paths (SSSP) problem we want to
maintain the distances between a given source node and every other node in
an -node -edge graph undergoing edge deletions. While its static
counterpart can be solved in near-linear time, this decremental problem is much
more challenging even in the undirected unweighted case. In this case, the
classic total update time of Even and Shiloach [JACM 1981] has been the
fastest known algorithm for three decades. At the cost of a
-approximation factor, the running time was recently improved to
by Bernstein and Roditty [SODA 2011]. In this paper, we bring the
running time down to near-linear: We give a -approximation
algorithm with expected total update time, thus obtaining
near-linear time. Moreover, we obtain time for the weighted
case, where the edge weights are integers from to . The only prior work
on weighted graphs in time is the -time algorithm by
Henzinger et al. [STOC 2014, ICALP 2015] which works for directed graphs with
quasi-polynomial edge weights. The expected running time bound of our algorithm
holds against an oblivious adversary.
In contrast to the previous results which rely on maintaining a sparse
emulator, our algorithm relies on maintaining a so-called sparse -hop set introduced by Cohen [JACM 2000] in the PRAM literature. An
-hop set of a graph is a set of weighted edges
such that the distance between any pair of nodes in can be
-approximated by their -hop distance (given by a path
containing at most edges) on . Our algorithm can maintain
an -hop set of near-linear size in near-linear time under
edge deletions.Comment: Accepted to Journal of the ACM. A preliminary version of this paper
was presented at the 55th IEEE Symposium on Foundations of Computer Science
(FOCS 2014). Abstract shortened to respect the arXiv limit of 1920 character