Sublinear-time Algorithms Research Articles

Sublinear-time algorithms for tournament graphs

We show that a random walk on a tournament on n vertices finds either a sink or a 3-cycle in expected time $O(\sqrt{n}\log n\sqrt{\log^{*}n})$ , that is, sublinear both in the size of the description of the graph as well as in the number of vertices.

Combinatorial Sublinear-Time Fourier Algorithms

We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N≫k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of $\hat{\mathbf{A}}$, and estimate their coefficients, in polynomial(k,log N) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002; Proceedings of SPIE Wavelets XI, 2005). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008).

Sublinear time algorithms in the theory of groups and semigroups

Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a small portion of the input. The most typical situation where sublinear time algorithms are considered is property testing. There are several interesting contexts where one can test properties in sublinear time. A canonical example is graph colorability. To tell that a given graph is not k-colorable, it is often sufficient to inspect just one vertex with incident edges: if the degree of a vertex is greater than k, then the graph is not k-colorable. It is a challenging and interesting task to find algebraic properties that could be tested in sublinear time. In this paper, we address several algorithmic problems in the theory of groups and semigroups that may admit sublinear time solution, at least for “most” inputs.

A randomized sublinear time parallel GCD algorithm for the EREW PRAM

We present a randomized parallel algorithm that computes the greatest common divisor of two integers of n bits in length with probability 1 − o ( 1 ) that takes O ( n log log n / log n ) time using O ( n 6 + ϵ ) processors for any ϵ > 0 on the EREW PRAM parallel model of computation. The algorithm either gives a correct answer or reports failure. We believe this to be the first randomized sublinear time algorithm on the EREW PRAM for this problem.

Sublinear estimation of entropy and information distances

In many data mining and machine learning problems, the data items that need to be clustered or classified are not arbitrary points in a high-dimensional space, but are distributions, that is, points on a high-dimensional simplex. For distributions, natural measures are not ℓ p distances, but information-theoretic measures such as the Kullback-Leibler and Hellinger divergences. Similarly, quantities such as the entropy of a distribution are more natural than frequency moments. Efficient estimation of these quantities is a key component in algorithms for manipulating distributions. Since the datasets involved are typically massive, these algorithms need to have only sublinear complexity in order to be feasible in practice. We present a range of sublinear-time algorithms in various oracle models in which the algorithm accesses the data via an oracle that supports various queries. In particular, we answer a question posed by Batu et al. on testing whether two distributions are close in an information-theoretic sense given independent samples. We then present optimal algorithms for estimating various information-divergences and entropy with a more powerful oracle called the combined oracle that was also considered by Batu et al. Finally, we consider sublinear-space algorithms for these quantities in the data-stream model. In the course of doing so, we explore the relationship between the aforementioned oracle models and the data-stream model. This continues work initiated by Feigenbaum et al. An important additional component to the study is considering data streams that are ordered randomly rather than just those which are ordered adversarially.

A sublinear-time approximation scheme for bin packing

The bin packing problem is defined as follows: given a set of n items with sizes 0 < w 1 , w 2 , … , w n ≤ 1 , find a packing of these items into a minimum number of unit-size bins possible. We present a sublinear-time asymptotic approximation scheme for the bin packing problem; that is, for any ϵ > 0 , we present an algorithm A ϵ that has sampling access to the input instance and outputs a value k such that C opt ≤ k ≤ ( 1 + ϵ ) ⋅ C opt + 1 , where C opt is the cost of an optimal solution. It is clear that uniform sampling by itself will not allow a sublinear-time algorithm in this setting; a small number of items might constitute most of the total weight and uniform samples will not hit them. In this work we use weighted samples, where item i is sampled with probability proportional to its weight: that is, with probability w i / ∑ i w i . In the presence of weighted samples, the approximation algorithm runs in O ̃ ( n ⋅ poly ( 1 / ϵ ) ) + g ( 1 / ϵ ) time, where g ( x ) is an exponential function of x . When both weighted sampling and uniform sampling are allowed, O ̃ ( n 1 / 3 ⋅ poly ( 1 / ϵ ) ) + g ( 1 / ϵ ) time suffices. In addition to an approximate value to C opt , our algorithm can also output a constant-size “template” of a packing that can later be used to find a near-optimal packing in linear time.

