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SIAM J. on Computing

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1993

Volume 22, Issue 6, pp. 1117-1349


Minimal NFA Problems are Hard

Tao Jiang and B. Ravikumar

SIAM J. Comput. 22, pp. 1117-1141 (25 pages) | Cited 11 times

Online Publication Date: July 31, 2006

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Finite automata (FA’s) are of fundamental importance in theory and in applications. The following basic minimization problem is studied: Given a DFA (deterministic FA), find a minimum equivalent nondeterministic FA (NFA). This paper shows that the natural decision problem associated with it is PSPACE-complete. More generally, let ${\text{A}} \to {\text{B}}$ denote the problem of converting a given FA of type A to a minimum FA of type B. This paper also shows that most of these problems are computationally hard. Motivated by the question of how much nondeterminism suffices to make the decision problem involving an NFA computationally hard, the authors study the complexity decision problems for FA’s and present several intractability results, even for cases in which the input is deterministic or nondeterministic with a very limited nondeterminism. For example, it is shown that it is PSPACE-complete to decide if $L(M_1 ) \cdot L(M_2 ) = L(M_3 )$, where $M_1 $, $M_2 $, and $M_3 $ are DFAs. These problems are related to some classical problems in automata theory (such as deciding whether an FA has the finite power property), as well as recent ones (such as determining the diversity of a given FA).

An $O(m\log n)$-Time Algorithm for the Maximal Planar Subgraph Problem

Jiazhen Cai, Xiaofeng Han, and Robert E. Tarjan

SIAM J. Comput. 22, pp. 1142-1162 (21 pages) | Cited 12 times

Online Publication Date: July 31, 2006

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Based on a new version of the Hopcroft and Tarjan planarity testing algorithm, this paper develops an $O(m\log n)$-time algorithm to find a maximal planar subgraph.

On the Existence of Pseudorandom Generators

Oded Goldreich, Hugo Krawczyk, and Michael Luby

SIAM J. Comput. 22, pp. 1163-1175 (13 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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Pseudorandom generators (suggested and developed by Blum and Micali and Yao) are efficient deterministic programs that expand a randomly selected $k$-bit seed into a much longer pseudorandom bit sequence that is indistinguishable in polynomial time from an (equally long) sequence of unbiased coin tosses. A fundamental question is to find simple conditions, as the existence of one-way functions, which suffice for constructing pseudorandom generators. This paper considers regular functions, in which every image of a $k$-bit string has the same number of preimages of length $k$. This paper shows how to construct pseudorandom generators from any regular one-way function.

A Generalized Suffix Tree and Its (Un)Expected Asymptotic Behaviors

Wojciech Szpankowski

SIAM J. Comput. 22, pp. 1176-1198 (23 pages) | Cited 14 times

Online Publication Date: July 31, 2006

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Suffix trees find several applications in computer science and telecommunications, most notably in algorithms on strings, data compressions, and codes. Despite this, very little is known about their typical behaviors. In a probabilistic framework, a family of suffix trees—further called $b$-suffix trees—built from the first $n$ suffixes of a random word is considered. In this family a noncompact suffix tree (i.e., such that every edge is labeled by a single symbol) is represented by $b = 1$, and a compact suffix tree (i.e., without unary nodes) is asymptotically equivalent to $b \to \infty $ as $n \to \infty $. Several parameters of $b$-suffix trees are studied, namely, the depth of a given suffix, the depth of insertion, the height and the shortest feasible path. Some new results concerning typical (i.e., almost sure) behaviors of these parameters are established. These findings are used to obtain several insights into certain algorithms on words, molecular biology, and universal data compression schemes.

Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths

David R. Karger, Daphne Koller, and Steven J. Phillips

SIAM J. Comput. 22, pp. 1199-1217 (19 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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The all-pairs shortest-paths problem in weighted graphs is investigated. An algorithm—the Hidden-Paths Algorithm—that finds these paths in time $O(m^ * n + n^2 \log n)$, where $m^ * $ is the number of edges participating in shortest paths, is presented. The algorithm is a practical substitute for Dijkstra’s algorithm. It is argued that $m^ * $ is likely to be small in practice since $m^ * = O(n\log n)$ with high probability for many probability distributions on edge weights. An $\Omega (mn)$ lower bound on the running time of any path-comparison-based algorithm for the all-pairs shortest-paths problem is also proved. Path-comparison-based algorithms form a natural class containing the Hidden-Paths Algorithm, as well as the algorithms of E. W. Dijkstra [Numer. Math., 1 (1959), pp. 269–271] and R. W. Floyd [Comm. ACM, 5 (1962), p. 345]. Lastly, generalized forms of the shortest-paths problem are considered, and it is shown that many of the standard shortest-paths algorithms are effective in this more general setting.

