Showing papers in "Journal of the ACM in 1978"

`` Strong '' NP-Completeness Results: Motivation, Examples, and Implications

Abstract: The NP-completeness of a computational problem ~s frequently taken to unply its \"mtractabthty\" However, there are certain NP-complete problems mvolvmg numbers, such as PARTITION and KNAPSACK, which are considered by many practitioners to be tractable The reason for this IS that, although no algontluns for solvmg them in tune bounded by a polynomial m the mput length are known, algorithms are known which solve them m tune bounded by a polynomial m the input length and the magmtude of the largest number an the given problem mstance. For other NP-complete problems mvolvmg numbers it can be shown that no such \"pseudopolynomml tune\" algonthra can exist unless P = NP. In this paper we provide a standard framework for stating and proving \"strong\" NP-completeness results of this sort, survey some of the strong NP-completeness results proved to date, and indicate some unphcauons of these results for both opumlzatlon and approximaUon algontluns

The Complexity of Some Problems on Subsequences and Supersequences

Abstract: The complexity of finding the Longest Common Subsequence (LCS) and the Shortest Common Supersequence (SCS) of an arbRrary number of sequences IS considered We show that the yes/no version of the LCS problem is NP-complete for sequences over an alphabet of size 2, and that the yes/no SCS problem is NPcomplete for sequences over an alphabet of size 5

Asynchronous Iterative Methods for Multiprocessors

Abstract: : A class of asynchronous iterative methods is presented for solving a system of equations. Existing iterative methods are identified in terms of asynchronous iterations, and new schemes are introduced corresponding to a parallel implementation on a multiprocessor system with no synchronization between cooperating processes. A sufficient condition is given to guarantee the convergence of any asynchronous iterations, and results are extended to include iterative methods with memory. Asynchronous iterative methods are then evaluated from a computational point of view, and bounds are derived for the efficiency. The bounds are compared with actual measurements obtained by running various asynchronous iterations on a multiprocessor, and the experimental results show clearly the advantage of purely asynchronous iterative methods. (Author)

Reversal-Bounded Multicounter Machines and Their Decision Problems

Abstract: Decidable and undecldable properties of various classes of two-way multlcounter machines (deterministic, nondetermmlstlc, multttape, pushdown store augmented) with reversal-bounded input and/or counters are investigated In particular It IS shown that the emptiness, infiniteness, dlsjointness, containment, universe, and equivalence problems are decidable for the class of deterministic two-way multlcounter machines whose input and counters are reversal-bounded

On the Average Number of Maxima in a Set of Vectors and Applications

Abstract: A maximal vector of a set ~s one which is not less than any other vector m all components We derive a recurrence relation for computing the average number of maxunal vectors m a set of n vectors m d-space under the assumpUon that all (nl) a relative ordermgs are equally probable. Solving the recurrence shows that the average number of maxmaa is O((ln n) a-~) for fixed d We use this result to construct an algorithm for finding all the maxima that have expected running tmae hnear m n (for sets of vectors drawn under our assumptions) We then use the result to find an upper bound on the expected number of convex hull points m a random point set

Fast Algorithms for Manipulating Formal Power Series

Abstract: The classical algorithms require order n ~ operations to compute the first n terms in the reversion of a power series or the composition of two series, and order nelog n operations if the fast Founer transform is used for power series multiplication In this paper we show that the composition and reversion problems are equivalent (up to constant factors), and we give algorithms which require only order (n log n) ~/2 operations In many cases of practical importance only order n log n operations are required, these include certain special functions of power series and power series solution of certain differential equations Applications to root-finding methods which use inverse mterpolauon and to queuemg theory are described, some results on multivariate power series are stated, and several open questions are mentioned

