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. In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of [SIAM J. Comput. , 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for 0-Extension . In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput. , 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least @math for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations. We propose to compress weighted graphs (networks), motivated by the observation that large networks of social, biological, or other relations can be complex to handle and visualize. In the process also known as graph simplification, nodes and (unweighted) edges are grouped to supernodes and superedges, respectively, to obtain a smaller graph. We propose models and algorithms for weighted graphs. The interpretation (i.e. decompression) of a compressed, weighted graph is that a pair of original nodes is connected by an edge if their supernodes are connected by one, and that the weight of an edge is approximated to be the weight of the superedge. The compression problem now consists of choosing supernodes, superedges, and superedge weights so that the approximation error is minimized while the amount of compression is maximized. In this paper, we formulate this task as the 'simple weighted graph compression problem'. We then propose a much wider class of tasks under the name of 'generalized weighted graph compression problem'. The generalized task extends the optimization to preserve longer-range connectivities between nodes, not just individual edge weights. We study the properties of these problems and propose a range of algorithms to solve them, with different balances between complexity and quality of the result. We evaluate the problems and algorithms experimentally on real networks. The results indicate that weighted graphs can be compressed efficiently with relatively little compression error. This paper addresses the optimal scaling of the ADMM method for distributed quadratic programming. Scaled ADMM iterations are first derived for generic equality-constrained quadratic problems and then applied to a class of distributed quadratic problems. In this setting, the scaling corresponds to the step-size and the edge-weights of the underlying communication graph. We optimize the convergence factor of the algorithm with respect to the step-size and graph edge-weights. Explicit analytical expressions for the optimal convergence factor and the optimal step-size are derived. Numerical simulations illustrate our results. Abstract Given a weighted graph G x , where ( x ( v ) : v ∈ V ) is a non-negative, real-valued weight assigned to the vertices of G , let B ( G x ) be an upper bound on the fractional chromatic number of the weighted graph G x ; so χ f ( G x ) ≤ B ( G x ) . We consider a particular upper bound B resulting from a generalization of the greedy coloring algorithm to weighted graphs. To investigate the worst-case performance of this upper bound, we study the graph invariant β ( G ) = sup x ≠ 0 B ( G x ) χ f ( G x ) . This graph invariant is shown to be equal to the size of the largest star subgraph in the graph. These results have implications for the design and performance of distributed communication networks. This work addresses graph-based semi-supervised classification and betweenness computation in large, sparse, networks (several millions of nodes). The objective of semi-supervised classification is to assign a label to unlabeled nodes using the whole topology of the graph and the labeling at our disposal. Two approaches are developed to avoid explicit computation of pairwise proximity between the nodes of the graph, which would be impractical for graphs containing millions of nodes. The first approach directly computes, for each class, the sum of the similarities between the nodes to classify and the labeled nodes of the class, as suggested initially in [1,2]. Along this approach, two algorithms exploiting different state-of-the-art kernels on a graph are developed. The same strategy can also be used in order to compute a betweenness measure. The second approach works on a trellis structure built from biased random walks on the graph, extending an idea introduced in [3]. These random walks allow to define a biased bounded betweenness for the nodes of interest, defined separately for each class. All the proposed algorithms have a linear computing time in the number of edges while providing good results, and hence are applicable to large sparse networks. They are empirically validated on medium-size standard data sets and are shown to be competitive with state-of-the-art techniques. Finally, we processed a novel data set, which is made available for benchmarking, for multi-class classification in a large network: the U.S. patents citation network containing 3M nodes (of six different classes) and 38M edges. The three proposed algorithms achieve competitive results (around 85 classification rate) on this large network-they classify the unlabeled nodes within a few minutes on a standard workstation. We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation &#xd5;(n)-time algorithms [2], [3], which are matched with &#x3a9;(n)-time lower bounds [4], [5], , &#xd5; and &#x3a9; hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios. . No &#x3a9;(n) lower bound or o(m) upper bound were known for exact computation.In this paper, we present an &#xd5;(n^ 5 4 )-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an &#xd5;(n^ 3 4 k^ 1 2 +n)-time algorithm for the k-source shortest paths problem where we want every node to know distances from k sources; this improves Elkins recent bound [6] when k= &#x3a9;(n^ 1 4 ).We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an &#xd5;(n&#x221a; r )-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in &#xd5;(n&#x221a; h ) time the knowledge about distances can be extended for additional h hops. For this, we use weight rounding to introduce small additive errors which can be later fixed. We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as , in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5. A weighted digraph is balanced if, for each node, the sum of the weights of the edges outgoing from that node is equal to the sum of the weights of the edges incoming to that node. Weight-balanced digraphs play a key role in a number of applications, including cooperative control, distributed optimization, and distributed averaging problems. We address the weight-balance