Linear Running Time Research Articles

Hardware SLE solvers: Efficient building blocks for cryptographic and cryptanalyticapplications

Solving systems of linear equations (SLEs) is a very common computational problem appearing in numerous research disciplines and in particular in the context of cryptographic and cryptanalytic algorithms. In this work, we present highly efficient hardware architectures for solving (small and medium-sized) systems of linear equations over F 2 k . These architectures feature linear or quadratic running times with quadratic space complexities in the size of an SLE, and can be clocked at high frequencies. Among the most promising architectures are one-dimensional and two-dimensional systolic arrays which we call triangular systolic and linear systolic arrays. All designs have been fully implemented for different sizes of SLEs and concrete FPGA implementation results are given. Furthermore, we provide a clear comparison of the presented SLE solvers. The significance of these designs is demonstrated by the fact that they are used in the recent literature as building blocks of efficient architectures for attacking block and stream ciphers (Bogdanov et al., 2007 [5]; Geiselmann et al., 2009 [17]) and for developing cores for multivariate signature schemes (Balasubramanian et al., 2008 [2]; Bogdanov et al., 2008 [6]).

PEGASUS: mining peta-scale graphs

In this paper, we describe PeGaSus, an open source Peta Graph Mining library which performs typical graph mining tasks such as computing the diameter of the graph, computing the radius of each node, finding the connected components, and computing the importance score of nodes. As the size of graphs reaches several Giga-, Tera- or Peta-bytes, the necessity for such a library grows too. To the best of our knowledge, PeGaSus is the first such library, implemented on the top of the Hadoop platform, the open source version of MapReduce. Many graph mining operations (PageRank, spectral clustering, diameter estimation, connected components, etc.) are essentially a repeated matrix-vector multiplication. In this paper, we describe a very important primitive for PeGaSus, called GIM-V (generalized iterated matrix-vector multiplication). GIM-V is highly optimized, achieving (a) good scale-up on the number of available machines, (b) linear running time on the number of edges, and (c) more than 5 times faster performance over the non-optimized version of GIM-V. Our experiments ran on M45, one of the top 50 supercomputers in the world. We report our findings on several real graphs, including one of the largest publicly available Web graphs, thanks to Yahoo!, with ≈ 6.7 billion edges.

Relaxed voting and competitive location under monotonous gain functions on trees

We examine competitive location problems where two competitors serve a good to users located in a network. Users decide for one of the competitors based on the distance induced by an underlying tree graph. The competitors place their server sequentially into the network. The goal of each competitor is to maximize his benefit which depends on the total user demand served. Typical competitive location problems include the ( 1 , X 1 ) -medianoid, the ( 1 , 1 ) -centroid, and the Stackelberg location problem. An additional relaxation parameter introduces a robustness of the model against small changes in distance. We introduce monotonous gain functions as a general framework to describe the above competitive location problems as well as several problems from the area of voting location such as Simpson, Condorcet, security, and plurality. In this paper we provide a linear running time algorithm for determining an absolute solution in a tree where competitors are allowed to place on nodes or on inner points. Furthermore we discuss the application of our approach to the discrete case.

Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently

In this paper, we introduce the notion of a constrained Minkowski sum: for two (finite) point-sets P,Q⊆ℝ2 and a set of k inequalities Ax≥b, it is defined as the point-set (P ⊕ Q) Ax≥b ={x=p+q∣p∈P,q∈Q,Ax≥b}. We show that typical interval problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O(Nlog N), where N=|P|+|Q| if k is fixed. For the special case $(P\oplus Q)_{x_{1}\geq \beta}$where P and Q consist of points with integer x 1-coordinates whose absolute values are bounded by O(N), we even achieve a linear running time O(N). We thereby obtain a linear running time for many interval problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of interval problems can be modeled and solved.

A note on the minimum bounded edge-partition of a tree

Minimum bounded edge-partition divides the edge set of a tree into the minimum number of disjoint connected components given a maximum weight for any component. It is an adaptation of the uniform edge-partition of a tree. An optimization algorithm is developed for this NP-hard problem, based on repeated bin packing of inter-related instances. The algorithm has linear running time for the class of ‘balanced trees’ common for the stochastic programming application which motivated investigation of this problem. Fast 2-approximation algorithms are formed for general instances by replacing the optimal bin packing with almost any bin packing heuristic. The asymptotic worst-case ratio of these approximation algorithms is never better than the absolute worst-case ratio of the bin packing heuristic used.

