9 research outputs found

    Noisy Boolean Hidden Matching with Applications

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    The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et al. [STOC\u2707], has played an important role in lower bounds for graph problems in the streaming model (e.g., subgraph counting, maximum matching, MAX-CUT, Schatten p-norm approximation). The BHM problem typically leads to ?(?n) space lower bounds for constant factor approximations, with the reductions generating graphs that consist of connected components of constant size. The related Boolean Hidden Hypermatching (BHH) problem provides ?(n^{1-1/t}) lower bounds for 1+O(1/t) approximation, for integers t ? 2. The corresponding reductions produce graphs with connected components of diameter about t, and essentially show that long range exploration is hard in the streaming model with an adversarial order of updates. In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain stronger than ?(?n) lower bounds for approximating a number of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant. We next introduce and study the graph classification problem, where the task is to test whether the input graph is isomorphic to a given graph. As a first step, we use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires ?(n) space, which seems challenging to show using either BHM or BHH

    Ruling Sets in Random Order and Adversarial Streams

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    Testable properties in general graphs and random order streaming

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    We present a novel framework closely linking the areas of property testing and data streaming algorithms in the setting of general graphs. It has been recently shown (Monemizadeh et al. 2017) that for bounded-degree graphs, any constant-query tester can be emulated in the random order streaming model by a streaming algorithm that uses only space required to store a constant number of words. However, in a more natural setting of general graphs, with no restriction on the maximum degree, no such results were known because of our lack of understanding of constant-query testers in general graphs and lack of techniques to appropriately emulate in the streaming setting off-line algorithms allowing many high-degree vertices. In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constant-query testers for general graphs, both, for one-sided and two-sided errors. Such canonizations were only known before (in the adjacency matrix model) for dense graphs (Goldreich and Trevisan 2003) and (in the adjacency list model) for bounded degree (di-)graphs (Goldreich and Ron 2011, Czumaj et al. 2016). Using the concept of canonical testers, we then prove that every property of general graphs that is constant-query testable with one-sided error can also be tested in constant-space with one-sided error in the random order streaming model. Our results imply, among others, that properties like (s,t) disconnectivity, k-path-freeness, etc. are constant-space testable in random order streams

    Testable bounded degree graph properties are random order streamable

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    We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constant-query testable in the adjacency list model can be tested with constant space in a single-pass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space. We then show that our approach can also be applied to constant time approximation algorithms for bounded degree graphs in the adjacency list model: As an example, we obtain a constant-space single-pass random order streaming algorithms for approximating the size of a maximum matching with additive error epsilon n (n is the number of nodes). Our result establishes for the first time that a large class of sublinear algorithms can be simulated in random order streams, while Omega(n) space is needed for many graph streaming problems for adversarial orders
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