GS] Goldberg, M., T. Spencer, A New Parallel Algorithm for the Maximal Independent [ Set Problem, SIAM J. Comp., 18 (1989), pp. 419 - 427.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 Introduction During the last years there is a growing interest in techniques for removing randomness from parallel (and sequential) algorithms. These techniques were originated by [7, 8] and generalized in [1, 2, 4, 6, 9, 10, 11]. The approach usually followed can be summarized as follows: The random variables which are considered are defined over a smaller probability space, specially designed, containing only a polynomial number of sample points. In that space, the random variables are only k wise independent (for ....

M. Goldberg and T. Spencer, A New Parallel Algorithm for the Maximal Independent Set Problem, SIAM J. on Computing, 18:2 (1989) 419-427.

....a subset S of V that does not contain any edge. S is a maximal independent set (MIS) if no proper superset of S is an IS. It is easy to find an MIS sequentially, but efficient parallel algorithms appear much harder. There has been much work on finding MISs in parallel in (hyper)graphs (see, e.g. [1, 4, 7, 9, 17, 15, 16, 18, 21]) Since an MIS can be much smaller than a maximum independent set (as in the case of a star graph) it is also of much interest to find ISs that have a guaranteed size. What is known in this context Recall that given a hypergraph H = V; E) the degree of a vertex is the number of edges that it ....

M. Goldberg and T. Spencer. A new parallel algorithm for the maximal independent set problem. SIAM J. Comput., 18:419--427, 1989.

.... in randomized parallel computation are various algorithms for finding maximal independent sets: the randomized algorithm of Luby [45] uses less operations than the deterministic algorithm of the same paper, and runs more quickly than the newer deterministic algorithm of Goldberg and Spencer [27]. Other problems with randomized algorithms more e#cient than their deterministic counterparts include finding connected components [24] and sorting integers [59] Finally, a randomized algorithm may have the same bounds as a deterministic algorithm, but may be much simpler. Also, randomized ....

M. Goldberg and T. Spencer, A New Parallel Algorithm for the Maximal Independent Set Problem. 28th Symp. Found. Comput. Sci., 1987, 161--165.

....(the 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 ....

M. Goldberg, T. Spencer, A new parallel algorithm for the maximal independent set problem, SIAM J. on Comp. 18 (1989), pp. 419-427.

....the size of the lexicographically first maximal independent set is also P complete; see Section 6.2. Karp and Wigderson gave the first NC algorithm for finding a maximal independent set [KW85] subsequently improved by Luby [Lub86] by Alon, Babai and Itai [ABI86] and by Goldberg and Spencer [GS89]. These algorithms do not compute the lexicographically first maximal independent set. Part II: P Complete Problems ffl 51 A.2.2 Lexicographically First Maximal Clique (LFMC) Given: An undirected graph G with an ordering on the vertices and a designated vertex v. Problem: Is vertex v in the ....

M. Goldberg and T. Spencer. A new parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 18:419--427, 1989.

....amounts to finding a maximal independent set in the independence system that we have just defined. Efficient (deterministic) parallel algorithms for finding a maximal independent set in an independence system are known for the special case where the size of a minimal dependent set is 2 or 3 ( 16] [9], 5] For the problems that are of interest to us minimal dependent sets may have nonconstant size and hence a different approach is needed for obtaining fast parallel algorithms. The minimal spanning subgraph problem has been studied earlier for the property of strong connectivity (transitive ....

M. Goldberg AND T. Spencer, A new parallel algorithm for the maximal independent set problem, SIAM J. Computing, vol. 18, 1989, pp.419-427.

....(MIS for short) can be trivially solved in linear time. However, the problem of constructing an efficient NC algorithm for MIS is non trivial. Karp and Widgerson were the first who proved MIS to be in NC [28] Presently, the most efficient deterministic NC algorithm is due to Goldberg and Spencer [21]. It runs in time O(log 4 n) and uses an EREW PRAM with a linear number of processors. Luby has constructed a randomized parallel algorithm for MIS that runs in O(log 2 n) time and uses a EREW PRAM with O(nm 2 ) processors [35] The following sequential algorithm constructs a maximal ....