On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming

Abstract: It IS shown that certain problems of optimal preemptive scheduling of unrelated parallel processors can be formulated and solved as hnear programming problems As a by-product of the linear programming formulaUons of these problems, upper bounds are obtained on the number of preempuons required for optimal schedules In particular it is shown that no more than O(m 2) preemptions are necessary, m order to schedule n jobs on m unrelated processors so as to minimize makespan

On Stable Parallel Linear System Solvers

Abstract: In this paper three stable parallel algorithms for solving dense and tndlagonai systems of lmear equations are discussed The algorithms are based on Givens' reduction of a matrix to the upper triangular form The algorithm for the dense case requires O(n) t ime steps compared to O(n log n) steps for Gausslan ehmmatlon with pivoting (in the absence of certain features of machine logic and hardware) For the trldlagonal case, one of the algorithms presented here is superior to the best previous algorithm in that with a modest increase in time It does not fall if any of the leading pnnclpal submatrlces is singular, the probablhty of overor underflow is minimized, and the error bound does not grow exponentially Furthermore, it is most statable when only a hmtted number of processors ts available.

On the Tape Complexity of Deterministic Context-Free Languages

Abstract: Let DSPACE(L(n)) denote the family of languages recognized by deterministic L(n)-tape bounded Turmg machines The pnnopal result described m this paper is the equivalence of the following statements (l) The determtmsttc context-free language L~ 2) (described m the paper) is m DSPACE(Iog(n)) (2) The simple LL(I) languages are m DSPACE(tog(n)) (3) The simple precedence languages are in DSPACE(Iog(n)). (4) DSPACE(Iog(n)) is identical to the famdy of languages recogmzed by deterministic two-way multlhead pushdown automata m polynomml tmae These results are obtained by constructing a determlmstlc context-free language L~ 2~ which is log(n)-complete for the family of determlmstlc context-free languages In other words, a tape hardest deterministic context-free language is described The best upper bound known on the tape complexity of (deterministic) context-free languages is (log(n)) 2

Preemptive Scheduling of Uniform Processor Systems

Abstract: AaSTRACT An O(n) t~me algorithm is presented to obtain an opt,mal fimsh time preemptive schedule for n independent tasks on m uniform processors This algorithm assumes that the tasks are lnmally ordered by task length and that the umform processors are ordered by processor speed

Separating Nondeterministic Time Complexity Classes

Abstract: AaSTancr. A recurslve padding technique is used to obtain conditions sufficient for separation of nondetermlmsttc multltape Turlng machine time complexity classes If T2 is a running time and Tl(n + 1) grows more slowly than T~(n), then there is a language which can be accepted nondetermmlstlcally within time bound T~ but which cannot be accepted nondetermlnlStlcally within time bound T1. If even T~(n + f(n)) grows more slowly than Tz(n), where f is the very slowly growing "rounded reverse" of some real-time countable function, then there is such a language over a single-letter alphabet. The strongest known dmgonalization results for both deterministic and nondetermlmstlc time complexity classes are reviewed and orgamzed for comparison with the results of the new padding technique

Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graph

Abstract: ^SSTnXcr. Gwen a graph G = (V, E) and four verttces s~, tx, s~, and t2, the problem of finding two disjoint paths, P~ from s~ to tt and P2 from s2 to t2, is considered This problem may arise as a transportation network problem and m printed clrcmts routing The relations between several vemons of the problem are discussed Efficient algorithms are gwen for the following special cases-acyche directed graphs and 3-connected planar and chordal graphs.