On Finding the Rectangular Duals of Planar Triangular Graphs

Xin He

SIAM J. Comput. 22, pp. 1218-1226 (9 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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This paper presents a new linear-time algorithm for finding rectangular duals of planar triangular graphs. The algorithm is conceptually simpler than the previously known algorithm. The coordinates of the rectangular dual constructed by the new algorithm are integers and carry clear combinatorial meaning.

Fast and Efficient Parallel Solution of Sparse Linear Systems

Victor Pan and John Reif

SIAM J. Comput. 22, pp. 1227-1250 (24 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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This paper presents a parallel algorithm for the solution of a linear system $A{\bf x} = {\bf b}$ with a sparse $n \times n$ symmetric positive definite matrix $A$, associated with the graph $G(A)$ that has $n$ vertices and has an edge for each nonzero entry of $A$. If $G(A)$ has an $s(n)$-separator family and a known $s(n)$-separator tree, then the algorithm requires only $O(\log ^3 n)$ time and $(|E| + {{M(s(n)))} / {\log n}}$ processors for the evaluation of the solution vector ${\bf x} = A^{ - 1} {\bf b}$, where $|E|$ is the number of edges in $G(A)$ and $M(n)$ is the number of processors sufficient for multiplying two $n \times n$ rational matrices in time $O(\log n)$. Furthermore, for this computational cost the algorithm computes a recursive factorization of $A$ such that the solution of any other linear system $A{\bf x} = {\bf b}'$ with the same matrix $A$ requires only $O(\log ^2 n)$ time and $({{|E|} / {\log n}}) + s(n)^2 $ processors.

On-Line Bin Packing of Items of Random Sizes, II

WanSoo T. Rhee and Michel Talagrand

SIAM J. Comput. 22, pp. 1251-1256 (6 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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This paper describes the construction of an on-line algorithm with the following property. There exist universal constants $K$, $\alpha $ such that given any probability measure $\mu $ on $[0,1]$ and a sequence $X_1 , \ldots ,X_n , \ldots $ of items independently and identically distributed according to $\mu $, the algorithm packs $X_1 , \ldots ,X_n $ into at most $T_n (X_1 , \ldots ,X_n ) + Kn^{{1 / 2}} (\log n)^{{3 / 4}} $ unit-size bins, where $T_n (X_1 , \ldots ,X_n )$ denotes the minimum number of bins needed to pack $X_1 , \ldots ,X_n $, and does this with probability greater than or equal to $1 - K\exp ( - \alpha (\log n)^{{3 / 2}} )$. In contrast with the authors’ previous work on this problem, the algorithm is now independent of $\mu $.

On the Computational Complexity of Small Descriptions

Ricard Gavaldà and Osamu Watanabe

SIAM J. Comput. 22, pp. 1257-1275 (19 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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For a set $L$ that is polynomial-time reducible (in short, $ \leqslant _{\text{T}}^{\text{P}} $-reducible) to some sparse set, the authors investigate the computational complexity of such sparse sets relative to $L$ and prove the first lower bounds on the complexity of recognizing such sets. Sets $A$ and $B$ are constructed such that both of them are $ \leqslant _{\text{T}}^{\text{P}} $-reducible to some sparse set, but $A$ (respectively, $B$) is $ \leqslant _{\text{T}}^{\text{P}} $-reducible to no sparse set in ${\text{P}}^A $ (respectively, ${\text{NP}}^B \cap {\text{co-NP}}^B $); that is, the complexity of sparse sets to which $A$ (respectively, $B$) is $ \leqslant _{\text{T}}^{\text{P}} $-reducible is more than ${\text{P}}^A $ (respectively, ${\text{NP}}^B \cap {\text{co-NP}}^B $). From these results or application of the proof technique the authors obtain (1) lower bounds for the relative complexity of generating polynomial-size circuits for some sets in ${{\text{P}} / {{\operatorname{poly}}}}$ and (2) separations of the classes of sets equivalent or reducible to sparse sets under various polynomial-time reducibilities.

