A fast algorithm for matrix balancing

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration can give quadratic convergence at low cost

A Heuristic Based Multi-Objective Approach for Network Reconfiguration of Distribution Systems

This paper presents an algorithm for network reconfiguration based on the heuristic rules and fuzzy multi-objective approach with an improved Fast Decoupled load flow algorithm. Multiple objectives are considered to minimize the real power loss, deviation in bus voltages, branch current violation and for load balancing among feeders, while subjected to a radial network structure in which all loads kept energized. These four objectives are modeled with fuzzy sets to evaluate their imprecise nature. Heuristic rules are also incorporated in the algorithm for drastically minimizing the number of tie-switch operations. An improved Fast Decoupled load flow algorithm with Single Matrix Model (FDC-SMM) has been proposed for distribution networks. The proposed algorithm is very effective in dealing with reconfiguration problems of single and multi-feeder networks Keywords: Multi-objective approach, Reconfiguration, Fuzzy set theory, Fast decoupled load flo

Distributed Simplicial Homology Based Load Balancing Algorithm for Cellular Networks

International audience—In this paper, we introduce a distributed load balancing algorithm for cellular networks. Traffic load in cellular networks is sometimes unbalanced. Some cells are overloaded, while others remain free. Simplicial homology is a tool from algebraic topology that allows to compute the coverage of a network by using only simple matrix computations. Our algorithm, which is based on simplicial homology, controls the transmission power of each cell in the network, not only to satisfy the coverage constraint, but also to redirect users from the overloaded cells to the underloaded ones. As a result, the traffic load of the cellular network is more balanced. The simulation results show that this algorithm improves the capacity of the whole network by 2.3% when the user demand is fast varying

Differential qd algorithm with shifts for rank-structured matrices

Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable. In particular, in the symmetric tridiagonal case where differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often faster convergence while preserving high relative accuracy (that is not guaranteed in QR algorithm). In eigenvalue computations for rank-structured matrices QR algorithm is also a popular choice since, in the symmetric case, the rank structure is preserved. In the unsymmetric case, however, QR algorithm destroys the rank structure and, hence, LR-type algorithms come to play once again. In the current paper we discover several variants of qd algorithms for quasiseparable matrices. Remarkably, one of them, when applied to Hessenberg matrices becomes a direct generalization of dqds algorithm for tridiagonal matrices. Therefore, it can be applied to such important matrices as companion and confederate, and provides an alternative algorithm for finding roots of a polynomial represented in the basis of orthogonal polynomials. Results of preliminary numerical experiments are presented

Fast Distributed PageRank Computation

Over the last decade, PageRank has gained importance in a wide range of applications and domains, ever since it first proved to be effective in determining node importance in large graphs (and was a pioneering idea behind Google's search engine). In distributed computing alone, PageRank vector, or more generally random walk based quantities have been used for several different applications ranging from determining important nodes, load balancing, search, and identifying connectivity structures. Surprisingly, however, there has been little work towards designing provably efficient fully-distributed algorithms for computing PageRank. The difficulty is that traditional matrix-vector multiplication style iterative methods may not always adapt well to the distributed setting owing to communication bandwidth restrictions and convergence rates. In this paper, we present fast random walk-based distributed algorithms for computing PageRanks in general graphs and prove strong bounds on the round complexity. We first present a distributed algorithm that takes O\big(\log n/\eps \big) rounds with high probability on any graph (directed or undirected), where $n$ is the network size and \eps is the reset probability used in the PageRank computation (typically \eps is a fixed constant). We then present a faster algorithm that takes O\big(\sqrt{\log n}/\eps \big) rounds in undirected graphs. Both of the above algorithms are scalable, as each node sends only small (\polylog n) number of bits over each edge per round. To the best of our knowledge, these are the first fully distributed algorithms for computing PageRank vector with provably efficient running time.Comment: 14 page

A Matrix Hyperbolic Cosine Algorithm and Applications

In this paper, we generalize Spencer's hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size $n$, it constructs an expanding Cayley graph of logarithmic degree in near-optimal O(n^2 log^3 n) time. For the latter case, we present a fast deterministic algorithm for spectral sparsification of positive semi-definite matrices, which implies an improved deterministic algorithm for spectral graph sparsification of dense graphs. In addition, we give an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrix sparsification. As a consequence, we obtain improved element-wise sparsification algorithms for diagonally dominant-like matrices.Comment: 16 pages, simplified proof and corrected acknowledging of prior work in (current) Section