A BRANCH AND BOUND ALGORITHM FOR FINDING THE MODES IN KERNEL DENSITY ESTIMATES

Kernel density estimators are established tools in nonparametric statistics. Due to their flexibility and ease of use, these methods are popular in computer vision and pattern recognition for tasks such as object tracking in video or image segmentation. The most frequently used algorithm for finding the modes in such densities (the mean shift) is a gradient ascent rule, which converges to local maxima. We propose a novel, globally optimal branch and bound algorithm for finding the modes in kernel densities. We show in experiments on datasets up to dimension five that the branch and bound method is faster than local optimization and observe linear runtime scaling of our method with sample size. Quantitative experiments on simulated data show that our method gives statistically significantly more accurate solutions than the mean shift. The mode localization accuracy is about five times more precise than that of the mean shift for all tested parameters. Applications to color image segmentation on an established benchmark test set also show measurably improved results when using global optimization.

A stochastic programming approach to cash management in banking

The treasurer of a bank is responsible for the cash management of several banking activities. In this work, we focus on two of them: cash management in automatic teller machines (ATMs), and in the compensation of credit card transactions. In both cases a decision must be taken according to a future customers demand, which is uncertain. From historical data we can obtain a discrete probability distribution of this demand, which allows the application of stochastic programming techniques. We present stochastic programming models for each problem. Two short-term and one mid-term models are presented for ATMs. The short-term model with fixed costs results in an integer problem which is solved by a fast (i.e. linear running time) algorithm. The short-term model with fixed and staircase costs is solved through its MILP equivalent deterministic formulation. The mid-term model with fixed and staircase costs gives rise to a multi-stage stochastic problem, which is also solved by its MILP deterministic equivalent. The model for compensation of credit card transactions results in a closed form solution. The optimal solutions of those models are the best decisions to be taken by the bank, and provide the basis for a decision support system.

Multilevel algorithms for linear ordering problems

Linear ordering problems are combinatorial optimization problems that deal with the minimization of different functionals by finding a suitable permutation of the graph vertices. These problems are widely used and studied in many practical and theoretical applications. In this paper, we present a variety of linear--time algorithms for these problems inspired by the Algebraic Multigrid approach, which is based on weighted-edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short (linear) running time of the algorithms enables testing very large graphs.

Time series data analysis in multiple granularity levels

A single-pass method for analysing time series data and extracting time-correlations (time-delayed relationships) among multiple time series data streams is described. The proposed method can detect and report time-delayed relationships among multiple time series data streams without having to transform the original data into another domain. Each time-correlation rule explains how the changes in the values of one set of time series data streams influence the values in another set of time series data streams. Those rules can be stored digitally and fed into various data analysis tools for further analysis. Performance experiments showed that the described method is 95% accurate and has a linear running time with respect to the amount of input data for pair-wise time series correlations.

Spectral Curvature Clustering (SCC)

This paper presents novel techniques for improving the performance of a multi-way spectral clustering framework (Govindu in Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol. 1, pp. 1150–1157, 2005; Chen and Lerman, 2007, preprint in the supplementary webpage) for segmenting affine subspaces. Specifically, it suggests an iterative sampling procedure to improve the uniform sampling strategy, an automatic scheme of inferring the tuning parameter from the data, a precise initialization procedure for K-means, as well as a simple strategy for isolating outliers. The resulting algorithm, Spectral Curvature Clustering (SCC), requires only linear storage and takes linear running time in the size of the data. It is supported by theory which both justifies its successful performance and guides our practical choices. We compare it with other existing methods on a few artificial instances of affine subspaces. Application of the algorithm to several real-world problems is also discussed.

A note on coloring sparse random graphs

Coja-Oghlan and Taraz [Amin Coja-Oghlan, Anusch Taraz, Exact and approximative algorithms for coloring G ( n , p ) , Random Structures and Algorithms 24 (3) (2004) 259–278] presented a graph coloring algorithm that has expected linear running time for random graphs with edge probability p satisfying n p ≤ 1.01 . In this work, we develop their analysis by exploiting generating function techniques. We show that, in fact, their algorithm colors G n , p with the minimal number of colors and has expected linear running time, provided that n p ≤ 1.33 .

Defining Statistical Timing Sensitivity for Logic Circuits With Large-Scale Process and Environmental Variations

The large-scale process and environmental variations for today's nanoscale ICs require statistical approaches for timing analysis and optimization. In this paper, we demonstrate why the traditional concept of slack and critical path becomes ineffective under large-scale variations and propose a novel sensitivity framework to assess the ldquocriticalityrdquo of every path, arc, and node in a statistical timing graph. We theoretically prove that the path sensitivity is exactly equal to the probability that a path is critical and that the arc (or node) sensitivity is exactly equal to the probability that an arc (or a node) sits on the critical path. An efficient algorithm with incremental analysis capability is developed for fast sensitivity computation that has linear runtime complexity in circuit size. The efficacy of the proposed sensitivity analysis is demonstrated on both standard benchmark circuits and large industrial examples.

