12,625 research outputs found
A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs
In undirected graphs with real non-negative weights, we give a new randomized
algorithm for the single-source shortest path (SSSP) problem with running time
in the comparison-addition model. This is
the first algorithm to break the time bound for real-weighted
sparse graphs by Dijkstra's algorithm with Fibonacci heaps. Previous undirected
non-negative SSSP algorithms give time bound of in comparison-addition model, where is the
inverse-Ackermann function and is the ratio of the maximum-to-minimum edge
weight [Pettie & Ramachandran 2005], and linear time for integer edge weights
in RAM model [Thorup 1999]. Note that there is a proposed complexity lower
bound of for hierarchy-based
algorithms for undirected real-weighted SSSP [Pettie & Ramachandran 2005], but
our algorithm does not obey the properties required for that lower bound. As a
non-hierarchy-based approach, our algorithm shows great advantage with much
simpler structure, and is much easier to implement.Comment: 17 page
Faster Parallel Algorithm for Approximate Shortest Path
We present the first work, time
algorithm in the PRAM model that computes -approximate
single-source shortest paths on weighted, undirected graphs. This improves upon
the breakthrough result of Cohen~[JACM'00] that achieves
work and time. While most previous approaches, including
Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent
developments in \emph{continuous optimization}, studying the shortest path
problem from the lens of the closely-related \emph{minimum transshipment}
problem. To obtain our algorithm, we demonstrate a series of near-linear work,
polylogarithmic-time reductions between the problems of approximate shortest
path, approximate transshipment, and -embeddings, and establish a
recursive algorithm that cycles through the three problems and reduces the
graph size on each cycle. As a consequence, we also obtain faster parallel
algorithms for approximate transshipment and -embeddings with
polylogarithmic distortion. The minimum transshipment algorithm in particular
improves upon the previous best work sequential algorithm of
Sherman~[SODA'17].
To improve readability, the paper is almost entirely self-contained, save for
several staple theorems in algorithms and combinatorics.Comment: 53 pages, STOC 202
Efficient Construction of Probabilistic Tree Embeddings
In this paper we describe an algorithm that embeds a graph metric
on an undirected weighted graph into a distribution of tree metrics
such that for every pair , and
. Such embeddings have
proved highly useful in designing fast approximation algorithms, as many hard
problems on graphs are easy to solve on tree instances. For a graph with
vertices and edges, our algorithm runs in time with high
probability, which improves the previous upper bound of shown by
Mendel et al.\,in 2009.
The key component of our algorithm is a new approximate single-source
shortest-path algorithm, which implements the priority queue with a new data
structure, the "bucket-tree structure". The algorithm has three properties: it
only requires linear time in the number of edges in the input graph; the
computed distances have a distance preserving property; and when computing the
shortest-paths to the -nearest vertices from the source, it only requires to
visit these vertices and their edge lists. These properties are essential to
guarantee the correctness and the stated time bound.
Using this shortest-path algorithm, we show how to generate an intermediate
structure, the approximate dominance sequences of the input graph, in time, and further propose a simple yet efficient algorithm to converted
this sequence to a tree embedding in time, both with high
probability. Combining the three subroutines gives the stated time bound of the
algorithm.
Then we show that this efficient construction can facilitate some
applications. We proved that FRT trees (the generated tree embedding) are
Ramsey partitions with asymptotically tight bound, so the construction of a
series of distance oracles can be accelerated
Bellman-Ford is optimal for shortest hop-bounded paths
This paper is about the problem of finding a shortest - path using at
most edges in edge-weighted graphs. The Bellman--Ford algorithm solves this
problem in time, where is the number of edges. We show that this
running time is optimal, up to subpolynomial factors, under popular
fine-grained complexity assumptions.
More specifically, we show that under the APSP Hypothesis the problem cannot
be solved faster already in undirected graphs with non-negative edge weights.
This lower bound holds even restricted to graphs of arbitrary density and for
arbitrary . Moreover, under a stronger assumption, namely
the Min-Plus Convolution Hypothesis, we can eliminate the restriction . In other words, the bound is tight for the entire space
of parameters , , and , where is the number of nodes.
Our lower bounds can be contrasted with the recent near-linear time algorithm
for the negative-weight Single-Source Shortest Paths problem, which is the
textbook application of the Bellman--Ford algorithm
Replacement Paths via Row Minima of Concise Matrices
Matrix is {\em -concise} if the finite entries of each column of
consist of or less intervals of identical numbers. We give an -time
algorithm to compute the row minima of any -concise matrix.
Our algorithm yields the first -time reductions from the
replacement-paths problem on an -node -edge undirected graph
(respectively, directed acyclic graph) to the single-source shortest-paths
problem on an -node -edge undirected graph (respectively, directed
acyclic graph). That is, we prove that the replacement-paths problem is no
harder than the single-source shortest-paths problem on undirected graphs and
directed acyclic graphs. Moreover, our linear-time reductions lead to the first
-time algorithms for the replacement-paths problem on the following
classes of -node -edge graphs (1) undirected graphs in the word-RAM model
of computation, (2) undirected planar graphs, (3) undirected minor-closed
graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete
Mathematic
A simpler and more efficient algorithm for the next-to-shortest path problem
Given an undirected graph with positive edge lengths and two
vertices and , the next-to-shortest path problem is to find an -path
which length is minimum amongst all -paths strictly longer than the
shortest path length. In this paper we show that the problem can be solved in
linear time if the distances from and to all other vertices are given.
Particularly our new algorithm runs in time for general
graphs, which improves the previous result of time for sparse
graphs, and takes only linear time for unweighted graphs, planar graphs, and
graphs with positive integer edge lengths.Comment: Partial result appeared in COCOA201
Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models
We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using to hide
polylogarithmic factors in (the number of nodes in the graph), our gradient
descent algorithm takes iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of . In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: -approximate SSSP using rounds, where is the (hop) diameter of the network.
(2) Broadcast congested clique model: -approximate
transshipment and SSSP using rounds. (3)
Multipass streaming model: -approximate transshipment and
SSSP using space and passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in ; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character
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