J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 332--343. Springer-Verlag, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. STOC 2000 Oregon, Portland Copyright 2000 ACM 0 89791 88 6 97 05 . 5.00 real world graphs. Indeed, Abello et al. [1] have shown that the degree sequence of so called call graphs is nicely approximated by a power law distribution. Call graphs are graphs of calls handled by some subset of telephony carriers for a specific time period. In addition, Faloutsos, et al. 9] have shown that the degree sequence of the ....

....model for graphs with a power law degree sequence. We then derive connectivity results which hold with high probability in various regimes of our parameters. And finally, we compare the results from the model with the exact connectivity structure for some call graphs computed by Abello et al. [1]. 1.1 Power Law Random Graphs The study of random graphs dates back to the work of Erdos and Renyi whose seminal papers [7; 8] laid the foundation for the theory of random graphs. There are three standard models for what we will call in this paper uniform random graphs [4] Each has two ....

J. Abello, A. Buchsbaum, and J. Westbrook, A functional approach to external graph algorithms, Proc. 6th European Symposium on Algorithms, pp. 332--343, 1998.

....show that techniques designed for massive amounts of data are not always a good choice for the more common case where the data size is only slightly larger than memory, e.g. by some small constant factor larger. Most theoretical approaches assume a large gap between data size and memory size; see [1] for an exception. There are many open challenges in the general area of link based ranking, and we expect many new schemes to be proposed. We are particularly interested in topic sensitive schemes such as those in [25, 38] that employ o# line preprocessing in order to allow higher throughput at ....

J. Abello, A. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In European Symposium on Algorithms, pages 332--343, 1998.

....equally well on unlabeled data, but we note that the hyperplanes are much smaller than the decision trees in representation. 5 Open Questions We flattened the graph structure completely. At the other extreme, there are nice algorithms for computing transitive closures of massive graphs [1]. Are there I Oefficient ways of capturing local unflattened structure (first and second neighborhoods) In practice, total disk space is limited and some data (i.e. arcs) must be discarded the first time they are seen. Without storing many days data, can one approximate the tendency of a ....

J. Abello and A. Buchsbaum and J. Westbrook. A Functional Approach to External Graph Algorithms. Proc. 6th Euro. Symp. on Algorithms, 1998, to appear.

....if their generator actually emulates other web properties. The power law distribution seems to be an ubiquitous property. The power law distri bution occurs in epidemics study [30] population study [28] genomes distribution [17, 29] various social phenomena [11, 26] and massive graphs [4, 6]. For the power law graphs in biological systems, the connectivity changes appear to be much more important than growth in size. 1.2 Properties for Graph Model Comparison Another important property that has been looked at is the diameters of web graphs. How ever, there are conflicting results ....

ABELLO, J., BUCHSBAUM, A., AND WESTBROOK, J. A functional approach to external graph algorithms. In Proceedings of 6th Eurpoean Syposium on Algorithms (1998), pp. 332-343. 8

....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 ....

....New Approach We show how to speed up the Munagala Ranade approach (MR BFS) of the previous section. We refer to the resulting algorithm as FAST BFS. We may assume w.l.o.g. that the input graph is connected (otherwise we may run the randomized O(sort(n m) I O connected components algorithm of [1] and only keep the nodes and edges of the component C s that contains the source node; all nodes outside of C s will be assigned BFS level infinity, and the BFS computation continues with C s ) We begin with the randomized preprocessing part of FAST BFS: 5.1 Partitioning a Graph into Small ....

J. Abello, A. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. Algorithmica, 32(3):437--458, 2002.

.... 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) memory transfers [30, 11] Both these algorithms are improvements of algorithms developed in [1, 19, 28]. 1.2 Our results The main result of this paper is an optimal cache oblivious priority queue. Our structure supports insert, delete, and ) amortized memory transfers and O(log N) amortized computation time it is described in Section 2. As discussed, even though several ecient priority queues ....