....problem is in NC 4 . They presented a parallel randomized algorithm with expected running time O(log 4 n) using O(n 2 ) processors on a EREW PRAM, and a deterministic algorithm with running time O(log 4 n) using O( n 3 log 3 n ) processors also on a EREW PRAM. Goldberg and Spencer [21] improved the result by presenting a deterministic parallel algorithm running in O(log 4 n) using only O(n m) EREW processors. Algorithm Parallel Maximal Independent Set(G) input : A graph G = V; E) output : A maximal independent set I on G. method : I ; A V while A 6= do C ....

[Article contains additional citation context not shown here]

M. Goldberg and T. Spencer, A New Parallel Algorithm for the Maximal Independent Set Problem. In Proc. 28th Symp. on Foundations of Computer Science, 1987.

....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. A New Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 18(2):419--427, 1989.

....other paths due to a shared edge in E sat and O(D p D) other paths due to shared unsaturated node endpoints. Thus, the graph of paths has O(D 3 ) vertices and O(D 5 ) edges. Finding a maximal independent set in this graph can be done in O(log 4 D) time using O(D 5 ) EREW PRAM processors [9, 11]. This results in the resource bounds stated in the theorem. 6 Final remark Theorem 5.4 presents a partial solution to the intriguing problem of whether a graph corresponding to a degree sequence can be constructed by an NC algorithm. The solution is rather involved and relies on Cohen s NC ....

M. Goldberg, T. Spencer. A new parallel algorithm for the maximal independent set problem. Proc. 28th FOCS, 1987.

....has attracted a great deal of attention in the theory of algorithms. Some problems, easy to solve sequentially in polynomial time, became non trivial questions in parallel computation. The maximal independent set problem was one of them. The problem is now known to be in NC ( KW] Lu] ABI] [GS]) Other important problems like maximum matching (matching of maximum size) are still not known to be in NC. There exist methods ( Lo1] KUW] MVV] which show that matching is in RNC (random parallel polylog time) in fact, in Las Vegas NC [Ka] One relaxation of the maximum matching problem ....

M. Goldberg and T. Spencer, A new parallel algorithm for the maximal independent set problem, Proc. 28th IEEE FOCS, 1987, pp. 161-165.

.... 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, A new parallel algorithm for the maximal independent set problem, SIAM J. Computing 18 (1989) 419-427.

....for which obtaining efficient sequential algorithms is easy, but obtaining efficient parallel algorithms is much harder. Karp and Wigderson [11] have shown that the problem is in NC. A lot of research was done to improve the processor and time complexity of parallel algorithms for the MIS problem [10, 13, 14]. Unfortunately, direct distributed implementation of these algorithms is inefficient. Intuitively, the main problem is that, at each iteration, the algorithm decides on the next step by collecting the information from every node in the graph to a single memory location and making a choice based ....

M. Goldberg and T. Spencer. A New Parallel Algorithm for the Maximal Independent Set Problem. In Proc. 28th IEEE Symp. on Foundations of Comp. Sci., pages 161--165, 1987.

....Research in Computer Science, Centre of the Danish National Research Foundation. 1 1 Introduction During the last years there is a growing interest in techniques for removing randomness from parallel (and sequential) algorithms. These techniques were originated by [7, 8] and generalized in [1, 2, 4, 6, 9, 10, 11]. The approach usually followed can be summarized as follows: The random variables which are considered are defined over a smaller probability space, specially designed, containing only a polynomial number of sample points. In that space, the random variables are only k wise independent (for ....

M. Goldberg and T. Spencer, A New Parallel Algorithm for the Maximal Independent Set Problem, SIAM J. on Computing 18:2 (1989) 419-427.

....the size of the lexicographically first maximal independent set is also P complete; see Section 6.2. Karp and Wigderson gave the first NC algorithm for finding a maximal independent set [KW85] subsequently improved by Luby [Lub86] by Alon, Babai and Itai [ABI86] and by Goldberg and Spencer [GS89]. These algorithms do not compute the lexicographically first maximal independent set. Part II: P Complete Problems ffl 51 A.2.2 Lexicographically First Maximal Clique (LFMC) Given: An undirected graph G with an ordering on the vertices and a designated vertex v. Problem: Is vertex v in the ....

