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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...
SublinearTime Parallel Algorithms for Matching and Related Problems
, 1988
"... This paper presents the first sublineartime deterministic parallel algorithms for bipartite matching and several related problems, including maximal nodedisjoint paths, depthfirst search, and flows in zeroone networks. Our results are based on a better understanding of the combinatorial struc ..."
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Cited by 33 (6 self)
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This paper presents the first sublineartime deterministic parallel algorithms for bipartite matching and several related problems, including maximal nodedisjoint paths, depthfirst search, and flows in zeroone networks. Our results are based on a better understanding of the combinatorial structure of the above problems, which leads to new algorithmic techniques. In particular, we show how to use maximal matching to extend, in parallel, a current set of nodedisjoint paths and how to take advantage of the parallelism that arises when a large number of nodes are "active" during an execution of a pushrelabel network flow algorithm. We also show how to apply our techniques to design parallel algorithms for the weighted versions of the above problems. In particular, we present sublineartime deterministic parallel algorithms for finding a minimumweight bipartite matching and for finding a minimumcost flow in a network with zeroone capacities, if the weights are polynomially ...
A Parallel Priority Queue with Constant Time Operations
 JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING
, 1998
"... We present a parallel priority queue that supports the following operations in constant time: parallel insertion of a sequence of elements ordered according to key, parallel decrease key for a sequence of elements ordered according to key, deletion of the minimum key element, as well as deletion ..."
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Cited by 15 (1 self)
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We present a parallel priority queue that supports the following operations in constant time: parallel insertion of a sequence of elements ordered according to key, parallel decrease key for a sequence of elements ordered according to key, deletion of the minimum key element, as well as deletion of an arbitrary element. Our data structure is the first to support multi insertion and multi decrease key in constant time. The priority queue can be implemented on the EREW PRAM, and can perform any sequence of n operations in O(n) time and O(m log n) work, m being the total number of keys inserted and/or updated. A main application is a parallel implementation of Dijkstra's algorithm for the singlesource shortest path problem, which runs in O(n) time and O(m log n) work on a CREW PRAM on graphs with n vertices and m edges. This is a logarithmic factor improvement in the running time compared with previous approaches.
Using Interior Point Methods for Fast Parallel Algorithms for Bipartite Matching and Related Problems
 SIAM J. Comput
, 1992
"... In this paper we use interiorpoint methods for linear programming, developed in the context of sequential computation, to obtain a parallel algorithm for the bipartite matching problem. Our algorithm finds a maximum cardinality matching in a bipartite graph with n nodes and m edges in O( p m l ..."
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Cited by 12 (2 self)
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In this paper we use interiorpoint methods for linear programming, developed in the context of sequential computation, to obtain a parallel algorithm for the bipartite matching problem. Our algorithm finds a maximum cardinality matching in a bipartite graph with n nodes and m edges in O( p m log 3 n) time on a CRCW PRAM. Our results extend to the weighted bipartite matching problem and to the zeroone minimumcost flow problem, yielding O( p m log 2 n log nC) algorithms, where C ? 1 is an upper bound on the absolute value of the integral weights or costs in the two problems, respectively. Our results improve previous bounds on these problems and introduce interiorpoint methods to the context of parallel algorithm design. 1 Introduction In this paper we use interiorpoint methods for linear programming, developed in the context of sequential computation, to obtain a parallel algorithm for the bipartite matching problem. Although Karp, Upfal, and Wigderson [6] have sho...
NC algorithms for comparability graphs, interval graphs, and unique perfect matching
 Proc. 5th Conf. Found. Software Technology and Theor. Comput. Sci., volume 206 of Lect. Notes in Comput. Sci
, 1985
"... Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval rep ..."
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Cited by 11 (0 self)
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Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for nding a maximum matching in an incomparability graph. 1
Randomized Parallel Algorithms
 IN SOLVING COMBINATORIAL PROBLEMS IN PARALLEL, VOLUME 1054 OF LNCS
, 1996
"... ..."
On Parallel Complexity of Maximum FMatching and the Degree Sequence Problem
, 1994
"... We present a randomized NC solution to the problem of constructing a maximum (cardinality) fmatching. As a corollary, we obtain a randomized NC algorithm for the problem of constructing a graph satisfying a sequence d 1 ; d 2 ;...; d n of equality degree constraints. We provide an optimal NC alg ..."
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We present a randomized NC solution to the problem of constructing a maximum (cardinality) fmatching. As a corollary, we obtain a randomized NC algorithm for the problem of constructing a graph satisfying a sequence d 1 ; d 2 ;...; d n of equality degree constraints. We provide an optimal NC algorithm for the decision version of the degree sequence problem and an approximation NC algorithm for the construction version of this problem. Our main result is an NC algorithm for constructing if possible a graph satisfying the degree constraints d 1 ; d 2 ;...; d n in case d i q P n j=1 d j =5 for i = 1; :::; n: 1 Introduction Finding a maximum (cardinality) matching in a graph is a fundamental problem in combinatorial optimization. It is a major open problem whether a maximum matching can be constructed by an NC algorithm. Achieving simultaneously a polylogtime and a polynomial number of processors is possible for this problem if random bits are used. Randomized NC algorithms ...