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1990

Volume 19, Issue 6, pp. 989-1161


An Improved Algorithm For Approximate String Matching

Zvi Galil and Kunsoo Park

SIAM J. Comput. 19, pp. 989-999 (11 pages) | Cited 21 times

Online Publication Date: July 13, 2006

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Given a text string, a pattern string, and an integer $k$, a new algorithm for finding all occurrences of the pattern string in the text string with at most $k$ differences is presented. Both its theoretical and practical variants improve upon the known algorithms.

Linear Programming with Two Variables Per Inequality in Poly-Log Time

George S. Lueker, Nimrod Megiddo, and Vijaya Ramachandran

SIAM J. Comput. 19, pp. 1000-1010 (11 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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The parallel time complexity of the linear programming problem with at most two variables per inequality is discussed. Let $n$ and $m$ denote the number of variables and the number of inequalities, respectively, in a linear programming problem. It is assumed that all inequalities are weak. Under the concurrent-read-exclusive-write PRAM model, an $O((\log m + \log^2 n) \log^2 n)$-time parallel algorithm for deciding feasibility is described. It requires $mn^{O(\log n)}$ processors in the worst case, though it is not known whether this bound is tight. When the problem is feasible, a solution can be computed within the same complexity. Moreover, linear programming problems with at most two nonzero coefficients in the objective function can be solved in poly-log time on a similar number of processors. Consequently, all these problems can be solved sequentially with only $O((\log m + \log ^2 n)^2 \log ^2 n)$ space. (These bounds assume that numbers take $O(1)$ space, and arithmetic on them takes $O(1)$ time; the problem can still be solved in poly-log space as a function of the input size even if a Turing machine model with rational input is used instead.) It is also shown that if the underlying graph has bounded tree-width and an underlying tree is given, then the feasibility problem is in the class NC.

A Time Complexity Gap for Two-Way Probabilistic Finite-State Automata

Cynthia Dwork and Larry Stockmeyer

SIAM J. Comput. 19, pp. 1011-1023 (13 pages) | Cited 6 times

Online Publication Date: July 13, 2006

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It is shown that if a two-way probabilistic finite-state automaton (2pfa) $M$ recognizes a nonregular language $L$ with error probability bounded below $\frac{1}{2}$, then there is a positive constant $b$ (depending on $M$) such that, for infinitely many inputs $x$, the expected running time of $M$ on input $x$ must exceed $2^{n^{b}}$ where $n$ is the length of $x$. This complements a result of Freivalds showing that 2pfa’s can recognize certain nonregular languages in exponential expected time. It also establishes a time complexity gap for 2pfa’s, since any regular language can be recognized by some 2pfa in linear time. Other results give roughly exponential upper and lower bounds on the worst-case increase in the number of states when converting a polynomial-time 2pfa to an equivalent two-way nondeterministic finite-state automaton or to an equivalent one-way deterministic finite-state automaton.

The Searchlight Scheduling Problem

Kazuo Sugihara, Ichiro Suzuki, and Masafumi Yamashita

SIAM J. Comput. 19, pp. 1024-1040 (17 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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The problem of searching for a mobile robber in a simple polygon by a number of searchlights is considered. A searchlight is a stationary point which emits a single ray that cannot penetrate the boundary of the polygon. The direction of the ray can be changed continuously, and a point is detected by a searchlight at a given time if and only if it is on the ray. A robber is a point that can move continuously with unbounded speed. First, it is shown that the problem of obtaining a search schedule for an instance having at least one searchlight on the polygon boundary can be reduced to that for instances having no searchlight on the polygon boundary. The reduction is achieved by a recursive search strategy called the one-way sweep strategy. Then various sufficient conditions for the existence of a search schedule are presented by using the concept of a searchlight visibility graph. Finally, a simple necessary and sufficient condition for the existence of a search schedule for instances having exactly two searchlights in the interior is presented.

Minimum Cuts for Circular-Arc Graphs

D. T. Lee, M. Sarrafzadeh, and Y. F. Wu

SIAM J. Comput. 19, pp. 1041-1050 (10 pages) | Cited 9 times

Online Publication Date: July 13, 2006

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The problem of finding a minimum cut of $n$ arcs on a unit circle is considered. It is shown that this problem can be solved in $\Theta (n \log n)$ time, which is optimal to within a constant factor. If the endpoints of the arcs are sorted, the problem can be solved in linear time. The solution to the minimum cut problem can be used to solve a minimum new facility problem in competitive location and a minimum partition set problem for the intersection model of a circle graph. As a by-product it is also shown that the maximum independent set of $n$ arcs can be obtained in linear time, assuming the endpoints are sorted, which is much simpler than the most recent result of Masuda and Nakajima [SIAMI. Comput., 17 (1988), pp. 41–52].

An Optimal $O(\log\log n)$ Time Parallel String Matching Algorithm

Dany Breslauer and Zvi Galil

SIAM J. Comput. 19, pp. 1051-1058 (8 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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An optimal $O(\log \log n)$ time parallel algorithm for string matching on CROW-PRAM is presented. It improves previous results of Galil [Inform, and Control, 67 (1985), pp. 144–157] and Vishkin [Inform, and Control, 67 (1985), pp. 91–113].

Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields

Dima Yu. Grigoriev, Marek Karpinski, and Michael F. Singer

SIAM J. Comput. 19, pp. 1059-1063 (5 pages) | Cited 9 times

Online Publication Date: July 13, 2006

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The authors consider the problem of reconstructing (i.e., interpolating) a $t$-sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field $GF[q]$ and the black box can evaluate the polynomial in the field $GF[q^{\ulcorner 2\log_{q}(nt)+3 \urcorner}]$, where $n$ is the number of variables, then there is an algorithm to interpolate the polynomial in $O(\log^3 (nt))$ boolean parallel time and $O(n^2 t^6 \log^2 nt)$ processors.
This algorithm yields the first efficient deterministic polynomial time algorithm (and moreover boolean $NC$-algorithm) for interpolating $t$-sparse polynomials over finite fields and should be contrasted with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in $GF[q]$ is not possible (cf. [M. Clausen, A. Dress, J. Grabmeier, and M. Karpinski, Theoret. Comput. Sci., 1990, to appear]). This algorithm, together with the efficient deterministic interpolation algorithms for fields of characteristic 0 (cf. [D. Yu. Grigoriev and M. Karpinski, in Proceedings of the 28th IEEE Symposium on the Foundations of Computer Science, 1987, pp. 166–172], [M. Ben-Or and P. Tiwari, in Proceedings of the 20th ACM Symposium on the Theory of Computing, 1988, pp. 301–309]), yields for the first time the general deterministic sparse conversion algorithm working over arbitrary fields. (The reason for this is that every field of positive characteristic contains a primitive subfield of this characteristic, and so this method can be applied to the slight extension of this subfield.) The method of solution involves the polynomial enumeration techniques of [D. Yu. Grigoriev and M. Karpinski, op. cit.] combined with introducing a new general method of solving the problem of determining if a $t$-sparse polynomial is identical to zero by evaluating it in a slight extension of the coefficient field (i.e., an extension whose degree over this field is logarithmic in nt).

Linear Circuits over $\operatorname{GF}(2)$

Noga Alon, Mauricio Karchmer, and Avi Wigderson

SIAM J. Comput. 19, pp. 1064-1067 (4 pages) | Cited 6 times

Online Publication Date: July 13, 2006

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For $n=2^k $, let $S$ be an $n \times n$ matrix whose rows and columns are indexed by $\operatorname{GF}(2)^k $ and, for $i, j \in \operatorname{GF}(2)^k , S_{i.j}=\langle i, j \rangle $, the standard inner product. Size-depth trade-oils are investigated for computing $S{\bf x}$ with circuits using only linear operations. In particular, linear size circuits with depth bounded by the inverse of an Ackerman function are constructed, and it is shown that depth two circuits require $\Omega (n \log n)$ size. The lower bound applies to any Hadamard matrix.

Random Polynomials and Approximate Zeros of Newton’s Method

Joel Friedman

SIAM J. Comput. 19, pp. 1068-1099 (32 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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In this paper the authors study the size of the set of “approximate zeros” for Newton’s method, for a randomly chosen polynomial over certain distributions. For a degree $d$ monic polynomial with coefficients chosen uniformly and independently in the unit ball, the results of this paper show, for example, that the set of approximate zeros is at least $Cd^{(-1.5-\varepsilon)}$ for any positive $\varepsilon$, with $C$ depending only on $\varepsilon$.

Category and Measure in Complexity Classes

Jack H. Lutz

SIAM J. Comput. 19, pp. 1100-1131 (32 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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This paper presents resource-bounded category and resource-bounded measure—two new tools for computational complexity theory—and some applications of these tools to the structure theory of exponential complexity classes.
Resource-bounded category, a complexity-theoretic generalization of the Baire category method, defines nontrivial ideals of meager subsets of E, ESPACE, and other complexity classes. Similarly, resource-bounded measure, a generalization of Lebesgue measure theory, defines the measure 0 subsets of complexity classes. Properties developed here include a useful characterization of meager sets in terms of resource-bounded Banach–Mazur games.
Resource-bounded category and measure are applied to the investigation of uniform versus nonuniform complexity. Kannan’s theorem that $\text{ESPACE} \nsubseteq \text{P}/\text{Poly}$ is extended by showing that ${{{\text{P}}} / {{\text{Poly}}}} \cap {\text{ESPACE}}$ is only a meager, measure 0 subset of ESPACE. A theorem of Huynh is extended similarly by showing that all but a meager, measure 0 subset of the languages in {\text{ESPACE}} have high space-bounded Kolmogorov complexity. A new hierarchy of exponential classes is introduced and used to refine known relationships between nonuniform complexity and time complexity.
Known properties of hard languages are also extended. Recent results of Schoning and Huynh state that any language $L$ that is $\leqq _m ^{\text{P}}$-hard for E or $\leqq _T ^{\text{P}}$-hard for {\text{ESPACE}} cannot be feasibly approximated. It is proven here that this conclusion in fact holds unless only a meager subset of E is $\leqq _m ^{\text{P}}$-reducible to $L$ and only a meager, measure 0 subset of {\text{ESPACE}} is $\leqq_{m}^{\text{PSPACE}}$ reducible to $L$. This suggests a new lower bound method which may be useful in interesting cases.

Computing Puiseux-Series Solutions to Determinantal Equations via Combinatorial Relaxation

Kazuo Murota

SIAM J. Comput. 19, pp. 1132-1161 (30 pages) | Cited 3 times

Online Publication Date: July 13, 2006

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Let $A(t,x) = (A_{ij} (t,x))$ be a square matrix with $A_{ij}$ being a polynomial in $t$ and $x$. This paper proposes an algorithm for computing the Puiseux (= fractional power) series solutions $x = x(t)$ to the equation ${\operatorname{det}}A(t, x) = 0$. The algorithm is based on an observation which links the Newton diagram (polygon) for det ${\operatorname{det}}A(t, x)$ with the perfect matchings of a bipartite graph associated with $A$. The algorithm is efficient, making full use of available fast network-type algorithms.
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