{"title":"A taxonomy of problems with fast parallel algorithms","authors":"Stephen A. Cook","doi":"10.1016/S0019-9958(85)80041-3","DOIUrl":null,"url":null,"abstract":"<div><p>The class <em>NC</em> consists of problems solvable very fast (in time polynomial in log <em>n</em>) in parallel with a feasible (polynomial) number of processors. Many natural problems in <em>NC</em> are known; in this paper an attempt is made to identify important subclasses of <em>NC</em> and give interesting examples in each subclass. The notion of <em>NC</em><sup>1</sup>-reducibility is introduced and used throughout (problem <em>R</em> is <em>NC</em><sup>1</sup>-reducible to problem <em>S</em> if <em>R</em> can be solved with uniform log-depth circuits using oracles for <em>S</em>). Problems complete with respect to this reducibility are given for many of the subclasses of <em>NC</em>. A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in <em>NC</em><sup>2</sup> (solvable by uniform Boolean circuits of depth <em>O</em>(log<sup>2</sup> <em>n</em>) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, <em>in</em> “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York).</p></div>","PeriodicalId":38164,"journal":{"name":"信息与控制","volume":"64 1","pages":"Pages 2-22"},"PeriodicalIF":0.0000,"publicationDate":"1985-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0019-9958(85)80041-3","citationCount":"662","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"信息与控制","FirstCategoryId":"1093","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019995885800413","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 662
Abstract
The class NC consists of problems solvable very fast (in time polynomial in log n) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC1-reducibility is introduced and used throughout (problem R is NC1-reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S). Problems complete with respect to this reducibility are given for many of the subclasses of NC. A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC2 (solvable by uniform Boolean circuits of depth O(log2n) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York).