Showing papers in "Journal of the ACM in 1980"

Reaching Agreement in the Presence of Faults

Abstract: The problem addressed here concerns a set of isolated processors, some unknown subset of which may be faulty, that communicate only by means of two-party messages. Each nonfaulty processor has a private value of information that must be communicated to each other nonfaulty processor. Nonfaulty processors always communicate honestly, whereas faulty processors may lie. The problem is to devise an algorithm in which processors communicate their own values and relay values received from others that allows each nonfaulty processor to infer a value for each other processor. The value inferred for a nonfaulty processor must be that processor's private value, and the value inferred for a faulty one must be consistent with the corresponding value inferred by each other nonfaulty processor.It is shown that the problem is solvable for, and only for, n ≥ 3m + 1, where m is the number of faulty processors and n is the total number. It is also shown that if faulty processors can refuse to pass on information but cannot falsely relay information, the problem is solvable for arbitrary n ≥ m ≥ 0. This weaker assumption can be approximated in practice using cryptographic methods.

Fast Probabilistic Algorithms for Verification of Polynomial Identities

Abstract: The s tar thng success o f the Rabm-S t ra s sen -So lovay p n m a h t y algori thm, together wi th the intr iguing foundat tonal posstbthty that axtoms of randomness may constttute a useful fundamenta l source o f m a t h e m a u c a l truth independent of the standard axmmaUc structure of mathemaUcs, suggests a wgorous search for probabdisuc algonthms In dlustratmn of this observaUon, vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials. Ancdlary fast algorithms for calculating resultants and Sturm sequences are given. Probabilistlc calculatton in real anthmetlc, prewously considered by Davis, is justified ngorously, but only in a special case. Theorems of elementary geometry can be proved much more efficiently by the techmques presented than by any known arttficml-mtelhgence approach

Mean-Value Analysis of Closed Multichain Queuing Networks

Abstract: It is shown that mean queue sizes, mean waiting times, and throughputs in closed multiple-chain queuing networks which have product-form solution can be computed recursively without computing product terms and normalization constants. The resulting computational procedures have improved properties (avoidance of numerical problems and, in some cases, fewer operations) compared to previous algorithms. Furthermore, the new algorithms have a physically meaningful interpretation which provides the basis for heuristic extensions that allow the approximate solution of networks with a very large number of closed chains, and which is shown to be asymptotically valid for large chain populations.

Parallel Prefix Computation

Abstract: The prefix problem is to compute all the products x t o x2 . . . . o xk for i ~ k .~ n, where o is an associative operation A recurstve construction IS used to obtain a product circuit for solving the prefix problem which has depth exactly [log:n] and size bounded by 4n An application yields fast, small Boolean ctrcmts to simulate fimte-state transducers. By simulating a sequentml adder, a Boolean clrcmt which has depth 2[Iog2n] + 2 and size bounded by 14n Is obtained for n-bit binary addmon The size can be decreased significantly by permitting the depth to increase by an addmve constant

Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems: Abstract Properties and Applications to Term Rewriting Systems

Abstract: This paper gives new results, and presents old ones m a umfied formahsm, concerning ChurchRosser theorems for rewrmng systems Abstract confluence propentes, depending solely on axioms for a binary relatton called reduction, are first presented Results of Newman and others are presented m a unified formahsm The systemattc use of a powerful mductmn pnnciple permRs the generahzauon of results of Sethl on reduction modulo eqmvalence. Simphficatton systems operating on terms of a first-order logic are then considered. Results by Rosen and Knuth and Bendix are extended to give several new crtteria for confluence of these systems It ts then shown how these criteria yield new methods for the mechanizaUon of equattonal theories

Fast Decision Procedures Based on Congruence Closure

Abstract: The notion of the congruence closure of a relation on a graph is defined and several algorithms for computing it are surveyed. A simple proof is given that the congruence closure algorithm provides a decision procedure for the quantifier-free theory of equality. A decision procedure is then given for the quantifier-free theory of LISP list structure based on the congruence closure algorithm. Both decision procedures determine the satisfiability of a conjunction of literals of length n in average time O(n log n) using the fastest known congruence closure algorithm. It is also shown that if the axiomatization of the theory of list structure is changed slightly, the problem of determining the satisfiability of a conjunction of literals becomes NP-complete. The decision procedures have been implemented in the authors' simplifier for the Stanford Pascal Verifier.

