Optimal Gossip Algorithms for Exact and Approximate Quantile Computations

This paper gives drastically faster gossip algorithms to compute exact and approximate quantiles. Gossip algorithms, which allow each node to contact a uniformly random other node in each round, have been intensely studied and been adopted in many applications due to their fast convergence and their robustness to failures. Kempe et al. [FOCS'03] gave gossip algorithms to compute important aggregate statistics if every node is given a value. In particular, they gave a beautiful $O(\log n + \log \frac{1}{\epsilon})$ round algorithm to $\epsilon$-approximate the sum of all values and an $O(\log^2 n)$ round algorithm to compute the exact $\phi$-quantile, i.e., the the $\lceil \phi n \rceil$ smallest value. We give an quadratically faster and in fact optimal gossip algorithm for the exact $\phi$-quantile problem which runs in $O(\log n)$ rounds. We furthermore show that one can achieve an exponential speedup if one allows for an $\epsilon$-approximation. We give an $O(\log \log n + \log \frac{1}{\epsilon})$ round gossip algorithm which computes a value of rank between $\phi n$ and $(\phi+\epsilon)n$ at every node.% for any $0 \leq \phi \leq 1$ and $0 < \epsilon < 1$. Our algorithms are extremely simple and very robust - they can be operated with the same running times even if every transmission fails with a, potentially different, constant probability. We also give a matching $\Omega(\log \log n + \log \frac{1}{\epsilon})$ lower bound which shows that our algorithm is optimal for all values of $\epsilon$

Distributed and Robust Support Vector Machine

In this paper, we consider the distributed version of Support Vector Machine (SVM) under the coordinator model, where all input data (i.e., points in R^d space) of SVM are arbitrarily distributed among k nodes in some network with a coordinator which can communicate with all nodes. We investigate two variants of this problem, with and without outliers. For distributed SVM without outliers, we prove a lower bound on the communication complexity and give a distributed (1-epsilon)-approximation algorithm to reach this lower bound, where epsilon is a user specified small constant. For distributed SVM with outliers, we present a (1-epsilon)-approximation algorithm to explicitly remove the influence of outliers. Our algorithm is based on a deterministic distributed top t selection algorithm with communication complexity of O(k log (t)) in the coordinator model. Experimental results on benchmark datasets confirm the theoretical guarantees of our algorithms

Protocolos para el Procesamiento Distribuido de Funciones de Umbral en Redes Inalámbricas de Sensores

Tesis (DCI)--FCEFN-UNC, 2015Determina el esquema de procesamiento más adecuado para la computación de funciones de tipo umbral en redes inalámbricas de sensores, se define un arquitectura determina el funcionamiento de cada nodo de la red y se desarrolla dos protocolos de procesamiento y filtrado de datos y de nodos vecinos

Tight Bounds for Distributed Selection

We revisit the problem of distributed k-selection where, given a general connected graph of diameter D consisting of n nodes in which each node holds a numeric element, the goal is to determine the k th smallest of these elements. In our model, there is no imposed relation between the magnitude of the stored elements and the number of nodes in the graph. We propose a randomized algorithm whose time complexity is O(D logD n) with high probability. Additionally, a deterministic algorithm with a worst-case time complexity of O(D log 2 D n) is presented which considerably improves the best known bound for deterministic algorithms. Moreover, we prove a lower bound of Ω(D logD n) for any randomized or deterministic algorithm, implying that the randomized algorithm is asymptotically optimal