Derandomizing local distributed algorithms under bandwidth restrictions

This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send $$O(\log n)$$-bit messages in each round of communication. Our framework mostly follows by the derandomization approach of Luby (J Comput Syst Sci 47(2):250–286, 1993) combined with the power of all to all communication. Our key results are as follows: first, we show that in the congested clique model, which allows all-to-all communication, there is a deterministic maximal independent set algorithm that runs in $$O(\log ^2 {\varDelta })$$ rounds, where $${\varDelta }$$ is the maximum degree. When $${\varDelta }=O(n^{1/3})$$, the bound improves to $$O(\log {\varDelta })$$. In addition, we deterministically construct a $$(2k-1)$$-spanner with $$O(kn^{1+1/k}\log n)$$ edges in $$O(k \log n)$$ rounds in the congested clique model.

We present a distributed randomized algorithm finding Minimum Spanning Tree (MST) of a given graph in O(1) rounds, with high probability, in the congested clique model. The input graph in the congested clique model is a graph of n nodes, where each node initially knows only its incident edges. The communication graph is a clique with limited edge bandwidth: each two nodes (not necessarily neighbours in the input graph) can exchange O(log n) bits. As in previous works, the key part of the MST algorithm is an efficient Connected Components (CC) algorithm. However, unlike the former approaches, we do not aim at simulating the standard Boruvka's algorithm, at least at initial stages of the CC algorithm. Instead, we develop a new technique which combines connected components of sample sparse subgraphs of the input graph in order to accelerate the process of uncovering connected components of the original input graph. More specifically, we develop a sparsification technique which reduces an initial CC problem in O(1) rounds to its two restricted instances. The former instance has a graph with maximal degree O(log log n) as the input - here our sample-combining technique helps. In the latter instance, a partition of the input graph into O(n/ log log n) connected components is known. This gives an opportunity to apply previous algorithms to determine connected components in O(1) rounds. Our result addresses a problem proposed by Lotker et al. [SPAA 2003; SICOMP 2005] and improves over previous O(log* n) algorithm of Ghaffari et al. [PODC 2016], and O(log log log n) algorithm of Hegeman et al. [PODC 2015]. It also determines Θ(1) round complexity in the congested clique for MST, as well as other graph problems, including bipartiteness, cut verification, s-t connectivity, and cycle containment.

The congested clique model is a message-passing model of distributed computation where k players communicate with each other over a complete network. Here we consider synchronous protocols in which communication happens in rounds (we allow them to be randomized with public coins). In the unicast communication mode, each player i has her own n-bit input \(x_i\) and may send \(k-1\) different b-bit messages through each of her \(k-1\) communication links in each round. On the other end is the broadcast communication mode, where each player can only broadcast a single message over all her links in each round. The goal of this paper is to complete our Brief Announcement at PODC 2015, where we initiated the study of the space that lies between the two extremes. For that purpose, we parametrize the congested clique model by two values: the range r, which is the maximum number of different messages a player is allowed to send in each round, and the bandwidth b, which is the maximum size of these messages. We show that the space between the unicast and broadcast congested clique models is very rich and interesting. For instance, we show that the round complexity of the pairwise set-disjointness function \(\textsc {pwdisj}\) is completely sensitive to the range r. This translates into a \(\varOmega (k)\) gap between the unicast (\(r=k-1\)) and the broadcast (\(r=1\)) modes. Moreover, provided that \(r \ge 2\) and \(rb/\log r = O(k)\), the round complexity of \(\textsc {pwdisj}\) is \(\varTheta (n/ k \log r )\). On the other hand, we also prove that the behavior of \(\textsc {pwdisj}\) is exceptional: almost every boolean function f has maximal round complexity \(\varTheta (n/b)\). Finally, we prove that \(\min \left( \left\lceil \frac{b'}{\lfloor \log r \rfloor } \right\rceil , \left\lceil \frac{r'}{r-1}\right\rceil \left\lceil \frac{b'}{b}\right\rceil \right) \) is an upper bound for the gap between the round complexities with parameters (b, r) and parameters \((b',r')\) of any boolean function.

