Alfred V. Aho, M. R. Garey, Jeffrey D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput, 1(2), pages 131--137, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Con ict of Interest Detection For every role r 2 fng [ R MaxRole Do If E ective(r) contains a pair of privileges which is in P Con icts Then abort (message: Privilege addition creates a con ict) end. Figure 3. Privilege insertion algorithm from [3] moved using an algorithm from [1]. After that, the step Adjust Direct and E ective of a ected roles would use any of these new edges to propagate new privileges to the e ective set of all roles senior in the graph, and also possibly remove privileges from a direct set. We now look at the revised algorithm which will take into ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal of Computing, 1(2):131-137, June 1972.

....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] ....

A. V. Aho, M. R. Garey and J. D. Ullman, The transitive reduction of a directed graph, SIAM J. Comput. 1 (1972) 131--137.

....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] ....

A. V. Aho, M. R. Garey and J. D. Ullman, The transitive reduction of a directed graph, SIAM J. Comput. 1 (1972) 131--137.

....e j in G if and only if there is a directed path from e i to e j in G . The transitive closure of G is the (unique) graph G such that, for any pair of nodes e i , e j E there is a directed path from e i to e j in G if and only if there is an edge (e i , e j ) in G . Aho et al. [1] show that every (directed) graph has a transitive reduction, which can be computed in polynomial time. They also show that such a reduction is unique in the case of directed acyclic graphs. Furthermore, they prove that the time needed to compute the transitive reduction of a graph di#ers from the ....

....and both of them are more e#cient than GO 0 . On the contrary, in case B4, the actual value of q and u must be taken into account, because n is small. However, if we do not know the actual value of q or u, we can compute the average value (n) of the function G(q, n) varying q over the interval [0, 1]. It holds that (n) 0 G(q, n)dq. We have that (n) n 1) for GO 0 , n) n 1) for GO 1 , and (n) n log n) for GO 2 . Hence, regarding to the average value of G(q, n) GO 1 is the best solution. The complexity bounds in case A2 are the following ....

A.V. Aho, M.R. Garey, and J.D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131--137, 1972.

....path from i to j in G if and only if there is a directed path from i to j in G . The transitive closure of G is the (unique) graph G with the property that, for any pair of nodes i, j E there is a directed path i to j in G if and only if there is an edge (i, j) in G . In [1], Aho et al..ii show that every (directed) graph has a transitive reduction, which can be computed in polynomial time. They also show that such a reduction is unique in the case of directed acyclic graphs. Furthermore, they prove that the time needed to compute the transitive reduction of a graph ....

A. V. Aho, M. R. Garey, and J. Ullman. The transitive reduction of a directed graph. SIAM Journal of Computing, 1(2):131--137, 1972.

....56 7.3.8 Analysis of procedure neutralNC . 57 7.3.9 Complexity of q deletions . 57 J.A. La Poutr and J. van Leeuwen 1 Introduction Let G = V, E be a directed graph, G = V, E its transitive closure and G = V, E its transitive reduction (cf. [1]) Let E and E be represented by incidence matrices. Suppose edges are inserted in and deleted from G one at a time. We consider the problem of efficiently updating G and G each time an edge is inserted or deleted. Ibaraki and Katoh [2] presented two algorithms that update G when edge ....

....as follows. In Section 2 we define some notions and in Section 3 we present some restricted algorithms, as an introduction to the ultimate algorithms. In Section 4 notions with respect to strongly connected components are introduced and the transitive reduction of a graph is defined as in [1]. Section 5 gives a precise description of the problems to solve. In Section 6 and Section 7 the procedures for edge insertions and edge deletions are presented, including correctness proofs and complexity considerations. Finally, in Section 8 the results are condensed in some theorems and some ....

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A.V. Aho, M.R. Garey and J.D. Ullman, The transitive reduction of a directed graph, SIAM Journal of Computing, vol.l, no.2 (June 1972) 131-137.

....that one would expect for lexicographic. The complete function close is given in Figure 9. Because this definition of close satisfies Lemmas 10 and 11, W os remains correct if this new close is used. One important aspect of close that has not been mentioned is its use of transitive reductions [1]. As we perform monotonicity based instantiations, we maintain the set of inclusion constraints E i in reduced form. This provides an efficient implementation of the guard of the while loop in the case where a variable ff must be shrunk: in this case, the only possible instantiation for ff is its ....

