H. Gazit and G. L. Miller. An improved parallel algorithm that computes the BFS numbering of a directed graph. Information Processing Letters 28:1, 61-65, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....mesh elements that induces a cycle for the shown angle (left) and its dependence graph for the angle shown (right) A sweeping method will deadlock when it encounters a cycle such as this. Gazit and Miller have an NC algorithm which can be used for locating SCCs that uses matrix multiplication [8]. Vishkin and Cole [5] and Amato [3] have proposed optimizations or extensions of this algorithm, but they still require O(n 2:376 ) processors and O(log n) time where n is the number of vertices in the graph. An NC algorithm developed by Kao for planar graphs was developed requiring O(log ....

H. Gazit and G. L. Miller, An improved parallel algorithm that computes the BFS numbering of a directed graph, Inform. Proces. Lett., 28 (1988), pp. 61-65.

....O(log n) time. A parallel algorithm for nding a depth rst search tree in an arbitrary graph was given by Aggarwal and Anderson [2] Their algorithm is randomized and uses O(nM(n) processors. A parallel algorithm for breadth rst search in an arbitrary graph was given by Gazit and Miller [20] that uses M(n) processors. Depth rst and breadth rst algorithms speci cally for chordal graphs have not previously appeared in the literature. To our knowledge, no previous NC algorithm was known for interval graph isomorphism. Recognition of interval graphs was previously shown to be in NC ....

H. Gazit, G. L. Miller, \An improved parallel algorithm that computes the BFS numbering of a directed graph," Inf. Proc. Let. 28 (1988), pp. 61-65.

....semiring. In fact, this algorithm can compute shortest paths. The technique is well known; see [1, 7] for more details. The problem with this algorithm is that it requires too many processors; to achieve O(log n) time requires about n processors. The processor bound has been improved somewhat [9] in the case of breadth first search. The most elementary parallel search technique is parallel breadth first search, in which the nodes are visited level by level as the search progresses. Level 0 consists of the starting node s, level 1 consists of the neighbors of s, level 2 consists of the ....

H. Gazit and G. L. Miller, "An improved parallel algorithm that computes the BFS numbering of a directed graph," Information Processing Letters 28 (1988), pp. 61-65.

....this just implies the existence of c 1 and N . If f(n) O(g(n) then c 2 and N exist. 1 2 Fleischer, Hendrickson and Pinar components problem (SCC) that avoid the use of depth first search. Gazit and Miller devised an NC algorithm for SCC, which is based upon matrix matrix multiplication [10]. This algorithm was improved by Cole and Vishkin [6] but still requires n 2:376 processors and O(log 2 n) time. Kao developed a more complicated NC algorithm for planar graphs that requires O(log 3 n) time and n= log n processors [12] More recently, Bader has an efficient parallel ....

....Despite our conviction that our divide and conquer approach is amenable to effective parallelization, we have not shown that it leads to an improved NC algorithm. Specifically, the reachability analyses at the core of our method can be solved in NC via Gazit and Miller s matrix product approach [10], but their technique can be used to find strongly connected components directly. Another question of interest to us is the potential for a linear time algorithm for strongly connected components that does not rely on depth first search. Any such algorithm could be a candidate for an improved ....

H. Gazit and G. L. Miller, An improved parallel algorithm that computes the BFS numbering of a directed graph, Inform. Process. Lett., 28 (1988), pp. 61--65.

....first search is P Complete [7] which in practical terms means it will not scale well on a parallel machine. 3 There are some parallel algorithms for detecting SCCs that do not rely on depth first search. Gazit and Miller have an NC algorithm for locating SCCs that uses matrix multiplication [10]. Vishkin and Cole further improved this algorithm [9] but it is still requires (O 2:376 ) processors and O(log 2 n) time where n is the number of vertices in the graph. An NC algorithm developed by Kao for planar graphs was developed requiring O(log 3 n) time and n=log n processors [11] ....

H. Gazit and G. L. Miller, An improved parallel algorithm that computes the BFS numbering of a directed graph, Inform. Proces. Lett., 28 (1988), pp. 61-65.

....are also available for this problem. However, they either need #(n 3 log n) operations or only work for the more narrow class of the input graphs and or digraphs (which have the edge weights bounded, say, by a constant or have a family of small separators available) AGM] DNS] DS] [GM], L] PK] PR1] PR2] S] The recent algorithm of [PP1] and [PP2] uses O(log 2.5 n) parallel time and O(n 3 ) operations in the case of a general graph with n vertices. In this paper we improve the latter time bound to the new record value of O(I (n) log n) still using O(n 3 ) ....