problem for a distributed system whose components (nodes) can exchange information via interconnection links (edges) that form an arbitrary, possibly directed, communication topology (digraph). We develop two iterative algorithms, a centralized one and a distributed one, both of which can be used to reach weight-balance, as long as the underlying communication topology forms a strongly connected digraph (or is a collection of strongly connected digraphs). The centralized algorithm is shown to reach weight-balance after a finite number of iterations (bounded by the number of nodes in the graph). The distributed algorithm operates by having each node adapt the weights on its outgoing edges and is shown to asymptotically lead to weight-balance. We also analyze the rate of convergence of the proposed distributed algorithm and obtain a (graph-dependent) worst-case bound for it. Finally, we provide examples to illustrate the operation, performance, and potential advantages of the proposed algorithms. Consider a complete communication network of @math nodes, where the nodes receive a common clock pulse. We study the synchronous @math -counting problem: given any starting state and up to @math faulty nodes with arbitrary behavior, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo @math in agreement. Thus, we are considering algorithms that are self-stabilizing despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilization time in @math (asymptotically optimal), (4) use a small number of states, and, consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomization, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, ... We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization. Consider a complete communication network of @math nodes, where the nodes receive a common clock pulse. We study the synchronous @math -counting problem: given any starting state and up to @math faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo @math in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilisation time in @math (asymptotically optimal), (4) use a small number of states, and consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomisation, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present two complementary approaches for reducing the number of bits communicated during and after stabilisation. Communities of vertices within a giant network such as the World-Wide Web are likely to be vastly smaller than the network itself. However, Fortunato and Barth ' e lemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than @math edges, where @math is the number of edges in the entire network. This resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks. Fortunato and Barth ' e lemy's argument can be extended to networks with weighted edges as well, and we derive this corollary argument. We conclude that weighted modularity algorithms may fail to resolve communities with fewer than @math total edge weight, where @math is the total edge weight in the network and @math is the maximum weight of an inter-community edge. If @math is small, then small communities can be resolved. Given a weighted or unweighted network, we describe how to derive new edge weights in order to achieve a low @math , we modify the CNM'' community detection algorithm to maximize weighted modularity, and show that the resulting algorithm has greatly improved accuracy. In experiments with an emerging community standard benchmark, we find that our simple CNM variant is competitive with the most accurate community detection methods yet proposed. Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn @math ) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn @math ) capable of returning in O(k) time an approximation @math of the distance δ(u,v) from u to v in G that satisfies @math , for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn @math log n). We present here a deterministic O(kmn @math ) time algorithm for constructing oracles of size O(kn @math ). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor. Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn @math ) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced. Given as input an edge-weighted graph, we analyze two algorithms for finding subgraphs with low total edge weight. The first algorithm finds a separator subgraph with a small number of components, and is analyzed for graphs with an arbitrary excluded minor. The second algorithm finds a spanner with small stretch factor, and is analyzed for graphs in a hereditary family G(k). These results imply (i) a QPTAS (quasi-polynomial time approximation scheme) for the TSP (traveling salesperson problem) in unweighted graphs with an excluded minor, and (ii) a QPTAS for the TSP in weighted graphs with bounded genus. We present a distributed algorithm that finds a maximal edge packing in O(Δ + log* W) synchronous communication rounds in a weighted graph, independent of the number of nodes in the network; here Δ is the maximum degree of the graph and W is the maximum weight. As a direct application, we have a distributed 2-approximation algorithm for minimum-weight vertex cover, with the same running time. We also show how to find an @math -approximation of minimum-weight set cover in O(f2k2 + fk log* W) rounds; here k is the maximum size of a subset in the set cover instance, f is the maximum frequency of an element, and W is the maximum weight of a subset. The algorithms are deterministic, and they can be applied in anonymous networks. We present a new way to encode weighted sums into unweighted pairwise constraints, obtaining the following results. - Define the k-SUM problem to be: given n integers in [-n^2k, n^2k] are there k which sum to zero? (It is well known that the same problem over arbitrary integers is equivalent to the above definition, by linear-time randomized reductions.) We prove that this definition of k-SUM remains W[1]-hard, and is in fact W[1]-complete: k-SUM can be reduced to f(k) * n^o(1) instances of k-Clique. - The maximum node-weighted k-Clique and node-weighted k-dominating set problems can be reduced to n^o(1) instances of the unweighted k-Clique and k-dominating set problems, respectively. This implies a strong equivalence between the time complexities of the node weighted problems and the unweighted problems: any polynomial improvement on one would imply an improvement for the other. - A triangle of weight 0 in a node weighted graph with m edges can be deterministically found in m^1.41 time.