Klaus Simon. An improved algorithm for transitive closure on acyclic digraphs. Technical Report A85/13, Universitat des Saarlandes, Fachbereich 10, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by a similar argument about the reachability relation. 24 1 2 k : Fig. 12. Embedded chain of length k. 8.1 Incremental Redundancy Elimination The enumeration algorithm in Section 5 has to maintain the transitive reduction of the dominance graph G. This can be done in time O(nm) see [29,30]) But for all recursive calls of the algorithm the reduction can be computed much faster, only the top level needs to do the full edged reduction. This is because the instances on which recursive calls work are just reduced graphs where one irredundant edge has been added. So we are faced with ....

K. Simon, An improved algorithm for transitive closure on acyclic digraphs, Theoretical Computer Science 58 (1-3) (1988) 325-346.

....equal to the product of the number of edges and vertices of D that are stored in p. 1 Introduction The problem of computing the transitive closure of a digraph was first considered in 1959 by B. Roy [15] and a variety of sequential algorithms to solve this problem have been proposed ever since [1, 2, 7, 8, 9, 14, 16, 19]. These sequential solutions, usually are based on the use of the adjacency matrix of the digraph, considered as a Boolean matrix or use the adjacency matrix in more directed terms as a problem representation [13] Parallel algorithms for this problem where presented by [10, 12] PRAM) 11] ....

....2 In the sequel, we evaluate the complexity of the method. Basically, the algorithm consists of at most 1 dlog pe parallel computations of a sequential transitive closure algorithm. We employ a sequential algorithm whose complexity is the product of the number of vertices and edges of the closure [9, 16]. Consider a worst case example, where D consists of a single path. Then D is a complete acyclic digraph. In this case, each processor j may compute the transitive closure D (S j ) of a digraph D(S j ) where jV (D(S j ) j = p and jE(D (S j ) j = n=p) n Gamma n=p) O(n =p) Since at ....

K. Simon, An improved algorithm for transitive closure on acyclic digraphs, Theoretical Computer Science 58 (1988) 325--346.

....in p. KEY WORDS Parallel algorithm, transitive closure, graph algorithm, CGM, BSP 1 Introduction The problem of computing the transitive closure of a digraph was first considered in 1959 by B. Roy [15] and a variety of sequential algorithms to solve this problem have been proposed ever since [1, 2, 7, 8, 9, 14, 16, 19]. These sequential solutions, usually are based on the use of the adjacency matrix of the digraph, considered as a Boolean matrix or use the adjacency matrix in more directed terms as a problem representation [13] Parallel algorithms for this problem where presented by [10, 12] PRAM) 11] ....

....In the sequel, we evaluate the complexity of the method. Basically, the algorithm consists of at most 1 dlog pe parallel computations of a sequential transitive closure algorithm. We employ a sequential algorithm whose complexity is the product of the number of vertices and edges of the closure [9, 16]. Consider a worst case example, where D consists of a single path. Then D is a complete acyclic digraph. In this case, each processor j may compute the transitive closure D (S j ) of a digraph D(S j ) where jV (D(S j ) j = p and jE(D (S j ) j = n=p) n n=p) O(n =p) Since at most ....

K. Simon, An improved algorithm for transitive closure on acyclic digraphs, Theoretical Computer Science 58 (1988) 325--346.

.... , the subgraph w p induced by the set of events that are relevant to p, that is, the events that initiate or terminate either p or a property incompatible with p, is extracted from w P , the transitive reduction w p of the graph w p is computed by using one of the standard algorithms, e.g. [11, 15]. computing w for each successor e ## of e 2 do computing w p for every property p for each (e # , e ## ) w do if is relevant to(e # ,p) and is relevant to(e ## ,p) then w p computing w p for every property p compute the transitive reduction w ....

K. Simon. An improved algorithm for transitive closure on acyclic digraphs. Theoretical Computer Science, 58(1-3):325--346, 1988.

.... by w P , the subgraph w p induced by the set of events a#ecting p, that is, the events that initiate or terminate either p or a property incompatible with p, is extracted from w P , the transitive reduction w p of the graph w p is computed by using one of the standard algorithms, e.g. [5, 9]. The set of MVIs for p includes all and only the p edges of w p . Hence, for every property p, query processing reduces to the retrieval of the p edges of w p . w p do [p [q e4 e1 e2 e3 [p [q e4 e1 e3 e2 e3 [p p] e4 e1 e3 e2 e3 [p p] e4 (A) B) C) D) ....