Equivalences Among Relational Expressions with the Union and Difference Operators

Abstract: Queries in relational databases can be formulated in terms of relational expressions using the relational operations select, project, join, union, and difference The equivalence problem for these queries is studied with query optimization m mind It ts shown that testmg eqmvalence of relational expressions with the operators select, project, join, and union is complete m the class FIt of the polynomial-time hierarchy A nonprocedural representation for queries formulated by these expressions is proposed This method of query representation can be viewed as a generahzatlon of tableaux or conjunctive queries (which are used to represent expressions with only select, project, and join) Furthermore, this method is extended to queries formulated by relatmnal expressions that also contain the difference operator, provided that the project operator is not applied to subexpresstons with the difference operator A procedure for testing eqmvalence of these queries is given It ts shown that testmg containment of tableaux is a necessary step in testing equivalence of queries with union and difference Three important cases m which containment of tableaux can be tested m polynomial time are described, although the containment problem is shown to be NP-complete even for tableaux that correspond to expressions with only one project and several join operators

Variations on the Common Subexpression Problem

Abstract: Let G be a directed graph such that for each vertex v in G, the successors of v are ordered Let C be any equivalence relation on the vertices of G. The congruence closure C* of C is the finest equivalence relation containing C and such that any two vertices having corresponding successors equivalent under C* are themselves equivalent under C* Efficient algorithms are described for computing congruence closures in the general case and in the following two special cases. 0) G under C* is acyclic, and (it) G is acychc and C identifies a single pair of vertices. The use of these algorithms to test expression eqmvalence (a problem central to program verification) and to test losslessness of joins in relational databases is described

New Algorithms for Bin Packing

Abstract: In the bin-packing problem a list L of n numbers are to be packed into unit-capacity bins. For any algorithm S, let r(S) be the maximum ratio S(L)/L* for large L*, where S(L) denotes the number of bins used by S and L* denotes the minimum number needed. An on-line O(n log n)-time algorithm RFF with r(RFF) = 5/3 and an off-line polynomial-time algorithm RFFD with r(RFFD) ≤ 11/9 - e for some fixed e > 0, are given. These are strictly better, respectively, than two prominent algorithms: the First-Fit (FF), which is on-line with r(FF) = 17/10, and the First-Fit-Decreasing (FFD) with r(FFD) = 11/9. Furthermore, it is shown that any on-line algorithm S must have r(S) ≥ 3/2. The question, “How well can an o(n log n)-time algorithm perform?” is also discussed. It is shown that in the generalized d-dimensional bin packing, any o(n log n)-time algorithm S must have r(S) ≥ d.

Flowshop scheduling with limited temporary storage

Abstract: We examine the problem of scheduling 2-machine flowshops in order to minimize makespan, using a limited amount of intermediate storage buffers. Although there are efficient algorithms for the extreme cases of zero and infinite buffer capacities, we show that all the intermediate (finite capacity) cases are NP-complete. We prove exact bounds for the relative improvement of execution times when a given buffer capacity is used. We also analyze an efficient heuristic for solving the 1-buffer problem, showing that it has a 3/2 worst-case performance. Furthermore, we show that the "no-wait" (i.e., zero buffer) flowshop scheduling problem with 4 machines is NP-complete. This partly settles a well-known open question, although the 3-machine case is left open here.

