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1980

Volume 9, Issue 4, pp. 665-855


Completeness with Finite Systems of Intermediate Assertions for Recursive Program Schemes

Krzysztof R. Apt and Lambert G. L. T. Meertens

SIAM J. Comput. 9, pp. 665-671 (7 pages)

Online Publication Date: July 13, 2006

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It is proved that in the general case of arbitrary context-free schemes a program is (partially) correct with respect to given initial and final assertions if and only if a suitable finite system of intermediate assertions can be found. Assertions are allowed from the extended state space $\mathcal {V} \times \mathcal {V}$. This result contrasts with the results of [2], where it is proved that if assertions are taken from the original state space $\mathcal {V}$, then in the general case an infinite system of intermediate assertions is needed. The extension of the state space allows a unification in the relational framework of [2], of the (essence of the) results of [2], and of [4], [5] and [6], and provides a semantic counterpart of the use of auxiliary variables.

A New Proof of the Linearity of the Boyer-Moore String Searching Algorithm

Leo J. Guibas and Andrew M. Odlyzko

SIAM J. Comput. 9, pp. 672-682 (11 pages) | Cited 13 times

Online Publication Date: July 13, 2006

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The Boyer-Moore algorithm searches for all occurrences of a specified string, the pattern, in another string, the text. We study the combinatorial structure of periodic strings and use these results to derive a new proof of the linearity of the Boyer-Moore algorithm in the worst case. Our proof reduces the previously best known bound of $7n$ to $4n$, where $n$ is the length of the text.

Compatible Orderings on the Metric Theory of Trees

Stephen L. Bloom and Ralph Tindell

SIAM J. Comput. 9, pp. 683-691 (9 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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In many studies of computation which make use of rooted labeled trees a partial ordering is usually imposed on the trees in the following way. A particular label, say $ \bot _0 $, is distinguished and identified with the atomic tree whose only vertex is a leaf labeled $ \bot _0 $. A tree $f$ is then defined to be less than a tree A tree $g$ if $g$ can be obtained from $f$ by attaching some new trees to leaves of $f$ labeled $ \bot _0 $.
This paper answers the following questions. What is the significance of the tree $ \bot _0 $ in this ordering? Can other nonatomic and perhaps infinite trees $ \bot $ be used to define a partial ordering on the trees in the same way? If so, what if anything distinguishes the partial ordering defined via the atomic tree $ \bot _0 $?

Approximate Solutions for the Bilinear Form Computational Problem

Dario Bini, Grazia Lotti, and Francesco Romani

SIAM J. Comput. 9, pp. 692-697 (6 pages) | Cited 7 times

Online Publication Date: July 13, 2006

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A set of bilinear forms can be evaluated with a multiplicative complexity lower than the rank of the associated tensor by allowing an arbitrarily small error. A topological interpretation of this fact is presented together with the error analysis. A complexity measure is introduced which takes into account the numerical stability of algorithms. Relations are established between the complexities of exact and approximate algorithms.

The Application of Multivariate Polynomials to Inference Rules and Partial Tests for Unsatisfiability

David A. Plaisted

SIAM J. Comput. 9, pp. 698-705 (8 pages)

Online Publication Date: July 13, 2006

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There are some relationships between unsatisfiability of sets of clauses, and properties of polynomials in several variables. These polynomials can be used to obtain a polynomial time solution to a certain problem involving sets of clauses. Using these polynomials, one can establish a correspondence between unsatisfiable sets of clauses and a convex region of Euclidean space. Also, some inference rules based on these polynomials may provide shorter proofs of inconsistency than are possible using other known inference rules.

Constant Time Generation of Rooted Trees

Terry Beyer and Sandra Mitchell Hedetniemi

SIAM J. Comput. 9, pp. 706-712 (7 pages) | Cited 10 times

Online Publication Date: July 13, 2006

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This paper generalizes a result of Ruskey [SIAM J. Comput., 7(1978), pp. 424–439] for generating $k$-ary trees lexicographically to generating all rooted trees with $n$ vertices. An algorithm is presented which generates canonical representations of these trees in a well-defined order. As in other works, the average number of steps per tree is constant.

On the Complexity of Bilinear Forms with Commutativity

Joseph JáJá

SIAM J. Comput. 9, pp. 713-728 (16 pages) | Cited 4 times

Online Publication Date: July 13, 2006

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We consider the general problem of computing sets of bilinear forms in commuting indeter-minates. We develop lower bound techniques which seem to be more powerful than those already known in the literature. We show that duality theory as it is known for bilinear forms with noncommuting indeter-minates does not hold in the commutative case; we prove that the multiplication of $2 \times n$ by $n \times 2$ matrices requires at least $\lceil 27n/8 \rceil $ multiplications while it is possible to multiply $2 \times 2$ by $2 \times n$ matrices using only $3n + 2$ multiplications. Moreover we settle the question of whether commutativity can reduce the number of multiplications by a factor of $\frac{1}{2}$, by showing that this can never happen. We also show that, over algebraically closed fields, the complexity of computing a pair of bilinear forms is the same whether or not commutativity is allowed.