Alfred V. Aho, Michael R. Garey, and Jeffrey D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131--137, June 1972.

....Floyd Warshall algorithm for all pairs shortest paths on arbitrary graphs. For dense graphs the third algorithm is the fastest; it relies on the fact that boolean matrix multiplication, transitive closure, and transitive reduction are equivalent problems with the same asymptotic running time [1]. Further reduction of edges is obtained by either edge concentration or search for complete bipartite subgraphs. We choose two arbitrary levels A and B and consider the induced bipartite subgraph G; V (G) A [ B: In both cases we search for a covering of G by complete bipartite subgraphs (A 1 ; ....

A. V. Aho, M. R. Garey, and J. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131--137, 1972.

....have one edge of type A for every edge of G, a complete graph on 2 indeg v nodes for every root v, and a graph of size 1 outdeg v for every leaf v. Thus the auxiliary graph A has n 0 = m nodes and m 0 = O(m) P v2V (2 indeg v ) 2 edges. The graph G can be reduced in time O(nm) see [AGU72]. Then we have no parallel edges, and hence a root r with indegree greater than n must have two dominance edges from di erent leaves of the same tree to r, which is trivial to recognize in time O(nm) So we can assume that the indegree of any root is at most n. Let us say that we have r n roots ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1:131-137, 1972.

....intended graph; an observer can infer her signatures on the remaining (inferred) edges. We note for the record that this minimum subset of edges having the same transitive closure is called the transitive reduction of a graph and can be computed eciently in both the undirected and directed cases[1]. Transitive Signature Schemes 5 With a transitive signature scheme, anyone can produce a short proof that a given edge is in the graph. Even if Alice didn t sign that edge explicitly, the proof is a signature that might as well been produced by Alice. Most conveniently, one does not need to ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput., 1:131-137, 1972.

....These results provide practical approximation algorithms for NP hard network design problems via an increased understanding of connectivity properties. Until now, the techniques developed have been applicable only to undirected graphs. We consider a basic network design problem in directed graphs [2, 12, 13, 18] which is as follows: given a digraph, find a smallest subset of the edges (forming a minimum equivalent graph (MEG) that maintains all reachability relations of the original graph. When the MEG problem is restricted to strongly connected graphs we call it the minimum SCSS (strongly connected ....

....algorithms for the restricted problems. Hsu [13] gives a polynomial time algorithm for the acyclic MEG problem. The related problem of finding a transitive reduction of a digraph a smallest set of edges yielding the same reachability relations is studied by Aho, Garey and Ullman [2]. Transitive reduction differs from the MEG problem in that the edges Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. Research supported by NSF Research Initiation Award CCR9307462. E mail : samir cs.umd.edu. y Computer ....

A. V. Aho, M. R. Garey and J. D. Ullman, The transitive reduction of a directed graph, SIAM Journal on Computing, 1 (2), pp. 131--137, (1972).

....component) Thus, the problem reduces in linear time to two cases: the graph is either acyclic or strongly connected. If the graph is acyclic, the MEG problem is equivalent to the transitive reduction problem, which was shown by Aho, Garey and Ullman to be equivalent to transitive closure [1]. Thus, we assume the graph is strongly connected, so that the problem is to find a small subset of the edges preserving the strong connectivity. We refer to this problem as the strongly connected spanning subgraph (SCSS) problem. Computer Science Department and Institute for Advanced Computer ....

A. V. Aho, M. R. Garey and J. D. Ullman, The transitive reduction of a directed graph, SIAM J. Comput., 1 (2), pp. 131--137, (1972).

....Indeed, one may use the transitive closure graph (i.e. G = X; E tc ) s.t. xy 2 E i x y) or the covering graph (i.e. D = X; E tr ) s.t. xy 2 E i y covers x) or any graph H whose transitive closure is G = X; E tc ) The algorithm that compute D or G from H is not linear in the general case, see [1]. From the original de nition of distributive lattices, one can produce a naive recognition algorithm running in O(n 3 ) when the input is a covering graph D = X; E tr ) with jXj = n. Under the same assumption, recently Bordat [5] has proposed an O(jE tr jloglog ) algorithm based upon ....

