P. Klein, Efficient Parallel Algorithms for Chordal Graphs, Proc. IEEE Symposium on Foundations of Computer Science, 1989, pp. 150--161.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....any graph in O(jV j jEj) time. The best known external memory results are O(jEj=jV jsort(jV j) log 2 (V=M) I Os deterministically and O(sort(jEj) I Os [1] A perfect elimination ordering of a chordal graph can be found in linear time using algorithms by [27, 29, 31] In the PRAM model, Klein [20] shows how to compute a perfect elimination ordering in O(log 2 n) time using O( n m) logn) processors. In external memory, the sequential approaches seem unfeasible, as they use search strategies similar to breadth first search, while a simulation of Klein s approach would lead to a suboptimal ....

Philip N. Klein. Efficient parallel algorithms for chordal graphs. In Proceedings of the 29th Symposium on foundations of Computer Science, pages 150--161, 1989.

....elimination and databases and have been the object of much algorithmic study. On the other hand, chordal graphs are also important from the graph theoretical point of view. The class of chordal graphs is subclass of the class of perfect graphs, and superclass of the class of interval graphs (see [20, 15] for details) It is natural to generalize about the length of the cycle on the chordal graph. We introduce a notion of a k chordal graph whose each cycle of length greater than k has a chord. An ordinary chordal graph is a 3 chordal graph, and so called chordal bipartite graph can be defined by a ....

....on a 3 chordal graph. For example, the maximum independent set problem, the chromatic number problem, and the maximum clique problem are NP complete on a general graph [10] and they are polynomial time solvable on a 3 chordal graph [11] These problems are even in NC on a 3 chordal graph [20, 15]. These facts motivate us to consider the case of general k as a function of n. We first show that the recognition problem for k chordalness is coNP complete for k = 2(n) We next show that, for any positive constant ffl, the NP complete problems above are also NP complete even on a k chordal ....

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P.N. Klein. Efficient Parallel Algorithms for Chordal Graphs. SIAM Journal on Computing, 25(4):797-- 827, 1996.

....Complexity of Elimination Ordering Procedures Elias Dahlhaus Basser Dept. of Computer Science University of Sydney, NSW 2006, Australia 1 Abstract We prove that lexicographic breadth first search is P complete and that a variant of the parallel perfect elimination procedure of P. Klein [24] is powerful enough to compute a semi perfect elimination ordering in sense of [23] if certain induced subgraphs are forbidden. We present an efficient parallel breadth first search algorithm for all graphs which have no cycle of length greater four and no house as an induced subgraph. A side ....

.... linear algorithms to recognize chordal graphs and to compute a perfect elimination ordering are the lexical breadth first search method of Rose, Tarjan, and Lueker [32] and the maximum cardinality search method of Tarjan and Yannakakis [36] An efficient parallel algorithm is due to Klein [24]. It is well known that a minimum coloring of a chordal graph can be obtained efficiently by coloring the vertices in reversal order with the color of the least number which is not the color of a greater neighbor. There is a more generalized result in behind. Chvatal [6] found out that one can ....

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P. Klein, Efficient Parallel Algorithms for Chordal Graphs, 29. IEEE-FOCS (1988), S. 150-161.

.... theory, and is easily solved in linear time sequentially [6] In parallel, performing ordered depth first search that visits vertices of a graph in a given order is known to be P complete [27] There are parallel algorithms for depth first search in various special graphs, such as chordal graphs [20], series parallel graphs [12] planar undirected graphs [10, 13, 15, 19, 29, 30] and planar directed graphs [16, 17, 18] In particular, Kao and Klein [18] gave an O(log 10 n) time, O(n log 10 n) work algorithm for depthfirst search in general planar directed graphs, and Kao [17] presented an ....

P. N. Klein, "Efficient parallel algorithms for chordal graphs," Proc. 29th Annual IEEE Symp. on Foundations of Computer Science, 1988, pp. 150--161.

....of G: The succeeding subsections give a more detailed analysis of this algorithm. 3.5.1. An Outline of the Chordal extension Algorithm. Here we present a Clique Separator Decomposition algorithm which combines methods introduced recently in [DK 88b] with certain extensions of P. Klein s FOCS 88 [Kl 88] method for chordal graphs. The algorithm first computes a chordal extension G 0 which preserves clique separators, together with a perfect elimination order on G 0 . The last step is to check for each clique separator of G 0 = V; E 0 ) V; E [ F ) whether it is a clique separator of ....

....sets C 1 ; C 2 , such that C C 1 C 2 and #(C 1 nC) #(C 1 nC 2 ) #(V nC 2 ) 2 3 #(V nC) Let [x; y] 2 F C iff x and y are adjacent to the same connected component of V n C, where C = C 1 or C = C 2 , and let E C : E [ F C . Procedures NONE and REF INE are based on P. Klein s ( Kl 88] new technique for chordal graphs. Observation. Whenever u and v 2 C i nC are in the same connected component with respect to EC i , they are in the same connected component of V nC with respect to E, and vice versa. 3.5.2. The Fine Structure of the Chordal Extension Algorithm. Procedure NONE ....

Klein, Ph. Efficient Parallel Algorithms on Chordal Graphs, Proc. 29 th IEEE FOCS (1988).

....computation of an MEO E of a given graph. Their sequential algorithm works in O(nm) time and O(n m) storage ( RTL 76] There are efficient parallel algorithms to recognize chordal graphs and to compute the perfect elimination ordering for chordal graphs ( Ed 87] NNS 87] DK 86] DK 87] Kl 88] In this paper we give a parallel solution to the MEO Problem by designing an algorithm computing an MEO for any given graph which works in O(log 3 n) parallel time and O(nm) processors on a CRCW PRAM. The MEO algorithm of this paper directly entails recent results on existence of ....