....We now consider algorithms for computing the minimal spanning forest (MSF) of an undirected weighted graph. In the I O model, a string of algorithms have been developed for the problem, culminating in an algorithm using O(sort(E) log 2 log 2 ( memory transfers developed by Arge et al. [19, 1, 28, 11]. This algorithm consists of two phases [11] In the rst phase an edge contraction algorithm inspired by PRAM MSF algorithms is used [20, 22] After the number of vertices has been reduced to O(E=B) a modi ed version of Prim s algorithm is then used in the second phase. Using our ....

J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. In Proc. Annual European Symposium on Algorithms, LNCS 1461, pages 332-343, 1998.

....assumption that the main memory is capable of holding B elements is made. memory algorithm takes 10 minutes to complete, the internal memory algorithm could use more than 150 hours, or equivalently, about 6 days I O efficient graph 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 ....

J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. In Proc. Annual European Symposium on Algorithms, LNCS 1461, pages 332--343, 1998.

....whether G has treewidth at most k and, if so, compute a nice treedecomposition of width at most k and size O(N) for G. Our tree decomposition algorithm uses the following results as subroutines: an O(sort(jV j jEj) I O algorithm for maximal matching in an arbitrary graph G = V; E) In [1], a deterministic O( jEj jV j sort(jV j) log 2 jV j jM j ) I O and a randomized O(sort(jEj) I O algorithm are presented. an O(sort(N) I O algorithm for computing a perfect elimination ordering for a triangulation with tree decomposition of width at most k, and a technique for ....

J. Abello, A.L. Buchsbaum, J.R. Westbrook. A functional approach to external graph algorithms. ESA'98, pp. 332-343, 1998.

....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 ....

J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. In Proc. Annual European Symposium on Algorithms, LNCS 1461, pages 332--343, 1998.

....are primitive operations often used in external memory algorithms. Their I O complexities are sort(N) Q( N= DB) log M=B (N=B) perm(N) Q(min(N;sort(N) and scan(N) Q(N= DB) respectively [32] 1. 2 Previous Results I O efficient graph algorithms have been studied in a number of papers [1, 2, 4, 5, 9, 11, 17, 20, 22, 23, 25, 27, 31]. We only discuss results on undirected BFS, DFS, single source shortest paths, planar embedding, and graph connectivity here. The best SSSP algorithm for arbitrary undirected graphs takes O(jV j (jEj=B)log 2 jEj) I Os [20] The best BFS algorithm for arbitrary undirected graphs takes O(jV j ....

J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th European Symposium on Algorithms, pages 332--343, 1998.

....planar graphs, provided that a BFS tree of the graph is given. In internal memory, a simple greedy algorithm computes a maximal matching for any graph in O(jV j jEj) time. The best known external memory results are O(jEj=jV jsort(jV j) log 2 (V=M) I Os deterministically and O(sort(jEj) I Os [1]. A perfect elimination ordering of a chordal graph can be found in linear time using algorithms by [27, 29, 31] In the PRAM model, Klein [20] shows how to compute a perfect elimination ordering in O(log 2 n) time using O( n m) logn) processors. In external memory, the sequential approaches ....

James Abello, Adam L. Buchsbaum, and Jeffery Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th European Symposium on Algorithms, pages 332--343, 1998.

....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 ....

J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. In Proc. Annual European Symposium on Algorithms, LNCS 1461, pages 332--343, 1998.

....data items are primitive operations often used in external memory algorithms. Their I O complexities are sort(N) Q( N=DB)log M=B (N=B) perm(N) Q(min(N;sort(N) and scan(N) O(N=DB) respectively [27] 1. 2 Previous Results Many authors have studied I O efficient graph algorithms [1, 2, 5, 6, 10, 18, 19, 22, 23, 25, 26]. We only discuss results on BFS, DFS, SSSP, and graph separators here. The best SSSP algorithm for arbitrary undirected graphs takes O(jV j (jEj=B)log 2 jEj) I Os [19] The best BFS algorithm for arbitrary undirected graphs takes O(jV j sort(jEj) I Os [26] Recently a BFS algorithm for ....

