Expected Time Algorithm Research Articles

Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

In this paper we provide an O ( m loglog O (1) n log (1/ϵ))-expected time algorithm for solving Laplacian systems on n -node m -edge graphs, improving upon the previous best expected runtime of \(O(m \sqrt {\log n} \mathrm{log log}^{O(1)} n \log (1/\epsilon)) \) achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of low spectral stretch graph approximations with improved stretch and sparsity bounds. As motivation for this work, we show that for every set of vectors in \(\mathbb {R}^d \) (not just those induced by graphs) and all integer k > 1 there exist an ultra-sparsifier with d − 1 + O ( d / k ) re-weighted vectors of relative condition number at most k 2 . For small k , this improves upon the previous best known multiplicative factor of \(k \cdot \tilde{O}(\log d) \) , which is only known for the graph case. Additionally, in the graph case we employ our low-stretch subgraph construction to obtain n − 1 + O ( n / k )-edge ultrasparsifiers of relative condition number k 1 + o (1) for k = ω (log δ n ) for any δ > 0: this improves upon the previous work for k = o (exp (log 1/2 − δ n )).

Distribution-Sensitive Construction of the Greedy Spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take varOmega (n^2) time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly. To find the usually few remaining ‘long’ edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives an O(n log ^2 n log ^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an O((n + |E|) log ^2 n log log n) expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E.

A fast algorithm for data collection along a fixed track

Recent research shows that significant energy saving can be achieved in wireless sensor networks (WSNs) with a mobile base station that collects data from sensor nodes via short-range communications. However, a major performance bottleneck of such WSNs is the significantly increased latency in data collection due to the low movement speed of mobile base stations. In this paper we study the problem of finding a data collection path for a mobile base station moving along a fixed track in a wireless sensor network to minimize the latency of data collection. The main contribution is an O(mnlog⁡n) expected time algorithm, where n is the number of sensors in the networks and m is the complexity of the fixed track.

Simulating Size-constrained Galton–Watson Trees

We discuss various methods for generating random Galton-Watson trees conditional on their sizes being equal to $n$. A linear expected time algorithm is proposed.

Fast Inference with Min-Sum Matrix Product

The MAP inference problem in many graphical models can be solved efficiently using a fast algorithm for computing min-sum products of n × n matrices. The class of models in question includes cyclic and skip-chain models that arise in many applications. Although the worst-case complexity of the min-sum product operation is not known to be much better than O(n(3)), an O(n(2.5)) expected time algorithm was recently given, subject to some constraints on the input matrices. In this paper, we give an algorithm that runs in O(n(2) log n) expected time, assuming that the entries in the input matrices are independent samples from a uniform distribution. We also show that two variants of our algorithm are quite fast for inputs that arise in several applications. This leads to significant performance gains over previous methods in applications within computer vision and natural language processing.

APPROXIMATE WEIGHTED FARTHEST NEIGHBORS AND MINIMUM DILATION STARS

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find approximate farthest neighbors that are farther than a factor (1 - ∊) of optimal in time O( log n) per query in D-dimensional Euclidean space for any constants D and ∊. As an application, we find an O(n log n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.

Finding a maximum matching in a sparse random graph in O ( n ) expected time

We present a linear expected time algorithm for finding maximum cardinality matchings in sparse random graphs. This is optimal and improves on previous results by a logarithmic factor.

Enclosing weighted points with an almost-unit ball

Given a set of n points with positive real weights in d-dimensional space, we consider an approximation to the problem of placing a unit ball, in order to maximize the sum of the weights of the points enclosed by the ball. Given an approximation parameter ε < 1 , we present an O ( n / ε d − 1 ) expected time algorithm that determines a ball of radius 1 + ε enclosing a weight at least as large as the weight of the optimal unit ball. This is the first approximate algorithm for the weighted version of the problem in d-dimensional space. We also present a matching lower bound for a certain class of algorithms for the problem.

