Emanuele G. Fusco | Searcharxiv

arXiv2015

Knowledge, Level of Symmetry, and Time of Leader Election

We study the time needed for deterministic leader election in the ${\cal LOCAL}$ model, where in every round a node can exchange any messages with its neighbors and perform any local computations. The topology of the network is unknown and nodes are unlabeled, but ports at each node have arbitrary fixed labelings which, together with the topology of the network, can create asymmetries to be exploited in leader election. We consider two versions of the leader election problem: strong LE in which exactly one leader has to be elected, if this is possible, while all nodes must terminate declaring that leader election is impossible otherwise, and weak LE, which differs from strong LE in that no requirement on the behavior of nodes is imposed, if leader election is impossible. We show that the time of leader election depends on three parameters of the network: its diameter $D$, its size $n$, and its level of symmetry $λ$, which, when leader election is feasible, is the smallest depth at which some node has a unique view of the network. It also depends on the knowledge by the nodes, or lack of it, of parameters $D$ and $n$.

cs.DC↗

arXiv2014

Clique counting in MapReduce: theory and experiments

We tackle the problem of counting the number of $k$-cliques in large-scale graphs, for any constant $k \ge 3$. Clique counting is essential in a variety of applications, among which social network analysis. Due to its computationally intensive nature, we settle for parallel solutions in the MapReduce framework, which has become in the last few years a {\em de facto} standard for batch processing of massive data sets. We give both theoretical and experimental contributions. On the theory side, we design the first exact scalable algorithm for counting (and listing) $k$-cliques. Our algorithm uses $O(m^{3/2})$ total space and $O(m^{k/2})$ work, where $m$ is the number of graph edges. This matches the best-known bounds for triangle listing when $k=3$ and is work-optimal in the worst case for any $k$, while keeping the communication cost independent of $k$. We also design a sampling-based estimator that can dramatically reduce the running time and space requirements of the exact approach, while providing very accurate solutions with high probability. We then assess the effectiveness of different clique counting approaches through an extensive experimental analysis over the Amazon EC2 platform, considering both our algorithms and their state-of-the-art competitors. The experimental results clearly highlight the algorithm of choice in different scenarios and prove our exact approach to be the most effective when the number of $k$-cliques is large, gracefully scaling to non-trivial values of $k$ even on clusters of small/medium size. Our approximation algorithm achieves extremely accurate estimates and large speedups, especially on the toughest instances for the exact algorithms. As a side effect, our study also sheds light on the number of $k$-cliques of several real-world graphs, mainly social networks, and on its growth rate as a function of $k$.

cs.DC↗