A. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. Westbrook. On external memory graph traversal. In Proc. 11th ACM-SIAM Symposium on Discrete Algorithms, pages 859--860. ACM-SIAM, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....algorithm performing o(V) I Os has been developed only recently [16] For directed graphs, even fewer results are known. The best known algorithms for directed SSSP, BFS, and topological sorting all perform C(V) I Os. More precisely, their I O complexity is O(min (V ) log V sort(E) V ) [5, 6, 13]. A number of improved algorithms have been developed for special classes of graphs. For trees for example, O(sort(N) I O algorithms are known for BFS and DFS numbering, Eulet tour computation, expression tree evaluation, topological sorting, as well as several other problems [5, 6] Most ....

....V ) 5, 6, 13] A number of improved algorithms have been developed for special classes of graphs. For trees for example, O(sort(N) I O algorithms are known for BFS and DFS numbering, Eulet tour computation, expression tree evaluation, topological sorting, as well as several other problems [5, 6]. Most problems on planar undirected graphs, including SSSP, BFS, and DFS, can also be solved in O(sort(N) I Os [3, 4, 6, 14] Almost all of these algorithms exploit the existence of small separators for planar graphs. More precisely, they use that for every planar graph G and any integer h 0, ....

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A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms, pages 859-860, 2000.

....yields an improved semi external BFS algorithm for sparse directed Eulerian graphs (Section 8) Finally, Section 9 provides some concluding remarks and open problems. 2 Previous Work and New Results Previous Work. I O efficient algorithms for graph traversal have been considered in, e.g. [1, 3, 4, 7 14]. In the following we will only discuss results related to BFS. The currently fastest BFS algorithm for general undirected graphs [14] requires (n sort(m) I Os. The best bound known for directed EM BFS is O(minfn M ; n ) log 2 g) I Os [7 9] This also yields an O(n ) I O ....

....been considered in, e.g. 1, 3, 4, 7 14] In the following we will only discuss results related to BFS. The currently fastest BFS algorithm for general undirected graphs [14] requires (n sort(m) I Os. The best bound known for directed EM BFS is O(minfn M ; n ) log 2 g) I Os [7 9]. This also yields an O(n ) I O algorithm for SEM BFS. Faster algorithms are only known for special types of graphs: O(sort(n) I Os are sufficient to solve EM BFS on trees [7] grid graphs [5] outer planar graphs [10] and graphs of bounded tree width [11] Slightly sublinear I O was known ....

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A. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. Westbrook. On external memory graph traversal. In Proc. 11th Symp. on Discrete Algorithms, pages 859--860. ACMSIAM, 2000.

.... a lazy (or batched) Problem Our cache oblivious result Previous best cache aware result Priority queue ) O( 8] List ranking O(sort(V ) O(sort(V ) 19, 8] Tree algorithms O(sort(V ) O(sort(V ) Directed BFS and DFS O( V E=B) log 2 V sort(E) O( V E=B) log 2 V sort(E) [18] O(V BM Undirected BFS O(V sort(E) O(V sort(E) 30] Minimal spanning forest O(sort(E) log 2 log 2 V ) O(sort(E) log 2 log 2 O(V sort(E) O(v sort(E) Figure 1: Summary of our results (Priority queue bounds are amortized) manner using main memory sized bu ers attached ....

.... ) algorithms for most problems on trees, such as computing an Euler Tour, Breadth First Search (BFS) Depth First Search (DFS) and computing a centroid decomposition [19] The best known DFS and BFS algorithms for general directed graphs use O(V BM ) 19] or O( V E=B) log 2 V sort(E) [18] memory transfers. For undirected graphs, an improved O(V sort(E) BFS algorithm has been developed [30] The best known algorithms for computing the connected components and the minimal spanning forest of a general undirected graph both use O(sort(E) log 2 log 2 ( E ) or O(V sort(E) ....