Equality Sets and Complexity Classes

Ronald V. Book and Franz–Josef Brandenburg

SIAM J. Comput. 9, pp. 729-743 (15 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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If $h_1 $, $h_2 $ are two homomorphisms, then the equality set$\operatorname{Eq}(h_1 ,h_2 )$ of $h_1 $, $h_2 $ is $\operatorname{Eq} (h_1 ,h_2 ) = \{ w | h_1 (w) = h_2 (w)\} $. In this paper it is shown how to characterize complexity classes of formal languages in terms of equality sets of pairs of homomorphisms with bounded balance. In addition the complete twin shuffle language is investigated, and it is shown that for alphabets with at least two letters, this language cannot be represented as the equality set of a pair of homomorphisms unless both homomorphisms are erasing and have linear bounded balance.

Finding Connected Components and Connected Ones on a Mesh-Connected Parallel Computer

David Nassimi and Sartaj Sahni

SIAM J. Comput. 9, pp. 744-757 (14 pages) | Cited 30 times

Online Publication Date: July 13, 2006

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Let $G = (V,E)$ be an undirected graph in which no vertex has degree more than $d$. Let $|V| = n^q = 2^q $ . In this paper we present an $O(q^3 (q + d)n\log n)$ algorithm to find the connected components of $G$ on a $q$-dimensional $n \times n \times \cdots \times n$ mesh-connected parallel computer. When $d = 2$, the connected components can be found in $O(q^4 n)$ time. We also show that the connected ones problem can be solved in $O(q^6 n)$ time.

Factorization of Symmetric Matrices and Trace-Orthogonal Bases in Finite Fields

Gadiel Seroussi and Abraham Lempel

SIAM J. Comput. 9, pp. 758-767 (10 pages) | Cited 12 times

Online Publication Date: July 13, 2006

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It is shown that every symmetric matrix $A$, with entries from a finite field $F$, can be factored over $F$ into $A = BB'$, where the number of columns of $B$ is bounded from below by either the rank $\rho (A)$ of $A$, or by $1 + \rho (A)$, depending on $A$ and on the characteristic of $F$ This result is applied to show that every finite extension $\Phi $ of a finite field $F$ has a trace-orthogonal basis over $F$. Necessary and sufficient conditions for the existence of a trace-orthonormal basis are also given. All proofs are constructive, and can be utilized to formulate procedures for minimal factorization and basis construction.

A Model and Proof Technique for Message-Based Systems

Jerome A. Feldman and Anil Nigam

SIAM J. Comput. 9, pp. 768-784 (17 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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Distributed computing with widely separated machines is a subject of growing theoretical and practical interest. This paper attempts to present a framework for the analysis of message-based distributed computations. This is done in the context of the classical critical section problem and the high-level language, PLITS. The proof techniques described are based on the use of finite-state machines which characterize the external behavior of each module in the distributed computation.

Efficient On-Line Construction and Correction of Position Trees

Mila E. Majster and Angelika Reiser

SIAM J. Comput. 9, pp. 785-807 (23 pages)

Online Publication Date: July 13, 2006

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This paper presents an on-line algorithm for the construction of position trees, i.e., an algorithm which constructs the position tree for a given string while reading the string from left to right. In addition, an on-line correction algorithm is presented which—upon a change in the string—can be used to construct the new position tree. Moreover, special attention is paid to computers with small memory. compactification of the trees and transport costs between main and secondary storage are discussed.

Performance Bounds for Level-Oriented Two-Dimensional Packing Algorithms

E. G. Coffman, Jr., M. R. Garey, D. S. Johnson, and R. E. Tarjan

SIAM J. Comput. 9, pp. 808-826 (19 pages) | Cited 64 times

Online Publication Date: July 13, 2006

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We analyze several “level-oriented” algorithms for packing rectangles into a unit-width, infinite-height bin so as to minimize the total height of the packing. For the three algorithms we discuss, we show that the ratio of the height obtained by the algorithm to the optimal height is asymptotically bounded, respectively, by 2, 1.7, and 1.5. The latter two improve substantially over the performance bounds for previously proposed algorithms. In addition, we give more refined bounds for special cases in which the widths of the given rectangles are restricted and in which only squares are to be packed.

A Polynomial Time Algorithm for Solving Systems of Linear Inequalities with Two Variables Per Inequality

Bengt Aspvall and Yossi Shiloach

SIAM J. Comput. 9, pp. 827-845 (19 pages) | Cited 3 times

Online Publication Date: July 13, 2006

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We present a constructive algorithm for solving systems of linear inequalities (LI) with at most two variables per inequality. The time complexity of the algorithm is $O(mn^3 I)$ on a random access machine, where $m$ is the number of inequalities, $n$ the number of variables, and $I$ the size of the binary encoding of the input. The LI problem is of importance in complexity theory because it is polynomial time (Turing) equivalent to linear programming. The subclass of LI treated in this paper is of practical interest in mechanical verification systems.

Orthogonal Packings in Two Dimensions

Brenda S. Baker, E. G. Coffman, Jr., and Ronald L. Rivest

SIAM J. Comput. 9, pp. 846-855 (10 pages) | Cited 98 times

Online Publication Date: July 13, 2006

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We consider problems of packing an arbitrary collection of rectangular pieces into an open-ended, rectangular bin so as to minimize the height achieved by any piece. This problem has numerous applications in operations research and studies of computer operation. We devise efficient approximation algorithms, study their limitations, and derive worst-case bounds on the performance of the packings they produce.
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