....from complexity point of view, the dominating step is the fourth one and yields the overall complexity of the algorithm. In this algorithm we have avoided the use of a general purpose algorithm for computation of the covering of G, since such an algorithm has not a linear time complexity [1]. But the complexity of the above algorithm is not 11 optimal, since it might be possible to produce an algorithm in O(jXj jEj) In fact using more sophisticated techniques we already have obtained an algorithm [16] that achieves O(jXj jEj jM(L)j) time complexity, where L is the lattice ....

A.V. Aho, M.R. Garey, and J.D Ullman. The transitive reduction of a directed graph. SIAM J. Computing, 1(2):131-137, 1972.

....# ) such that for every choice of vertices x, y # V there is a directed path from x to y in D if and only if D # has such a path. The minimum equivalent digraph problem and its generalizations to higher degrees of connectivity has practical applications and has been studied extensively, see e.g. [1, 10, 13, 14, 17]. Furthermore, for a given class of digraphs, which is closed under the operation of taking induced subdigraphs, one can find the minimum equivalent digraph in polynomial time if and only if one can solve the MSSS problem in polynomial time for that class. Hence it is of interest to find classes ....

A.V. Aho, M.R. Garey and J.D. Ullman. The transitive reduction of a directed graph, SIAM J. Computing 1(2) (1972) 131-137.

....to acknowledge financial support from the Danish Research Council (under grant 9800435) 1 has as few arcs as possible. This problem, which generalizes the hamiltonian cycle problem and hence is NP hard, is of practical interest and has been considered several times in the literature, see e.g. [1, 12, 15, 16, 17, 18]. The MSSS problem is an essential subproblem of the so called minimum equivalent digraph problem. Here one is seeking a spanning subgraph with the minimum number of arcs in which the reachability relation is the same as in the original graph (i.e. there is a path from x to y if and only if the ....

A. V. Aho, M. R. Garey and J. D. Ullman. The transitive reduction of a directed graph, Siam J. Computing 1(2) (1972) 131-137.

....r j , if r i :rpset ae r j :rpset, then there must be a path from r i to r j . This implies that there is a path from MinRole to every r i , and there is a path from every r i to MaxRole. The role graphs are drawn without redundant edges, i.e. the graph is represented by its transitive reduction [AGU72]. The nodes are arranged on the page so that all is Gamma junior edges go up the page. In addition, for every role, one can distinguish between its effective privileges, and its direct privileges. The direct privileges of role r are those which are not contained in the rpset of any of r s ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal of Computing, 1(2):131--137, June 1972.

....than H(G) that satisfies condition (i) Intuitively, we can describe a Hasse graph as a graph with no short cuts. If there is a path from u to v containing at least one node different from u or v, then no (u,v) edge exists in the Hasse graph. The Hasse graph of an directed acyclic graph is unique [1]. The parse will be found using the Hasse graph corresponding to the input DAG. 2.2 Clans The central focus of our decomposition algorithm is the clan. Let G be a dependency graph. A subset X G is a clan iff for all x, y X and all z G X, a) z is an ancestor of x iff z is an ancestor of y, or ....

A. V. Aho, R. Garey, and J. D. Ullman, Transitive Reduction of a Directed Graph, SIAM J. Comput. 1 (1972), 131-137.

....m 0 if and only if message m 0 is sent in the context of message m; i.e. the process that sent m 0 had either sent m or already received m. Let G OE denote the directed acyclic graph representation of OE. A context graph, denoted G = M; E) is taken to be the transitive reduction of G OE [2]. That is, G contains all the vertices and none of the redundant edges of G OE , where edge (m; m 0 ) is redundant if G OE also contains a path from m to m 0 of length greater than one. Figure 1 gives G OE and G for a conversation in which m 1 was the initial message of the conversation; m 2 ....

A. Aho, M. Garey, and J. Ullman. The transitive reduction of a directed graph. SIAM J. Computing (1972), 131-137.

....product of a lower triangular matrix L (with unit diagonal) times an upper triangular matrix U consists of n major steps. In the outer product formulation of Gaussian elimination, A is transformed into the product L [k] A [k] U [k] after step k Gamma 1 and just before step k (1 k n; A [1] = A; L [n 1] L; U [n 1] U ) Throughout this paper, the notation X [k] will refer to the state of X just before step k, where X is a matrix, set, graph, scalar, etc. The state of X after LU factorization is complete is denoted by X [n 1] The active submatrix, A [k] k: n;k: n , ....