M. Goldberg and T. Spencer. A new parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 18:419--427, 1989.

....subgraph with property P amounts to finding a maximal independent set in the independence system we just defined. Efficient parallel algorithms for finding a maximal independent set in an independence system are known for the special case where the size of a minimal dependent set is 2 or 3 ( 11] [5], 2] 8] For the problems that are of interest to us minimal dependent sets may have nonconstant size and hence, a different approach is needed. The minimal spanning subgraph problem has been studied earlier for the property of strong connectivity (or transitive compaction [4] and for ....

M. Goldberg, T. Spencer, A new parallel algorithm for the maximal independent set problem, SIAM J. Computing, vol. 18, 1989, pp.419-427.

....an efficient deterministic NC algorithm for finding a maximal independent set in a hypergraph of dimension 3: the algorithm runs in time O(log 4 n) time on n m processors of an EREW PRAM and is optimal up to a polylogarithmic factor. Our algorithm adapts the technique of Goldberg and Spencer ([5]) for finding a maximal independent set in a graph (or hypergraph of dimension 2) It is the first efficient NC algorithm for finding a maximal independent set in a hypergraph of dimension greater than 2. 1 Department of Computer Science, University of Bonn, 5300 Bonn 1. 2 Department of ....

....hyperedges. 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 ....

[Article contains additional citation context not shown here]

M. Goldberg, T. Spencer, A new parallel algorithm for the maximal independent set problem, SIAM J. Computing, vol. 18, 1989, pp.419-427.

....Lemma 9 (G) 2 f Delta(G) Delta(G) 1g: A coloring with Delta(G) 1 colors can be constructed in O(jV j Delta jEj) time. Now consider the complete graph K d with d vertices. We show that for this particular case there is a simpler and more efficient scheme to color the edges O(d 2 ) [6]. Let V = f0; 1; d Gamma1g. If d is odd then the edges (i; j) are colored with color i j mod d. Note that adjacent edges get different colors and that for each vertex i the color 2 Delta i mod d is unused. d colors are clearly optimal in this case. In the case that d is even, we remove ....

Goldberg, M., Th. Spencer, `A New Parallel Algorithm for the Maximal Independent Set Problem,' Siam Journal on Computing, 18, pp. 419--427, 1989.

....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. A New Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 18(2):419--427, 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. A New Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 18(2):419--427, 1989.

....linear recurrences, polynomial arithmetic (evaluation, multiplication, division, GCD) evaluation of elementary functions (exp, ln, sin, etc. Matrices [47, 67, 80, 149, 224, 270] Matrix multiplication, determinant, rank, inverse, solution of linear system, Cholesky factorisation. Graphs [28, 58, 68, 80, 89, 92, 99, 101, 133, 149, 151, 162, 178, 224, 230, 240, 248]: Algebraic path problem (transitive closure, shortest paths, minimum cost spanning tree, topological ordering of dag) transitive reduction, connected components, biconnectivity, triconnectivity, Euler tours, ear decomposition, maximal independent set, symmetry breaking, lowest common ancestors, ....

M Goldberg and T Spencer. A new parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 18(2):419--427, April 1989. McCOLL : GENERAL PURPOSE PARALLEL COMPUTING

....value is A(11 , 4) for which it is known only that 72 A(11 , 4) 79 . H(11 , 4) is a graph with only 2048 vertices: would some graph theorist please determine the size of a largest clique in it (A number of algorithms for clique finding have been published in recent years [1] 2] 14] [15], 26] Every entry in Table 1 not followed by a period is an unsolved problem of this type. If a reader should find a larger clique than is presently known in any of these problems, please write down the vertices of the clique (which form a record breaking code) and notify the author TABLE ....

M. Goldberg and T. Spencer, "A new parallel algorithm for the maximal independent set problem", SIAM J. Comput., vol. 18 (1989), pp. 419-427.

....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. A New Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 18(2):419--427, 1989.