On Finding a Fixed Point in a Boolean Network with Maximum Indegree 2

Finding fixed points in discrete dynamical systems is important because fixed points correspond to steady-states. The Boolean network is considered as one of the simplest discrete dynamical systems and is often used as a model of genetic networks. It is known that detection of a fixed point in a Boolean network with n nodes and maximum indegree K can be polynomially transformed into (K+1)-SAT with n variables. In this paper, we focus on the case of K = 2 and present an O(1.3171n) expected time algorithm, which is faster than the naive algorithm based on a reduction to 3-SAT, where we assume that nodes with indegree 2 do not contain self-loops. We also show an algorithm for the general case of K = 2 that is slightly faster than the naive algorithm.

An optimal randomized algorithm for d-variate zonoid depth

A randomized linear expected-time algorithm for computing the zonoid depth [R. Dyckerhoff, G. Koshevoy, K. Mosler, Zonoid data depth: Theory and computation, in: A. Prat (Ed.), COMPSTAT 1996—Proceedings in Computational Statistics, Physica-Verlag, Heidelberg, 1996, pp. 235–240; K. Mosler, Multivariate Dispersion, Central Regions and Depth. The Lift Zonoid Approach, Lecture Notes in Statistics, vol. 165, Springer-Verlag, New York, 2002] of a point with respect to a fixed dimensional point set is presented.

Minimum dilation stars

The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R d . In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O ( n log n ) -time algorithm for evaluating the dilation of a given star; a randomized O ( n log n ) expected-time algorithm for finding an optimal center in R d ; and for the case d = 2 , a randomized O ( n 2 α ( n ) log 2 n ) expected-time algorithm for finding an optimal center among the input points.

Modular exponentiation via the explicit Chinese remainder theorem

Fix pairwise coprime positive integers p 1 ,P 2 ,..., p a . We propose representing integers u modulo m, where m is any positive integer up to roughly p 1 p 2 ... P s , as vectors (u mod p 1 , u mod p 2 ,.., u mod p s ). We use this representation to obtain a new result on the parallel complexity of modular exponentiation: there is an algorithm for the Common CRCW PRAM that, given positive integers x, e, and m in binary, of total bit length n, computes x e mod m in time O(n/lglgn) using n O(1) processors. For comparison, a parallelization of the standard binary algorithm takes superlinear time; Adleman and Kompella gave an O((lg n) 3 ) expected time algorithm using exp(O(n lg n)) processors; von zur Gathen gave an NC algorithm for the highly special case that m is polynomially smooth.

The fragment assembly string graph

We present a concept and formalism, the string graph, which represents all that is inferable about a DNA sequence from a collection of shotgun sequencing reads collected from it. We give time and space efficient algorithms for constructing a string graph given the collection of overlaps between the reads and, in particular, present a novel linear expected time algorithm for transitive reduction in this context. The result demonstrates that the decomposition of reads into kmers employed in the de Bruijn graph approach described earlier is not essential, and exposes its close connection to the unitig approach we developed at Celera. This paper is a preliminary piece giving the basic algorithm and results that demonstrate the efficiency and scalability of the method. These ideas are being used to build a next-generation whole genome assembler called BOA (Berkeley Open Assembler) that will easily scale to mammalian genomes.

A fast expected time algorithm for the 2-D point pattern matching problem

Point set pattern matching is an integral part of many pattern recognition problems. We study a randomized algorithm for the alignment approach to model-based recognition. Under certain mild assumptions we show that if our scene is a set of n points and our model is a set of m< n points our algorithm has expected running time O(n( log m) 3/2) for finding an occurrence of the model in the scene. This is significantly faster than any existing algorithms in the literature. We then describe some experimental results on randomly generated data using a practical version of our algorithm. These results agree well with the theoretical analysis.