Two-Dimensional Voronoi Diagrams in the Lp-Metric

Abstract: The Voronoi diagram, also known as the Thiessen diagram, for a set of N points in the Cartesian plane in which the L,-metnc is the distance measure, where p is a real number between 1 and 0o inclusive, is defined, and an algorithm for constructing the dmgram m O(NlogN) tune is presented This algonthm uses the divide-and-conquer technique. Many proximity problems revolving a set of points, such as finding the nearest neighbor of a given point, finding the minimum spamung tree, findmg the smallest circle (m the Lp-metric) enclosing the point set, etc., can be solved very efficiently via the diagram The running time of the algorithm presented is also shown to be optimal to within a constant factor. KEY WORDS AND PHRASES Voronoi diagram, L,-metric, computational geometry, computational complexity, analysis of algorithm, divide-and-conquer CR CATEGORIES 4 49, 5 25, 5 32 1. Introduction

A Polynomial Solution to the Undirected Two Paths Problem

Abstract: Given an undirected graph G = (V, E) and vertices s, , t t ; s2,t2, the problem is to determine whether or not G admits two vertex disjoint paths P, and P2 connecting s, with t, and s2 with t2, respectively. This problem is solved by an O ( n . m ) algorithm (n = IVJ, m = JElL An important by-product of the paper is a theorem stating that if G is 4-connected and nonplanar, then such paths P, and P2 exist for any choice of s,, s2, h, and t2 (as conjectured by Watkins).

Sharing a Processor Among Many Job Classes

Abstract: . A single-server processor-sharing system with M job classes is analyzed in the steady state. The scheduling strategy considered divides the total processor capacity in unequal fractions among the different job classes. More precisely, if there are N~jobs of classj in the system, j = 1, 2 ..... M, each class k job receives a fraction gh/(~M.~ giN~) of the processor capacity. Earlier analyses of this system are shown to be incorrect and new expressions for the conditional expected response times Wk(t) of class k jobs with required service time t are obtained (for general required service time distributions). These yield the asymptotic behavior of W~(t) as t ~ oo and rather simple formulas in the exponential case. The unconditional average response times are also obtained. KEY WORDS AND PHRASES: processor sharing, strategy, egalitarian, discriminatory job classes, Laplace transform, Fourier-Stiehjes transform CR CATEGORIES: 4.35.4.6, 5.5, 6.2

Equality and Domain Closure in First-Order Databases

Abstract: A class of first-order databases with no function signs is considered. A closed database DB is one for which the only existing individuals are those explicitly referred to in the formulas of DB. Formally, this is expressed by including in DB a domain closure axiom (x)x = c1 ∨···∨ x = cp, where c1,…,cp are all of the constants occurring in DB. It is shown how to completely capture the effects of this axiom by means of suitable generalizations of the projection and division operators of relational algebra, thereby permitting the underlying theorem prover used for query evaluation to ignore this axiom.A database is E-saturated if all of its constants denote distinct individuals. It is shown that such databases circumvent the usual problems associated with equality, which arise in more general databases.Finally, it is proved for Horn databases and positive queries that only definite answers are obtained, and for databases with infinitely many constants that infinitely long indefinite answers can arise.

GO Is Polynomial-Space Hard

Abstract: It is shown that, given an arbitrary GO position on an n × n board, the problem of determining the winner is Pspace hard. New techniques are exploited to overcome the difficulties arising from the planar nature of board games. In particular, it is proved that GO is Pspace hard by reducing a Pspace-complete set, TQBF, to a game called generalized geography, then to a planar version of that game, and finally to GO.

Consistency in Hierarchical Database Systems

Abstract: The problems of locking and consistency m database systems are examined It is assumed that each transacuon, when executed alone, transforms a consistent state into a consistent state A set of conditions is derived to guarantee that when transactions are processed concurrently, the results are the same as would be obtained by processing the transactmns serially These conditions are used to estabhsh a locking protocol in Merarchmal database systems The locking protocol allows transaeuons to request new locks after releasing a lock. However, a data item may be locked at most once as a result of each transacUon It ~s shown that the protocol ensures consistency and that tt ts deadlock free.