....contribution. That is, F [k] F [k] L [ F [k] U F [k] L = n ht; row ii j i 2 L [k] t o F [k] U = n ht; column ji j j 2 U [k] t o : These edges are referred to as active L edges and active U edges, respectively. 4. 1 The basic graph Before factorization starts, the dag G [1] is empty and A [1] is the bipartite graph of the original matrix, A. The pivot search selects a pivot and permutes it to the first row and column (renaming them as row and column one, in our notation) Consecutive pivots with identical pattern are included in the first frontal matrix E 1 ....

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A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput., 1:131--137, 1972.

....can also represent reference parameter alias pairs, for example. Although the compact representation is not graph based, it can easily be mapped to and from graph based representations [Jones and Muchnick 1981] The compact representation is not equivalent to performing a transitive reduction [Aho et al. 1972] over the alias graph during alias analysis in that transitive edges are not discarded. We elaborate on the transfer function presented in (2) and (3) of Section 2 using the compact representation. Consider the assignment statement p i = q j , where p i is a pointer expression with i levels of ....

Aho, A. V., Garey, M. R., and Ullman, J. D. 1972. The transitive reduction of a directed graph. SIAM Journal on Computing 1, 2, 131--137.

....actually permits the effects of processing one element in a row to be visible at the time of processing another element in that row, but the correctness is not compromised. Effectively, the marking optimization is equivalent to considering only the immediate successors in the transitive reduction [3] of the given graph) The Blocked Hybrid Algorithm The processing constraint is satisfied since elements in the lower triangular half are processed block row wise, with the off diagonal block of each block row being fully computed prior to the computation of the diagonal block. The result ....

A. V. Aho, M. R. Garey, and J. D. Ullman, "The Transitive Reduction of a Directed Graph," SIAM J. Computing, 1(2), June 1972, pp. 131-137.

....D = V; A) find a spanning subdigraph D 0 = V; A 0 ) such that for every choice of vertices x; y 2 V there is a directed path from x to y in D if and only if D 0 has such a path. The minimum equivalent digraph problem has practical applications and has been studied extensively, see e.g. [1, 13, 14, 17]. Furthermore, for a given class of digraphs, which is closed under the operation of taking induced subdigraphs, one can find the minimum equivalent digraph in polynomial time if and only if one can solve the MSSS problem in polynomial time for that class. Hence it is of interest to find classes ....

A.V. Aho, M.R. Garey and J.D. Ullman. The transitive reduction of a directed graph, SIAM J. Computing 1(2) (1972) 131-137.

....i.e. ha; bi is written as (a; b; P ) where P signifies the (may) alias is possible. The compact representation can also represent reference parameter alias pairs. The compact representation, like the points to representation, is not equivalent to performing a transitive reduction [Aho et al. 1972] over the alias graph during alias analysis. The compact and points to representations both consider path length information, i.e. number of dereferences, when combining alias relations. A transitive reduction representation does not capture this information. In classical data flow analysis ....

Aho, A. V., Garey, M. R., and Ullman, J. D. 1972. The transitive reduction of a directed graph. SIAM Journal on Computing 1, 2, 131--137.

....closure of the original graph, the arc is deleted; otherwise, it is kept. The algorithm of Figure 6.17 has complexity O(mn 4 ) where m and n are the number of arcs and nodes in graph G, respectively. More efficient algorithms for computing the transitive reduction in O(n 2:8 ) time exist [108]. 120 procedure Reduction(G) f ( G is a directed graph, G = X; A) H = G foreach (x y) 2 A) f G = G Gamma (x y) if G 6= H f G = G (x y) g g g Figure 6.17 Algorithm to compute the transitive reduction of a graph G. The direct benefit of eliminating redundant dependences is ....

A. V. Aho, M. R. Garey, and J. D. Ullman, "The transitive reduction of a directed graph," SIAM Journal of Computing, vol. 1, pp. 131--137, June 1972.