H. Gazit and G. L. Miller. An improved parallel algorithm that computes the BFS numbering of a directed graph. Inform. Process. Lett. 28(1):61--65, 1988.

....depth rst search is P Complete [7] which in practical terms means it will not scale well on a parallel machine. There are some parallel algorithms for detecting SCCs that do not rely on 3 depth rst search. Gazit and Miller have an NC algorithm for locating SCCs that uses matrix multiplication [10]. Vishkin and Cole further improved this algorithm [9] but it is still requires (O 2:376 ) processors and O(log 2 n) time where n is the number of vertices in the graph. An NC algorithm developed by Kao for planar graphs was developed requiring O(log 3 n) time and n=log n processors [11] ....

H. Gazit and G. L. Miller, An improved parallel algorithm that computes the BFS numbering of a directed graph, Inform. Proces. Lett., 28 (1988), pp. 61-65.

....approximate estimation on the path lengths. That is, if the (i; j) element of A 2 r becomes 1 for the first time, we can say that the shortest path length from i to j is between 2 r01 1 and 2 r for r 1. As r gets large, the gap between 2 r01 1 and 2 r gets large. As Gazit and Miller [6] observe we can fill the gap in increasing order of r in the following way. Let shortest paths up to 2 r01 be computed already. Then a shortest path from i to j, whose length is between 2 r01 1 and 2 r consists of a shortest path from i to k whose length is up to 2 r01 and a path from k ....

H. Gazit and G. Miller, An improved parallel algorithm that computes the bfs numbering of a directed graph, Info. Proc. Lett. 28 (1988) pp. 61--65.

....in G. 2: Compute maxfr v j v in Gg. The classic naive parallel algorithm can compute the first step as a kind of graph reachability problem. It computes the transitive closure of the adjacency Boolean matrix of G, and r v is given by the number of 1s of the vth row (the details are discussed in [4, 16]) Thus, the first step is computed by using O(log n) time and n 3 processors on a CRCW PRAM. Clearly, the resources are sufficient to compute the second step. Thus, the problem is in NC 2 . 3 Lexicographically first maximal independent set problem 3.1 An algorithm for the LFMIS problem The ....

H. Gazit and L. Miller. An Improved Parallel Algorithm That Computes the BFS Numbering of a Directed Graph. Imformation Processing Letters, 28:61--65, 1988.

....most specific p edges w.r.t. the object o. The algorithm is based on digraph reachability, which is implemented by computing the transitive closure of the digraph 14 induced by the WIN. The reachability process is reduced to a breadth first search (BFS) numbering of a directed acyclic graph. In [9], a parallel algorithm is given that solves this problem in O(log 2 n) time using O(M(n) processors. After computing a single source BFS numbering of the nodes, we can decide whether a node is reachable from the source by examining its distance from the source. The directed acyclic graph ....

....ffl Subset Of(S) This operation computes all the subsets of the set S. It is used in the algorithm to activate appropriate sets of processors. First, the algorithm identifies the active nodes and edges (line 3) This is effected by a single source multiple sink reachability process taken from [9]. From that stage on, the predicate Is Active(o) that is subsequently used, is well defined and computable in constant time. In lines 5 through 7, all the plus weights, except those of the form ( p) are converted to actual edges. Then the algorithm locates all the p edges (lines 8 through 10) ....

H. Gazit and G.L. Miller, An Improved Parallel Algorithm that Computes the BFS Numbering of a Directed Graph, Information Processing Letters, 28:61--65, 1988.

....algorithm cannot be used for our problem. Alternatively, there exist several parallel algorithms for the strongly connected components problem (SCC) that avoid the use of depth first search. Gazit and Miller devised an NC algorithm for SCC, which is based upon matrix matrix multiplication [9]. This algorithm was improved by Cole and Vishkin [6] but still requires n 2:376 processors and O(log 2 n) time. Kao developed a more complicated NC algorithm for planar graphs that requires O(log 3 n) time and n= log n processors [11] More recently, Bader has an efficient parallel ....

H. Gazit and G. L. Miller, An improved parallel algorithm that computes the BFS numbering of a directed graph, Inform. Process. Lett., 28 (1988), pp. 61--65.

....computation steps of algorithms. The applications we present focus on the parallel complexity of breadth first search (BFS) 10] and depth first search (DFS) 28] BFS is in NC. Gazit and Miller presented an algorithm for assigning BFS level numbers of a directed graph using the EREW PRAM model [11]. Their algorithm runs in time O(log 2 n) and uses M(n) processors, where M(n) denotes the number of processors required to multiply two n Theta n matrices in log n time on an EREW PRAM. Throughout the remainder of this paper, M(n) denotes this value. The best known bound for M(n) is O(n ....

....takes as input G, produces the BFS level number for each node, and assigns each node its ordered BFS number indicating the order the vertex was visited within its level. procedure REDUCE 1. Using the algorithm presented by Gazit and Miller for computing BFS level numbers of a directed graph [11], label all nodes in G with their level numbers. 2. In parallel for all edges in G, determine if an edge connects a vertex at level i to a vertex at level i 1. If it does not then delete it from E. Call the newly formed subgraph of G, H = V; E 0 ) 3. Use an oracle call to obtain an ordered ....

H. Gazit and G. L. Miller. An improved parallel algorithm that computes the bfs numbering of a directed graph. Information Processing Letters, 28(2):61--65, 1988.

....on the choice of F 1 . If we choose the topological ordering top 1 , this step takes O(log 2 n) time with O(n) processors by using the algorithm in [12] If we choose level 1 , this step can be implemented by using the breadth first search in G 1 in O(log 2 n) time with M(n) processors [8]. A more compact rectangular dual will be computed in this case) x 1 (v) x 2 (v) for each v 2 V can be computed in O(1) time with O(n) processors. The implementation of Step (3) is similar. Step (4) takes O(1) time with O(n) processors. Thus the central part of Algorithm 3 is Step (1) This ....

H. Gazit and G. L. Miller, An Improved Parallel Algorithm That Computes the BFS Numbering of a Directed Graph, Info. Proc. Lett. 28 (1988), pp. 61-65.

.... e) 99] where n and e are the number of nodes and edges, respectively. Therefore, the processor time product for the parallel algorithm is far in excess of the time required to solve the problem serially. The best known bounds for a BFS algorithm are O(log 2 n) using O(n 2:376 ) processors [51, 37]. This means BFS is placed in NC 2 . Techniques related to transitive closure are used to exhibit the algorithm. However, since BFS can be solved sequentially in time O(n e) 99] this algorithm is not efficient for the same reason as given above. Reghbati and Corneil [82, 80] describe an ....

GAZIT, H. and MILLER, G. L. An Improved Parallel Algorithm that Computes the BFS Numbering of a Directed Graph. Inf. proc. Letters 28 (1988), 61--65.

....of processors. Kucera[1] s classic algorithm employs matrix powering over the semiring (N,min, using O(logn) time and O(n 3 ) processors on the CRCW PRAM, or using O(log 2 n) time and n 3 =logn processors on the EREW PRAM. Using improved matrix multiplication algorithm, Gazit and Miller[2] provided an O(log 2 n) time and M (n) processor EREW PRAM algorithm for solving the above problem, M (n) being the number of processors needed for matrix multiplication of two n Theta n integer matrices in O(logn) time. Currently, the best known value of M (n) is O(n 2:376 ) The algorithm ....

H. Gazit and G. L. Miller,An improved parallel algorithm that computes the BFS numbering of a directed graph.IPL 28(1988),61-65.

....in G. 2: Compute maxfr v j v in Gg. The classic naive parallel algorithm can compute the first step as a kind of graph reachability problem. It computes the transitive closure of the adjacency Boolean matrix of G, and r v is given by the number of 1s of the vth row (the details are discussed in [5, 16]) Thus, the first step is computed by using O(log n) time and n 3 processors on a CRCW PRAM. Clearly, the resources are sufficient to compute the second step. Thus, the problem is in NC 2 . ut 3 Lexicographically first maximal independent set problem 3.1 An algorithm for the LFMIS problem ....

H. Gazit and L. Miller. An Improved Parallel Algorithm That Computes the BFS Numbering of a Directed Graph. Information Processing Letters, 28:61--65, 1988.

....are in the min plus semiring. In fact, this algorithm can compute shortest paths (see [2,25] for more details) The problem with this algorithm is that it requires too many processors; to achieve O(log n) time requires about n 3 processors. The processor bound has been improved somewhat [41] in the case of breadth first search. The most elementary parallel search technique is parallel breadth first search. In this technique the nodes are visited level by level as the search progresses. Level 0 consists of the starting node s, level 1 consists of the neighbors of s, level 2 consists ....

H. Gazit and G. L. Miller, "An improved parallel algorithm that computes the BFS numbering of a directed graph," Information Processing Letters 28 (1988), 61--65.

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H. Gazit and G. L. Miller. An improved parallel algorithm that computes the BFS numbering of a directed graph. Information Processing Letters 28:1, 61-65, 1988.

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