Separating sublinear time computations by approximate diameter

We study the problem of separating sublinear time computations via approximating the diameter for a sequence S=p 1 p 2 ⋅⋅⋅ p n of points in a metric space, in which any two consecutive points have the same distance. The computation is considered respectively under deterministic, zero error randomized, and bounded error randomized models. We obtain a class of separations using various versions of the approximate diameter problem based on restrictions on input data. We derive tight sublinear time separations for each of the three computation models via proving that computation with O(n r ) time is strictly more powerful than that with O(n r−e ) time, where r and e are arbitrary parameters in (0,1) and (0,r) respectively. We show that, for any parameter r∈(0,1), the bounded error randomized sublinear time computation in time O(n r ) cannot be simulated by any zero error randomized sublinear time algorithm in o(n) time or queries; and the same is true for zero error randomized computation versus deterministic computation.

Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons

We present the first sublinear-time algorithms for computing order statistics in the Farey sequence and for the related problem of ranking. Our algorithms achieve a running times of nearly O(n 2/3), which is a significant improvement over the previous algorithms taking time O(n).We also initiate the study of a more general problem: counting primitive lattice points inside planar shapes. For rational polygons containing the origin, we obtain a running time proportional to D 6/7, where D is the diameter of the polygon.

Sublinear time width-bounded separators and their application to the protein side-chain packing problem

Given d>2 and a set of n grid points Q in ℜ d , we design a randomized algorithm that finds a w-wide separator, which is determined by a hyper-plane, in $O(n^{2\over d}\log n)$ sublinear time such that Q has at most $({d\over d+1}+o(1))n$ points on either side of the hyper-plane, and at most $c_{d}wn^{d-1\over d}$ points within $\frac{w}{2}$ distance to the hyper-plane, where c d is a constant for fixed d. In particular, c 3=1.209. To our best knowledge, this is the first sublinear time algorithm for finding geometric separators. Our 3D separator is applied to derive an algorithm for the protein side-chain packing problem, which improves and simplifies the previous algorithm of Xu (Research in computational molecular biology, 9th annual international conference, pp. 408–422, 2005).

Sublinear-time approximation algorithms for clustering via random sampling

We present a novel analysis of a random sampling approach for four clustering problems in metric spaces: k-median, k-means, min-sum k-clustering, and balanced k-median. For all these problems, we c...

Triple product wavelet integrals for all-frequency relighting

This paper focuses on efficient rendering based on pre-computed light transport, with realistic materials and shadows under all-frequency direct lighting such an environment maps. The basic difficulty is representation and computation in the 6D space of light direction, view direction, and surface position. While image-based and synthetic methods for real-time rendering have been proposed, they do not scale to high sampling rates with variation of both lighting and viewpoint. Current approaches are therefore limited to lower dimensionality (only lighting or viewpoint variation, not both) or lower sampling rates (low frequency lighting and materials). We propose a new mathematical and computational analysis of pre-computed light transport. We use factored forms, separately pre-computing and representing visibility and material properties. Rendering then requires computing triple product integrals at each vertex, involving the lighting, visibility and BRDF. Our main contribution is a general analysis of these triple product integrals, which are likely to have broad applicability in computer graphics and numerical analysis. We first determine the computational complexity in a number of bases like point samples, spherical harmonics and wavelets. We then give efficient linear and sublinear-time algorithms for Haar wavelets, incorporating non-linear wavelet approximation of lighting and BRDFs. Practically, we demonstrate rendering of images under new lighting and viewing conditions in a few seconds, significantly faster than previous techniques.

