H. S. Warren. A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations. Communications of the ACM, 18(4):218--220, April 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....considerations can be done for the numeric constraint propagation technique. Even bigger is the room for improvements of transitive closure (now absorbing 60 of the prototype s execution time) First of all we are currently using a naive variation of Warshall Warren algorithm for graph closure [26, 25], since it Program Analysis Time Benefits fib 1640 ms . 1 future redundant constraints (N 1) deterministic if 1st argument definite a ; partially deterministic if 2nd argument definite; 1 simplified procedure call. a The success pattern N # 2 would have been found even ....

H. S. Warren. A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations. Communications of the ACM, 18(4):218--220, April 1975.

....These can be classified in two broad classes: the iterative methods such as [AgJa87, BaRa86, ChMa89, HAC90, HaQaCh88, IOA86, LU87, VaBo86] These methods compute the TC of a relation with successive iterations until no new tuples are generated. Other algorithms are called direct methods such as [AgJa87, AgJa90, IoRa88, WAR62, and WAR75]. These techniques compute the TC of a relation in a fixed number of passes over the relation. More closely related to our efforts is the work reported by Ullman and Yannakakis [UlYa90] They show that standard algorithms, those that (possibly depending on the data) examine every triplet ....

....then no matter how fine the partition, the paths between nodes can consist only of paths between buckets, or paths within buckets. Only diagonal buckets have paths within them; hence the internal closure operation B kk in the algorithm. 2 a version of our algorithm based on Warren s method [WAR75] can be implemented as well. 3 internal TC of a bucket B ij is used to refer to the TC relation which would result if only the tuples in B ij were used. 1: for k = 1 to N 2: for i = 1 to N 3: for j = 1 to N 4.1: B ij = B ij [ 4.2: B ik ffi B kj ) 4.3: B ik ffi (B kk ) ffi B kj ) ....

Henry S. Warren, "A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations" CACM April 1975 p 218-220

....Direct algorithms, on the other hand, process each element (a node or an edge) a constant number of times (usually once) and terminate after such processing is complete independent of the underlying data. In direct algorithms, there are two families. Matrix based direct algorithms, such as in [2, 25, 26], are best understood in terms of a matrix representation and manipulation. Graph based direct algorithms, such as in [7, 8, 11, 18, 20] are best understood in terms of a graph traversal. Graph based algorithms often coalesce nodes belonging to the same strongly connected component into one node ....

....a i j , iff k j, and 2. For all i, j, k, processing of the element a jk precedes the processing of the element a i j , iff k j. Various processing orders can be derived subject to these two constraints, giving rise to a whole family of Warshall derived algorithms. The Warren algorithm [25] that processes matrix elements in row order but in two passes 1 : i = 1 n j = 1 i 1 process a i j i = 1 n j = i 1 n process a i j can be viewed as a Warshall derived algorithm, since it observes the two precedence constraints listed above. The ....

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H. S. Warren, "A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations," Commun. ACM, 18(4), April 1975, pp. 218-220.

....algorithm became very pronounced as the number of vertices increased beyond 50. Furthermore, Warshall s algorithm is very memory efficient, especially if the matrix does not need to be unpacked. Warren described a modification of Warshall s algorithm to optimize it for a paging environment [Warren75]. Warshall s algorithm scans a boolean matrix by columns in order to use boolean OR operations on the rows of the matrix. Warren s algorithm scans by rows and uses boolean OR operations on the rows. Whereas Warshall s algorithm has one main scan, Warren s algorithm has two. If each row of the ....

Warren, Henry S. A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations. Communications of the ACM (1975) vol. 18, no. 4, pp. 218-220.

....between the two approaches illustrates that the encoded path view can handle more users per time period. 6 RELATED RESEARCH Path problems in the transitive closure have long been a rich subject of investigation in the literature. Classic algorithms such as the Warshall and Warren 0 s algorithms [23, 22], Dijkstra 0 s algorithm [9] breadth first search algorithm [8] depth first search algorithm [8] and heuristic A algorithm [8] have been the subjects of studies and targets for comparison. For ITS route guidance, the most critical path problem is the shortest path problem. ITS maps are ....

