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1975

Volume 4, Issue 4, pp. 397-545


Complexity Results for Multiprocessor Scheduling under Resource Constraints

M. R. Garey and D. S. Johnson

SIAM J. Comput. 4, pp. 397-411 (15 pages) | Cited 53 times

Online Publication Date: July 13, 2006

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We examine the computational complexity of scheduling problems associated with a certain abstract model of a multiprocessing system. The essential elements of the model are a finite number of identical processors, a finite set of tasks to be executed, a partial order constraining the sequence in which tasks may be executed, a finite set of limited resources, and, for each task, the time required for its execution and the amount of each resource which it requires. We focus on the complexity of algorithms for determining a schedule which satisfies the partial order and resource usage constraints and which completes all required processing before a given fixed deadline. For certain special cases, it is possible to give such a scheduling algorithm which runs in low order polynomial time. However, the main results of this paper imply that almost all cases of this scheduling problem, even with only one resource, are NP-complete and hence are as difficult as the notorious traveling salesman problem.

Proving Theorems with the Modification Method

D. Brand

SIAM J. Comput. 4, pp. 412-430 (19 pages) | Cited 12 times

Online Publication Date: July 13, 2006

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A method for proving theorems in first order predicate calculus theories with equality is described and proven complete. Completeness of this “Modification Method” implies completeness of Paramodulation without the functionally reflexive axioms, thus proving a conjecture of Wos and Robinson (1969). Moreover, completeness holds with some other restrictions, such as limiting paramodulation into variables. Experimental results using the Modification Method are included.

Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question

Theodore Baker, John Gill, and Robert Solovay

SIAM J. Comput. 4, pp. 431-442 (12 pages) | Cited 49 times

Online Publication Date: July 13, 2006

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We investigate relativized versions of the open question of whether every language accepted nondeterministically in polynomial time can be recognized deterministically in polynomial time. For any set $X$, let $\mathcal{P}^X (\text{resp. }\mathcal{NP}^X )$ be the class of languages accepted in polynomial time by deterministic (resp. nondeterministic) query machines with oracle $X$. We construct a recursive set $A$ such that $\mathcal{P}^A = \mathcal{NP}^A $. On the other hand, we construct a recursive set $B$ such that $\mathcal{P}^B \ne \mathcal{NP}^B $. Oracles $X$ are constructed to realize all consistent set inclusion relations between the relativized classes $\mathcal{P}^X $, $\mathcal{NP}^X $, and co $\mathcal{NP}^X $, the family of complements of languages in $\mathcal{NP}^X $. Several related open problems are described.

Preserving Proximity in Arrays

Arnold L. Rosenberg

SIAM J. Comput. 4, pp. 443-460 (18 pages) | Cited 12 times

Online Publication Date: July 13, 2006

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Efficiency of storage management in algorithms which use arrays is often enhanced if the arrays are stored in a proximity-preserving manner; that is, array positions which are close to one another in the array are stored close to one another. This paper is devoted to studying certain qualitative and quantitative questions concerning preservation of proximity by array storage schemes (or,realizations). It is shown that fully extendible array realizations cannot preserve proximity in any global sense, not even proximity along a single direction (say, along rows). They can, however, preserve proximity in certain local senses; and realizations which are optimal in various senses of local preservation are exhibited. Partially extendible array storage schemes can preserve proximity in a global way; bounds on their “diameters of preservation” are derived, and optimal schemes are exhibited.

Response Time of a Fixed-Head Disk to Transfers of Variable Length

Erol Gelenbe, Jacques Lenfant, and Dominique Potier

SIAM J. Comput. 4, pp. 461-473 (13 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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Due to the practical complexity of addressing variable length records placed in arbitrary locations of a fixed-head disk (or drum), and because of difficulty of managing secondary memory space in such cases, variable length records are often stored with their first address at a fixed location of the magnetic support. We present a queuing model of such a scheme, assuming a Poisson arrival stream and arbitrary distributed record lengths. The stationary probability distribution of the number of transfer requests in queue and the expected response time are obtained. Numerical examples illustrating the results are presented.

The Enumeration of Generalized Double Stochastic Nonnegative Integer Square Matrices

D. M. Jackson and G. H. J. Van Rees

SIAM J. Comput. 4, pp. 474-477 (4 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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The problem of enumerating generalized double stochastic integer square matrices is considered. The superposition theorem is used in conjunction with Schur functions to obtain the counting series for the $5 \times 5$ and $6 \times 6$ cases.

