Showing papers by "Vladimir Braverman published in 2013"

How hard is counting triangles in the streaming model

Abstract: The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. Specifically, the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of Ω(m) for graphs G with m edges. If a constant number of passes is allowed, we show a lower bound of Ω(m/T), T the number of triangles. We match, in some sense, this lower bound with a 2-pass O(m/T1/3)-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least T triangles. We present a new graph parameter ρ(G) --- the triangle density, and conjecture that the space complexity of the triangles problem is Θ(m/ρ(G)). We match this by a second algorithm that solves the distinguishing problem using O(m/ρ(G))-memory.

How Hard is Counting Triangles in the Streaming Model

Abstract: The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of $\Omega(m)$ for graphs $G$ with $m$ edges on $n$ vertices. If a constant number of passes is allowed, we show a lower bound of $\Omega(m/T)$, $T$ the number of triangles. We match, in some sense, this lower bound with a 2-pass $O(m/T^{1/3})$-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least $T$ triangles. We present a new graph parameter $\rho(G)$ -- the triangle density, and conjecture that the space complexity of the triangles problem is $\Omega(m/\rho(G))$. We match this by a second algorithm that solves the distinguishing problem using $O(m/\rho(G))$-memory.

Generalizing the Layering Method of Indyk and Woodruff: Recursive Sketches for Frequency-Based Vectors on Streams

Abstract: In their ground-breaking paper, Indyk and Woodruff (STOC 05) showed how to compute the k-th frequency moment F k (for k > 2) in space O(poly − log(n,m) · n\(^{1-{2} \over{k}})\), giving the first optimal result up to poly-logarithmic factors in n and m (here m is the length of the stream and n is the size of the domain.) The method of Indyk and Woodruff reduces the problem of F k to the problem of computing heavy hitters in the streaming manner. Their reduction only requires polylogarithmic overhead in term of the space complexity and is based on the fundamental idea of “layering.” Since 2005 the method of Indyk and Woodruff has been used in numerous applications and has become a standard tool for streaming computations.

Approximating Large Frequency Moments with Pick-and-Drop Sampling

Abstract: Given data stream D = {p 1,p 2,…,p m } of size m of numbers from {1,…, n}, the frequency of i is defined as f i = |{j: p j = i}|. The k-th frequency moment of D is defined as \(F_k = \sum_{i=1}^n f_i^k\). We consider the problem of approximating frequency moments in insertion-only streams for k ≥ 3. For any constant c we show an O(n 1 − 2/k log(n)log(c)(n)) upper bound on the space complexity of the problem. Here log(c)(n) is the iterative log function. Our main technical contribution is a non-uniform sampling method on matrices. We call our method a pick-and-drop sampling; it samples a heavy element (i.e., element i with frequency Ω(F k )) with probability Ω(1/n 1 − 2/k ) and gives approximation \(\tilde{f_i} \ge (1-\epsilon)f_i\). In addition, the estimations never exceed the real values, that is \( \tilde{f_j} \le f_j\) for all j. For constant e, we reduce the space complexity of finding a heavy element to O(n 1 − 2/k log(n)) bits. We apply our method of recursive sketches and resolve the problem with O(n 1 − 2/k log(n)log(c)(n)) bits. We reduce the ratio between the upper and lower bounds from O(log2(n)) to O(log(n)log(c)(n)). Thus, we provide a (roughly) quadratic improvement of the result of Andoni, Krauthgamer and Onak (FOCS 2011).

How to Catch L 2 -Heavy-Hitters on Sliding Windows

Abstract: Finding heavy-elements (heavy-hitters) in streaming data is one of the central, and well-understood tasks. Despite the importance of this problem, when considering the sliding windows model of streaming (where elements eventually expire) the problem of finding L 2-heavy elements has remained completely open despite multiple papers and considerable success in finding L 1-heavy elements.