Properties of Conflict-Free and Persistent Petri Nets

Abstract: Petri nets have been extensively studied because of their suitability as models for asynchronous computing Despite this effort, the mathematical properties of Petrl nets are not very well understood In this paper we investigate two unportant special types of Petn nets, the conflict-free nets and the persistent nets, the former being a proper subset of the latter. Our results completely characterize the sets of reachable markings attainable by such nets Reachabihty sets of persistent nets are shown to be semllmear A stronger result is obtamed for conflict-free nets which results m an exponential time algorithm for deciding boundedness of such nets The best known upper bound for deciding boundedness of arbitrary nets is exponential space We conclude with a proof that all reachablhty sets of Petri nets may be realized with a "small amount" of nonperslstence

Some Complexity Results for Matrix Computations on Parallel Processors

Abstract: In this pape r it is shown how da ta movemen t , r a the r than ar i thmet ic opera t ions , can be the hml tmg fac tor in the pe r fo rmance o f paral lel compu te r s on matr ix computa t ions In par t icular it is p roved that for machines with two-dimens iona l rec tangula r grid connect ivi ty (such as I L L I A C IV), mui t tphcat lon and inversion o f NxN matrices inherent ly requi re O(N) steps, even if the process ing e lements are no t cons t ra ined to execute identical instruct ions.

k + 1 Heads Are Better than k

Abstract: : There are languages which can be recognized by a deterministic (k+1)-headed one-way finite automaton but which cannot be recognized by a k-headed one-way (deterministic or non-deterministic) finite automaton. Furthermore, there is a language accepted by a k-headed nondeterministic finite automaton which is accepted by no k-headed deterministic finite automaton. (Author)

Renaming a Set of Clauses as a Horn Set

Abstract: An efflcment algorithm is given for determining whether a set of clauses can be transformed into a Horn set by a predicate renaming

Self-Organizing Binary Search Trees

Abstract: We consider heuristics which attempt to maintain a binary search tree in a near optimal form, assuming that elements are requested with fixed, but unknown, independent probabilities. A "move to root" heuristic is shown to yield an expected search time within a constant factor of that of an optimal static binary search tree. On the other hand, a closely related "simple exchange" technique is shown not to have this property. The rate of convergence of the "move to root" heuristic is discussed. We also consider the more general case in which elements not in the tree may have non-zero probability of being requested.

Some Matching Problems for Bipartite Graphs

Abstract: Certam apphcatlons require finding a perfect matching in a bipartite graph that satisfies some additional properties For one such type of restncUon the problem is proven to be NP-complete. If for a single subset of edges no more than r edges may be included m the matching then an O(ne) algorithm is given An efficient algorithm for finding all perfect matchmgs is presented It requires O(e) time per matching and a total of O(e) space This algorithm may be used to calculate the permanent of a matrix Finally, the algorithm ts generahzed to find all maximum matchmgs

All Algebraic Functions Can Be Computed Fast

Abstract: : The expansions of algebraic functions can be computed 'fast' using the Newton Polygon Process and any 'normal' iteration. Let M(j) be the number of operations sufficient to multiply two jth degree polynomials. It is shown that the first N terms of an expansion of any algebraic function defined by an nth degree polynomial can be computed in O(n(M(N)) operations, while the classical method needs O(N sup n) operations. Among the numerous applications of algebraic functions are symbolic mathematics and combinatorial analysis. Reversion, reciprocation, and nth root of a polynomial are all special cases of algebraic functions.

Foundations of Probabilistic and Utility-Theoretic Indexing

Abstract: One of the most perplexing problems of reformation retrieval has been the estabhshment of rational criteria for deciding what index terms or descriptors to assign to a unit of stored information for purposes of later retrieval Both probablhstJc and utlhty-theoretlc criteria have m the past been proposed for thts purpose. The present paper derives explicit decision rules of both kinds from a common conceptual and mathematical foundation The result IS a unified theory of indexing

Satisfiability Is Quasilinear Complete in NQL

Abstract: Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL

Two-Commodity Flow

Abstract: An algorithm is given to fmd maxtmum two-commodity flow m an undirected graph The algorithm is an improvement on Hu's two-commodity flow algorithm using the methods of Dmlc's single-commodity flow algorithm Karzanov's Improvement of Dmlc's algorithm can be applied to yield an O(I V ) 3) algorithm. It is shown that finding maximum two-commodity flow m a dwected graph is much more dffficuh, in fact it Is as difficult as hnear programming FmaUy, the problem of finding feasible flow m an undirected graph with lower and upper bounds on the edges is shown to be NP-complete even for a single commodity

