Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1+o(1))-approximate shortest paths in O(D+√n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC’04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC’17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, [FOCS’17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we provide an alternative single-source shortest path algorithm with complexity O(n3 4D1 4). For polylogarithmic D, this improves on Elkin’s O(n5 6) bound and gets closer to the Ω(n1 2) lower bound of Elkin [STOC’04]. For larger values of D, we present an improved variant of our algorithm which achieves complexity O(max n3 4+o(1) , n3 4D1 6 + D ), and thus compares favorably with Elkin’s bound of O(max n5 6, n2 3D1 3 + D ) in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graph (i.e., graphs where the two directions of an edge can have different weights), constituting the first sublinear-time algorithm for directed graphs. Our algorithm also extends to the case of exact r-source shortest paths, in which we provide the fastest algorithm for moderately small r and D, improving on those of
In this paper, we present algorithms that improve on this state of the art both quantitatively and qualitatively for integer-weighted graphs. We will soon discuss the quantitative aspect, i.e., the improvements in the bounds. The qualitative aspect is the fact that our algorithms extend to networks with asymmetric weights, i.e., when the two directions of an edge can have different weights. Although this asymmetric case is relevant for practical networks, it has gained a far more important motivation recently even when working on graphs with symmetric weights, due to the introduction of the @cite_17 to distributed computing. The scaling framework massages the weights in a way that simplifies the problem considerably, modulo the potential of making the weights asymmetric. We will discuss this framework in the next section.
@cite_17
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We propose a class of distributed iterative algorithms that enable the asymptotic scaling of a primitive column stochastic matrix, with a given sparsity structure, to a doubly stochastic form. We also demonstrate the application of these algorithms to the average consensus problem in networked multi-component systems. More specifically, we consider a setting where each node is in charge of assigning weights on its outgoing edges based on the weights on its incoming edges. We establish that, as long as the (generally directed) graph that describes the communication links between components is strongly connected, each of the proposed matrix scaling algorithms allows the system components to asymptotically assign, in a distributed fashion, weights that comprise a primitive doubly stochastic matrix. We also show that the nodes can asymptotically reach average consensus by executing a linear iteration that uses the time-varying weights (as they result at the end of each iteration of the chosen matrix scaling algorithm). In this paper, we study the impact of edge weights on distances in sparse random graphs. We interpret these weights as delays and take them as independent and identically distributed exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some standard regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of @math when the size of the graph @math tends to infinity. We also derive the exact value for the prefactor. These results allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average. This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP. We bound the performance guarantees that follow from Turan-like bounds for unweighted and weighted independent sets in bounded-degree graphs. In particular, a randomized approach of Boppana forms a simple 1-round distributed algorithm, as well as a streaming and preemptive online algorithm. We show it gives a tight (( +1) 2 )-approximation in unweighted graphs of maximum degree ( ), which is best possible for 1-round distributed algorithms. For weighted graphs, it gives only a (( +1) )-approximation, but a simple modification results in an asymptotic expected (0.529 ( +1) )-approximation. This compares with a recent, more complex ( )-approximation [6], which holds deterministically.