34 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
Asymptotically Faster Quantum Distributed Algorithms for Approximate Steiner Trees and Directed Minimum Spanning Trees
The CONGEST and CONGEST-CLIQUE models have been carefully studied to
represent situations where the communication bandwidth between processors in a
network is severely limited. Messages of only bits of information
each may be sent between processors in each round. The quantum versions of
these models allow the processors instead to communicate and compute with
quantum bits under the same bandwidth limitations. This leads to the following
natural research question: What problems can be solved more efficiently in
these quantum models than in the classical ones? Building on existing work, we
contribute to this question in two ways. Firstly, we present two algorithms in
the Quantum CONGEST-CLIQUE model of distributed computation that succeed with
high probability; one for producing an approximately optimal Steiner Tree, and
one for producing an exact directed minimum spanning tree, each of which uses
rounds of communication and messages,
where is the number of nodes in the network. The algorithms thus achieve a
lower asymptotic round and message complexity than any known algorithms in the
classical CONGEST-CLIQUE model. At a high level, we achieve these results by
combining classical algorithmic frameworks with quantum subroutines. An
existing framework for using distributed version of Grover's search algorithm
to accelerate triangle finding lies at the core of the asymptotic speedup.
Secondly, we carefully characterize the constants and logarithmic factors
involved in our algorithms as well as related algorithms, otherwise commonly
obscured by notation. The analysis shows that some improvements are
needed to render both our and existing related quantum and classical algorithms
practical, as their asymptotic speedups only help for very large values of .Comment: 23 pages, 0 figure
Computing Power of Hybrid Models in Synchronous Networks
During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of the relative powers of these models. That is, for every pair of such models, we determine whether one is (strictly) stronger than the other, or whether the two models are incomparable. The separation results are obtained by approaching communication complexity through an original angle, which may be of an independent interest. The two players are not bounded to compute the value of a binary function, but the combined outputs of the two players are constrained by this value. In particular, we introduce the XOR-Index problem, in which Alice is given a binary vector x ? {0,1}? together with an index i ? [n], Bob is given a binary vector y ? {0,1}? together with an index j ? [n], and, after a single round of 2-way communication, Alice must output a boolean out_A, and Bob must output a boolean out_B, such that out_A ? out_B = x_j? y_i. We show that the communication complexity of XOR-Index is ?(n) bits
Computing Power of Hybrid Models in Synchronous Networks
During the last two decades, a small set of distributed computing models for
networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested
Clique (BCC) play a prominent role. We consider hybrid models resulting from
combining these three models. That is, we analyze the computing power of models
allowing to, say, perform a constant number of rounds of CONGEST, then a
constant number of rounds of LOCAL, then a constant number of rounds of BCC,
possibly repeating this figure a constant number of times. We specifically
focus on 2-round models, and we establish the complete picture of the relative
powers of these models. That is, for every pair of such models, we determine
whether one is (strictly) stronger than the other, or whether the two models
are incomparable. The separation results are obtained by approaching
communication complexity through an original angle, which may be of independent
interest. The two players are not bounded to compute the value of a binary
function, but the combined outputs of the two players are constrained by this
value. In particular, we introduce the XOR-Index problem, in which Alice is
given a binary vector together with an index , Bob is
given a binary vector together with an index , and,
after a single round of 2-way communication, Alice must output a boolean
, and Bob must output a boolean , such that
\mbox{out}_A\land\mbox{out}_B = x_j\oplus y_i. We show that the communication
complexity of XOR-Index is bits
Quantum Distributed Algorithms for Detection of Cliques
The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case.
In this paper we present a framework for fast quantum distributed clique detection. This improves upon the state-of-the-art for the triangle case, and is also more general, applying to larger clique sizes.
Our main technical contribution is a new approach for detecting cliques by encapsulating this as a search task for nodes that can be added to smaller cliques. To extract the best complexities out of our approach, we develop a framework for nested distributed quantum searches, which employ checking procedures that are quantum themselves.
Moreover, we show a circuit-complexity barrier on proving a lower bound of the form ?(n^{3/5+?}) for K_p-detection for any p ? 4, even in the classical (non-quantum) distributed CONGEST setting
On the Distributed Complexity of Large-Scale Graph Computations
Motivated by the increasing need to understand the distributed algorithmic
foundations of large-scale graph computations, we study some fundamental graph
problems in a message-passing model for distributed computing where
machines jointly perform computations on graphs with nodes (typically, ). The input graph is assumed to be initially randomly partitioned among
the machines, a common implementation in many real-world systems.
Communication is point-to-point, and the goal is to minimize the number of
communication {\em rounds} of the computation.
Our main contribution is the {\em General Lower Bound Theorem}, a theorem
that can be used to show non-trivial lower bounds on the round complexity of
distributed large-scale data computations. The General Lower Bound Theorem is
established via an information-theoretic approach that relates the round
complexity to the minimal amount of information required by machines to solve
the problem. Our approach is generic and this theorem can be used in a
"cookbook" fashion to show distributed lower bounds in the context of several
problems, including non-graph problems. We present two applications by showing
(almost) tight lower bounds for the round complexity of two fundamental graph
problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our
approach, as demonstrated in the case of PageRank, can yield tight lower bounds
for problems (including, and especially, under a stochastic partition of the
input) where communication complexity techniques are not obvious.
Our approach, as demonstrated in the case of triangle enumeration, can yield
stronger round lower bounds as well as message-round tradeoffs compared to
approaches that use communication complexity techniques