Consider n nodes connected to a single coordinator. Each node receives an individual online data stream of numbers and, at any point in time, the coordinator has to know the k nodes currently observing the largest values, for a given k between 1 and n. We design and analyze an algorithm that solves this problem while bounding the amount of messages exchanged between the nodes and the coordinator. Our algorithm employs the idea of using filters which, intuitively speaking, leads to few messages to be sent, if the new input is "similar" to the previous ones. The algorithm uses a number of messages that is on expectation by a factor of O((log + k) log n) larger than that of an offline algorithm that sets filters in an optimal way, where is upper bounded by the largest value observed by any node.
Following the idea of studying competitive algorithms for monitoring streams and the notion of filters, @cite_1 present an algorithm for online dominance tracking of distributed streams. In this problem a coordinator always has to be informed about the dominance relationship between @math distributed nodes each observing an online stream of @math -dimensional values. Here, the dominance relation of two nodes @math and @math currently observing @math and @math , respectively, both from @math , is defined as @math dominates @math if @math for some @math and @math for all other @math . Their algorithm is based on the idea of assigning filters to the nodes and they show that a mid-point strategy, which basically sets filters to be the mid-point between neighboring nodes, is @math -competitive with respect to the number of messages sent in comparison to an offline algorithm that sets filters optimally.
@cite_1
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In this paper, we study the dominance relation under a stochastic setting. Let @math be a set of @math colored stochastic points in @math , each of which is associated with an existence probability. We investigate the problem of computing the probability that a realization of @math contains inter-color dominances, which we call the (CSD) problem. We propose the first algorithm to solve the CSD problem for @math in @math time. On the other hand, we prove that, for @math , even the CSD problem with a restricted color pattern is #P-hard. In addition, even if the existence probabilities are restricted to be @math , the problem remains #P-hard for @math . A simple FPRAS is then provided to approximate the desired probability in any dimension. We also study a variant of the CSD problem in which the dominance relation is considered with respect to not only the standard basis but any orthogonal basis of @math . Specifically, this variant, which we call the free-basis colored stochastic dominance (FBCSD) problem, considers the probability that a realization of @math contains inter-color dominances with respect to any orthogonal basis of @math . We show that the CSD problem is polynomial-time reducible to the FBCSD problem in the same dimension, which proves the #P-hardness of the latter for @math . Conversely, we reduce the FBCSD problem in @math to the CSD problem in @math , by which an @math time algorithm for the former is obtained. Influence problem is one of the central problems in the study of online social networks, the goal of which is to influence all nodes with the minimum number of seeds. However, in the real world, it might be too expensive to influence all nodes. In many cases, it is satisfactory to influence nodes only up to some percent p. In this paper, we study the minimum partial positive influence dominating set (MPPIDS) problem. In fact, we presented an approximation algorithm for a more general problem called minimum partial set multicover problem. As a consequence, the MPPIDS problem admits an approximation with performance ratio @math źH(Δ), where @math H(·) is the Harmonic number, @math ź=1 (1-(1-p)ź),źźΔ2 ź, and @math Δ,ź are the maximum degree and the minimum degree of the graph, respectively. For power-law graphs, we show that our algorithm has a constant performance ratio. Let @math , @math for @math , @math bounding the bit length of the coefficients of the @math 's, and let @math be a quantifier-free @math -formula defining a convex semialgebraic set. We design an algorithm returning a rational point in @math if and only if @math . It requires @math bit operations. If a rational point is outputted, its coordinates have bit length dominated by @math . Using this result, we obtain a procedure for deciding whether a polynomial @math is a sum of squares of polynomials in @math . Denote by @math the degree of @math , @math the maximum bit length of the coefficients in @math , @math , and @math . This procedure requires @math bit operations, and the coefficients of the outputted polynomials have bit length dominated by @math . A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, @math the minimum degree, and @math the maximum degree. We show that every graph has a domatic partition with @math dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a @math -approximation algorithm for domatic number, since the domatic number is always at most @math . We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every @math , a @math -approximation implies that @math . This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with @math dominating sets, where the "o(1)" term goes to zero as @math increases. This can be turned into an efficient algorithm that produces a domatic partition of @math sets.