M. Goldberg, T. Spencer. Constructing a maximal independent set in parallel. SIAM J. Dis. Math., Vol 2, No. 3, 322-328(Aug. 1989). 12  Home/Search   Document Details and Download   Summary   Related Articles   Check

....MIS problem) then the problem becomes polynomial. In fact, a maximal independent set can be easily found sequentially in a linear time. Karp and Wigderson [15] showed that MIS is in NC. Starting with their work, a number of parallel algorithms have been proposed to solve this problem [2] 10] [11], 17] 18] Currently, the most ecient algorithm is presented in [11] it runs in O(log 3 n) time on O( n m) log n) processors. A common drawback of all NC algorithms for MIS mentioned above is that occasionally they can nd too small a set. Any graph with a large independent set and a ....

....independent set can be easily found sequentially in a linear time. Karp and Wigderson [15] showed that MIS is in NC. Starting with their work, a number of parallel algorithms have been proposed to solve this problem [2] 10] 11] 17] 18] Currently, the most ecient algorithm is presented in [11]; it runs in O(log 3 n) time on O( n m) log n) processors. A common drawback of all NC algorithms for MIS mentioned above is that occasionally they can nd too small a set. Any graph with a large independent set and a vertex adjacent to all other vertices is a potential example of such a ....

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M. Goldberg, T. Spencer, Constructing a maximal independent set in parallel, SIAM J. on Discr. Math., 2 (1989), pp. 322-328.

....problem is a typical maximality problem, that is to find a maximal vertex induced subgraph that satisfies a specified graph property. Since Karp and Wigderson showed that the MIS problem is in the class NC [11] much work has been devoted to the study of parallel complexity of maximality problems [13, 1, 6, 5, 3, 17, 18]. On the other hand, the lexicographically first maximal independent set (LFMIS) problem is a typical P complete problem [2] and P completeness of the lexicographically first maximal subgraph (LFMS) problems for some properties was shown [14, 17] see also [15, 7] for a comprehensive reference) ....

....1 reducibility in [2] we use the log space reducibility simply as in [14] A function F 0 is said to be P complete if F 0 is in P and for each F in P there are log space computable functions f and g such that F (x) g(F 0 (f(x) for all inputs. It is well known that the MIS problem is in NC [11, 1, 13, 6, 5, 12], and the LFMIS problem is one of the fundamental P complete problems [14, 15, 7] Recall that the EREW PRAM is the parallel model where the processors operate synchronously and share a common memory, but no two of them are allowed simultaneous access to a memory cell (whether the access is for ....

M. Goldberg and T. Spencer. Constructing a Maximal Independent Set in Parallel. SIAM J. Disc. Math., 2(3):322--328, 1989.

....procedure. Here we have to solve the problem to compute an inclusion maximal clique in parallel. As a side result, we found out that an inclusion maximal clique can be computed in polylogarithmic time with a linear processor bound in nearly the same manner as an inclusion maximal independent set [22]. In the fifth section, we discuss an efficient parallel breadth first search algorithm which is not only applicable for chordal graphs but for graphs not containing the house or a cycle of length greater than four as an induced subgraph. 2 Notation A graph G = V; E) consists of a vertex set V ....

....statement, we have to consider the complexity to compute an inclusion maximal clique. Theorem 2 A maximal clique can be computed by an EREW PRAM in O(log 3 n) time using O(n m) processors. Proof: We proceed in a similar way as in the maximal independent set al..gorithm of Goldberg and Spencer [22]. To make the paper more self contained, we repeat the algorithm of Goldberg and Spencer. The main subprocedure is FINDSET which computes an independent set I such that jI [ N(I)j (1=2 Gamma o(1) n m) We apply FINDSET to the original graph, remove all vertices of I [ N(I) and apply again ....

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M. Goldberg, T. Spencer, Constructing a Maximal Independent Set in Parallel, SIAM Journal on Discrete Mathematics 2 (1989), pp. 322-328. 13

.... be eciently solved in parallel: Theorem 1 MIS(A; I) 2 NC for dim(A) 3, and MIS(A; I) 2 RNC for dim(A) 4; 5; The statements of Theorem 1 were previously known [4, 23] only for I = when MIS(A; I) turns into the classical problem of computing a single maximal independent set for A (see [1, 12, 15, 16, 20, 21, 22, 25]) We show that conversely, MIS(A; I) can be reduced to the above special case. Theorem 2 If dim(A) const, then problem MIS(A; I) is NC reducible 1 to problem MIS(A 0 ; where A 0 is some induced partial hypergraph of A. Given a hypergraph A 2 V , a subfamily A 0 A is called ....

