Abstract

This paper initiates formal analysis of a simple, distributed algorithm for community detection on networks. We analyze an algorithm that we calltextsc {Max-LPA}, both in terms of its convergence time and in terms of the" quality" of the communities detected.textsc {Max-LPA} is an instance of a class of community detection algorithms calledtextit {label propagation} algorithms. As far as we know, most analysis of label propagation algorithms thus far has been empirical in nature and in this paper we seek a theoretical understanding of label propagation algorithms. In our main result, we define a clustered version ofer random graphs with clusters where the probability , of an edge connecting nodes within a cluster is higher than , the probability of an edge connecting nodes in distinct clusters. We show that even with fairly general restrictions on and ( for any , , where is the number of nodes),textsc {Max-LPA} detects the clusters in just two rounds. Based on this and on empirical results, we conjecture thattextsc {Max-LPA} can correctly and quickly identify communities on clustereder graphs even when the clusters are much sparser, ie, with for some .

Abstract

Graphics processing units (GPUs) provide large computational power at a very low price, which position GPUs well as an ubiquitous accelerator. However, GPUs are space constrained, and hence applications developed for GPUs are space sensitive. Space-constrained computational devices such as GPUs can greatly benefit from representations that reduce space consumption drastically. One such representation is the succinct representation of trees. Succinct representation of trees generally allows for operations such as parent queries, least common ancestor queries, and so on. Mapping such a robust representation to the GPU for targeted applications can lead to substantial improvement in problem sizes that are processed at a given point of time. Space-saving methods such as succinct data structures remain largely unexplored on the GPU. In this work, a succinct representation of ordered trees on the GPU is explored, with application to discrete range searching (DRS). Based on the succinct representations found applicable, a space--saving solution for DRS is presented here. In our method, DRS is mapped to a least common ancestor query on a Cartesian tree. For space-efficient DRS queries, we store the succinct representation of the Cartesian tree of an array. Our method uses a maximum of 7.5 bits of additional space per element. Furthermore, the speed-up achieved by our method is in the range of 20--25 for preprocessing and 25--35 for batch querying over a sequential implementation. Compared to an 8-threaded implementation, our preprocessing and querying methods obtain a speed-up of 6--8. We also study the applications of the DRS on the GPU. Efficient primitives expand the range of applications performed on the GPU. DRS is one such primitive with direct applications to string processing, document and text retrieval systems, and least common ancestor queries. We suggest that graph algorithms that use the least common ancestor, can be enabled on the GPU based on DRS primitive. We also show some applications of DRS in tree queries and string querying.

Abstract

Sparse matrix-sparse/dense matrix multiplications, spgemm and csrmm, respectively, among other applications find usage in various matrix formulations of graph problems. Considering the difficulties in executing graph problems and the duality between graphs and matrices, computations such as spgemm and csrmm have recently caught the attention of HPC community. These computations pose challenges such as load balancing, irregular nature of the computation, and difficulty in predicting the output size. It is even more challenging when combined with the GPU architectural constraints such as memory accesses, limited shared memory, strict SIMD and thread execution. To address these challenges on a GPU, we evaluate three possible variations of matrix multiplication (Row-Column, Column-Row, Row-Row) and perform suitable optimizations targeted at sparse matrices. Our experiments indicate that the Row-Row formulation, which mostly outperforms the other formulations, is 3.5x faster on average compared to an optimized multi-core implementation in the Intel MKL library. We extend the Row-Row formulation to a CPU+GPU hybrid algorithm that simultaneously utilizes the CPU also. In this direction, we present heuristics to find the right amount of work division between the CPU and the GPU. Our hybrid row-row formulation of the spgemm operation performs 5.5x faster on average when compared to the optimized multi-core implementation in the Intel MKL library. Our experience indicates that it is difficult to identify right amount of work division between the CPU and the GPU. We therefore investigate a subclass of sparse matrices, band matrices, and present an analytical method to identify a good work division when multiplying two band matrices. Our GPU csrmm operation performs 2.5x faster on average when compared to a corresponding implementation in the cusparse library, which outperforms the Intel MKL library implementation.

Abstract

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a (G), is the minimum number of colors required for acyclic vertex coloring of graph G. For a family F of graphs, the acyclic chromatic number of F, denoted by a (F), is defined as the maximum a (G) over all the graphs G∈ F. In this paper we show that a (F)= 8 where F is the family of graphs of maximum degree 5 and give a linear time algorithm to achieve this bound.