K. Simon. An improved algorithm for transitive closure on acyclic digraphs. Theoretical Computer Science, 58(1-3):325--346, 1988.

....(resp. accessible from e 2 ) Then, for every property p P , the subgraph w p is computed by extracting the edges of w (e 1 , e 2 ) such that both their endpoints interfere with p. Finally, the transitive reduction w p of w p is computed by using one of the standard algorithms, e.g. [4, 6]. put in I P red(e 1 ) the predecessors of e 1 put in I Succ(e 2 ) the successors of e 2 I P red(e 1 ) do I Succ(e 2 ) do for each (e # , e ## ) w do if both e # and e ## interfere with p then compute the transitive reduction w p The set of MVIs for a ....

K. Simon. An improved algorithm for transitive closure on acyclic digraphs. Theoretical Computer Science, 58(1-3):325--346, 1988.

....is covered by a set of k known paths, as it is the case for alignment graphs, it is possible to maintain the transitive closure eciently in both time and space. Various works in di erent areas independently noticed that a path cover can be used to materialize the transitive closure in O(kn) space [Simon 1988; Fidge 1988; Mattern 1989; Jagadish 1990] Simon [Simon 1988] presented a (static) algorithm to compute the transitive closure of an acyclic graph, its worst case execution time is O(km r ) where m r is the size of the transitive reduction of the graph. Jagadish [Jagadish 1990] gives a procedure ....

....for alignment graphs, it is possible to maintain the transitive closure eciently in both time and space. Various works in di erent areas independently noticed that a path cover can be used to materialize the transitive closure in O(kn) space [Simon 1988; Fidge 1988; Mattern 1989; Jagadish 1990] Simon [Simon 1988] presented a (static) algorithm to compute the transitive closure of an acyclic graph, its worst case execution time is O(km r ) where m r is the size of the transitive reduction of the graph. Jagadish [Jagadish 1990] gives a procedure to maintain the O(kn) space materialized transitive closure ....

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Simon, K. 1988. An improved algorithm for transitive closure on acyclic digraphs. Theor. Comput. Sci. 58, 325-346.

....1. Y : 2. for every vertex a of G do 3. if there are no outcoming edges from a 4. then Y : Y [ fag; 5. return(Y ) Table 2.6: Finding minimal elements of partially ordered set represented by the graph G transitive edges. An efficient transitive reduction algorithm is presented in [K. Simon, 1985] Its worst case time is O(V Delta E red ) where V is the number of vertices in the graph, and E red is the number of edges after the transitive reduction. The average case running time of the algorithm is O(V 2 Delta log log V ) 5. Transitive closure Given an acyclic directed graph, we ....

....closure Given an acyclic directed graph, we wish to find its transitive closure, that is to add all possible transitive edges to the graph. For every two vertices a and b such that there is a path from a to b, we add an edge from a to b. The algorithm that solves this problem is presented in [K. Simon, 1985] The running time of the algorithm is the same as the running time of the transitive reduction algorithm: the average case running time is O(V 2 Delta log log V ) and the worst case running time is O(V Delta E red ) where V is the number of vertices in the graph, and E red is the number of ....

Klaus Simon. An improved algorithm for transitive closure on acyclic digraphs. Technical Report A85/13, Universitat des Saarlandes, Fachbereich 10, 1985.

....the Bernoulli graphs, p is a measure for the number of edges and, therefore, it is often considered as a function of n. This interpretation of p is particularly relevant for the study of threshold functions (see [2, 11] or [12] Evolution of random graphs) and for the analysis of algorithms (see [4, 13]) Moreover, it implies that for the random graphs exact closed formulae are much more useful than approximations. Since Erd os and R eny have introduced the concept of random graph [6] the problems are usually considered asymptotically, determining for example Poisson approximations [1] ....

Simon, Improved Algorithm for Transitive Closure on Acyclic Digraphs, Theor. Comp. Science, Vol. 58, pp. 325-346, 1988.

....2 1 Introduction Some points: a) What is transitive reduction (b) Why do we need that (e.g. lub and glb in AI applications, and as an opposite extreme of transitive closure) c) Dynamic maintain of transitive reduced graph. d) Previous work in transitive reduction, e.g. [1, 3, 4, 5]. some of them work only for special cases, like strong connected) Given a finite or infinite set U in which a partial ordering # is defined. Let V be a finite subset of U. According to the relation #, assume that a directed acyclic graph (dag) G = V, E) is given and furthermore G is ....