....B V whose induced subgraph is bipartite. In this paper we present efficient sequential (linear time) and parallel (NC) algorithms for constructing an MBS. 1 Introduction In the last few years several efficient parallel algorithms have appeared for the maximal independent set (MIS) problem [GS, KW, Luby]. This may be thought of as the problem of finding a maximal 1 colorable set in a given graph. In this paper we consider the natural extension to finding a maximal 2 colorable set. Supported by a General Electric Foundation Graduate Fellowship y Supported in part by NSF Grant DCR 85 52938 and ....

M. GOLDBERG and T. SPENCER, A New Parallel Algorithm for the Maximal Independent Set Problem, SIAM J. Computing 18 (1989), 419-427.

.... heuristics 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 ....

M. Goldberg and T. Spencer, A new parallel algorithm for the maximal independent set problem, SIAM J. Comput., Vol. 18: 419-427, 1989.

No context found.

GS] Goldberg, M., T. Spencer, A New Parallel Algorithm for the Maximal Independent [ Set Problem, SIAM J. Comp., 18 (1989), pp. 419 - 427.

....might have many candidate nodes to add to the independent set. However, due to adjacency constraints, not all of these nodes can be added simultaneously. A symmetry breaking technique is therefore needed to find a large set of nodes to add, as has been done in previous parallel MIS algorithms [GS87,KW85,Lub86]. Previous symmetry breaking techniques have focused on randomization. It is often desirable, however, to have a deterministic algorithm. Karp and Wigderson [KW85] and Luby [Lub86] proposed methods to convert certain randomized algorithms into deterministic ones. Their methods, however, ....

....and in O(lg 2 n) time on an EREW PRAM. ffl We give an O(lg n lg n) time CRCW PRAM algorithm for 5 coloring an embedded planar graph. The above results improve the running time and processor bounds for the respective problems. The best deterministic linear processor algorithm for finding MIS [GS87] runs in O(lg 4 n) time on constant degree graphs, compared to O(lg n) time of our algorithm. 2 The 5 coloring algorithms for planar graphs described in [BK87,Nao86] use O(lg 3 n) time and the same (large) number of processors as needed by Luby s MIS subroutine [Lub86] The O(lg 3 n) ....

M. Goldberg and T. Spencer. A new parallel algorithm for the maximal independent set problem. In Proc. 28th IEEE Symp. on Foundations of Computer Science, 1987. (To appear).

....for which obtaining efficient sequential algorithms is easy, but obtaining efficient parallel algorithms is much harder. Karp and Wigderson [10] have shown that the problem is in NC. A lot of research was done to improve the processor and time complexity of parallel algorithms for the MIS problem [9, 12, 13]. Unfortunately, direct distributed implementation of these algorithms is inefficient. Intuitively, the main problem is that, at each iteration, these algorithms decide on the next step by collecting the information from every node in the graph to a single memory location and making a choice based ....

M. Goldberg and T. Spencer. A New Parallel Algorithm for the Maximal Independent Set Problem. In Proc. 28th IEEE Annual Symposium on Foundations of Computer Science, pages 161--165, 1987.

No context found.

M. Goldberg and T. Spencer. A new parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 18(2):419--427, 1989.

No context found.

M. Goldberg, T. Spencer. A new parallel algorithm for the maximal independent set problem. SIAM J. Comput., Vol. 18, No. 2, pp. 419-427(April 1989).

No context found.

M. Goldberg, T. Spencer. A new parallel algorithm for the maximal independent set problem. SIAM J. Comput., Vol. 18, No. 2, pp. 419-427(April 1989).

No context found.

Mark Goldberg and Thomas Spencer. A new parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 18(2):419--427, April 1989.

No context found.

M. Goldberg and T. Spencer, " A New Parallel Algorithm for the Maximal Independent Set Problem", Proc. 28th Symp. on Foundations of Computer Science, 1987.

No context found.

Goldberg, M., T. Spencer, "A New Parallel Algorithm for the Maximal Independent Set Problem", a preliminary version appears in the Proceedings of 28 th Annual Symposium on Foundations of Computer Science, 1987, pp. 161--165, to appear in SIAM J. on Computing.

No context found.

Goldberg, M., Spencer, T., "A New Parallel Algorithm for the Maximal Independent Set Problem," FOCS 1987.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC