Francesco Romani

Ranking a stream of news

According to a recent survey made by Nielsen NetRatings, searching on news articles is one of the most important activity online. Indeed, Google, Yahoo, MSN and many others have proposed commercial search engines for indexing news feeds. Despite this commercial interest, no academic research has focused on ranking a stream of news articles and a set of news sources. In this paper, we introduce this problem by proposing a ranking framework which models: (1) the process of generation of a stream of news articles, (2) the news articles clustering by topics, and (3) the evolution of news story over the time. The ranking algorithm proposed ranks news information, finding the most authoritative news sources and identifying the most interesting events in the different categories to which news article belongs. All these ranking measures take in account the time and can be obtained without a predefined sliding window of observation over the stream. The complexity of our algorithm is linear in the number of pieces of news still under consideration at the time of a new posting. This allow a continuous on-line process of ranking. Our ranking framework is validated on a collection of more than 300,000 pieces of news, produced in two months by more then 2000 news sources belonging to 13 different categories (World, U.S, Europe, Sports, Business, etc). This collection is extracted from the index of comeToMyHead, an academic news search engine available online.

O ( n 2.7799 ) complexity for n × n approximate matrix multiplication

The problem of multiplying (m X n) X (n X k) matrices (the (m, n, k) problem) is an interesting case of the bilinear forms computation theory. We call EC-algorit’hms (Exactly-Computing) those solving exactly the problem in the real arithmetic. They are usually estimated by the number of multiplications required. The state of the art is the O(n2*7g51) ECalgorithm given by [3] for the (n, n, n) problem. We introduce here a new class of algorithms approximating the result with arbitrary precision. We call them’ APA-algorithms (Arbitrary-Precision-Approximating). A bilinear EC-algorithm to compute the set of bilinear forms

Fast PageRank Computation Via a Sparse Linear System

Recently, the research community has devoted increased attention to reducing the computational time needed by web ranking algorithms. In particular, many techniques have been proposed to speed up the well-known PageRank algorithm used by Google. This interest is motivated by two dominant factors: (1) the web graph has huge dimensions and is subject to dramatic updates in terms of nodes and links, therefore the PageRank assignment tends to became obsolete very soon; (2) many PageRank vectors need to be computed according to different choices of the personalization vectors or when adopting strategies of collusion detection. In this paper, we show how the PageRank computation in the original random surfer model can be transformed in the problem of computing the solution of a sparse linear system. The sparsity of the obtained linear system makes it possible to exploit the effectiveness of the Markov chain index reordering to speed up the PageRank computation. In particular, we rearrange the system matrix according to several permutations, and we apply different scalar and block iterative methods to solve smaller linear systems. We tested our approaches on web graphs crawled from the net. The largest one contains about 24 millions nodes and more than 100 million links. Upon this web graph, the cost for computing the PageRank is reduced by 65% in terms of Mflops and by 92% in terms of time respect to the power method commonly used.

Approximate Solutions for the Bilinear Form Computational Problem

A set of bilinear forms can be evaluated with a multiplicative complexity lower than the rank of the associated tensor by allowing an arbitrarily small error. A topological interpretation of this fact is presented together with the error analysis. A complexity measure is introduced which takes into account the numerical stability of algorithms. Relations are established between the complexities of exact and approximate algorithms.

Some Properties of Disjoint Sums of Tensors Related to Matrix Multiplication

Let t be a disjoint sum of tensors associated to matrix multiplication. The rank of the tensorial powers of t is bounded by an expression involving the elements of t and an exponent for matrix multiplication. This relation leads to a trascendental equation defining a new exponent for matrix multiplication.

ON BAND MATRICES AND THEIR INVERSES

Abstract Structural properties of the inverses of band matrices are discussed. The definition of semiseparable matrices is given, and the theorem is proved that the inverse of a strict band matrix is a semiseparable matrix and vice versa. Finally, a recurrence algorithm is recommended for computing the blocks of the inverses of strict band matrices.

Algorithm 691: Improving QUADPACK automatic integration routines

Two automatic adaptive integrators from QUADPACK (namely, QAG, and QAGS) are modified by substituting the Gauss-Kronrod rules used for local quadrature with recursive monotone stable (RMS) formulas. Extensive numerical tests, both for one-dimensional and two-dimensional integrals, show that the resulting programs are faster, perform less functional evaluations, and are more suitable

Parallel solution of block tridiagonal linear systems

Abstract The explicit structure of the inverse of block tridiagonal matrices is presented in terms of blocks defined by linear recurrence relations. Parallel algorithms are shown which solve block second order linear recurrences without using commutativity. Moreover we investigate the parallel solution of the associated block tridiagonal linear system. Using this theoretical background, the implementation of the algorithms is analyzed both on a small number of processors and on a hypercube. The resulting complexity is given in terms of parallel steps, each consisting of block operations, and the cost due to interprocessor communications is taken into account, too.

On the asymptotic complexity of rectangular matrix multiplication

Abstract. The numher of essential multiplications required to multiply matrices of size N x N and N x N’ is studied as a function f(s 1. Bounds to f(x) sharper than trivial ones are presented and the asymptotic behawour of f(s 1 is studied. An analogous investigation is performed for the problem of multiplying matrices of size N x N’ and N’ x N ‘. 1. Introduction and preliminaries The complexity of square N x N matrix multiplication has been investigated by several authors [l-lo]. A lower bound to the number of operations is N’ and the best known upper bound is O(N2~“16h) [8]. Some results on rectangular matrix multiplication have been obtained by Brockett and Dobkin [a] who have shown the upper bound N* +o(N*) to the non-scalar complexity of N X N by N X log N matrix multiplication. Recently Coppersmith [S] found the upper bound O(N’ log” N) for the N x N by N x N” matrix multipli- cation prnblem with n s (2 log 2)/(5 log 5) = 0.1722.

Interpolatory integration formulas for optimal composition

A set of symmetric, closed, interpolatory integration formulas on the interval [-1, 1] with positive weights and increasing degree of precision is introduced. These formulas, called recursive monotone stable (RMS) formulas, allow applying higher order or compound rules without wasting previously computed functional values. An exhaustive search shows the existence of 27 families of RMS formulas, stemming from the simple trapezoidal rule.