Every t-dynamic proper n-coloring of a graph G describes a shadow allocation of any (n,t+1)-threshold secret sharing scheme based on G, so that, after just one round of communication, each participant can either reconstruct the secret, or obtain a different shadow from each one of his/her neighbors. Thus, for just one round of communication, this scheme is fair if and only if the threshold is either less than or equal to the minimum degree of G, or greater than or equal to its maximum degree. Despite that the dynamic coloring problem has widely been dealt with in the literature, a comprehensive study concerning this implementation in cryptography is still required. This paper delves into this topic by focusing on the use of extended neighborhood coronas for modeling communication networks whose average path lengths are small even after an asymptotic growth of their center and/or outer graphs. Particularly, the dynamic coloring problem is solved for any extended neighborhood corona with center path or star, for which we establish optimal shadow allocations of any (fair) threshold secret sharing scheme based on them. Some bounds are also established for the dynamic chromatic number of any extended neighborhood corona.

The paper is devoted to Time Division Multiple Access Link Scheduling Protocols in wireless sensor networks for full duplex (two-way) communication, where each sensor is scheduled on an incident link as a transmitter and as a receiver in two different time slots. We formulate the full duplex link scheduling problem (FDLSP) as distance-2 edge coloring in bidirected graphs. We proves that there exists a Δ-approximation algorithm for FDLSP (Δ being the maximum node degree in the network). Then, we present two distributed algorithms. The first is a synchronous algorithm based on finding maximal independent sets. The second is an asynchronous depth first search (DFS) algorithm. The maximal independent set based algorithm requires only O(Δlog*n) communication rounds (where n is the number of processors in the network) for growth bounded graphs, which is a realistic geometric model of sensor networks. For general graphs, the maximal independent set based algorithm requires O(Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> + Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> log*n) communication rounds, improving upon the previous best algorithm with O(nΔ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> m) communication rounds (where m is the number of links in the network). The asynchronous DFS based algorithm requires only O(n) communication rounds for both general and growth bounded graphs. The simulations show that the proposed algorithms assign on average equal or fewer number of time slots compared to the best known distributed algorithm while being significantly faster.

Consider an n-vertex graph G = (V,E) of maximum degree Δ, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds.

The reconfigurable mesh is a processor array that consists processors arranged in 1-dimensional or 2- dimensional grids with a reconfigurable bus system. The main contribution of this paper is to show initialization algorithms on the 1-dimensional reconfigurable mesh with n processors. We assume that processors are identical, and does not have unique IDs. Initialization is a task that assigns sequential IDs to processors in the reconfigurable mesh. We first show a simple deterministic initialization algorithm for the I-dimensional reconfigurable mesh that runs in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n) time. This deterministic algorithm is optimal, because no deterministic solution can perform initialization in less than <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n) time. Quite surprisingly, we show that expected sublinear-time initialization is possible if we use randomized techniques. Our initialization algorithm runs in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ((log n + log f) log log n) time with probability at least 1-1/4 for every real number f ges 1. It follows that the initialization algorithm runs in expected O(log n log log n) time. We also proved that any randomized initialization need to run in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (log n) time. Thus, our randomized initialization algorithm running in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (log n log log n) time is very close to a theoretical lower bound Omega(log n) time.

We study the edge-coloring problem in the message-passing model of distributed computing. This is one of the most fundamental problems in this area. Currently, the best-known deterministic algorithms for (2Δ-1)-edge-coloring requires O(Δ) + log* n time [23], where Δ is the maximum degree of the input graph. Also, recent results of [5] for vertex-coloring imply that one can get an O(Δ)-edge-coloring in O(Δµ" log n) time, and an O(Δ1 + µ)-edge-coloring in O(log Δ log n) time, for an arbitrarily small constant µ > 0.