A fast haplotype inference method for large population genotype data

With the rapid progress of genotyping techniques, many large-scale, genome-wide disease studies are now under way. One of the challenges of large disease-association studies is developing a fast and accurate computing method for haplotype inference from genotype data. In this paper, a new computing method for population-based haplotype inference problem is proposed. The designed method does not assume haplotype blocks in the population and allows each individual haplotype to have its own structure, and thus is able to accommodate recombination and obtain higher adaptivity to the genotype data, specifically in the case of long marker maps. This method develops a dynamic programming algorithm, which is theoretically guaranteed to find exact maximum likelihood solutions of the variable order Markov chain model for haplotype inference problem within linear running time. Hence, it is fast and, as a result, practicable for large genotype datasets. Through extensive computational experiments on large-scale real genotype data, the proposed method is shown to be fast and efficient.

Converting suffix trees into factor/suffix oracles

Several methods to compress suffix trees were defined, most of them with the aim of obtaining compact (that is, space economical) index structures. Besides this practical aspect, a compression method can reveal structural properties of the resulting data structure, allowing a better understanding of it and a better estimation of its performances. In this paper, we propose a simple method to compress suffix trees by merging couples of nodes. This idea was already used in the literature in a context different from ours. The originality of our approach is that the nodes we merge are not chosen with respect to their subtrees (which is difficult to test algorithmically), nor with respect to the words spelled along branches (which usually requires testing several branches before finding the good one) but with respect to their position in the tree (which is easy to compute). Another particularity of our method is it needs to read no edge label: it is exclusively based on the topology of the suffix tree. The compact structure resulting after compression is the factor/suffix oracle introduced by Allauzen, Crochemore and Raffinot whose accepted language includes the accepted language of the corresponding suffix tree. The interest of our paper is therefore threefold: 1. A topology-based compression method is defined for (compact) suffix trees. 2. A new property of a factor/suffix oracle is established, that is, like a DAG, it results from the corresponding suffix tree after a linear number of appropriate node mergings; unlike a DAG, the merged nodes do not necessarily have isomorphical subtrees. 3. A new algorithm to transform a suffix tree into a factor/suffix oracle is given, which has linear running time and thus improves the quadratic complexity previously known for the same task.

Computing the Soft Error Rate of a Combinational Logic Circuit Using Parameterized Descriptors

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Soft errors have emerged as an important reliability challenge for nanoscale very large scale integration designs. In this paper, we present a fast and efficient soft error rate (SER) analysis methodology for combinational circuits. We first present a novel parametric waveform model based on the Weibull function to represent particle strikes at individual nodes in the circuit. We then describe the construction of the descriptor object that efficiently captures the correlation between the transient waveforms and their associated rate distribution functions. The proposed algorithm consists of operations to inject, propagate, and merge these descriptors while traversing forward along the gates in a circuit. The parameterized waveforms enable an efficient static approach to calculate the SER of a circuit. We exercise the proposed approach on a wide variety of combinational circuits and observe that our algorithm has linear runtime with the size of the circuit. The runtimes for soft error estimation were observed to be in the order of about 1 s, compared to several minutes or even hours for previously proposed methods. </para>

Efficient algorithms for the d-dimensional rigidity matroid of sparse graphs

Let R d ( G ) be the d-dimensional rigidity matroid for a graph G = ( V , E ) . Combinatorial characterization of generically rigid graphs is known only for the plane d = 2 [W. Whiteley, Rigidity and scene analysis, in: J.E. Goodman, J. O'Rourke (Eds.), Handbook of Discrete and Computational Geometry, CRC Press LLC, Boca Raton, FL, 2004, pp. 1327–1354, Chapter 60]. Recently Jackson and Jordán [B. Jackson, T. Jordán, The d-dimensional rigidity matroid of sparse graphs, Journal of Computational Theory (B) 95 (2005) 118–133] derived a min-max formula which determines the rank function in R d ( G ) when G is sparse, i.e. has maximum degree at most d + 2 and minimum degree at most d + 1 . We present efficient algorithms for sparse graphs G in higher dimensions d ⩾ 3 that (i) detect if E is independent in the rigidity matroid for G, and (ii) construct G using vertex insertions preserving if G is isostatic, and (iii) compute the rank of R d ( G ) . The algorithms have linear running time assuming that the dimension d ⩾ 3 is fixed.

