We design fast deterministic algorithms for distance computation in the congested clique model. Our key contributions include: -- A @math -approximation for all-pairs shortest paths in @math rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model. -- A @math -approximation for multi-source shortest paths from @math sources in @math rounds on weighted undirected graphs. This is the first sub-polynomial algorithm obtaining this approximation for a set of sources of polynomial size. Our main techniques are new distance tools that are obtained via improved algorithms for sparse matrix multiplication, which we leverage to construct efficient hopsets and shortest paths. Furthermore, our techniques extend to additional distance problems for which we improve upon the state-of-the-art, including diameter approximation, and an exact single-source shortest paths algorithm for weighted undirected graphs in @math rounds.
The distributed CONGEST model is identical to the model, with the difference that the communication network is identical to the input graph @math , and nodes can communicate only with their neighbours in each round. Distance computation is extensively studied in the CONGEST model. The study of APSP in weighted graphs has been the focus of many recent papers @cite_23 @cite_47 @cite_30 @cite_9 , culminating in a near tight @math algorithm @cite_23 . Such results were previously known in unweighted graphs @cite_32 @cite_43 @cite_28 or for approximation algorithms @cite_38 @cite_17 . Approximate and exact algorithms for SSSP are studied in @cite_2 @cite_17 @cite_33 @cite_15 @cite_57 @cite_67 @cite_9 . While near-tight algorithms exist for approximating SSSP @cite_2 , there is currently a lot of interest in understanding the complexity of SSSP and SSSP @cite_41 @cite_67 @cite_9 . The problem is studied in @cite_21 , demonstrating the applicability of this tool for many distance problems such as APSP and diameter approximation in unweighted graphs. An extension for the weighted case is studied in @cite_38 . Algorithms and lower bounds for approximating the diameter are studied in @cite_21 @cite_55 @cite_28 @cite_31 @cite_10 .
@cite_30 @cite_47 @cite_41 @cite_43 @cite_2 @cite_15 @cite_10 @cite_38 @cite_67 @cite_21 @cite_23 @cite_17 @cite_28 @cite_55 @cite_32 @cite_57 @cite_33 @cite_9 @cite_31
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We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size @math can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized @math -approximation algorithm to compute all pairs shortest paths (APSP) in time @math on weighted graphs, where we use the convention that @math is essentially @math polylog @math and @math is essentially @math polylog @math . We complement this result by proving that any randomized @math -approximation of APSP and @math -approximation of the diameter of a graph takes @math time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a @math -approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST CLIQUE model and contrasts a recent @math -approximation for APSP that runs in time @math in the unicast version of the CONGEST CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's @math -approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model presented by Lenzen and Peleg at PODC'13. In the distributed all-pairs shortest paths problem (APSP), every node in the weighted undirected distributed network (the CONGEST model) needs to know the distance from every other node using least number of communication rounds (typically called time complexity ). The problem admits @math -approximation @math -time algorithm and a nearly-tight @math lower bound [Nanongkai, STOC'14; Lenzen and Patt-Shamir PODC'15] @math , @math and @math hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios LenzenP_podc13,HolzerW12,PelegRT12,Nanongkai-STOC14 . . For the exact case, Elkin [STOC'17] presented an @math time bound, which was later improved to @math [Huang, Nanongkai, Saranurak FOCS'17]. It was shown that any super-linear lower bound (in @math ) requires a new technique [Censor-Hillel, Khoury, Paz, DISC'17], but otherwise it remained widely open whether there exists a @math -time algorithm for the exact case, which would match the best possible approximation algorithm. This paper resolves this question positively: we present a randomized (Las Vegas) @math -time algorithm, matching the lower bound up to polylogarithmic factors. Like the previous @math bound, our result works for directed graphs with zero (and even negative) edge weights. In addition to the improved running time, our algorithm works in a more general setting than that required by the previous @math bound; in our setting (i) the communication is only along edge directions (as opposed to bidirectional), and (ii) edge weights are arbitrary (as opposed to integers in 1, 2, ... poly(n) ). ... A distributed network is modeled by a graph having n nodes (processors) and diameter D. We study the time complexity of approximating weighted (undirected) shortest paths on distributed networks with a O (log n) bandwidth restriction on edges (the standard synchronous CONGEST model). The question whether approximation algorithms help speed up the shortest paths and distance computation (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is well-understood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both single-source shortest paths (SSSP) and all-pairs shortest paths (APSP) in the weighted case. Our main result is an algorithm for SSSP. Previous results are the classic O(n)-time Bellman-Ford algorithm and an O(n1 2+1 2k + D)-time (8k⌈log(k + 1)⌉ --1)-approximation algorithm, for any integer k ≥ 1, which follows from the result of Lenzen and Patt-Shamir (STOC 2013). (Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use O(·) to hide the O(poly log n) term.) We present an O (n1 2D1 4 + D)-time (1 + o(1))-approximation algorithm for SSSP. This algorithm is sublinear-time as long as D is sublinear, thus yielding a sublinear-time algorithm with almost optimal solution. When D is small, our running time matches the lower bound of Ω(n1 2 + D) by Das (SICOMP 2012), which holds even when D=Θ(log n), up to a poly log n factor. As a by-product of our technique, we obtain a simple O (n)-time (1+ o(1))-approximation algorithm for APSP, improving the previous O(n)-time O(1)-approximation algorithm following from the results of Lenzen and Patt-Shamir. We also prove a matching lower bound. Our techniques also yield an O(n1 2) time algorithm on fully-connected networks, which guarantees an exact solution for SSSP and a (2+ o(1))-approximate solution for APSP. All our algorithms rely on two new simple tools: light-weight algorithm for bounded-hop SSSP and shortest-path diameter reduction via shortcuts. These tools might be of an independent interest and useful in designing other distributed algorithms. We study approximate distributed solutions to the weighted all-pairs-shortest-paths (APSP) problem in the CONGEST model. We obtain the following results. @math A deterministic @math -approximation to APSP in @math rounds. This improves over the best previously known algorithm, by both derandomizing it and by reducing the running time by a @math factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and require that these names are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of @math . In the relabeling model, we obtain the following results. @math A randomized @math -approximation to APSP, for any integer @math , running in @math rounds, where @math is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approximation ratio from @math to @math . Also, the new algorithm uses uses labels of asymptotically optimal size, namely @math bits. @math A randomized @math -approximation to APSP, for any integer @math , running in time @math and producing compact routing tables of size @math . The node lables consist of @math bits. This improves on the approximation ratio of @math for tables of that size achieved by the best previously known algorithm, which terminates faster, in @math rounds.