M. Goldberg, T. Spencer, Constructing a maximal independent set in parallel, SIAM J. Disc. Math. 2 (1989) 322-328.

.... O(log H) Unfortunately, even the fastest known deterministic parallel algorithm for maximal matching with n items is O(log 2 n) with a quartic number of processors for dense graphs (which we have) Luba] or O(log 3 n) with a quadratic number of processors [IsS] or n 2 = log n processors [GoS]. Since we have n = H 0 , this means that the fastest known algorithm is Theta(log 2 H) and still doesn t work since we have only H = Theta( H 0 ) 2 log 2 H 0 ) processors. Subsection 4.3.2 addresses the subject of how to do the matching efficiently. For the current purposes it is ....

Mark Goldberg and Thomas Spencer, "Constructing a Maximal Independent Set in Parallel," SIAM J. Discrete Math 2, 322--328.

....The dimension of H is the maximum size of a hyperedge in E. An independent set in H is a subset of V that does not contain any hyperedge of E; an independent set is maximal if it is not properly contained in another independent set. Although several efficient parallel algorithms ( 5] [6], 1] 10] are known for computing a maximal independent set in ordinary graphs (i.e. hypergraphs of dimension 2) the question of whether there is an NC algorithm for arbitrary hypergraphs is still open ( 7] In this paper we present an efficient NC algorithm for the special case where the ....

M. Goldberg, T. Spencer, Constructing a Maximal Independent Set in Parallel, SIAM J. Disc. Math., vol. 2, 1989, pp. 322-328.

....a typical maximality problem, that is to find a maximal vertex induced subgraph that satisfies a specified graph property. Karp and Wigderson first showed that the MIS problem is in the class NC [15] Since then, much work has been devoted to the study of parallel complexity of maximality problems [17, 1, 9, 8, 7, 20, 21]. On the other hand, the lexicographically first maximal independent set (LFMIS) problem is a typical P complete problem [5] and P completeness of the lexicographically first maximal subgraph (LFMS) problems for some properties was shown [18, 20] see also [10] for a comprehensive reference) ....

....1 reducibility in [5] we use the log space reducibility simply as in [18] A problem F 0 is said to be P complete if F 0 is in P and for each F in P there are log space computable functions f and g such that F (x) g(F 0 (f(x) for all inputs. It is well known that the MIS problem is in NC [15, 1, 17, 9, 8, 16], and the LFMIS problem is one of the fundamental P complete problems [5, 18, 10] Recall that the EREW PRAM is the parallel model where the processors operate synchronously and share a common memory, but no two of them are allowed simultaneous access to a memory cell (whether the access is for ....

M. Goldberg and T. Spencer. Constructing a Maximal Independent Set in Parallel. SIAM J. Disc. Math., 2(3):322--328, 1989.

....problem is a typical maximality problem, that is to find a maximal vertex induced subgraph that satisfies a specified graph property. Since Karp and Wigderson showed that the MIS problem is in the class NC [12] much work has been devoted to the study of parallel complexity of maximality problems [14, 1, 7, 6, 4, 17, 18]. On the other hand, the lexicographically first maximal independent set (LFMIS) problem is a typical P complete problem [3] and P completeness of the lexicographically first maximal subgraph (LFMS) problems for some properties was shown [15, 17] see also [8] for a comprehensive reference) As ....

....1 reducibility in [3] we use the log space reducibility simply as in [15] A function F 0 is said to be P complete if F 0 is in P and for each F in P there are log space computable functions f and g such that F (x) g(F 0 (f(x) for all inputs. It is well known that the MIS problem is in NC [12, 1, 14, 7, 6, 13], and the LFMIS problem is one of the fundamental P complete problems [15, 8] Recall that the EREW PRAM is the parallel model where the processors operate synchronously and share a common memory, but no two of them are allowed simultaneous access to a memory cell (whether the access is for ....