....of the triangular matrix and the right hand side vector. In Section 3, we define elimination dags. They are simply the smallest structures that preserve the set of paths in the graphs of the lower and upper triangular factor matrices. In graph theoretic terms, the edags are transitive reductions [1] of the graphs of L and U . We show that if the matrix is symmetric, elimination dags are simply elimination trees. We prove some graph theoretic results about the path structure of the graph of a matrix, its triangular factors, and its elimination dags. Section 4 contains the main results, which ....

....dags for sparse LU factorization. 3.1. Transitive reduction of a directed graph. Theorem 2. 1 relates the structure of the solution vector x to information about paths in the directed graph G(L) One economical way to represent path information for a directed graph is by its transitive reduction [1]. The idea is to find another directed graph with fewer edges than the given graph G(L) but with the same path structure. In such a graph, fewer edges would need to be traversed to generate Struct(x) for a given Struct(b) A graph G ffi is a transitive reduction of a given directed graph G if ....

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A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1:131--137, 1972. 19

....functional dependency. For the remaining operators, operations of type (b) have the same cost in both representations. c) Operations that update the schema: operations which add delete edges levels to from the schema, and compute the transitive reduction (i.e. the graph without transitive edges) [2] of the updated schema. They have the same cost for all the operators. d) Operations that update the instance: in general, they are additions deletions of roll up functions associated to edges added deleted by means of the operations described in (c) Their execution cost is higher for the ....

A. V. Aho, M. Garey, and J. Ullman. The transitive reduction of a directed graph. SIAM J. Computing 1:2, pgs. 131-137, 1972.

....out in linear time in the number of nodes and edges in the original graph. 2.2 Transitive reduction For the alarm placement problem, we are only interested in knowing whether there is path from one node to another in the graph. The transitive reduction of a directed acyclic graph, defined in [1], is unique and the time complexity of this reduction is the same as that of computing the transitive closure. 3 Preliminary results Let G = V; E) be a condensed, transitively reduced, directed acyclic graph with jV j = n. Proofs for lemmas are omitted. Definition 3.1: The level of a node in a ....

A. Aho, M. R. Garey, and J. D. Ullman, "The transitive reduction of a directed graph," SIAM J. Comput., vol. 1, no. 2, pp. 131-137, June 1972.

....another in the graph. A directed graph G t is said to be a transitive reduction of the directed graph G provided there is a directed path from node u to node v in G t if and only if there is such a path in G, and that there is no graph with fewer edges than G t satisfying the above property [1]. The transitive reduction of a directed acyclic graph is unique and the time complexity of obtaining the transitive reduction is the same as that of computing the transitive closure. 3 Preliminary Results Let G = V; E) be a condensed, transitively reduced, directed acyclic graph with jV j = n. ....

A. Aho, M. R. Garey, and J. D. Ullman, The transitive reduction of a directed graph, SIAM J. Comput. 1 (1972) 131--137.

....we introduce a formal definition on nonredundant graphs. Definition 6.1 A non redundant graph is a directed graph such that there is no direct edge between two nodes if there exists a path between the two nodes. In graph theory, there is a similar concept, transitive reduction of a directed graph [5]. Definition 6.2 The transitive reduction of a directed graph G is the directed graph G with the smallest number of edges such that for every path between vertices in G, G has a path between those vertices. We will show that given a graph G, a non redundant graph G is a G s transitive ....

....minimal edges. There are two approaches to build a non redundant DDG. The first is a reduction based approach, in which a DDG is built first, and a transitive reduction algorithm can then be used to eliminate redundancies [27] It has shown that the time complexity of this approach is O(n 3 ) [5, 55]. The second is a direct approach in which a non redundant DDG is built directly. In our case, since we our graph is a DAG, better algorithms can be designed if we can take advantage of the leveldirected graph representation. The two algorithms that we will introduce are based on the direct ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131--137, June 1972.

....( of Lemma 5.3, can be done in O(n d 3 ) time. Observe that Theorem 5.4 yields that Delta 0 = Delta(G) Delta. Subsequently, by Lemma 5.5, the partial order ( Delta; can be computed in O(n 2d 3 ) time. The transitive reduction ( Delta; 0 ) can be found in O(n 2:376(d 1) time [1, 8]. The steps 4 to 6 clearly take less than this time, while step 7 needs O(n 2d 4) time. By Theorem 5.7, a solution for Minimum Fill In or, as Corollary 5.8 implies, for Interval Completion corresponds exactly to a maximal chain in ( Delta; with a minimum number of new edges in G C . This ....

