Distributed Symmetry Breaking in Hypergraphs

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in $O(\log^2 n)$ rounds ($n$ is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an $O(\Delta^{\epsilon}\text{polylog}(n))$-round hypergraph MIS algorithm in the CONGEST model where $\Delta$ is the maximum node degree of the hypergraph and $\epsilon > 0$ is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every p∈N , a family of L p instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by O~(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1] d , for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3⋅ϕ8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3−1/d⋅ϕ8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4−1/d⋅ϕ) . In addition, we prove an upper bound of O(ϕ√d) on the expected approximation factor with respect to all L p metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ d

Pac-learning unambiguous nts languages

Abstract. Non-terminally separated (NTS) languages are a subclass of deterministic context free languages where there is a stable relationship between the substrings of the language and the non-terminals of the grammar. We show that when the distribution of samples is generated by a PCFG, based on the same grammar as the target language, the class of unambiguous NTS languages is PAC-learnable from positive data alone, with polynomial bounds on data and computation.

Less hashing, same performance: Building a better bloom filter

ABSTRACT: A standard technique from the hashing literature is to use two hash functions h1(x) and h2(x) to simulate additional hash functions of the form gi(x) = h1(x) + ih2(x). We demonstrate that this technique can be usefully applied to Bloom filters and related data structures. Specifically, only two hash functions are necessary to effectively implement a Bloom filter without any loss in the asymptotic false positive probability. This leads to less computation and potentially less need fo

Concentration Bounds for Unigrams Language Model

We show several PAC-style concentration bounds for learning unigrams language model. One interesting quantity is the probability of all words appearing exactly k times in a sample of size m. A standard estimator for this quantity is the Good-Turing estimator. The existing analysis on its error shows a PAC bound of approximately O . We improve its dependency on k to O 4 # k . We also analyze the empirical frequencies estimator, showing that its PAC error bound is approximately . We derive a combined estimator, which has , for any k. A standard measure..

The fourth moment in Luby's distribution

Luby (1988) proposed a way to derandomize randomized computations which is based on the construction of a small probability space whose elements are 3-wise independent. In this paper we prove some new properties of Luby's space. More precisely, we analyze the fourth moment and prove an interesting technical property which helps to understand better Luby's distribution. As an application, we study the behavior of random edge cuts in a weighted graph. (orig.)SIGLEAvailable from TIB Hannover: RR 1912(95-1-019) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekCommission of the European Communities, Brussels (Belgium). European Programme for Research and Development in Information Technology (ESPRIT); Ministry of Education, Athens (Greece)DEGerman