Binary Searching with Nonuniform Costs and Its Application to Text Retrieval

We study the problem of minimizing the expected cost of binary searching for data where the access cost is not fixed and depends on the last accessed element, such as data stored in magnetic or optical disk. We present an optimal algorithm for this problem that finds the optimal search strategy in O(n 3 ) time, which is the same time complexity of the simpler classical problem of fixed costs. Next, we present two practical linear expected time algorithms, under the assumption that the access cost of an element is independent of its physical position. Both practical algorithms are online, that is, they find the next element to access as the search proceeds. The first one is an approximate algorithm which minimizes the access cost disregarding the goodness of the problem partitioning. The second one is a heuristic algorithm, whose quality depends on its ability to estimate the final search cost, and therefore it can be tuned by recording statistics of previous runs. We present an application for our algorithms related to text retrieval. When a text collection is large it demands specialized indexing techniques for efficient access. One important type of index is the suffix array, where data access is provided through an indirect binary search on the text stored in magnetic disk or optical disk. Under this cost model we prove that the optimal algorithm cannot perform better than Ω(1/ log n) times the standard binary search. We also prove that the approximate strategy cannot, on average, perform worse than 39% over the optimal one. We confirm the analytical results with simulations, showing improvements between 34% (optimal) and 60% (online) over standard binary search for both magnetic and optical disks.

A fast bit-vector algorithm for approximate string matching based on dynamic programming

The approximate string matching problem is to find all locations at which a query of length m matches a substring of a text of length n with k -or-fewer differences. Simple and practical bit-vector algorithms have been designed for this problem, most notably the one used in agrep . These algorithms compute a bit representation of the current state-set of the k -difference automaton for the query, and asymptotically run in either O ( nm/w ) or O ( nm log σ/ w ) time where w is the word size of the machine (e.g., 32 or 64 in practice), and σ is the size of the pattern alphabet. Here we present an algorithm of comparable simplicity that requires only O ( nm/w) time by virtue of computing a bit representation of the relocatable dynamic programming matrix for the problem. Thus, the algorithm's performance is independent of k , and it is found to be more efficient than the previous results for many choices of k and small m . Moreover, because the algorithm is not dependent on k , it can be used to rapidly compute blocks of the dynamic programming matrix as in the 4-Russians algorithm of Wu et al.(1996). This gives rise to an O(kn/w) expected-time algorithm for the case where m may be arbitrarily large. In practice this new algorithm, that computes a region of the dynamic progr amming (d.p.) matrx w entries at a time using the basic algorithm as a subroutine is significantly faster than our previous 4-Russians algorithm, that computes the same region 4 or 5 entries at a time using table lookup. This performance improvement yields a code that is either superior or competitive with all existing algorithms except for some filtration algorithms that are superior when k/m is sufficiently small.

Computing Many Faces in Arrangements of Lines and Segments

We present randomized algorithms for computing many faces in an arrangement of lines or of segments in the plane, which are considerably simpler and slightly faster than the previously known ones. The main new idea is a simple randomized $O(n \log n)$ expected time algorithm for computing $\sqrt{n}$ cells in an arrangement of n lines.

Efficient Randomized Algorithms for the Repeated Median Line Estimator

The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Recently there has been a great deal of interest is robust estimators, because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is its breakdown point, that is, the fraction (up to 50%) of outlying data points that can corrupt the estimator. One problem with robust estimators is that achieving high breakdown points (near 50%) has proved to be computationally demanding. In this paper we present the best known theoretical algorithm and a practical subquadratic algorithm for computing a 50% breakdown point line estimator, the Siegel or repeated median line estimator. We first present an O(n\log n) randomized expected-time algorithm, where n is the number of given points. This algorithm relies, however, on sophisticated data structures. We also present a very simple O(n log 2 n) randomized algorithm for this problem, which uses no complex data structures. We provide empirical evidence that, for many realistic input distributions, the running time of this second algorithm is actually O(n log n) expected time.

Unoriented $Theta$-Maxima in the Plane: Complexity and Algorithms

We introduce the unoriented $\Theta$-maximum as a new criterion for describing the shape of a set of planar points. We present efficient algorithms for computing the unoriented $\Theta$-maximum of a set of planar points. We also propose a simple linear expected time algorithm for computing the unoriented $\Theta$-maximum of a set of planar points when $\Theta=\pi/2$.

An approximation algorithm for least median of squares regression

Least median of squares (LMS) regression is a robust method to fit equations to observed data (typically in a linear model). This paper describes an approximation algorithm for LMS regression. The algorithm generates a regression solution with median residual no more than twice the optimal median residual. Random sampling is used to provide a simple O( n log 2 n) expected time algorithm in the two-dimensional case that is successful with high probability. This algorithm is also extended to arbitrary dimension d with O( n d − 1 log n) worst-case complexity for fixed d > 2.