Richard Peng

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model [12]. The key idea of our algorithm is to combine the sampling algorithm of [31,32] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [1], treating each set appropriately. We obtain a running time \(O \left( m + \frac{m^{3/2} \Delta \log{n} }{t \epsilon^2} \right)\) and an e approximation (multiplicative error), where n is the number of vertices, m the number of edges and Δ the maximum number of triangles an edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage \(O\left(m^{1/2}\log{n} + \frac{m^{3/2} \Delta \log{n}}{t \epsilon^2} \right)\) and a constant number of passes (three) over the graph stream. We apply our methods in various networks with several millions of edges and we obtain excellent results. Finally, we propose a random projection based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.

Uniform Sampling for Matrix Approximation

Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time significantly. For theoretical performance guarantees, each row must be sampled with probability proportional to its statistical leverage score. Unfortunately, leverage scores are difficult to compute. A simple alternative is to sample rows uniformly at random. While this often works, uniform sampling will eliminate critical row information for many natural instances. We take a fresh look at uniform sampling by examining what information it does preserve. Specifically, we show that uniform sampling yields a matrix that, in some sense, well approximates a large fraction of the original. While this weak form of approximation is not enough for solving linear regression directly, it is enough to compute a better approximation. This observation leads to simple iterative row sampling algorithms for matrix approximation that run in input-sparsity time and preserve row structure and sparsity at all intermediate steps. In addition to an improved understanding of uniform sampling, our main proof introduces a structural result of independent interest: we show that every matrix can be made to have low coherence by reweighting a small subset of its rows.

Iterative Row Sampling

There has been significant interest and progress recently in algorithms that solve regression problems involving tall and thin matrices in input sparsity time. Given a n * d matrix where n ≥ d, these algorithms find an approximation with fewer rows, allowing one to solve a poly(d) sized problem instead. In practice, the best performances are often obtained by invoking these routines in an iterative fashion. We show these iterative methods can be adapted to give theoretical guarantees comparable to and better than the current state of the art. Our approaches are based on computing the importances of the rows, known as leverage scores, in an iterative manner. We show that alternating between computing a short matrix estimate and finding more accurate approximate leverage scores leads to a series of geometrically smaller instances. This gives an algorithm whose runtime is input sparsity plus an overhead comparable to the cost of solving a regression problem on the smaller approximation. Our results build upon the close connection between randomized matrix algorithms, iterative methods, and graph sparsification.

Fully Dynamic (1+ e)-Approximate Matchings

We present the first data structures that maintain near optimal maximum cardinality and maximum weighted matchings on sparse graphs in sub linear time per update. Our main result is a data structure that maintains a (1+ϵ) approximation of maximum matching under edge insertions/deletions in worst case Õ(√mϵ-2) time per update. This improves the 3/2 approximation given by Neiman and Solomon [20] which runs in similar time. The result is based on two ideas. The first is to re-run a static algorithm after a chosen number of updates to ensure approximation guarantees. The second is to judiciously trim the graph to a smaller equivalent one whenever possible. We also study extensions of our approach to the weighted setting, and combine it with known frameworks to obtain arbitrary approximation ratios. For a constant ϵ and for graphs with edge weights between 1 and N, we design an algorithm that maintains an (1+ϵ) approximate maximum weighted matching in Õ(√m log N) time per update. The only previous result for maintaining weighted matchings on dynamic graphs has an approximation ratio of 4.9108, and was shown by An and et al. [2], [3].

An efficient parallel solver for SDD linear systems

We present the first parallel algorithm for solving systems of linear equations in symmetric, diagonally dominant (SDD) matrices that runs in polylogarithmic time and nearly-linear work. The heart of our algorithm is a construction of a sparse approximate inverse chain for the input matrix: a sequence of sparse matrices whose product approximates its inverse. Whereas other fast algorithms for solving systems of equations in SDD matrices exploit low-stretch spanning trees, our algorithm only requires spectral graph sparsifiers.

Solving SDD linear systems in nearly m log 1/2 n time

We show an algorithm for solving symmetric diagonally dominant (SDD) linear systems with m non-zero entries to a relative error of ε in O(m log1/2 n logc n log(1/ε)) time. Our approach follows the recursive preconditioning framework, which aims to reduce graphs to trees using iterative methods. We improve two key components of this framework: random sampling and tree embeddings. Both of these components are used in a variety of other algorithms, and our approach also extends to the dual problem of computing electrical flows. We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. In the graph setting, this leads to ultra-sparsifiers that have optimal behavior in expectation. The improved running time makes previous low stretch embedding algorithms the running time bottleneck in this framework. In our analysis, we relax the requirement of these embeddings to snowflake spaces. We then obtain a two-pass approach algorithm for constructing optimal embeddings in snowflake spaces that runs in O(m log log n) time. This algorithm is also readily parallelizable.