SIMON. An Improved Algorithm for Transitive Closure on Acyclic Digraphs. Theoretical Computer Science 58 (1988) 325--346.

....we will consider some special choices of transition probabilities for which the random variable described by the process assumes a concrete interpretation. The motivation behind this work arises from the average case analysis of combinatorial algorithms, in particular the Simon algorithm [Sim88, CS94] for computing the transitive closure on acyclic digraphs and the so called Approximate Counting Algorithm (see [Fla85, Mor78] a method for keeping approximate counts of large numbers in small registers . In [Sim88] two variants of the above process have been analyzed, namely: n; 1 ....

....analysis of combinatorial algorithms, in particular the Simon algorithm [Sim88, CS94] for computing the transitive closure on acyclic digraphs and the so called Approximate Counting Algorithm (see [Fla85, Mor78] a method for keeping approximate counts of large numbers in small registers . In [Sim88] two variants of the above process have been analyzed, namely: n; 1 Gamma q and n; q : For the first transition probability, the underlying random variable models the size of the transitive closure of a node in the random acyclic digraph G n;p . Closed forms for its distribution ....

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K. Simon. Improved Algorithm for Transitive Closure on Acyclic Digraphs. Theoretical Computer Science, 58:325--346, 1988.

....of chains present in the successor set. As the interval representation, the chain representation can be computed during a single depth rst traversal of the input graph. We developed the chain 2 representation from the method for computing the transitive closure of an acyclic graph by Simon [11]. To compare the dioeerent representations, we studied their average case behavior, since the results of a worst case analysis are often overly pessimistic and even misleading. For instance, many transitive closure algorithms have approximately the same worst case execution time, but behave ....

....the same worst case execution time, but behave dioeerently with typical inputs. Unfortunately, the mathematical average case analysis is much more diOEcult than the worst case analysis. Only a couple of previous articles on transitive closure computation contain an average case analysis [12, 13, 14, 11], and the approach used in these articles cannot be generalized to other methods for computing the transitive closure. Our choice, therefore, was to study the average case performance experimentally. A bene t of this choice was that we could mechanically apply the same technique for all ....

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K. Simon. An improved algorithm for transitive closure on acyclic digraphs. Theoretical Computer Science, 58(1-3):325346, 1988. 26

....we will consider some special choices of transition probabilities for which the random variable described by the process assumes a concrete interpretation. The motivation behind this work arises from the average case analysis of combinatorial algorithms, in particular the Simon algorithm [Sim88, CS94] for computing the transitive closure on acyclic digraphs and the so called Approximate Counting Algorithm (see [Fla85, Mor78] a method for keeping approximate counts of large numbers in small registers . In [Sim88] two variants of the above process have been analyzed, namely: n; 1 ....

....analysis of combinatorial algorithms, in particular the Simon algorithm [Sim88, CS94] for computing the transitive closure on acyclic digraphs and the so called Approximate Counting Algorithm (see [Fla85, Mor78] a method for keeping approximate counts of large numbers in small registers . In [Sim88] two variants of the above process have been analyzed, namely: n; 1 Gamma q For the first transition probability, the underlying random variable models the size of the transitive closure of a node in the random acyclic digraph G n;p . Closed forms for its distribution and moments ....

[Article contains additional citation context not shown here]

K. Simon. Improved Algorithm for Transitive Closure on Acyclic Digraphs. Theoretical Computer Science, 58:325--346, 1988.

....of an event A and Pr ( A j B ) for the conditional probability of A given B. Let X be a random variable, then E(X) will denote the expected value of X and Var (X) its variance. Furthermore, we will use the shortening q = 1 Gamma p in the rest of the paper. The following was proved by Simon [Sim88]. Theorem 1. Let fl n (1) fl r n (1) be the size of Gamma (1) Gamma r (1) in the G n;p model of a random acyclic digraph. Then it holds: 4 1. E(fl r n (1) log n 2 2. E(fl n (1) n Gamma j log pj 1 p 3. E(fl r n (1) p q (n Gamma 1 Gamma E(fl n (1) ....