A decision tree-based attribute weighting filter for naive Bayes

The naive Bayes classifier continues to be a popular learning algorithm for data mining applications due to its simplicity and linear run-time. Many enhancements to the basic algorithm have been proposed to help mitigate its primary weakness – the assumption that attributes are independent given the class. All of them improve the performance of naive Bayes at the expense (to a greater or lesser degree) of execution time and/or simplicity of the final model. In this paper we present a simple filter method for setting attribute weights for use with naive Bayes. Experimental results show that naive Bayes with attribute weights rarely degrades the quality of the model compared to standard naive Bayes and, in many cases, improves it dramatically. The main advantages of this method compared to other approaches for improving naive Bayes is its run-time complexity and the fact that it maintains the simplicity of the final model.

On the Construction of a Strongly Connected Broadcast Arborescence with Bounded Transmission Delay

Energy conservation is an important concern in wireless networks. Many algorithms for constructing a broadcast tree with minimum energy consumption and other goals have been developed. However, no previous research work considers the total energy consumption and transmission delays of the broadcast tree simultaneously. In this paper, based on an (alpha, beta)-tree, a novel concept to wireless networks, we define a new strongly connected broadcast arborescence with bounded transmission delay (SBAT) problem and design the strongly connected broadcast arborescence (SBA) algorithm with linear running time to construct a strongly connected broadcast tree with bounded total power, while satisfying the constraint that the transmission delays between the source and the other hosts are also bounded. We also propose the distributed version of the SBA algorithm. The theoretical analysis and simulation results show that the SBA algorithm gives a proper solution to the SBAT problem

Fast index based algorithms and software for matching position specific scoring matrices.

BackgroundIn biological sequence analysis, position specific scoring matrices (PSSMs) are widely used to represent sequence motifs in nucleotide as well as amino acid sequences. Searching with PSSMs in complete genomes or large sequence databases is a common, but computationally expensive task.ResultsWe present a new non-heuristic algorithm, called ESAsearch, to efficiently find matches of PSSMs in large databases. Our approach preprocesses the search space, e.g., a complete genome or a set of protein sequences, and builds an enhanced suffix array that is stored on file. This allows the searching of a database with a PSSM in sublinear expected time. Since ESAsearch benefits from small alphabets, we present a variant operating on sequences recoded according to a reduced alphabet. We also address the problem of non-comparable PSSM-scores by developing a method which allows the efficient computation of a matrix similarity threshold for a PSSM, given an E-value or a p-value. Our method is based on dynamic programming and, in contrast to other methods, it employs lazy evaluation of the dynamic programming matrix. We evaluated algorithm ESAsearch with nucleotide PSSMs and with amino acid PSSMs. Compared to the best previous methods, ESAsearch shows speedups of a factor between 17 and 275 for nucleotide PSSMs, and speedups up to factor 1.8 for amino acid PSSMs. Comparisons with the most widely used programs even show speedups by a factor of at least 3.8. Alphabet reduction yields an additional speedup factor of 2 on amino acid sequences compared to results achieved with the 20 symbol standard alphabet. The lazy evaluation method is also much faster than previous methods, with speedups of a factor between 3 and 330.ConclusionOur analysis of ESAsearch reveals sublinear runtime in the expected case, and linear runtime in the worst case for sequences not shorter than ||m + m - 1, where m is the length of the PSSM and a finite alphabet. In practice, ESAsearch shows superior performance over the most widely used programs, especially for DNA sequences. The new algorithm for accurate on-the-fly calculations of thresholds has the potential to replace formerly used approximation approaches. Beyond the algorithmic contributions, we provide a robust, well documented, and easy to use software package, implementing the ideas and algorithms presented in this manuscript.

ICCAP-a linear time sparsification and reordering algorithm for 3-D BEM capacitance extraction

This paper presents an efficient, simple, hierarchical, and sparse three-dimensional capacitance extraction algorithm, i.e., ICCAP. Most previous capacitance extraction algorithms, such as FastCap and HiCap, introduce intermediate variables to facilitate the hierarchical potential calculation, but still preserve the basic panels as basis. In this paper, we discover that those intermediate variables are a fundamentally much better basis than leaf panels. As a result, we are able to explicitly construct the sparse potential coefficient matrix and solve it with linear memory and linear run time in comparison with the most recent hierarchical O(nlogn) approach in PHiCap. Furthermore, the explicit sparse formulation of a potential matrix not only enables the usage of preconditioned Krylov subspace iterative methods, but also the reordering technique. A new reordering technique, i.e., level-oriented reordering (LOR), is proposed to further reduce over 20% of memory consumption and run time compared with no reordering techniques applied. In fact, LOR is even better than the state-of-the-art minimum degree reordering and more efficient. Without complicated orthonormalization matrix computation, ICCAP is very simple, efficient, and accurate. Experimental results demonstrate the superior run time and memory consumption over previous approaches while achieving similar accuracy.