Testing membership in parenthesis languages

AbstractWe continue the investigation of properties defined by formal languages. This study was initiated by Alon et al. [1], who described an algorithm for testing properties defined by regular languages. Alon et al. also considered several context free languages, and in particular Dyck languages, which contain strings of properly balanced parentheses. They showed that the first Dyck language, which contains strings over a single type of pairs of parentheses, is testable in time independent of n, where n is the length of the input string. However, the second Dyck language, defined over two types of parentheses, requires Ω (log n) queries. Here we describe a sublinear‐time algorithm for testing all Dyck languages. Specifically, the running time of our algorithm is Õ(n2/3/ϵ3), where ϵ is the given distance parameter. Furthermore, we improve the lower bound for testing Dyck languages to \documentclass{article}\usepackage{amsfonts,amsmath}\pagestyle{empty}\begin{document}$\tilde{\Omega}$\end{document}(n1/11) for constant ϵ. We also describe a testing algorithm for the context free language LREV = {uurvvr : u, v ∈ Σ*}, where Σ is a fixed alphabet. The running time of our algorithm is \documentclass{article} \usepackage{amsfonts,amsmath}\pagestyle{empty} \begin{document}$\tilde{O}(\sqrt{n/\epsilon}$ \end{document}, which almost matches the lower bound given by Alon et al. [1]. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 98–138, 2003

Constant-space string-matching in sublinear average time

Given two strings: pattern P of length m and text T of length n. The string-matching problem is to find all occurrences of the pattern P in the text T. We present a string-matching algorithms which works in o( n) average time and constant additional space for one-dimensional texts and two-dimensional arrays. This is a first attempt to the small-space string-matching problem in which sublinear time algorithms are achieved. We show that all occurrences of one- or two-dimensional patterns can be found in O( n r ) average time with constant memory, where r is the repetition size of P (size of the longest repeated subword of P).

Embedding Torus Automata into a Ring of Automata

In this paper, we deal with two-dimensional cellular automata. Our goal is to present a linear space and sub-linear time algorithm to embed a bounded finite torus, considered as a cellular space, into a ring of automata. Such an embedding has to preserve the neighborhood of the cells and to keep the periodicities of the torus. Finally, we generalize this result to torus of greater dimension.

An efficient parallel algorithm for the minimal elimination ordering (MEO) of an arbitrary graph

We design the first efficient parallel algorithm for computing the minimal elimination ordering (MEO) of an arbitrary graph. The algorithm works in O(log 3 n) parallel time and O( nm) processors on CREW PRAM, for an n-vertex, m-edge graph, and is optimal up to a polylogarithmic factor with respect to the best sequential algorithm of Rose, et al. (1976). The MEO problem for arbitrary graphs arises in a number of combinatorial optimization problems, as well as in database applications, scheduling problems, and the sparse Gaussian elimination on symmetric matrices. It was believed before to be inherently sequential, and strongly resisting sublinear parallel time (sublinear sequential storage) algorithms. As an application, this paper gives the first efficient parallel solutions to the problem of minimal fill-in for arbitrary graphs and connected combinatorial optimization problems (see e.g. Rose et al., 1976; Tarjan, 1985), and to the problem of the Gaussian elimination of sparse symmetric matrices (Rose, 1970; 1973) (The problem of computing a minimum fill-in is known to be NP-complete (Yannakakis, 1981). The method of solution involves a development of new techniques for solving connected minimal set system problem, and combining it with some new divide-and-conquer methods.

Parallel Algorithms for Dynamic Programming Recurrences with More Than O(1) Dependency

We study the parallel computation of dynamic programming. We consider four important dynamic programming problems which have wide application, and that have been studied extensively in sequential computation: (1) the 1D problem, (2) the gap problem, (3) the parenthesis problem, and (4) the RNA problem. The parenthesis problem has fast parallel algorithms; almost no work has been done for parallelizing the other three. We present a unifying framework for the parallel computation of dynamic programming recurrences with more than O(1) dependency. We use two well-known methods, the closure method and the matrix product method, as general paradigms for developing parallel algorithms. Combined with various techniques, they lead to a number of new results. Our main results are optimal sublinear-time algorithms for the 1D, parenthesis, and RNA problems.