J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th European Symposium on Algorithms, pages 332--343, 1998.

....of memory, Link1 allows experiments on collections of 100 million Web pages, which contained much of the Web in 1998. To scale with the growth of the Web since 1998, we considered switching the Link Database to a disk based approach. Although there is active research in disk based graph algorithms [CGG95, KS96, ABW98], memory based graph algorithms tend to be simpler, easier to understand, and easier to implement. Therefore, we developed compression techniques to keep the Link Database in memory. Many compression techniques currently exist. These include both symbolwise methods, such as Huffman codes, that ....

J. Abello, A. Buchsbaum, and J. Westbrook. A Functional Approach to External Graph Algorithms. Proc. Annual European Symposium on Algorithms, pages 332-343, 1998.

.... researchers to build and analyze call graphs ; these are massive, sparse multigraphs in which each node represents a phone number and each arc A B represents a call from A to B, and their structural properties are both challenging to compute and useful for marketing and infrastructure planning [ABW98, ACL99, F99] Call detail data feeds must be processed as streams because of their size: The network handles between 200M and 300M calls a day and generates over 7GB of call detail data. 1.2.2 Stream Checking and Testing Because network data collection is currently dynamic, distributed, and ....

....grouped data streams facilitate reporting most frequently accessed web pages, most frequently called 800 numbers, most frequently used services, etc. Finally, we remark that grouping algorithms are a key enabler of recent improvements in the I O complexity of semi external graph algorithms [ABW98] Semi external computation of basic graph structures, e.g. connected components, minimum spanning trees, and matchings, is necessary when dealing with call graphs, web graphs, and other massive, sparse graphs arising from communications networks. 2 Definitions and Computational Models Our ....

J. Abello, A. Buchsbaum, and J. Westbrook. A Functional Approach to External Graph Algorithms In Proc. of 6'th European Symp. on Algorithms, LNCS Vol. 1461, Springer, Berlin, 1998, pages 332-434. Expanded version submitted for journal publication.

....notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. STOC 2000 Oregon, Portland Copyright 2000 ACM 0 89791 88 6 97 05 . 5.00 real world graphs. Indeed, Abello et al. [1] have shown that the degree sequence of so called call graphs is nicely approximated by a power law distribution. Call graphs are graphs of calls handled by some subset of telephony carriers for a specific time period. In addition, Faloutsos, et al. 9] have shown that the degree sequence of the ....

....model for graphs with a power law degree sequence. We then derive connectivity results which hold with high probability in various regimes of our parameters. And finally, we compare the results from the model with the exact connectivity structure for some call graphs computed by Abello et al. [1]. 1.1 Power Law Random Graphs The study of random graphs dates back to the work of Erdos and R enyi whose seminal papers [7; 8] laid the foundation for the theory of random graphs. There are three standard models for what we will call in this paper uniform random graphs [4] Each has two ....

J. Abello, A. Buchsbaum, and J. Westbrook, A functional approach to external graph algorithms, Proc. 6th European Symposium on Algorithms, pp. 332--343, 1998.

....of internal storage is not sufficient, we speak of external computation and algorithms. The simplest instance is that one has a system consisting of one conventional computer with one hard disk connected to it. Theoretically external list ranking and connected components has been solved optimally [8, 9], but again much work must be done to achieve acceptable practical performance [10] Nevertheless, for many problems, including ours, even the best external implementation will be much slower than if the problems were solved on a system with sufficient internal memory. A solution might be to ....

Abello, J., A.L. Buchsbaum, J. R. Westbrook, A Functional Approach to External Graph Algorithms, Proc. 6th European Symposium on Algorithms, LNCS 1461, Springer-Verlag, 1998, 332-343.