Minimum Covers in Relational Database Model

Abstract: Numerous algonthms concernmg relahonal databases use a cover for a set of funcUonal dependencies as all or part of their input Examples are Been and Bernsteln's synthesis algorithm and the tableau modtfication algorithm of Aho et al The performance of these algorahms may depend on both the number of funcuonal dependencies m the cover and the total size of the cover Starting with a smaller cover wdl make such algorithms run faster After Bernstem, many researchers beheve that the problem of finding a minimum cover is NPcomplete It as shown here that minimum covers can be found m polynomial time, using the nouon of dwect determmatwn The proofdetads the structure ofmmtmum covers, refining the structure Bernstem and Been show for nonredundant covers The kernel algorithm of Lewis, Sekino, and TIng is improved using these results

An optimal routing algorithm for mesh-connected Parallel computers

Abstract: An opUmal algorithm to route data in a mesh-connected parallel computer is presented This algorithm can be used to perform any data routing that can be specified by the permutation and complementing of the bits in a PE address Matrix transpose, bit reversal, vector reversal, and perfect shuffle are examples of data permutations that can be specified in this way The algorithm presented uses the minimum number of unit distance routing steps for every data permutation that can be specified as above K~EV WORDS ANY PrmASES parallel algorithm, mesh-connected computer, ILLIAC IV, permutation, complexity, data routing CR CATEGORIES 5 25, 5 31, 6 22

Database Location in Computer Networks

Abstract: Recent years have wRnessed an increasing number of systems of computers that are distributed geographically and connected by high-capacRy commumcauons channels in designing and managing such a network one must decide where to place copies of the various databases available to the users of the system This decision must trade off the cost of accessing a database, which ~s reduced by additional copies, against the cost of storing and updating the additional copies An optimizing algorithm for a general model of this problem ms described, and successful computational experience with large real examples is reported

Continuation-Based Program Transformation Strategies

Abstract: Program transformations often revolve the generahzation of a function to take additional arguments It is shown that m many eases such an addit ional variable arises as a representation of the continuation or global context m which the function is evaluated. By considering continuations, local transformation strategies can take advantage of global knowledge The general results are followed by two examples' the a-fl tree pruning algorithm and an algorithm for the conversion of a propositional formula to conjunctive normal form

Lower Bounds on Information Transfer in Distributed Computations

Abstract: Lower bounds on the interprocessor communication required for computing a differentiable real-valued function in a distributed network are derived These bounds are independent of the network interconnection configuration, and they impose no assumptions other than differentiability constraints on the computations performed by individual processors As a sample application, lower bounds on information transfer in the distributed computation of some-typical matrix operations are exhibited

An Algorithm to Enumerate All Cutsets of a Graph in Linear Time per Cutset

Abstract: Thts paper deals wRh the problem of enumerating all the cutsets or all the s-t cutsets separatmg two spectfied verttces s and t m an undirected graph A vanety of approaches have been proposed for this problem, among which one based on the partmon e ra set of veruces rote two sets is the most effi¢ienL It is first shown that an algorithm of this type has time complexity O((n + m)(n log2#)#), and two new algorithms with ume complexity O((n + m)O + I)) are then proposed One of these new algorithms has space complexity O(nZ), and the other has space complexity O(n + m), where n and m are the numbers of veraces and edges, respectively, and ta ts the number ofs-t cutsets m a given graph The results of some computatmnal experiments are also described. An mvest~gaUon ~s made of the extent to whtch the new algorithms are better, and how good the performance of the old algorithm is, especmlly when a given graph is \"dense,\" t e , 2m/(n(n 1)) _> 0.4.

Queuing Network Models with State-Dependent Routing

Abstract: A model of a closed queuing network within which customer routing between queues may depend on the state of the network is presented. The routing functions allowed may be rational functions of the queue lengths of various downstream queues which reside within special subnetworks called p-subnetworks. If a network with no state-dependent routing has a product-form joint equilibrium distribution of the queue lengths, then the introduction of these routing functions will preserve the product form of the equilibrium distribution. An example to illustrate the applicability of the model to the problem of analyzing a load balancing strategy is presented. It is also indicated how the parametric analysis of a network with routing functions can be simplified through the analysis of a simpler “equivalent” network.