....graph of the collection SG of subtrees of T . For v 2 V let S v be the corresponding subtree in SG . For t 2 VG let c t be the set fS 2 SG j t 2 Sg. We may assume that the maximal cliques of G are exactly the sets c t : fv j S v 2 c t g ( Ga 72] Bu 74] Klein proved the following result ( Kl 88] Theorem 4 There is a parallel algorithm for computing for each chordal graph G a perfect elimination ordering and the subtree representation (TG ; SG ) in time O(log 2 n) and O(n m) processors on a CRCW PRAM. Consider any ordering on the vertex set V of the graph G = V; E) Then the ....

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Klein, Ph., Efficient Parallel Algorithms on Chordal Graphs, Proc. 29 th IEEE FOCS (1988).

....are paths, then the sparse tree representation is of size O(n) In so far optimal parallel algorithms for chordal graphs in their sparse tree representation induce also optimal parallel algorithms for path graphs and interval graphs in their path or interval representation respectively. As [12] we even consider the problem of the computation of a depth first search and a breadth first search tree. Afterwards we consider the problems to find a maximum independent set, a minimum clique cover, and a minimum coloring. Finally we consider the problem to find a minimum dominating set. This ....

....ancestor of a pair of vertices in L v with the smallest distance to the root of T . In that case we add the root of T v to L v . Such a procedure can be implemented in O(log m) time using O(m= log m) processors. 3 Breadth first and Depth First Search We first review the algorithm of P. Klein [12] to compute a breadth first search and a depth first search tree. We assume that a perfect elimination ordering on the vertices of G is known, i.e. if x y,x z, xy 2 E, and xz 2 E then yz 2 E. A depth first search tree is determined by the parent function P (x) min fyjxy 2 E; x yg ....

P. Klein, Efficient Parallel Algorithms for Chordal Graphs, 29 th Symposium on Foundation of Computer Science (1988), pp. 150-161.

....result of this section is the characterization of clique composable graphs as the well known class of chordal graphs. The class of chordal graphs is a strict superset of the class of interval graphs [14] and like interval graphs, chordal graphs have also been the subject of a lot of research [12, 14, 16]. Problems that are NP complete for arbitrary graphs (for example, graph coloring, maximum clique) can be solved in polynomial time for chordal graphs [12] and the problem of recognizing chordal graphs can also be solved in linear time. 2 The structure of an algebraic compiler An algebraic ....

Philip N. Klein. Efficient parallel algorithms for chordal graphs. In Proceedings of the 25th Annual IEEE Symposium on Foundations of Computer Science, pages 150--161, 1988.

.... by Lawler [210] Other special classes of graphs where the maximum clique independent set problem have been studied in the literature can be found in [23] 48] 49] 79] 83] 88] 89] 91] 90] 108] 115] 129] 130] 147] 148] 154] 171] 180] 181] 182] 198] 197] [203], 224] 228] 229] 236] 243] 249] 257] 282] 290] 291] 304] 86] 103] and [325] We should note here that the weighted or unweighted version of the maximum clique problem, the maximum independent set problem, and the minimum vertex cover problem may, with respect to hardness, ....

P.N. Klein, Efficient parallel algorithms for chordal graphs, Proc. 29th Ann. Symp. Found. Computer Sci.: 150--161, 1988.

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P. Klein, Efficient Parallel Algorithms for Chordal Graphs, Proc. IEEE Symposium on Foundations of Computer Science, 1989, pp. 150--161.

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P. Klein. "EfficientParallel Algorithms for Chordal Graphs". Proc. 29th Symp. Found. of Comp. Sci., FOCS 1989, pp. 150--161.

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P. Klein, Efficient Parallel Algorithms for Chordal Graphs, Proc. IEEE Symposium on Foundations of Computer Science, 1989, pp. 150--161.

No context found.

P. Klein. "EfficientParallel Algorithms for Chordal Graphs". Proc. 29th Symp. Found. of Comp. Sci., FOCS 1989, pp. 150--161.

No context found.

P. Klein, Efficient Parallel Algorithms for Chordal Graphs, Proc. IEEE Symposium on Foundations of Computer Science, 1989, pp. 150--161.

No context found.

P. Klein. "EfficientParallel Algorithms for Chordal Graphs". Proc. 29th Symp. Found. of Comp. Sci., FOCS 1989, pp. 150--161.

No context found.

P. Klein. "Efficient Parallel Algorithms for Chordal Graphs". Proc. 29th Symp. Found. of Comp. Sci., FOCS 1989, pp. 150--161.

No context found.

P. Klein, Efficient Parallel Algorithms for Chordal Graphs, Proc. IEEE Symposium on Foundations of Computer Science, 1989, pp. 150--161.

No context found.

P. Klein. "EfficientParallel Algorithms for Chordal Graphs". Proc. 29th Symp. Found. of Comp. Sci., FOCS 1989, pp. 150--161.

No context found.

P. Klein. "EfficientParallel Algorithms for Chordal Graphs". Proc. IEEE Symposium on Foundations of Computer Science, 1989, pp. 150--161.

No context found.

P. Klein, Efficient Parallel Algorithms for Chordal Graphs, 29. IEEE-FOCS (1988), pp. 150-161.

No context found.

P. Klein, Efficient Parallel Algorithms for Chordal Graphs, 29 th Symposium on Foundation of Computer Science (1988), pp. 150-161.

No context found.

P. Klein. "Efficient Parallel Algorithms for Chordal Graphs". Proc. 29th Symp. Found. of Comp. Sci., FOCS 1989, pp. 150--161.

No context found.

P. Klein. "Efficient Parallel Algorithms for Chordal Graphs". Proc. 29th Symp. Found. of Comp. Sci., FOCS 1989, pp. 150--161.

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