.... 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 ....

....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 randomized algorithms [12, 1]. Improved algorithms have been developed for several special classes of graphs. For trees, O(sort(N) algorithms are known for BFS and DFS numbering, Euler tour computation, expression tree evaluation, topological sorting, as well as several other problems [10, 12] For planar graphs, O(sort(N) ....

J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. In Proc. Annual European Symposium on Algorithms, LNCS 1461, pages 332--343, 1998.

....assumption that the main memory is capable of holding B 2 elements is made. 2 memory algorithm takes 10 minutes to complete, the internal memory algorithm could use more than 150 hours, or equivalently, about 6 days I O efficient graph 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 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 ....

J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. In Proc. Annual European Symposium on Algorithms, LNCS 1461, pages 332--343, 1998.

.... researchers to build and analyze call graphs ; these are massive, sparse multigraphs in which each node represents a phone number and each arc A B represents a call from A to B, and their structural properties are both challenging to compute and useful for marketing and infrastructure planning [ABW98, ACL99, F99] Call detail data feeds must be processed as streams because of their size: The network handles between 200M and 300M calls a day and generates over 7GB of call detail data. 1.2.2 Stream Checking and Testing Because network data collection is currently dynamic, distributed, and ....

....grouped data streams facilitate reporting most frequently accessed web pages, most frequently called 800 numbers, most frequently used services, etc. Finally, we remark that grouping algorithms are a key enabler of recent improvements in the I O complexity of semi external graph algorithms [ABW98] Semi external computation of basic graph structures, e.g. connected components, minimum spanning trees, and matchings, is necessary when dealing with call graphs, web graphs, and other massive, sparse graphs arising from communications networks. 2 Definitions and Computational Models Our ....

J. Abello, A. Buchsbaum, and J. Westbrook. A Functional Approach to External Graph Algorithms In Proc. of 6'th European Symp. on Algorithms, LNCS Vol. 1461, Springer, Berlin, 1998, pages 332-434. Expanded version submitted for journal publication.

.... Delta E V sort(V ) Our algorithm is much faster than this, since M BD V BD=E, and in fact optimal when the average degree of the graph is Omega Gamma BD) Earlier a O(log V M Delta Sort(E) algorithm was given by Chiang et al. [2] and this bound was matched recently by Abello et al. [4] using a simple recursive algorithm. For sparse graphs(E = O(V ) closed under edge contraction, the algorithm in [2] is shown to work in time O(Sort(V ) and is hence optimal. Our algorithm also has this property, and is of course far better for general sparse graphs. Chiang et al. [2] also give ....

James Abello, Adam Buchsbaum, and Jeffery Westbrook, A Functional Approach to External Graph Algorithms, in Proc. 6th ESA, 1998.

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J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 332--343. Springer-Verlag, 1998.

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J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 332--343. Springer-Verlag, 1998.

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J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 332--343. Springer-Verlag, 1998.

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J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 332--343. Springer-Verlag, 1998.

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J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 332--343. Springer-Verlag, 1998.

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J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. Algorithmica, 32(3):437--458, 2002.

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J. Abello, A. L. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In Proceedings of the 6th Annual European Symposium on Algorithms, pages 332--343. Springer-Verlag, 1998.

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J. Abello, A. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. Algorithmica, 32(3):437--458, 2002.

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J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. Algorithmica, 32(3):437--458, 2002.

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J. Abello, A.L. Buchsbaum, J.R. Westbrook. A Functional Approach to External Graph Algorithms. Algorithmica, 32(3): 437-458, 2002.

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J. Abello, A. Buchsbaum, and J. Westbrook, A functional approach to external graph algorithms, Proc. 6th European Symposium on Algorithms, pp. 332-343, 1998.

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J. Abello, A. L. Buchsbaum, and J. R. Westbrook. A functional approach to external graph algorithms. Algorithmica, 32:437-458, 2002.

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