On Scheduling Chains of Jobs on One Processor with Limited Preemption

John Bruno and Micha Hofri

SIAM J. Comput. 4, pp. 478-490 (13 pages) | Cited 8 times

Online Publication Date: July 13, 2006

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A scheduling rule is given for determining the processing order of tasks which have the precedence structure of chains. It is assumed that the service times follow known distributions, that they are all independent, that costs are accrued by tasks at a constant rate until their service requirements are satisfied, that all the tasks are available at time 0 and that the service is interruptible at task-specific sets of points. The rule consists of computing for each chain an “optimal assignment” for its tasks and a rank function which depends on this assignment. Choosing at each point in time the chain with the smallest rank produces an optimal schedule. It is proved that the “optimal assignments” have the desirable property that as long as a task does not exceed its allotted service time, no preemption should take place.

A Convergence Theorem for Hierarchies of Model Neurones

M. D. Alder

SIAM J. Comput. 4, pp. 491-506 (16 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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The threshold logic unit (T.L.U.) has been proposed as a model for a single neurone; other substantially cognate terms are “perceptron” and “adaline”. Networks of these elements have been advanced as tentative models of some aspects of brain functioning. In particular, hierarchical nets appear to exhibit a sufficient flexibility to make them interesting both as plausible models of learning in the central nervous system and also as general objects of study in connection with pattern recognition and artificial intelligence.
In this paper, we discuss the well-known “perceptron convergence theorem” in a fairly general setting, and consider variations appropriate to nets of such units. A certain familiarity with the relevant chapters of Nilsson’s Learning Machines [1] and also with current mathematical formalism is presupposed.

Network Flow and Testing Graph Connectivity

Shimon Even and R. Endre Tarjan

SIAM J. Comput. 4, pp. 507-518 (12 pages) | Cited 69 times

Online Publication Date: July 13, 2006

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An algorithm of Dinic for finding the maximum flow in a network is described. It is then shown that if the vertex capacities are all equal to one, the algorithm requires at most $O(|V|^{1/2} \cdot |E|)$ time, and if the edge capacities are all equal to one, the algorithm requires at most $O(|V|^{2/3} \cdot |E|)$ time. Also, these bounds are tight for Dinic’s algorithm.
These results are used to test the vertex connectivity of a graph in $O(|V|^{1/2} \cdot |E|^2 )$ time and the edge connectivity in $O(|V|^{5/3} \cdot |E|)$ time.

A Simple Algorithm for Global Data Flow Analysis Problems

Matthew S. Hecht and Jeffrey D. Ullman

SIAM J. Comput. 4, pp. 519-532 (14 pages) | Cited 20 times

Online Publication Date: July 13, 2006

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A simple, iterative bit propagation algorithm for solving global data flow analysis problems such as “available expressions” and “live variables” is presented and shown to be quite comparable in speed to the corresponding interval analysis algorithm. This comparison is facilitated by a result relating two parameters of a reducible flow graph (rfg). Namely, if $G$ is an rfg, $d$ is the largest number of back edges found in any cycle-free path in $G$, and $k$ is the length of the interval derived sequence of $G$, then $k \geqq d$. (Intuitively, $k$ is the maximum nesting depth of loops in a computer program, while $d$ is a measure of the maximum loop-interconnectedness.) The node ordering employed by the simple algorithm is the reverse of the order in which a node is last visited while growing any depth-first spanning tree of the flow graph. In addition, a dominator algorithm for an rfg is presented which takes $O$(edges) bit vector steps.

Evaluating Polynomials at Fixed Sets of Points

A. V. Aho, K. Steiglitz, and J. D. Ullman

SIAM J. Comput. 4, pp. 533-539 (7 pages) | Cited 9 times

Online Publication Date: July 13, 2006

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We investigate the evaluation of an $(n - 1)$st degree polynomial at a sequence of $n$ points. It is shown that such an evaluation reduces directly to a simple convolution if and only if the sequence of points is of the form $b, ba,ba^2 , \cdots ,ba^{n - 1} $ for complex numbers $a$ and $b$ (the so-called “chirp transform”). By more complex reductions we develop an $O(n\log n)$ evaluation algorithm for sequences of points of the form \[ b + c + d,\quad ba^2 + ca + d,\quad ba^4 + ca^2 + d, \cdots \] for complex numbers $a$, $b$, $c$ and $d$. Finally we show that the evaluation of an $(n - 1)$st-degree polynomial and all its derivatives at a single point requires at most $O(n\log n)$ steps.

An Elementary Solution of the Queuing System G/G/1

Alan G. Konheim

SIAM J. Comput. 4, pp. 540-545 (6 pages) | Cited 5 times

Online Publication Date: July 13, 2006

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In this note we give an elementary method for calculating the stationary distribution of waiting time in a G/G/1 queue.
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