Linear-work greedy parallel approximate set cover and variants

We present parallel greedy approximation algorithms for set cover and related problems. These algorithms build on an algorithm for solving a graph problem we formulate and study called Maximal Nearly Independent Set (MaNIS)---a graph abstraction of a key component in existing work on parallel set cover. We derive a randomized algorithm for MaNIS that has <i>O</i>(<i>m</i>) work and <i>O</i>(log² <i>m</i>) depth on input with <i>m</i> edges. Using MaNIS, we obtain RNC algorithms that yield a (1+ε)<i>H_n</i>-approximation for set cover, a (1 - 1/<i>e</i> -ε)-approximation for max cover and a (4 + ε)-approximation for min-sum set cover all in linear work; and an <i>O</i>(log* <i>n</i>)-approximation for asymmetric <i>k</i>-center for <i>k</i> ≤ log^<i>O</i>(1) <i>n</i> and a (1.861+ε)-approximation for metric facility location both in essentially the same work bounds as their sequential counterparts.

Faster Spectral Sparsification and Numerical Algorithms for SDD Matrices

We study algorithms for spectral graph sparsification. The input is a graph <i>G</i> with <i>n</i> vertices and <i>m</i> edges, and the output is a sparse graph <i>&Gtilde;</i> that approximates <i>G</i> in an algebraic sense. Concretely, for all vectors <i>x</i> and any ε > 0, the graph <i>&Gtilde;</i> satisfies (1-ε )<i>x^<i>T</i> <i>L</i>_<i>G</i>^<i>x</i></i> ≤ <i>x</i>^<i>T</i> <i>L</i><i>&Gtilde;</i>^<i>x</i> ≤ (1+ε)<i>x</i>^<i>T</i> <i>L</i>_<i>G</i>^<i>x</i>, where <i>L_G</i> and <i>&Gtilde;</i> are the Laplacians of <i>G</i> and <i>&Gtilde;</i> respectively. The first contribution of this article applies to all existing sparsification algorithms that rely on solving solving linear systems on graph Laplacians. These algorithms are the fastest known to date. Specifically, we show that less precision is required in the solution of the linear systems, leading to speedups by an <i>O</i>(log <i>n</i>) factor. We also present faster sparsification algorithms for slightly dense graphs: — An <i>O</i>(<i>m</i>log <i>n</i>) time algorithm that generates a sparsifier with <i>O</i>(<i>n</i>log ³<i>n</i>/ε²) edges. — An <i>O</i>(<i>m</i>log log <i>n</i>) time algorithm for graphs with more than <i>n</i>log ⁵<i>n</i>log log <i>n</i> edges. — An <i>O</i>(<i>m</i>) algorithm for graphs with more than <i>n</i>log ¹⁰<i>n</i> edges. — An <i>O</i>(<i>m</i>) algorithm for unweighted graphs with more than <i>n</i>log ⁸<i>n</i> edges. These bounds hold up to factors that are in <i>O</i>(<i>poly</i>(log log <i>n</i>)) and are conjectured to be removable.

Parallel graph decompositions using random shifts

We show an improved parallel algorithm for decomposing an undirected unweighted graph into small diameter pieces with a small fraction of the edges in between. These decompositions form critical subroutines in a number of graph algorithms. Our algorithm builds upon the shifted shortest path approach introduced in [Blelloch, Gupta, Koutis, Miller, Peng, Tangwongsan, SPAA 2011]. By combining various stages of the previous algorithm, we obtain a significantly simpler algorithm with the same asymptotic guarantees as the best sequential algorithm.

Scalable Large Near-Clique Detection in Large-Scale Networks via Sampling

Extracting dense subgraphs from large graphs is a key primitive in a variety of graph mining applications, ranging from mining social networks and the Web graph to bioinformatics [41]. In this paper we focus on a family of poly-time solvable formulations, known as the k-clique densest subgraph problem (k-Clique-DSP) [57]. When k=2, the problem becomes the well-known densest subgraph problem (DSP) [22, 31, 33, 39]. Our main contribution is a sampling scheme that gives densest subgraph sparsifier, yielding a randomized algorithm that produces high-quality approximations while providing significant speedups and improved space complexity. We also extend this family of formulations to bipartite graphs by introducing the (p,q)-biclique densest subgraph problem ((p,q)-Biclique-DSP), and devise an exact algorithm that can treat both clique and biclique densities in a unified way. As an example of performance, our sparsifying algorithm extracts the 5-clique densest subgraph --which is a large-near clique on 62 vertices-- from a large collaboration network. Our algorithm achieves 100% accuracy over five runs, while achieving an average speedup factor of over 10,000. Specifically, we reduce the running time from ∼2 107 seconds to an average running time of 0.15 seconds. We also use our methods to study how the k-clique densest subgraphs change as a function of time in time-evolving networks for various small values of k. We observe significant deviations between the experimental findings on real-world networks and stochastic Kronecker graphs, a random graph model that mimics real-world networks in certain aspects. We believe that our work is a significant advance in routines with rigorous theoretical guarantees for scalable extraction of large near-cliques from networks.