A taxonomy of problems with fast parallel algorithms

Q4 Mathematics
Stephen A. Cook
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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(log2 n) 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).

快速并行算法问题的分类
NC类由可快速解决的问题(在log n的时间多项式内)与可行的(多项式)处理器数量并行组成。NC的许多自然问题是已知的;本文试图找出NC的重要子类,并在每个子类中给出有趣的例子。nc1可约性的概念被引入并贯穿始终(问题R是nc1可约为问题S,如果R可以用S的一致对数深度电路来解决)。关于这种可约性的完整问题给出了NC的许多子类。一种通用的技术,“并行贪婪算法”,被识别并用于证明寻找图的最小生成森林可简化为图可达性问题,因此在NC2中(可通过深度为O(log2 n)和多项式大小的一致布尔电路解决)。从电路族的角度对LOGCFL类进行了新的表征。定义了可约为整数行列式的DET类问题,并给出了许多例子。给出了在确定多项式时间内完成的一个新问题,即寻找图中字典顺序上的第一个极大团。本文是S. a . Cook(1983)在《Proceedings 1983 Intl》中的修订版。发现。Comut。科学。Conf.,“计算机科学讲义卷158,第78-93页,Springer-Verlag,柏林/纽约)。
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来源期刊
信息与控制
信息与控制 Mathematics-Control and Optimization
CiteScore
1.50
自引率
0.00%
发文量
4623
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