A.V. Aho, M.R. Garey, and J.D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput., 1:131--137, 1972.

....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 ....

A. V. AHO AND M. R. GAREY AND J. D. ULLMAN. The transitive reduction of a directed graph, SIAM Journal of Computing 1(2), 1972.

....be a tautology, P 1 : P k ) must be unsatisfiable. These two conditions can be checked efficiently by using a graph representation for the conjunction (Ishakbeyoglu and Ozsoyoglu 1992) Constraints should not contain redundant predicates either. Using the transitive reduction of a directed graph (Aho et al. 1972), a non redundant subset of a conjunction of inequalities can be found in polynomial time (Yu and Ozsoyoglu 1984) In this paper, without loss of generality, we assume that constraints are not tautologies, and also do not contain redundant predicates. We also assume that the constraint base is ....

Aho, A. V., Garey, M. R., Ullman, J. D., The Transitive Reduction of a Directed graph, SIAM Journal of Computing, 1:2, 1972, 31-7.

....the squares of directed acyclic graphs Clearly, the square of DAG is a DAG. All square roots of a DAG G 2 contain the transitive reduction of G 2 as a subgraph. The time complexity of finding both the transitive closure and reduction of a digraph is equivalent to boolean matrix multiplication [1]. 7 Acknowledgments We thank Gene Stark for providing a simpler proof of Theorem 5.7, and Alan Tucker for valuable discussions. ....

A. Aho, M. Garey, and J. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1:131--137, 1972.

....u; v such that u terminates before v starts, add the edge (u; v) to E. 3. Remove from E the edges that appear in both directions. 4. Compute the transitive reduction 2 of G. 2 The transitive reduction of a directed graph G is the smallest subgraph of G that has the same closure as G [AGU72] A DAG has a unique transitive reduction. A B D C E E A B D C Figure 3: Example 6 5. Return (V; E) Theorem 4 Given a log of m executions of a given process having n activities, Algorithm 1 computes the minimal conformal graph in O(n 2 m) time. Proof: First we show that after step 3, G is ....

....2, the minimal dependency graph. Lemma 3 shows that this graph is also conformal and, since a conformal graph has to be a dependency graph, it is the minimal conformal graph. Since m AE n, the second step clearly dominates the running time. The running time of step 4 is O(jV jjEj) O(n 3 ) AGU72] A simpler algorithm to compute the transitive reduction is given in the Appendix. 2 Example 6 Consider the log fABCDE, ACDBE, ACBDEg. After step 3 of the algorithm, we obtain the first graph of Figure 3 (the dashed edges are the edges that are removed at step 3) from which the next underlying ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal of Computing, 1(2), 1972.

....Danish Research Council (under grant 9800435) spanning subgraph D 0 of D such that D 0 has as few arcs as possible. This problem, which generalizes the hamiltonian cycle problem and hence is NP hard, is of practical interest and has been considered several times in the literature, see e.g. [1, 12, 15, 16, 17, 18]. The MSSS problem is an essential subproblem of the so called minimum equivalent digraph problem (in fact, these two problems can be reduced to each other in polynomial time) Here one is seeking a spanning subgraph with the minimum number of arcs in which the reachability relation is the same as ....

A.V. Aho, M. R. Garey and J. D. Ullman. The transitive reduction of a directed graph, Siam J. Computing 1(2) (1972) 131-137.

....These results provide practical approximation algorithms for NP hard network design problems via an increased understanding of connectivity properties. Until now, the techniques developed have been applicable only to undirected graphs. We consider a basic network design problem in directed graphs [2, 12, 13, 18] which is as follows: given a digraph, find a smallest subset of the edges (forming a minimum equivalent graph (MEG) that maintains all reachability relations of the original graph. When the MEG problem is restricted to strongly connected graphs we call it the minimum SCSS (strongly connected ....

