242 research outputs found
A Faster Distributed Single-Source Shortest Paths Algorithm
We devise new algorithms for the single-source shortest paths (SSSP) problem
with non-negative edge weights in the CONGEST model of distributed computing.
While close-to-optimal solutions, in terms of the number of rounds spent by the
algorithm, have recently been developed for computing SSSP approximately, the
fastest known exact algorithms are still far away from matching the lower bound
of rounds by Peleg and Rubinovich [SIAM
Journal on Computing 2000], where is the number of nodes in the network
and is its diameter. The state of the art is Elkin's randomized algorithm
[STOC 2017] that performs rounds. We
significantly improve upon this upper bound with our two new randomized
algorithms for polynomially bounded integer edge weights, the first performing
rounds and the second performing rounds. Our bounds also compare favorably to the
independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a
-approximation -round algorithm for directed SSSP and a new work/depth trade-off for exact
SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2018
Decoherence and Full Counting Statistics in a Mach-Zehnder Interferometer
We investigate the Full Counting Statistics of an electrical Mach-Zehnder
interferometer penetrated by an Aharonov-Bohm flux, and in the presence of a
classical fluctuating potential. Of interest is the suppression of the
Aharonov-Bohm oscillations in the distribution function of the transmitted
charge. For a Gaussian fluctuating field we calculate the first three
cumulants. The fluctuating potential causes a modulation of the conductance
leading in the third cumulant to a term cubic in voltage and to a contribution
correlating modulation of current and noise. In the high voltage regime we
present an approximation of the generating function.Comment: 10 pages, 6 figure
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
Faster Cut Sparsification of Weighted Graphs
A cut sparsifier is a reweighted subgraph that maintains the weights of the
cuts of the original graph up to a multiplicative factor of .
This paper considers computing cut sparsifiers of weighted graphs of size
. Our algorithm computes such a sparsifier in time
, both for graphs with polynomially
bounded and unbounded integer weights, where is the functional
inverse of Ackermann's function. This improves upon the state of the art by
Bencz\'ur and Karger (SICOMP 2015), which takes time. For
unbounded weights, this directly gives the best known result for cut
sparsification. Together with preprocessing by an algorithm of Fung et al.
(SICOMP 2019), this also gives the best known result for polynomially-weighted
graphs. Consequently, this implies the fastest approximate min-cut algorithm,
both for graphs with polynomial and unbounded weights. In particular, we show
that it is possible to adapt the state of the art algorithm of Fung et al. for
unweighted graphs to weighted graphs, by letting the partial maximum spanning
forest (MSF) packing take the place of the Nagamochi-Ibaraki (NI) forest
packing. MSF packings have previously been used by Abraham at al. (FOCS 2016)
in the dynamic setting, and are defined as follows: an -partial MSF packing
of is a set , where is a maximum
spanning forest in . Our method for
computing (a sufficient estimation of) the MSF packing is the bottleneck in the
running time of our sparsification algorithm.Comment: To be presented at the 49th EATCS International Colloquium on
Automata, Languages and Programming (ICALP 2022
An Improved Random Shift Algorithm for Spanners and Low Diameter Decompositions
Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given k ? 2, can be used to compute a spanner of stretch 2k-1 and expected size O(n^{1+1/k}) in k rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG\u2719]) by making the bounds on both running time and stretch independent of the random choices of the algorithm, whereas they only hold with high probability in previous results. Spanners are used in certain synchronizers, thus our improvement directly carries over to such synchronizers. Furthermore, for keeping the total number of inter-cluster edges small in low diameter decompositions, our clustering algorithm provides the following guarantees. Given ? ? (0,1], we compute a low diameter decomposition with diameter bound O((log n)/?) such that each edge e ? E is an inter-cluster edge with probability at most ?? w(e) in O((log n)/?) rounds in the CONGEST model. Again, this improves upon the state of the art (by Miller, Peng, and Xu [SPAA\u2713]) by making the bounds on both running time and diameter independent of the random choices of the algorithm, whereas they only hold with high probability in previous results
Fast Algorithms for Energy Games in Special Cases
In this paper, we study algorithms for special cases of energy games, a class
of turn-based games on graphs that show up in the quantitative analysis of
reactive systems. In an energy game, the vertices of a weighted directed graph
belong either to Alice or to Bob. A token is moved to a next vertex by the
player controlling its current location, and its energy is changed by the
weight of the edge. Given a fixed starting vertex and initial energy, Alice
wins the game if the energy of the token remains nonnegative at every moment.
If the energy goes below zero at some point, then Bob wins. The problem of
determining the winner in an energy game lies in . It is a long standing open problem whether a polynomial time
algorithm for this problem exists.
We devise new algorithms for three special cases of the problem. The first
two results focus on the single-player version, where either Alice or Bob
controls the whole game graph. We develop an
time algorithm for a game graph controlled by Alice, by providing a reduction
to the All-Pairs Nonnegative Prefix Paths problem (APNP), where is the
maximum weight and is the best exponent for matrix multiplication.
Thus we study the APNP problem separately, for which we develop an
time algorithm. For both problems, we improve
over the state of the art of for small . For the APNP
problem, we also provide a conditional lower bound, which states that there is
no time algorithm for any , unless the APSP
Hypothesis fails. For a game graph controlled by Bob, we obtain a near-linear
time algorithm. Regarding our third result, we present a variant of the value
iteration algorithm, and we prove that it gives an time algorithm for
game graphs without negative cycles
Deterministic Incremental APSP with Polylogarithmic Update Time and Stretch
We provide the first deterministic data structure that given a weighted
undirected graph undergoing edge insertions, processes each update with
polylogarithmic amortized update time and answers queries for the distance
between any pair of vertices in the current graph with a polylogarithmic
approximation in time.
Prior to this work, no data structure was known for partially dynamic graphs,
i.e., graphs undergoing either edge insertions or deletions, with less than
update time except for dense graphs, even when allowing
randomization against oblivious adversaries or considering only single-source
distances
Fast Deterministic Fully Dynamic Distance Approximation
In this paper, we develop deterministic fully dynamic algorithms for
computing approximate distances in a graph with worst-case update time
guarantees. In particular, we obtain improved dynamic algorithms that, given an
unweighted and undirected graph undergoing edge insertions and
deletions, and a parameter , maintain
-approximations of the -distance between a given pair of
nodes and , the distances from a single source to all nodes
("SSSP"), the distances from multiple sources to all nodes ("MSSP"), or the
distances between all nodes ("APSP").
Our main result is a deterministic algorithm for maintaining
-approximate -distance with worst-case update time
(for the current best known bound on the matrix multiplication
exponent ). This even improves upon the fastest known randomized
algorithm for this problem. Similar to several other well-studied dynamic
problems whose state-of-the-art worst-case update time is , this
matches a conditional lower bound [BNS, FOCS 2019]. We further give a
deterministic algorithm for maintaining -approximate
single-source distances with worst-case update time , which also
matches a conditional lower bound.
At the core, our approach is to combine algebraic distance maintenance data
structures with near-additive emulator constructions. This also leads to novel
dynamic algorithms for maintaining -emulators that improve
upon the state of the art, which might be of independent interest. Our
techniques also lead to improved randomized algorithms for several problems
such as exact -distances and diameter approximation.Comment: Changes to the previous version: improved bounds for approximate st
distances using new algebraic data structure
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