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A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proc. ACM-SIAM Symp. on Discrete Algorithms,

.... have been considered by a number of authors [10, 11, 17, 33, 25, 20, 31, 7, 6, 30, 22, 26] If V is the number of vertices and E the number of edges in a graph, the best known algorithms for depth first search, depending on the exact relationship between V and E, use O( V ) 17] or O [25, 16] I Os. For breadth first search an O(V sort(E) algorithm has been developed for the undirected case [30] while the best known algorithm for the directed case uses O [16] The best known algorithms for connected components, minimum spanning tree, and single source shortest path all ....

....for depth first search, depending on the exact relationship between V and E, use O( V ) 17] or O [25, 16] I Os. For breadth first search an O(V sort(E) algorithm has been developed for the undirected case [30] while the best known algorithm for the directed case uses O [16]. The best known algorithms for connected components, minimum spanning tree, and single source shortest path all work on undirected graphs and use O sort(E) log log V B [30] O (sort(E) log B scan(E) log V ) 25] and O [25] I Os, respectively. For the special case of grid graphs ....

A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proc. ACM-SIAM Symp. on Discrete Algorithms, 2000. (to appear).

.... N B ) 3] the sorting bound) In practice the difference between an algorithm doing N I Os and one doing scan(N) or sort(N) I Os can be significant [9] problem general undirected graphs DFS O Gamma V M E B V Delta [13] O Gamma (V scan(E) Delta log V B sort(E) Delta [23, 11] BFS O(V E B Delta sort(V ) 26] CC O Gamma sort(E) Delta log log V B E Delta [26] MST O Gamma log 2 V M Delta sort(E) Delta [13] O (sort(E) Delta log B scan(E) Delta log V ) 23] SSSP O Gamma V E B Delta log 2 V B Delta [23] Table 1: Best known ....

....V M Delta sort(E) Delta [13] O (sort(E) Delta log B scan(E) Delta log V ) 23] SSSP O Gamma V E B Delta log 2 V B Delta [23] Table 1: Best known upper bounds for basic graph theoretic problems. I O efficient graph algorithms have been considered by a number of authors [5, 6, 13, 29, 23, 17, 27, 2, 1, 26, 20, 25, 11]. Table 1 reviews the best known algorithms for basic graph theoretic problems on general undirected graphs. For directed graphs the best known algorithm for breadth first search (BFS) and depth first search (DFS) use O Gamma (V E B ) Delta log V B sort(E) Delta I Os [11] Lower ....

[Article contains additional citation context not shown here]

A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 859--860, 2000.

.... be designed using the distributionsweeping technique [86] Several other data structures can be constructed efficiently using buffers, and the buffer tree technique has been used to develop several other data structures which in turn have been used to develop algorithms in many different areas [25, 29, 20, 21, 107, 15, 76, 46, 144, 145, 96, 44, 136]. Priority queues. External buffered priority queues have been extensively researched because of their applications in graph algorithms. Arge showed how to perform deletemin operations on a basic buffer tree in amortized O( 1 B log M=B N B ) I Os [14] Note that in this case the deletemin ....

....external priority queue. Using the buffer tree technique on a tournament tree, Kumar and Schwabe [107] developed a priority queue supporting update operations in O( 1 B log N B ) I Os. They also showed how to use their structure in several efficient external graph algorithms (see e. g [2, 7, 18, 22, 27, 46, 59, 81, 97, 107, 110, 111, 116, 118, 122, 142, 156] for other results on external graph algorithms and data structures) Note that if the priority of an element is known, an update operation can be performed in O( 1 B log M=B N B ) I Os on a buffer tree using a delete and an insert operation. 4 3 sided planar range searching In internal ....

A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 859--860, 2000.

....known general BFS algorithm uses only O(jV j (jEj=jV j)sort(jV j) O(jV j sort(jEj) I Os [17] this suggests that on undirected graphs DFS might be harder than BFS. For directed graphs the best known algorithms for BFS and DFS both use O( jV j jEj=B) Delta log(jV j=B) sort(jEj) I Os [6]. In general we cannot hope to design algorithms that perform less than Omega (min(jV j; sort(jV j) I Os for either of the two problems [2, 8, 17] As mentioned above, in practice O(min(jV j; sort(jV j) O(sort(jV j) Still, all of the above algorithms use Omega (jV j) I Os. For planar ....