We study the complexity of local search in the max-cut problem with FLIP neighborhood, in which exactly one node changes the partition. We introduce a technique of constructing instances which enforce certain sequences of improving steps. Using our technique we can show that already graphs with maximum degree four satify the following two properties.There are instances with initial solutions for which every local search takes exponential time to converge to a local optimum. The problem of computing a local optimum reachable from a given solution by a sequence of improving steps is PSPACE-complete. Schaffer and Yannakakis (JOC '91) showed via a so called “tight” PLS-reduction that the properties (1) and (2) hold for graphs with unbounded degree. Our improvement to the degree four is the best possible improvement since Poljak (JOC '95) showed for cubic graphs that every sequence of improving steps has polynomial length, whereby his result is easily generalizable to arbitrary graphs with maximum degree three. In his paper Poljak also asked whether (1) holds for graphs with maximum degree four, which is settled by our result. Many tight PLS-reductions in the literature are based on the max-cut problem. Via some of them our constructions carry over to other problems and show that the properties (1) and (2) already hold for very restricted sets of feasible inputs of these problems. Since our paper provides the two results that typically come along with tight PLS-reductions it does naturally put the focus on the question whether it is even PLS-complete to compute a local optimum on graphs with maximum degree four – a question that was recently asked by Ackermann et al. We think that our insights might be helpful for tackling this question.

Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In contrast, only few analytical results are known. We show that MinGreedy cannot improve on the trivial approximation ratio of $$\frac{1}{2}$$12 whp., even for bipartite graphs. Our hard inputs seem to require a small number of high-degree nodes. This motivates an investigation of greedy algorithms on graphs with maximum degree $$\varDelta $$Δ: we show that MinGreedy achieves a $${\frac{{\varDelta }-1}{2{\varDelta }-3}} $$Δ-12Δ-3-approximation for graphs with $${\varDelta } {=} 3$$Δ=3 and for $$\varDelta $$Δ-regular graphs, and a guarantee of $${\frac{{\varDelta }-1/2}{2{\varDelta }-2}} $$Δ-1/22Δ-2 for graphs with maximum degree $${\varDelta } $$Δ. Interestingly, our bounds even hold for the deterministic MinGreedy that breaks all ties arbitrarily. Moreover, we investigate the limitations of the greedy paradigm, using the model of priority algorithms introduced by Borodin, Nielsen, and Rackoff. We study deterministic priority algorithms and prove a $${\frac{{\varDelta }-1}{2{\varDelta }-3}}$$Δ-12Δ-3-inapproximability result for graphs with maximum degree $${\varDelta } $$Δ; thus, these greedy algorithms do not achieve a $$\frac{1}{2} {+} \varepsilon $$12+?-approximation and in particular the $$\frac{2}{3}$$23-approximation obtained by the deterministic MinGreedy for $${\varDelta } {=} 3$$Δ=3 is optimal in this class. For k-uniform hypergraphs we show a tight $$\frac{1}{k}$$1k-inapproximability bound. We also study fully randomized priority algorithms and give a $$\frac{5}{6}$$56-inapproximability bound. Thus, they cannot compete with matching algorithms of other paradigms.

We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schaffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Roglin, and Vocking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local MAX-CUT problem with FLIP-neighborhood is PLS-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is PLS-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless PLS ⊆ P). On the other side, we also prove that on graphs with degree O(log n) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance &sigma2, every FLIP local search terminates in time polynomial in n and σ-1, with probability 1-n-Ω(1). Putting both results together, we may conclude that although local MAX-CUT is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.