....algorithms for the restricted problems. Hsu [13] gives a polynomial time algorithm for the acyclic MEG problem. The related problem of finding a transitive reduction of a digraph a smallest set of edges yielding the same reachability relations is studied by Aho, Garey and Ullman [2]. Transitive reduction differs from the MEG problem in that the edges 3 Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. Research supported by NSF Research Initiation Award CCR9307462. E mail : samir cs.umd.edu. y ....

A. V. Aho, M. R. Garey and J. D. Ullman, The transitive reduction of a directed graph, SIAM Journal on Computing, 1 (2), pp. 131--137, (1972).

....happen at the same geographic location or at a close geographic distance. 6 Eliminating transitive links is the problem of finding the minimum equivalent graph (MEG) For directed acyclic graphs, MEG is also transitive reduction and vice versa. This operation can be done in less than O(n 3 ) [Aho et al. 1972]. Algorithm Optimization Criterion Complexity Greedy 1 [Gupta et al. 1979] None O(n) Greedy min Min. minimal distance O(n) Greedy max Max. minimal distance O(n) Minimal change Min. change of location O(k:n 2 ) Table 1: Four algorithms for building compact summarizations. The complexity ....

Aho, A. V., Garey, M. R., and Ullman, J. D. (1972). "The Transitive Reduction of a Directed Graph". SIAM J. Comput., 1, No 2:131--136.

....by just taking the edges that remain in the graph representation for K. We can also do better, and first extract a minimal set of constraints equivalent to K in polynomial time, i.e. the natural extension of the notion of transitive reduction to weighted directed (in fact, acyclic) digraphs [AGU72] Another improvement is to quantify existentially on the time constants occurring in K but not in B: on the graph representation of B, this means repeatedly replacing any couple of edges c n Gamma c 0 n 0 Gamma c 00 by c n n 0 Gamma c 00 , where c 0 is existentially ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal of Computing, 1:131--137, 1972.

....For a given access path, the corresponding path(s) in the directed graph is (are) traversed to determine the named object (s) to which it is aliased. During this traversal, the number of de references encountered must be counted. Note that this is equivalent to performing transitive reduction [1] over the directed graph of alias relations during alias analysis, and later computing full alias information on demand. With this compact representation, alias information of (7) in Section 3 will become: f p; cell1 ; q; cell2 ; cell1:lef t) cell3 ; cell1:right) cell2 g; 26) and ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131--137, 1972.

....objects (malloc sites) in [10] any naming method may be employed. The exhaustive list of aliases holding for the example is f p; q , q; r , p; r , p; q g. Notice that p; r and p; q can be inferred from f p; q , q ; r g, which corresponds to the transitive reduction [1] of the directed graph. This compact representation [10] which we utilize in our pointer induced alias analysis method, combines two techniques to reduce the size of the alias sets. It discards alias pairs that do not have at least one named object or that involve more than one level of ....

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131--137, 1972.

No context found.

Alfred V. Aho, M. R. Garey, Jeffrey D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput, 1(2), pages 131--137, 1972.

No context found.

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal of Computing, 2, 1972, pages 131-137.

No context found.

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput., 1(2):131-137, 1972.

No context found.

A. V. Aho, M. R. Garey, and J. D. Ullman. 1972. The transitive reduction of a directed graph. SIAM Journal of Computing, 1:131-137.

No context found.

A. V. Aho, M. R. Garey, J. D. Ullman, The transitive reduction of a directed graph, SIAM J. Comput. 1 (1972) 131--137.

No context found.

A. V. Aho, M. R. Garey, and J. D. Ullman. The transitive reduction of a directed graph. SIAM Journal on Computing, 1(2):131--137, 1972.

No context found.

A. V. Aho, M. R. Garey and J. D. Ullman. The transitive reduction of a directed graph, Siam J. Computing 1(2) (1972) 131-137.

No context found.

Alfred V. Aho, M. R. Garey, Jeffrey D. Ullman. The transitive reduction of a directed graph. SIAM J. Comput, 1(2), pages 131--137, 1972.

No context found.

A. V. Aho, M. R. Garey and J. D. Ullman, The transitive reduction of a directed graph, SIAM Journal on Computing, 1 (2), pp. 131--137, (1972).

No context found.

Aho, A. V., Garey, M. R., and Ullmann, J. D. The transitive reduction of a directed graph. SIAM J. Comput., 1:131--137, 1972.

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