J. Ebert, "A Sensitive Transitive Closure Algorithm," Information Processing Letters, 12, 1981, pp. 255-258.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 since these nodes will have identical successors, and process nodes of the condensed acyclic graph so obtained in a reverse topological ....

....in each strongly connected component into a single node is acyclic. Tarjan [22] developed an O(e) algorithm for determining strongly connected components of a graph by means of a depthfirst search, which also produces as a by product a topological sort on the components. It has been observed [7, 8, 11, 20] that it is possible to modify Tarjan s algorithm in a way that the successor lists are also expanded as the strongly connected components are being determined, and thus compute the transitive closure. 3. HYBRID ALGORITHMS The hybrid algorithms we propose in this section are best described ....

J. Ebert, "A Sensitive Transitive Closure Algorithm," Information Processing Letters, 12, 1981, pp. 255-258.

....the literature that focus on minimizing the I O costs of path computation in a database setting that assumes a fixed size main memory I O buffer. Most of such research has proposed solutions to solve recursive query problems for general databases that focused on pure transitive closure computation [1, 4, 7, 13, 27, 28, 29, 35]. In our work, rather than aiming for generality, we now take an application driven stance by proposing different disk page clustering algorithms for optimizing path query processing for GIS type of applications and then experimentally evaluating their relative advantages and disadvantages. Two ....

Ebert, J., 1981. "A Sensitive Transitive Closure Algorithm", Information Processing Letters, 12. , 255 - 258.

....His 4 algorithm detects the strong components, builds the condensation graph, computes the transitive closure of the condensation graph, and finally computes the transitive closure of the input graph from the transitive closure of the condensation graph. Later, Eve and Kurki Suonio [16] Ebert [15], Schmitz [41] Ioannidis et al. 26] and Nuutila and Soisalon Soininen [34, 35, 36, 37] presented transitive closure algorithms that use the strong components without generating the condensation graph. In [35] we showed that algorithm comp tc [34] has the smallest worst case execution time of ....

J. Ebert. A sensitive transitive closure algorithm. Information Processing Letters, 12:255-- 258, 1981.

....(23) end; 24) begin Main Program (25) unmark all nodes in V; 26) initialize stack to empty; 27) for each node v in V do (28) if v is not visited then (29) simpletc(v) 30) end. Figure 3: A simple one pass transitive closure algorithm based on Tarjan s algorithm [8] Ebert s algorithm [1] is a slight modification of simple tc. When processing an edge (v; w) Ebert s algorithm adds the successor set of w to the successor set of v only if edge (v; w) is either a tree edge or an intercomponent cross edge. The processing of a fully detected strong component is similar to simple tc. The ....

....: n. When exactly this happens depends on the order in which the edges are processed. In any case, newtc adds the root of component C 2 into all the sets adjacent component roots[i] 1 i n. When processing set adjacent component roots[n] newtc inserts the nodes of component C 2 into tc[1]. When newtc processes the adjacent component roots sets of the other nodes of C 1 , tc[1] already contains the root of C 2 and no set operations need to be done. newtc does n set insertions when detecting component C 2 and no unions. When detecting component C 1 newtc does n set insertions and ....

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J. Ebert. A sensitive transitive closure algorithm. Information Processing Letters, 12:255--258, 1981.

....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. ....

....components of the input graph [6] since all nodes in a strong component have the same successor set. For example, in Figure 1 nodes f , g, h, and i are in the same strong component. Some transitive closure algorithms presented in the literature are based on the detection of the strong components [4, 5, 7, 8, 12, 14, 15], but these algorithms do not efficiently avoid all redundancies caused by strong components. Some algorithms generate a partial successor set for each node of a component. Other scan the input graph more than once. In computing the successor sets, the algorithms perform unnecessary set ....

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J. Ebert. A sensitive transitive closure algorithm. Information Processing Letters, 12:255--258, 1981.