....space, for any 0 fl 1=2. Further improved algorithms have been developed for special classes of planar graphs. For trees, O(sort(N) I O algorithms are known for both BFS and DFS as well as for Euler tour computation, expression tree evaluation, topological sorting, and several other problems [6, 8, 3]. BFS and DFS can also be solved in O(sort(N) I Os on outerplanar graphs [13] and on k outerplanar graphs [14] Developing O(sort(N) I O DFS and BFS algorithms for arbitrary planar graphs remains a challenging open problem. 1.2 Our Results The contribution of this paper is two fold. In Sec. 2 ....

A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 859--860, 2000.

.... Gamma sort(E) Delta log log V B E Delta [25] MST O Gamma sort(E) Delta log V M Delta [12] O (sort(E) Delta log B scan(E) Delta log V ) 22] SSSP O Gamma V E B Delta log V B Delta [22] I O efficient graph algorithms have been considered by a number of authors [1, 2, 5, 6, 10, 12, 16, 19, 22, 24 26, 29]. Table 1 reviews the best known algorithms for basic graph theoretic problems on general undirected graphs. For directed graphs the best known algorithm for breadth first search (BFS) and depth first search (DFS) use O Gamma (V scan(E) Delta log V B sort(E) Delta I Os [10] Lower ....

....24 26, 29] Table 1 reviews the best known algorithms for basic graph theoretic problems on general undirected graphs. For directed graphs the best known algorithm for breadth first search (BFS) and depth first search (DFS) use O Gamma (V scan(E) Delta log V B sort(E) Delta I Os [10]. Lower bound results were proved in [6, 12, 25] Note that no O(sort(E) deterministic) algorithm is known for any of the problems, and that the best known algorithms for DFS, BFS and SSSP require Omega (V ) I Os. MST and connected components (CC) can be solved in O(sort(E) I Os with ....

[Article contains additional citation context not shown here]

A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 859--860, 2000.

.... 11, 17, 33, 25, 20, 31, 7, 6, 30, 22, 26] If V is the number of vertices and E the number of edges in a graph, the best known algorithms for depth first search, depending on the exact relationship between V and E, use O( V M E B V ) 17] or O Gamma (V E B ) log V B sort(E) Delta [25, 16] I Os. For breadth first search an O(V sort(E) algorithm has been developed for the undirected case [30] while the best known algorithm for the directed case uses O Gamma (V E B ) log V B sort(E) Delta [16] The best known algorithms for connected components, minimum spanning ....

....E B V ) 17] or O Gamma (V E B ) log V B sort(E) Delta [25, 16] I Os. For breadth first search an O(V sort(E) algorithm has been developed for the undirected case [30] while the best known algorithm for the directed case uses O Gamma (V E B ) log V B sort(E) Delta [16]. The best known algorithms for connected components, minimum spanning tree, and single source shortest path all work on undirected graphs and use O Gamma sort(E) log log V B E Delta [30] O (sort(E) log B scan(E) log V ) 25] and O Gamma V E B log 2 V B Delta [25] I Os, ....

A. L. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. R. Westbrook. On external memory graph traversal. In Proc. ACM-SIAM Symp. on Discrete Algorithms, 2000. (to appear).

No context found.

A. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. Westbrook. On external memory graph traversal. In Proc. 11th ACM-SIAM Symposium on Discrete Algorithms, pages 859--860. ACM-SIAM, 2000.

No context found.

Adam L. Buchsbaum, Michael Goldwasser, Suresh Venkatasubramanian, and Jeffery Westbrook. On external memory graph traversal. In Symposium on Discrete Algorithms, pages 859--860, 2000.

No context found.

Adam L. Buchsbaum, Michael Goldwasser, Suresh Venkatasubramanian, and Je#ery Westbrook. On external memory graph traversal. In Symposium on Discrete Algorithms, pages 859--860, 2000.

No context found.

A. Buchsbaum, M. Goldwasser, S. Venkatasubramanian, and J. Westbrook. On external memory graph traversal. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 859--860, 2000.

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