Generation of Binary Trees from Ballot Sequences

Abstract: An efficient algorithm for generating and indexing all shapes of n-noded binary trees is described The algorithm is based on a correspondence between binary trees and the class of stack-sortable permutations. together with a representation of such permutatmns as ballot sequences Justification for the related procedures is given, and their efficiency estabhshed by comparison to other approaches

Some Properties of Predicate Transformers

Abstract: This paper defines some "weakest precondltmn'" predicate transformers, Investigates their "healthi- ness" properties, and apphes them to Dljkstra's language of guarded commands It shows that Dljkstra's w~ function is not the weakest healthy one, but it ~s clearly the best one for practical programming, because it proves the absence of bhnd alleys from a nondetermmlsac program A previous paper (10) has suggested that a computer programming language should be given several separate formal definitions, which should be proved consistent with each other. The definitions should be complementary, m the sense that one of them should satisfy the need of the programmer to prove the correctness of his program, and another should assist in the construction and proof of a correct implementation. The suggestion was illustrated by a simple deterministic language, for which the axioms of conditional correctness were proved consistent with an abstract model of ~ts implementation. The results of that previous paper are here extended and improved m the following ways:

A Global Bisection Algorithm for Computing the Zeros of Polynomials in the Complex Plane

Abstract: A numerical method for solving polynomial equations ms presented Its basic module is a method. using Sturm sequences, for counting the zeros which lie m a given rectangle m the complex plane The method ms deflation-free, handles multiple complex zeros, and contains budt-m safeguards against buildup of roundoff error

Pivot Strategies for Primal-Simplex Network Codes

Abstract: AaSTRACT. Techniques are presented for improving the efficiency ofpnmal-stmplex network codes An adaptive candidate hst, enumerating the pivot variables, is provided. Proper use of this list greatly reduces computation tune (espectaUy m large-scale network problems) and experiential data Js included It is again shown that the number of iterations, i e pivots, ~s a poor surrogate for measunng the performance of primal-simplex network algorithms.

A New Incompleteness Result for Hoare's System

Abstract: A structure A is presented for which Hoare's formal system for partial correctness is incomplete, even if the entire first-order theory of A is included among the axioms. It follows that the language of first-order logic is insufficient to express all loop invariants. The implications of this result for program-proving are discussed.

The Bottleneck Traveling Salesman Problem: Algorithms and Probabilistic Analysis

Abstract: The bottleneck traveling salesman problem seeks to minimize the maximum length arc over all Hamlltonlan cycles in a graph A probabdlStlC analysis is presented for random problems It is shown that the optimal ob.lectwe value can be closely approximated by a beta function Finally, effective solution techmques are developed and computational experience is reported

Exact Aggregation in Exponential Queueing Networks

Abstract: Aggregation as often used for the mvestlgat~on of large systems This techmqu¢ is pamculafly attracuve for nearly completely decomposable systems It as essentmlly an approxunate method, but at remains attractave since the error ,s small and bounds for at can be obtamed We are concerned here w,th aggregauon m exponential queuemg network analysas, and more precisely wath the condmons under which aggregation will yaeld exact results

A Linear-Time On-Line Recognition Algorithm for ``Palstar''

Abstract: Let P1 = {w ~ X*:w = w R, [w I > 1} be the set of all nontnvial pahndromes over X A hneartime on-hne recogmtmn algorithm is presented for P~ ("palstar") on a random-access machine with addmon and umform cost criterion Also presented are a hnear-tlme on-line recognmon algorithm for P~ on a muitltape Turmg machine and a recognition algorithm for Pt 2 on a two-way deterministic pushdown automaton. The correctness of these algorithms is based on new "cancellation iemmas" for the languages P~ and P~