....to avoid redundant computations. memtc belongs to a class of practical and efficient transitive closure algorithms, which are based on strong component detection. Purdom [16] presented the first of these algorithms. Later better variants have been suggested by Eve and Kurki Suonio [6] Ebert [5], Schmitz [17] Ioannidis et al. 8] and Nuutila and Soisalon Soininen [12, 13, 14, 15] In [13] we showed that algorithm comptc [12] has the smallest worst case execution time of these algorithms. Several performance evaluations of transitive closure algorithms have been presented in the ....

J. Ebert. A sensitive transitive closure algorithm. Information Processing Letters, 12:255--258, 1981.

....data that is being compacted. The results in Figure 18 show that the BSR C algorithm has significantly improved performance over the BSR algorithm. Sensitive dynamic reordering helps to further reduce I O. 7 Related Work Many transitive closure algorithms have been presented in the literature [2, 3, 4, 6, 7, 8, 11, 12, 13, 16, 19, 23, 22]. While most of the presented algorithms emphasize the problem of reachability, others are not applicable to the shortest path problem for cyclic graphs [7, 8, 16, 19] because of the node collapsing technique [20] they adopt. Among the ones that address the shortest path problem for cyclic graphs, ....

....I O. 7 Related Work Many transitive closure algorithms have been presented in the literature [2, 3, 4, 6, 7, 8, 11, 12, 13, 16, 19, 23, 22] While most of the presented algorithms emphasize the problem of reachability, others are not applicable to the shortest path problem for cyclic graphs [7, 8, 16, 19] because of the node collapsing technique [20] they adopt. Among the ones that address the shortest path problem for cyclic graphs, there is a lack of performance studies for the shortest path problem for cyclic graphs with the exception of [2, 13] For this reason, we implemented a variation of ....

Ebert, J., "A Sensitive Transitive Closure Algorithm", Information Processing Letters, 12. , 1981, pp. 255 -- 258.

....sets of path equivalent vertices. These sets are called the strong components of the graph. All vertices in a strong component have the same successor sets. Several transitive closure algorithms presented in literature use graph search and strong component detection to avoid redundant computations [2, 3, 4, 8, 9]. These algorithms use Tarjan s algorithm [10] or some related depth first search based algorithm for strong component detection. The algorithms can be classified with respect to the time point at which the successor sets are constructed during the computation [4] Eve s and Kurki Suonio s ....

....algorithm [10] or some related depth first search based algorithm for strong component detection. The algorithms can be classified with respect to the time point at which the successor sets are constructed during the computation [4] Eve s and Kurki Suonio s algorithm [3] Ebert s algorithm [2] and the algorithm gdftc by Ioannidis et al. 4] construct the successor sets during the strong component detection. Purdom s algorithm [8] and the algorithm btc by Ioannidis et al. 4] first detect all strong components and after that construct the successor sets. Schmitz s algorithm [9] ....

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J. Ebert. A sensitive transitive closure algorithm. Information Processing Letters, 12:255--258, 1981.

....) space in the worst case, it typically is compact; the transitive closure takes much less space than the input graph. Further, the representation can be computed eOEciently during a single depth rst traversal of the input graph, and it can be used with eOEcient transitive closure algorithms like [5, 6, 7, 8, 9] that are based on strong component detection. We developed the representation from a method for compressing the transitive closure of an acyclic graph by Agrawal et al. 10] The second representation is based on a chain decomposition of the condensation graph induced by the input graph. A chain ....

....Their algorithms detect the strong components, build the condensation graph, compute the transitive closure of the condensation graph, and nally compute the transitive closure of the input graph from the transitive closure of the condensation graph. Later, Eve and Kurki Suonio [6] Ebert [5], Schmitz [9] Ioannidis et al. 7] and Nuutila and Soisalon Soininen [8, 18, 25, 26] presented transitive closure algorithms that use the strong components without generating the condensation graph. All these algorithms are based on Tarjan s strong component algorithm [27] In [18] we showed ....

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J. Ebert. A sensitive transitive closure algorithm. Information Processing Letters, 12:255 258, 1981.

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