H.S. Warren, "A Modification of Warshall's Algorithm for the Transitive Closure of Biary Relations," Commun. ACM, 18, 4, 1975, pp. 218 -- 220

....(a ik a kj ) For i = 1 to v For k = i 1 to v For j = 1 to v a ij = a ij (a ik a kj ) Figure 6: Warren s algorithm 3.2. 1 Warshall and Warren algorithms Algorithms to compute the transitive closure of the adjacency matrix of a graph have been initially proposed by Warshall and Warren in [65, 64]. In their original formulation they are tuple oriented and depth first. Since Warren s algorithm is generally more efficient than Warshall s, we illustrate the former algorithm only. The computation can be expressed as in Fig. 6 (this formulation is due to [1] The initial relation is ....

H.S. WARREN, "A modification of Warshall's algorithm for the transitive closure of binary relations," Communications of the ACM, vol. 18, no. 4, 1975, pp. 218-220.

....on the other hand, process each element (tuple, node or edge) a constant number of times and terminate after processing is complete. The number of iterations they perform is independent of the underlying database. Transitive closure algorithms are furthermore distinguished in matrix (e.g. War75] graph [Ita88] or hybrid [AJ90] algorithms depending on the representation and the techniques used. Hybrid algorithms utilize the optimizing features of graph algorithms in a matrix framework. Most algorithms on graphs solve a variety of problems involving path computations such as, critical ....

H. Warren. A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations. Communications of the ACM, 18(4):218--220, 1975.

....considerations can be done for the numeric constraint propagation technique. Even bigger is the room for improvements of transitive closure (now absorbing 60 of the prototype s execution time) First of all we are currently using a naive variation of Warshall Warren algorithm for graph closure [26, 25], since it Program Analysis Time Benefits fib 1640 ms ffl 1 future redundant constraints (N 1) ffl deterministic if 1st argument definite a ; ffl partially deterministic if 2nd argument definite; ffl 1 simplified procedure call. a The success pattern N 2 would have been found even ....

H. S. Warren. A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations. Communications of the ACM, 18(4):218--220, April 1975.

....making use of wavefront relations (saving previous processing results to avoid redundant processing) and grouping those joins which reduce the size of intermediate results are demonstrated to be beneficial on a WISS (Wisconsin Storage System) platform. In [7] an adaptation of Warren s algorithm [12] is proposed, its performance is compared with that of an iterative algorithm and an improved version of the iterative algorithm (logarithmic) They evaluate the performance of the algorithms for different source relation sizes, available memory sizes, join selectivities, and maximum path length. ....

H.S. Warren. A modification of warshall's algorithm for the transitive closure of binary relations. CACM, 18(4):218--220, 1975.

....reasons. First, the size of the inputs seems to grow in proportion to the growth of the memory capacity. Since the CPU speed has grown at the same rate as the memory capacity, only linear algoritms retain their execution times on typical inputs. Traditional transitive closure algorithms, such as [4, 5, 12, 14, 15, 17, 18], are not linear. Second, typical inputs (and outputs) in the area of databases do not fit into the main memory. Traditional transitive closure algorithms are mostly designed for main memory operation. In computing the transitive closure, a major problem is the avoidance of redundant computations. ....

H.S. Warren. A modification of warshall's algorithm for the transitive closure of binary relations. Communications of the ACM, 18(4):218--220, 1975.

....Algorithms for ITS We use the shortest path transitive closure concept to model the path view. Because ITS graphs are cyclic, algorithms solving the ITS shortest path problem must be cycle indifferent. Many such shortest path transitive closure algorithms have been presented in the literature [2, 3, 6, 12, 13, 22, 23]. Classic ones such as the Warshall and Warren 0 s algorithms, that are matrix based [22, 23] and the Dijkstra 0 s algorithm, that traverses the graph based on priorities [6] work better in main memory environments. Recently proposed transitive closure algorithms [2, 3, 12, 13] provide ....

....are cyclic, algorithms solving the ITS shortest path problem must be cycle indifferent. Many such shortest path transitive closure algorithms have been presented in the literature [2, 3, 6, 12, 13, 22, 23] Classic ones such as the Warshall and Warren 0 s algorithms, that are matrix based [22, 23], and the Dijkstra 0 s algorithm, that traverses the graph based on priorities [6] work better in main memory environments. Recently proposed transitive closure algorithms [2, 3, 12, 13] provide disk based solutions that can be adopted to solve linear recursive queries in relational databases ....

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Warren, H.S. "A Modification of Warshall's Algorithm for the Transitive Closure of Biary Relations," Commun. ACM, 18, 4, 1975, pp. 218 -- 220

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