....t 0 or 62 f1; tg, P 1;1 = 1 and P t; 1 Gamma ) P t Gamma1; Gamma1 P t Gamma1; Gamma1 otherwise; this means that the process starts at epoch 1 from state 1, direct transition from state is only possible to state 1, and this transition has probability . In Simon [Sim88] it is shown that = 1 Gamma q holds for fl n (1) and therefore the quantities P n;h satisfy the recurrence P 1;1 = 1 and P n;h = q h Pn Gamma1;h (1 Gamma q h Gamma1 ) Pn Gamma1;h Gamma1 : 6) We will prove our theorem by induction on n. For n = 1 the hypothesis is trivially ....

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K. Simon. Improved Algorithm for Transitive Closure on Acyclic Digraphs. Theoretical Computer Science, 58, 1988.

.... X j=1 L j (q) q n Gammaj Gamma1 (q) j h n Gamma1 j i : From expression (44) we get an intuitive proof for the following asymptotic result, which follows from (34) and the relation E Theta fl r n = p q Gamma n Gamma E Theta fl n Delta ; 48) which had been proved in [Sim88]: Corollary 3.7 lim n 1 E Theta fl r n = p q X 1 q 1 Gamma q = p q S 0 (q) 49) Proof. By interchanging the order of the summations we obtain from (44) E Theta fl r n = p q n X j=2 j Gamma1 X =1 q 1 Gamma q q n Gammaj j Gamma1 Y i=1 (1 ....

K. Simon. Improved Algorithm for Transitive Closure on Acyclic Digraphs. Theoretical Computer Science, 58:325--346, 1988.

....probability p j 1 Gamma q, we find the transition probability (50) 3 A node is called a source resp. a sink if it does not have any predecessor resp. successor. Next let us consider n; q : 60) This distribution plays a central role in many different algorithmic analyses, such as [Sim88], SCC93] Mor78] Fla85] Ros84] and [GGMM85] In the theory of random graphs it is commonly observed in conjunction with greedy structures. Let us consider for instance the construction of a greedy stable set 4 S in a random undirected labeled graph with n nodes. We start by setting S = f1g ....

K. Simon. Improved Algorithm for Transitive Closure on Acyclic Digraphs. Theoretical Computer Science, 58:325--346, 1988.

....we will consider some special choices of transition probabilities for which the random variable described by the process assumes a concrete interpretation. The motivation behind this work arises from the average case analysis of combinatorial algorithms, in particular the Simon algorithm [Sim88, CS94] for computing the transitive closure on acyclic digraphs and the so called Approximate Counting Algorithm (see [Fla85, Mor78] a method for keeping approximate counts of large numbers in small registers . In [Sim88] two variants of the above process have been analyzed, namely: n; 1 ....

....analysis of combinatorial algorithms, in particular the Simon algorithm [Sim88, CS94] for computing the transitive closure on acyclic digraphs and the so called Approximate Counting Algorithm (see [Fla85, Mor78] a method for keeping approximate counts of large numbers in small registers . In [Sim88] two variants of the above process have been analyzed, namely: n; 1 Gamma q and n; q : For the first transition probability, the underlying random variable models the size of the transitive closure of a node in the random acyclic digraph G n;p . Closed forms for its distribution ....

[Article contains additional citation context not shown here]

K. Simon. Improved Algorithm for Transitive Closure on Acyclic Digraphs. Theoretical Computer Science, 58:325--346, 1988.

....to [8] In Bernoulli graphs p is a measure for the number of edges and is normally considered as a function of n. In particular, the interpretation p = p(n) is essential for studying threshold functions (see [5, 12] or [16] Evolution of random graphs) and for the analysis of algorithms (see [6, 19]) Of course, this meaning of p influences significantly the treatment of our problems. According to the disintegration process, in physics the parameter p stands for a small constant which allows simple approaches. For p = p(n) more adequate techniques are necessary, e.g. the second moment ....

K. Simon. Improved algorithm for transitive closure on acyclic digraphs. Theor. Comp. Science, 58:325--346, 1988.

No context found.

Klaus Simon. An improved algorithm for transitive closure on acyclic digraphs. Technical Report A85/13, Universitat des Saarlandes, Fachbereich 10, 1985.

No context found.

K. Simon. An improved algorithm for transitive closure on acyclic digraphs. Tech. Report A85/13, Universitat des Saarlandes, 1985.

No context found.

Simon, K. (1988). An improved algorithm for transitive closure on acyclic digraphs. Theoret. Comput. Sci., 58:325--346.

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