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

The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model. The key idea of our algorithm is to combine the sampling algorithm of Tsourakakis et al. and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in the Alon, Yuster and Zwick work 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 $\epsilon$ approximation (multiplicative error), where $n$ is the number of vertices, $m$ the number of edges and $\Delta$ 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.Comment: 1) 12 pages 2) To appear in the 7th Workshop on Algorithms and Models for the Web Graph (WAW 2010

Random projections for high-dimensional curves

Modern time series analysis requires the ability to handle datasets that are inherently high-dimensional; examples include applications in climatology, where measurements from numerous sensors must be taken into account, or inventory tracking of large shops, where the dimension is defined by the number of tracked items. The standard way to mitigate computational issues arising from the high-dimensionality of the data is by applying some dimension reduction technique that preserves the structural properties of the ambient space. The dissimilarity between two time series is often measured by ``discrete'' notions of distance, e.g. the dynamic time warping, or the discrete Fr\'echet distance, or simply the Euclidean distance. Since all these distance functions are computed directly on the points of a time series, they are sensitive to different sampling rates or gaps. The continuous Fr\'echet distance offers a popular alternative which aims to alleviate this by taking into account all points on the polygonal curve obtained by linearly interpolating between any two consecutive points in a sequence. We study the ability of random projections \`a la Johnson and Lindenstrauss to preserve the continuous Fr\'echet distance of polygonal curves by effectively reducing the dimension. In particular, we show that one can reduce the dimension to $O(\epsilon^{-2} \log N)$, where $N$ is the total number of input points while preserving the continuous Fr\'echet distance between any two determined polygonal curves within a factor of $1\pm \epsilon$. We conclude with applications on clustering.Comment: 22 page

Lattice-based locality sensitive hashing is optimal

Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC ‘98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes “nearby” points to the same bucket and “far away” points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps determine both query time and space usage. In a seminal work, Andoni and Indyk (FOCS ‘06) constructed an LSH family based on random ball partitionings of space that achieves an LSH exponent of 1/c2 for the ℓ2 norm, which was later shown to be optimal by Motwani, Naor and Panigrahy (SIDMA ‘07) and O’Donnell, Wu and Zhou (TOCT ‘14). Although optimal in the LSH exponent, the ball partitioning approach is computationally expensive. So, in the same work, Andoni and Indyk proposed a simpler and more practical hashing scheme based on Euclidean lattices and provided computational results using the 24-dimensional Leech lattice. However, no theoretical analysis of the scheme was given, thus leaving open the question of finding the exponent of lattice based LSH. In this work, we resolve this question by showing the existence of lattices achieving the optimal LSH exponent of 1/c2 using techniques from the geometry of numbers. At a more conceptual level, our results show that optimal LSH space partitions can have periodic structure. Understanding the extent to which additional structure can be imposed on these partitions, e.g. to yield low space and query complexity, remains an important open problem

State-of-the-art in aerodynamic shape optimisation methods

Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners

Lattice-based locality sensitive hashing is optimal

Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC ‘98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes “nearby” points to the same bucket and “far away” points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps determine both query time and space usage. In a seminal work, Andoni and Indyk (FOCS ‘06) constructed an LSH family based on random ball partitionings of space that achieves an LSH exponent of 1/c2 for the l2 norm, which was later shown to be optimal by Motwani, Naor and Panigrahy (SIDMA ‘07) and O’Donnell, Wu and Zhou (TOCT ‘14). Although optimal in the LSH exponent, the ball partitioning approach is computationally expensive. So, in the same work, Andoni and Indyk proposed a simpler and more practical hashing scheme based on Euclidean lattices and provided computational results using the 24-dimensional Leech lattice. However, no theoretical analysis of the scheme was given, thus leaving open the question of finding the exponent of lattice based LSH. In this work, we resolve this question by showing the existence of lattices achieving the optimal LSH exponent of 1/c2 using techniques from the geometry of numbers. At a more conceptual level, our results show that optimal LSH space partitions can have periodic structure. Understanding the extent to which additional structure can be imposed on these partitions, e.g. to yield low space and query complexity, remains an important open problem