VC Dimension and Uniform Learnability of Sparse Polynomials and Rational Functions

Marek Karpinski and Thorsten Werther

SIAM J. Comput. 22, pp. 1276-1285 (10 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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The authors prove upper and lower bounds on the VC dimension of sparse univariate polynomials over reals and apply these results to prove uniform learnability of sparse polynomials and rational functions. As an application the solution to the open problem of Vapnik [in Estimation of Dependences Based on Empirical Data, Springer-Verlag, Berlin, 1982] on computational approximation of the regression in a class of polynomials used in the theory of empirical data dependences is given.

Computing a Face in an Arrangement of Line Segments and Related Problems

Bernard Chazelle, Herbert Edelsbrunner, Leonidas Guibas, Micha Sharir, and Jack Snoeyink

SIAM J. Comput. 22, pp. 1286-1302 (17 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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This paper presents a randomized incremental algorithm for computing a single face in an arrangement of $n$ line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is $O(n\alpha (n)\log n)$. The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson–Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387–421]. A few extensions of the technique, obtaining efficient randomized incremental algorithms for constructing the entire arrangement of a collection of line segments and for computing a single face in an arrangement of Jordan arcs are also presented.

A Better Heuristic for Preemptive Parallel Machine Scheduling with Batch Setup Times

Bo Chen

SIAM J. Comput. 22, pp. 1303-1318 (16 pages) | Cited 12 times

Online Publication Date: July 31, 2006

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This paper addresses the problem of scheduling $N$ jobs on $M$ identical parallel machines with the objective of minimizing the makespan. Jobs are divided into $B$ batches. A sequence-independent batch setup time on a machine is incurred whenever the machine starts its processing on or switches it from a job in one batch to a job in another batch. On the basis of a heuristic for this NP-hard problem proposed by Monma and Potts, a modified heuristic that requires the same implementing time $O(N + (M + B)\log (M + B))$ and is asymptotically optimal is presented. Furthermore, for a certain class of problems, which includes the case in which each batch contains a single job, it has the worst-case performance ratio $\tau _M = \max \{ {\frac{{3M}}{{2M + 1}},\frac{{3M - 4}}{{2M - 2}}} \}$.

A New Method for Computing Page-Fault Rates

Jiang-Hsing Chu and Gary D. Knott

SIAM J. Comput. 22, pp. 1319-1330 (12 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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When a program is executed in a caching environment, the caching algorithm can be modeled by an associated finite-state automaton. It is assumed that the finite automaton will reach a steady state after processing a long string. By considering the finite automaton, a formula is obtained for the expected page-fault rate in terms of the steady-state probabilities of the automaton. It is possible to derive the steady-state probabilities for the least-recently-used (LRU) algorithm with order-0 and order-1 programs based on a method that describes the page reference strings as regular expressions. The steady-state behavior for caching algorithms with order-1 programs has never been reported before. This analysis method is then applied to obtain an analysis of the caching behavior of a practical storage-and-retrieval algorithm.

Learning Decision Trees Using the Fourier Spectrum

Eyal Kushilevitz and Yishay Mansour

SIAM J. Comput. 22, pp. 1331-1348 (18 pages) | Cited 10 times

Online Publication Date: July 31, 2006

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This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over $GF(2)$).
This paper shows how to learn in polynomial time any function that can be approximated (in norm $L_2 $) by a polynomially sparse function (i.e., a function with only polynomially many nonzero Fourier coefficients). The authors demonstrate that any function $f$ whose $L_1 $-norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial $L_1 $-norm can be learned deterministically.
The algorithm can also exactly identify a decision tree of depth $d$ in time polynomial in $2^d $ and $n$. This result implies that trees of logarithmic depth can be identified in polynomial time.

Correction: Parallel Merge Sort

Richard Cole

SIAM J. Comput. 22, pp. 1349-1349 (1 page)

Online Publication Date: July 31, 2006

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