M. Goldberg and T. Spencer. Constructing a Maximal Independent Set in Parallel. SIAM J. Disc. Math., 2(3):322--328, 1989.

....but it is unlikely to be of much practical value. In the last few years, substantial progress has been made on overcoming this bottleneck. A number of important new PRAM algorithms for sparse matrix and graph problems have been developed which are very efficient in their use of processors [68, 98, 99, 100, 114, 132, 144, 160, 162, 203, 244, 252]. As we have seen, the robustness of the PRAM model and the class NC has permitted the development of a rich theory of parallel algorithms and their complexity. However, the McCOLL : GENERAL PURPOSE PARALLEL COMPUTING above considerations show that a naive preoccupation with NC may not result in ....

M Goldberg and T Spencer. Constructing a maximal independent set in parallel. SIAM Journal of Discrete Mathematics, 2(3):322--328, August 1989.

....a specified graph property has been widely investigated. Karp and Wigderson first showed that the typical maximality problem, the maximal independent set (MIS) problem, is in the class NC [1] Since then, much work has been devoted to the study of parallel complexity of maximality problems [2, 3, 4, 5, 6, 7, 8]. On the other hand, the lexicographically first maximal independent set (LFMIS) problem is a typical P complete problem [9] and P completeness of the lexicographically first maximal subgraph (LFMS) problems for some graph properties has been shown [10, 7] see also [11] for a comprehensive ....

....1 reducibility in [9] we use the log space reducibility simply as in [10] A problem F 0 is said to be P complete if F 0 is in P and for each F in P there are log space computable functions f and g such that F (x) g(F 0 (f (x) for all inputs. It is well known that the MIS problem is in NC [1, 3, 2, 4, 5, 20], and the LFMIS problem is one of the fundamental P complete problems [9, 10, 11] Recall that the EREW PRAM is the parallel model where the processors operate synchronously and share a common memory, but no two of them are allowed simultaneous access to a memory cell (whether the access is for ....

M. Goldberg and T. Spencer. Constructing a Maximal Independent Set in Parallel. SIAM J. Disc. Math., 2(3):322--328, 1989.

....restricted MLF problem. It remains to analyze the complexity of Algorithm 1. Steps 1 and 2. 3 can be done in O(log n) expected time with n 2 processor on a CRCW PRAM [14] in O(log 2 n) time with n 4 processor on an EREW PRAM [14] or in O(log 3 n) time with n 2 processor on an EREW PRAM [9]. It is not so difficult to see that the other steps can be done in O(log n) time using n 2 processors on an EREW PRAM. This establishes the theorem. In the next sections, we present parallel algorithms for solving the restricted MLF problem. 3 An RNC algorithm and its derandomization 3.1 ....

M. Goldberg and T. Spencer. Constructing a Maximal Independent Set in Parallel. SIAM J. Disc. Math., 2(3):322--328, 1989.

.... for the maximum clique independent set and related problems on arbitrary or special class of graphs can be found in [89] 91] 94] 117] Among others, further (partly randomized) parallel algorithms for the (weighted) maximum clique problem are proposed in [196] 221] 102] 144] [145], 81] 264] 6] and [100] 7 Selected Applications In many applications, the underlying problem can be formulated as a maximum clique problem while in others a subproblem of the solution procedure consists of finding a maximum clique. This necessitates the development of fast exact and ....

M. Goldberg and T. Spencer, Constructing a maximal independent set in parallel, SIAM J. Discr. Math., Vol. 2: 322--328, 1989.

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M. Goldberg, T. Spencer. Constructing a maximal independent set in parallel. SIAM J. Dis. Math., Vol 2, No. 3, 322-328(Aug. 1989). 12

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M. Goldberg, T. Spencer. Constructing a maximal independent set in parallel. SIAM J. Dis. Math., Vol 2, No. 3, 322-328(Aug. 1989). 22

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M. Goldberg, T. Spencer. Constructing a maximal independent set in parallel. SIAM J. Dis. Math., Vol 2, No. 3, 322-328(Aug. 1989).

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M. Goldberg and T. Spencer, Constructing a maximal independent set in parallel, SIAM J. Discrete Math., Vol. 2, 1989, 322-328.

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