Fixed Point Languages, Equality Languages, and Representation of Recursively Enumerable Languages

Abstract: Fixed point languages and equality languages of homomorphisms and dgsm mappings are consid- ered. Some basic properties of these classes of languages are proved, and it is shown how to use them to represent recursively enumerable sets. In particular, very simple languages are introduced which play the same role for the class of recursively enumerable languages that the Dyck languages play for the class of context-free languages. Finally, a new type of acceptor for defining equality languages is introduced. KEY WOADS AND PHRASES: equality language, fLxed point language, recursively enumerable language, determin- istic sequential machine, Turing machine, Post correspondence problem, shuffle, AFL generator, representation of languages

Reasoning About Recursively Defined Data Structures

Abstract: A decision algorithm is given for the quantifier-free theory of recursively defined data structures which, for a conjunction of length n, decides its satisfiability in time linear in n. The first-order theory of recursively defined data structures, in particular the first-order theory of LISP list structure (the theory of CONS, CAR and CDR), is shown to be decidable but not elementary recursive.

Polling in a Loop System

Abstract: A commumeatton system conststmg of n buffered input terminals connected to a computer by a single channel is analyzed The terminals are polled m sequence Data are removed one umt at a tune from the termmal's buffer. When the buffer has been emptied, the channel is used for system overhead for a randomly determined length of tune The system continues wRh a poll of the next terminal. The stationary d~stributmns of waning tunes and queueing delay are determined for independent input processes The queuemg delay is mmtmized by proper selectton of the polling order rK~Y WORDS AND PHRASES commumcatlon networks, queuemg theory, polling, statistical multiplexing CR CATEGORIES: 3 8, 5 5, 8.1

On the Generation of Binary Trees

Abstract: A binary tree may be uniquely represented by a code reflecting traversal of the corresponding extended binary tree m a given monotomc order A general algorithm for constructing codes of all binary trees with n vertices is presented Different orders of traversal yield different ordermgs of the generated trees The algorithm IS illustrated with an example of the sequence of binary trees obtained from ballot sequences

Verification Decidability of Presburger Array Programs

Abstract: : A program annotated with inductive assertions is said to be verification decidable if all of the verification conditions generated from the program and assertions are formulas in a decidable theory. The Presburger array theory, is defined, containing two logical sorts: integer and array-of-integer. Addition, subtraction, and comparisons are permitted for integers. Array contents and assign functions, and, since the elements of the arrays are integers, array accesses may be nested. The first result is that the validity of unquantified formulas in Presburger array theory is decidable, yet quantified formulas in general are undecidable.

Augmenting Parsers to Support Incrementality

Abstract: The concept of incremental parsing is briefly introduced and motivated. A general shift-reduce incremental parser is presented and compared with the corresponding conventional parser in terms of speed of analysis and storage requirements. It is then shown that additional speed-up can be obtained in the particular case of LR parsing. It is suggested that the approach can be applied to other parsing algorithms and generalized to the whole compiling or interpreting process.

A Polynomial Algorithm for the Two-Variable Integer Programming Problem

Abstract: A polynomial time algorithm is presented for solving the following two-variable integer programming problem maximize ClXl + c2x2 subject to a, lxl + a,2x2 = O, integers, where a,j, cj, and b, are assumed to be nonnegattve integers This generahzes a result of Htrschberg and Wong, who developed a polynomial algorithm for the same problem with only one constraint (l e, where n = 1) However, the techniques used here are quite different KEY WORDS AND PHRASES integer programming, knapsack problem, polynomial algorithm, coefficient size, feasible region decomposition CR CATEGORIES 3 15, 5 25, 5 30, 5 40 Introduction We consider the following integer programming problem: (1) maximize clxa + czx2 subject to aaxl + a,2xz ~ b,, a= 1,2 .... n, and xl, x2 => 0, integers, where au, G, and b, are assumed to be nonnegative integers. We first show that the solution to (1) can be obtained easily from the solutions to at most n problems, each of which is of the form (2) maximize