Multi hop links are used to communicate between nodes in a Mobile Ad Hoc Network (MANET). Data packets are routed through intermediate nodes in the network to the destination node since each node functions as a router. Dynamic routing protocols are being developed for network communication efficiency. Routing protocols in MANETS face problems such as versatility, scalability, and restricted bandwidth. A fuzzy logic approach is suggested as part of the study. In this work, a fuzzy logic methodology is planned to choose the best routes in order to better QoS End-to-end delay (Ae), when a node leaves the network (Nl), A fuzzy rule-based scheme NR includes the number of packets dropped (ND) and the number of path error RRER caused parameters. Simulations are carried out in this study using the proposed fuzzy approach process Simulations are carried out in this study using the future process fuzzy approach. The measurement criteria are throughput and the number of packets dropped. The tentative results obtained show that the proposed fuzzy approach improves the QoS better than existing regular Ad hoc On-demand Distance Vector ( AODV) routing protocol. In the final stage of investigation, the use of a hybrid optimization approach to choose the best route is suggested. The proposed hybrid optimization is based on the Genetic Algorithm (GA) and the Hill Climbing algorithm. The hybrid optimization aids in the resolution of the local minimum problem Hill-climbing begins with an infeasible solution and continues until a viable solution is found, at which point the viable solution is returned to GA. The simulation results show that the throughput is high. When compared to the AODV protocol, the proposed Fuzzy-Hybrid GA method yields about 29.56 percent more and 6.33 percent more when compared to the Fuzzy-GA solution.

Clinical databases require accurate entity resolution (ER). One approach is to use algorithms that assign questionable cases to manual review. Few studies have compared the performance of common algorithms for such a task. Furthermore, previous work has been limited by a lack of objective methods for setting algorithm parameters. We compared the performance of common ER algorithms: using algorithmic optimization, rather than manual parameter tuning, and on two-threshold classification (match/manual review/non-match) as well as single-threshold (match/non-match). We manually reviewed 20,000 randomly selected, potential duplicate record-pairs to identify matches (10,000 training set, 10,000 test set). We evaluated the probabilistic expectation maximization, simple deterministic and fuzzy inference engine (FIE) algorithms. We used particle swarm to optimize algorithm parameters for a single and for two thresholds. We ran 10 iterations of optimization using the training set and report averaged performance against the test set. The overall estimated duplicate rate was 6%. FIE and simple deterministic algorithms allowed a lower manual review set compared to the probabilistic method (FIE 1.9%, simple deterministic 2.5%, probabilistic 3.6%; p<0.001). For a single threshold, the simple deterministic algorithm performed better than the probabilistic method (positive predictive value 0.956 vs 0.887, sensitivity 0.985 vs 0.887, p<0.001). ER with FIE classifies 98.1% of record-pairs correctly (1/10,000 error rate), assigning the remainder to manual review. Optimized deterministic algorithms outperform the probabilistic method. There is a strong case for considering optimized deterministic methods for ER.

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: Fast Δ-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al. [6], we prove that the randomized complexity of Δ-coloring a tree with maximum degree Δ is θ(logΔ log n), for any Δ ≥ 55, whereas its deterministic complexity is θ(logΔn) for any Δ ≥ 3. This also establishes a large separation between the deterministic complexity of Δ-coloring and (Δ+1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(logΔn) rounds can be transformed to run in O(log* n -- log*Δ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(logΔn) time deterministically. (This gives an alternate proof that deterministically Δ-coloring a tree with small Δ takes Ω(logΔn) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √log n}. This shows that a deterministic Ω(logΔn) lower bound for any problem (Δ-coloring a tree, for example) implies a randomized Ω(logΔ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2O(√log log n) terms in the complexities of the best MIS and (Δ+1)-coloring algorithms without also improving the 2O(√log n})-round Panconesi-Srinivasan algorithms.

Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area, most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Therefore, an interesting fundamental problem in this area is whether randomization is inherently necessary for obtaining good approximation ratios.In this work, we give evidence that randomization is not necessary for obtaining good algorithms by presenting a new technique for derandomization of algorithms for submodular function maximization. Our high level idea is to maintain explicitly a (small) distribution over the states of the algorithm, and carefully update it using marginal values obtained from an extreme point solution of a suitable linear formulation. We demonstrate our technique on two recent algorithms for unconstrained submodular maximization and for maximizing a submodular function subject to a cardinality constraint. In particular, for unconstrained submodular maximization we obtain an optimal deterministic 1/2-approximation showing that randomization is unnecessary for obtaining optimal results for this setting.