Space-efficient Local Computation Algorithms

Alon, Noga; Rubinfeld, Ronitt; Vardi, Shai; Xie, Ning

doi:10.1137/1.9781611973099.89

Cited by 45 publications

(92 citation statements)

References 27 publications

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“…A k-wise independent hash function whose range is polynomial in n can be constructed using Θ(k log n) mutually independent random bits [4]. Therefore, this implies that Θ(log 2 n) mutually independent random bits are sufficient to generate h and the Θ(log n) gr's.…”

Section: Linear Sketches Of a Graphmentioning

confidence: 99%

Toward Optimal Bounds in the Congested Clique

Hegeman

Pandurangan

Pemmaraju

et al. 2015

Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing

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We study two fundamental graph problems, Graph Connectivity (GC) and Minimum Spanning Tree (MST), in the well-studied Congested Clique model, and present several new bounds on the time and message complexities of randomized algorithms for these problems. No non-trivial (i.e., super-constant) time lower bounds are known for either of the aforementioned problems; in particular, an important open question is whether or not constant-round algorithms exist for these problems. We make progress toward answering this question by presenting randomized Monte Carlo algorithms for both problems that run in O(log log log n) rounds (where n is the size of the clique). Our results improve by an exponential factor on the long-standing (deterministic) time bound of O(log log n) rounds for these problems due to Lotker et al. (SICOMP 2005). Our algorithms make use of several algorithmic tools including graph sketching, random sampling, and fast sorting.The second contribution of this paper is to present several almosttight bounds on the message complexity of these problems. Specifically, we show that Ω(n 2 ) messages are needed by any algorithm (including randomized Monte Carlo algorithms, and regardless of the number of rounds) that solves the GC (and hence also the MST) problem if each machine in the Congested Clique has initial knowledge only of itself (the so-called KT0 model). In contrast, if the machines have initial knowledge of their neighbors' IDs (the so-called KT1 model), we present a randomized Monte Carlo algorithm for MST that uses O(n polylog n) messages and runs in O(polylog n) rounds. To complement this, we also present a lower bound in the KT1 model that shows that Ω(n) messages are required by any al- * gorithm that solves GC, regardless of the number of rounds used. Our results are a step toward understanding the power of randomization in the Congested Clique with respect to both time and message complexity.

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Section: Linear Sketches Of a Graphmentioning

confidence: 99%

Toward Optimal Bounds in the Congested Clique

Hegeman

Pandurangan

Pemmaraju

et al. 2015

Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing

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“…Other Graph Problems. The model of local computation algorithms as considered in this work, was defined by Rubinfeld, Tamir, Vardi, and Xie [36] (see also Alon, Rubinfeld, Vardi, and Xie [2]). Such algorithms for maximal independent set, hypergraph coloring, k-CNF and maximum matching are given in [2,25,26,36].…”

Section: Local Algorithms Formentioning

confidence: 99%

“…The model of local computation algorithms as considered in this work, was defined by Rubinfeld, Tamir, Vardi, and Xie [36] (see also Alon, Rubinfeld, Vardi, and Xie [2]). Such algorithms for maximal independent set, hypergraph coloring, k-CNF and maximum matching are given in [2,25,26,36]. This model generalizes other models that have been studied in various contexts, including locally decodable codes (e.g., [24]), local decompression [14], and local filters/reconstructors [1,9,12,17,18,37].…”

Section: Local Algorithms Formentioning

confidence: 99%

“…• There is f (t) = ((log t) −1 (log log t) −2 ) such that if G is f -non-expanding then one can locally construct a sparse spanning subgraph of G. The algorithm is deterministic. • There is g(t) = O((log t) −1 (log log t) 2 ) and a family of 3-regular graphs G n that are gnon-expanding such that every (possibly randomized) local algorithm for constructing a sparse spanning subgraph of G n must accept every edge of G n .…”

Section: Introductionmentioning

confidence: 99%

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Constructing near spanning trees with few local inspections

Levi

Moshkovitz

Ron

et al. 2016

Random Struct Algorithms

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Constructing a spanning tree of a graph is one of the most basic tasks in graph theory.Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph G consisting of (1+ )n edges (for a prespecified constant > 0), where the decision for different edges should be consistent with the same subgraph G . Can this task be performed by inspecting only a constant number of edges in G? Our main results are:• We show that if every t-vertex subgraph of G has expansion 1/(log t) 1+o(1) then one can (deterministically) construct a sparse spanning subgraph G of G using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm.• We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion 1/(log t) 1−o(1) . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.

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“…Property-preserving data reconstruction [1] is a direction of research in sublinear algorithms that has its roots in property testing [14,10]. Some related notions include locally decodable codes [12], program checking [7] and, more generally, local computation [15,2].…”

Section: Introductionmentioning

confidence: 99%

Limitations of Local Filters of Lipschitz and Monotone Functions

Awasthi

Jha

Molinaro

et al. 2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. We study local filters for two properties of functions f : {0, 1} d → R: the Lipschitz property and monotonicity. A local filter with additive error a is a randomized algorithm that is given black-box access to a function f and a query point x in the domain of f . Its output is a value F (x), such that (i) the reconstructed function F (x) satisfies the property (in our case, is Lipschitz or monotone) and (ii) if the input function f satisfies the property, then for every point x in the domain (with high constant probability) the reconstructed value F (x) differs from f (x) by at most a. Local filters were introduced by Saks and Seshadhri (SICOMP 2010) and the relaxed definition we study is due to Bhattacharyya et al. (RANDOM 2010), except that we further relax it by allowing additive error. Local filters for Lipschitz and monotone functions have applications to areas such as data privacy. We show that every local filter for Lipschitz or monotone functions runs in time exponential in the dimension d, even when the filter is allowed significant additive error. Prior lower bounds (for local filters with no additive error, i.e., with a = 0) applied only to more restrictive class of filters, e.g., nonadaptive filters. To prove our lower bounds, we construct families of hard functions and show that lookups of a local filter on these functions are captured by a combinatorial object that we call a c-connector. Then we present a lower bound on the maximum outdegree of a c-connector, and show that it implies the desired bounds on the running time of local filters. Our lower bounds, in particular, imply the same bound on the running time for a class of privacy mechanisms.

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Space-efficient Local Computation Algorithms

Cited by 45 publications

References 27 publications

Toward Optimal Bounds in the Congested Clique

Toward Optimal Bounds in the Congested Clique

Constructing near spanning trees with few local inspections

Limitations of Local Filters of Lipschitz and Monotone Functions

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