Results 1  10
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17
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2350 (12 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...
Nondeterministic Space is Closed Under Complementation
, 1988
"... this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4 ..."
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Cited by 236 (15 self)
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this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4] for a discussion of the history and importance of this problem, and Hopcroft and Ullman [5] for all relevant background material and definitions. The history behind the proof is as follows. In 1981 we showed that the set of firstorder inductive definitions over finite structures is closed under complementation [6]. This holds with or without an ordering relation on the structure. If an ordering is present the resulting class is P. Many people expected that the result was false in the absence of an ordering. In 1983 we studied firstorder logic, with ordering, with a transitive closure operator. We showed that NSPACE[log n] is equal to (FO + pos TC), i.e. firstorder logic with ordering, plus a transitive closure operation, in which the transitive closure operator does not appear within any negation symbols [7]. Now we have returned to the issue of complementation in the light of recent results on the collapse of the log space hierarchies [10, 2, 14]. We have shown that the class (FO + pos TC) is closed under complementation. Our
Languages That Capture Complexity Classes
 SIAM Journal of Computing
, 1987
"... this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first ..."
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Cited by 230 (21 self)
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this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first order language of graphs (respectively groups, binary strings, etc.) together with a least fixed point operator. As another example, a property is in logspace if and only if it is expressible in first order logic together with a deterministic transitive closure operator. The roots of our approach to complexity theory go back to 1974 when Fagin showed that the NP properties are exactly those expressible in second order existential sentences. It follows that second order logic expresses exactly those properties which are in the polynomial time hierarchy. We show that adding suitable transitive closure operators to second order logic results in languages capturing polynomial space and exponential time, respectively. The existence of such natural languages for each important complexity class sheds a new light on complexity theory. These languages reaffirm the importance of the complexity classes as much more than machine dependent issues. Furthermore a whole new approach is suggested. Upper bounds (algorithms) can be produced by expressing the property of interest in one of our languages. Lower bounds may be demonstrated by showing that such expression is impossible.
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Optimal and Sublogarithmic Time Randomized Parallel Sorting Algorithms
 SIAM Journal on Computing
, 1989
"... .We assume a parallel RAM model which allows both concurrent reads and concurrent writes of a global memory. Our main result is an optimal randomized parallel algorithm for INTEGER SORT (i.e., for sorting n integers in the range [1; n]). Our algorithm costs only logarithmic time and is the first kno ..."
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Cited by 64 (12 self)
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.We assume a parallel RAM model which allows both concurrent reads and concurrent writes of a global memory. Our main result is an optimal randomized parallel algorithm for INTEGER SORT (i.e., for sorting n integers in the range [1; n]). Our algorithm costs only logarithmic time and is the first known that is optimal: the product of its time and processor bounds is upper bounded by a linear function of the input size. We also give a deterministic sublogarithmic time algorithm for prefix sum. In addition we present a sublogarithmic time algorithm for obtaining a random permutation of n elements in parallel. And finally, we present sublogarithmic time algorithms for GENERAL SORT and INTEGER SORT. Our sublogarithmic GENERAL SORT algorithm is also optimal. Key words. Randomized algorithms, parallel sorting, parallel random access machines, random permutations, radix sort, prefix sum, optimal algorithms. AMS(MOS) subject classifications. 68Q25. 1 A preliminary version of this paper ...
Languages which capture complexity classes
 SIAM J. on Computing
, 1987
"... We present in this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a graph property is in polynomial time if and only if it is ..."
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Cited by 52 (5 self)
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We present in this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a graph property is in polynomial time if and only if it is
Symmetric Logspace is Closed Under Complement
 CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE
, 1994
"... We present a Logspace, manyone reduction from the undirected stconnectivity problem to its complement. This shows that SL = co  SL. ..."
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Cited by 26 (1 self)
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We present a Logspace, manyone reduction from the undirected stconnectivity problem to its complement. This shows that SL = co  SL.
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
The complexity of graph connectivity
, 1992
"... In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1 ..."
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Cited by 23 (1 self)
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In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1
The Complexity of Reconfiguring Network Models
, 1992
"... This paper concerns some of the theoretical complexity aspects of the reconfigurable network model. The computational power of the model is investigated under several variants, depending on the type of switches (or switch operations) assumed by the network nodes. Computational power is evaluated by ..."
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Cited by 19 (5 self)
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This paper concerns some of the theoretical complexity aspects of the reconfigurable network model. The computational power of the model is investigated under several variants, depending on the type of switches (or switch operations) assumed by the network nodes. Computational power is evaluated by focusing on the set of problems computable in constant time in each variant. A hierarchy of such problem classes corresponding to different variants is shown to exist and is placed relative to traditional classes of complexity theory. Department of Mathematics and Computer Science, The Haifa University, Haifa, Israel. Email: yosi@mathcs2.haifa.ac.il y Department of Computer Science, Technische Universitat Munchen, 80290 Munchen, Germany. Email: lange@informatik.tumuenchen.de z Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehovot 76100, Israel. Email: peleg@wisdom.weizmann.ac.il. Supported in part by an Allon Fellowship, by a Bantrell Fellowship an...