Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference

Local Approximation Schemes for Ad Hoc and Sensor Networks

We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs

Leveraging Physical Layer Capabilites: Distributed Scheduling in Interference Networks with Local Views

In most wireless networks, nodes have only limited local information about the state of the network, which includes connectivity and channel state information. With limited local information about the network, each node's knowledge is mismatched; therefore, they must make distributed decisions. In this paper, we pose the following question - if every node has network state information only about a small neighborhood, how and when should nodes choose to transmit? While link scheduling answers the above question for point-to-point physical layers which are designed for an interference-avoidance paradigm, we look for answers in cases when interference can be embraced by advanced PHY layer design, as suggested by results in network information theory. To make progress on this challenging problem, we propose a constructive distributed algorithm that achieves rates higher than link scheduling based on interference avoidance, especially if each node knows more than one hop of network state information. We compare our new aggressive algorithm to a conservative algorithm we have presented in [1]. Both algorithms schedule sub-networks such that each sub-network can employ advanced interference-embracing coding schemes to achieve higher rates. Our innovation is in the identification, selection and scheduling of sub-networks, especially when sub-networks are larger than a single link.Comment: 14 pages, Submitted to IEEE/ACM Transactions on Networking, October 201

Distributed and Parallel Algorithms for Set Cover Problems with Small Neighborhood Covers

In this paper, we study a class of set cover problems that satisfy a special property which we call the {\em small neighborhood cover} property. This class encompasses several well-studied problems including vertex cover, interval cover, bag interval cover and tree cover. We design unified distributed and parallel algorithms that can handle any set cover problem falling under the above framework and yield constant factor approximations. These algorithms run in polylogarithmic communication rounds in the distributed setting and are in NC, in the parallel setting.Comment: Full version of FSTTCS'13 pape

An Improved Distributed Algorithm for Maximal Independent Set

The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some recent techniques, also leads to a near-optimal global complexity. Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability, all nodes terminate after $O(\log n)$ rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node $v$ terminates after $O(\log \mathsf{deg}(v)+\log 1/\epsilon)$ rounds, with probability at least $1-\epsilon$. The guarantee holds even if the randomness outside $2$-hops neighborhood of $v$ is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high probability global complexity of $O(\log \Delta) + 2^{O(\sqrt{\log \log n})}$, where $\Delta$ denotes the maximum degree. This improves over the $O(\log^2 \Delta) + 2^{O(\sqrt{\log \log n})}$ result of Barenboim et al., and gets close to the $\Omega(\min\{\log \Delta, \sqrt{\log n}\})$ lower bound of Kuhn et al. Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lov\'asz Local Lemma

Towards Optimal Distributed Node Scheduling in a Multihop Wireless Network through Local Voting

In a multihop wireless network, it is crucial but challenging to schedule transmissions in an efficient and fair manner. In this paper, a novel distributed node scheduling algorithm, called Local Voting, is proposed. This algorithm tries to semi-equalize the load (defined as the ratio of the queue length over the number of allocated slots) through slot reallocation based on local information exchange. The algorithm stems from the finding that the shortest delivery time or delay is obtained when the load is semi-equalized throughout the network. In addition, we prove that, with Local Voting, the network system converges asymptotically towards the optimal scheduling. Moreover, through extensive simulations, the performance of Local Voting is further investigated in comparison with several representative scheduling algorithms from the literature. Simulation results show that the proposed algorithm achieves better performance than the other distributed algorithms in terms of average delay, maximum delay, and fairness. Despite being distributed, the performance of Local Voting is also found to be very close to a centralized algorithm that is deemed to have the optimal performance

An improved connectionist activation function for energy minimization

Symmetric networks that are based on energy minimization, such as Boltzmann machines or Hopfield nets, are used extensively for optimization, constraint satisfaction, and approximation of NP-hard problems. Nevertheless, finding a global minimum for the energy function is not guaranteed, and even a local minimum may take an exponential number of steps. We propose an improvement to the standard activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm is uniform and does not assume that the network is a tree. It performs no worse than the standard algorithms for any network topology. In the case where there are trees growing from a cyclic subnetwork, the new algorithm performs better than the standard algorithms by avoiding local minima along the trees and by optimizing the free energy of these trees in linear time. The algorithm is self-stabilizing for trees (cycle-free undirected graphs) and remains correct under various scheduling demons. However, no uniform protocol exists to optimize trees under a pure distributed demon and no such protocol exists for cyclic networks under central demon