Abstract

General purpose programming on the graphics processing units(GPGPU) has received a lot of attention in the parallel computing community as it promises to offer a large computational power at a very low price. GPGPU is best suited for regular data parallel algorithms. They are not directly amenable for algorithms which have irregular data access patterns such as convex hull, list ranking etc. In this paper, we present a GPU-optimized implementation for finding the convex hull of a two dimensional point set. Our implementation tries to minimize the impact of irregular data access patterns. Our implementation can find the convex hull of 10 million random points in less than 0.2 seconds and achieves a speedup of up to 14 over the standard sequential CPU implementation. We also discuss some of the practical issues relating to the implementation of convex hull algorithms on massively multi-threaded architectures like that of the GPU.

Abstract

Graphics Processing Units provide a large computational power at a very low price which position them as an ubiquitous accelerator. GPGPU is accelerating general purpose computations using GPU's. GPU's have been used to accelerate many Linear Algebra routines and Numerical Methods. Lanczos is an iterative method well suited for finding the extreme eigenvalues and the corresponding eigenvectors of large sparse symmetric matrices. In this paper, we present an implementation of Lanczos Algorithm on GPU using the CUDA programming model and apply it to two important problems : graph bisection using spectral methods, and image segmentation. Our GPU implementation of spectral bisection performs better when compared to both an Intel Math Kernel Library implementation and a Matlab implementation. Our GPU implementation shows a speedup up to 97.3 times over Matlab Implementation and 2.89 times over the Intel Math Kernel Library implementation on a Intel Core i7 920 Processor, which is a quad-core CPU. Similarly, our image segmentation implementation achieves a speed up of 3.27 compared to a multicore CPU based implementation using Intel Math Kernel Library and OpenMP. Through this work, we therefore wish to establish that the GPU may still be a better platform for also highly irregular and computationally intensive applications.

Abstract

This paper considers the question of how many colors a distributed graph coloring algorithm would need to use if it had only k rounds available, for any positive integer k. In our main result, we present an algorithm that runs in O(k) rounds for any k bounded below by ©(log log n) and bounded above by O(√log n), and uses O(a Å n1/k) colors to color a graph with arboricity a. This result is optimal since the palette size matches the lower bound of Barenboim and Elkin (PODC 2008). This result is achieved via the use of several new results developed in this paper on coloring graphs whose edges have been acyclically oriented. For example, suppose that G is an n-vertex, acyclically oriented graph with maximum out-degree Δo. We present an algorithm that, for any k ≥ 2 loglog n, runs in O(k) rounds on G to produce an (i) O(Δo)-coloring when Δo Δ ©(maxkn2/k2log1+1/kn, 2k) and an (ii) O(Δo Å n2/k2)-coloring when Δo ∈ Ω(maxk log1+1/kn, 2k). These results are useful in any setting where it is possible to efficiently compute acyclic orientations of a graph with Δo << Δ. We derive non-trivial bounds on the palette size even when k < 2 loglog n. Our main technical contributions can be summarized as: (i) developing a k-round version of the algorithm of Kothapalli et al. (IPDPS 2006) which computes an O(?)-coloring of a graph in O(√log n) rounds, and (ii) developing an oriented version of the Brooks-Vizing coloring result of Grable and Panconesi (SODA 1998).

Abstract

We consider the dominating point set reporting problem in two-dimension. We propose a data structure for finding the set of dominating points inside a given orthogonal query rectangle. Given a set of n points in the plane, it supports 4-sided queries in O (log n+ k), where k is size of the output, using O (n log n) space. This work can be of application when range queries are generated using mobile devices with limited display capacity.

Abstract

In this work we study the problem of finding axesparallel regions of maximum density for weighted point sets in IR2 and IR3. The 2-d variant is motivated by applications in thermal analysis of VLSI chips.

Abstract

Multiplying a sparse matrix with a vector (spmv for short) is a fundamental operation in many linear algebra kernels. Having an efficient spmv kernel on modern architectures such as the GPUs is therefore of principal interest. The computational challenges that spmv poses are significantlydifferent compared to that of the dense linear algebra kernels. Recent work in this direction has focused on designing data structures to represent sparse matrices so as to improve theefficiency of spmv kernels. However, as the nature of sparseness differs across sparse matrices, there is no clear answer as to which data structure to use given a sparse matrix. In this work, we address this problem by devising techniques to understand the nature of the sparse matrix and then choose appropriate data structures accordingly. By using our technique, we are able to improve the performance of the spmv kernel on an Nvidia Tesla GPU (C1060) by a factor of up to80% in some instances, and about 25% on average compared to the best results of Bell and Garland [3] on the standard dataset (cf. Williams et al. SC'07) used in recent literature. We also use our spmv in the conjugate gradient method and show an average 20% improvement compared to using HYB spmv of [3], on the dataset obtained from the The University of Florida Sparse Matrix Collection [9].