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A Scalable Distributed Parallel BreadthFirst Search Algorithm on BlueGene/L.
 In Proceedings of IEEE/ACM Supercomputing ’05.
, 2005
"... Abstract Many emerging largescale data science applications require searching large graphs distributed across multiple memories and processors. This paper presents a distributed breadthfirst search (BFS) scheme that scales for random graphs with up to three billion vertices and 30 billion edges. ..."
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Cited by 58 (3 self)
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Abstract Many emerging largescale data science applications require searching large graphs distributed across multiple memories and processors. This paper presents a distributed breadthfirst search (BFS) scheme that scales for random graphs with up to three billion vertices and 30 billion edges. Scalability was tested on IBM BlueGene/L with 32,768 nodes at the Lawrence Livermore National Laboratory. Scalability was obtained through a series of optimizations, in particular, those that ensure scalable use of memory. We use 2D (edge) partitioning of the graph instead of conventional 1D (vertex) partitioning to reduce communication overhead. For Poisson random graphs, we show that the expected size of the messages is scalable for both 2D and 1D partitionings. Finally, we have developed efficient collective communication functions for the 3D torus architecture of BlueGene/L that also take advantage of the structure in the problem. The performance and characteristics of the algorithm are measured and reported.
A Parallelization of Dijkstra's Shortest Path Algorithm
 IN PROC. 23RD MFCS'98, LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously workefficient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We giv ..."
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Cited by 38 (6 self)
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The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously workefficient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random edge weights uniformly distributed in [0, 1]. We use
Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms
, 1995
"... We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a consta ..."
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Cited by 37 (4 self)
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We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(ff(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(n fi ), for any constant 0 ! fi ! 1. Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time.
Subcubic Cost Algorithms for the All Pairs Shortest Path Problem
 Algorithmica
, 1995
"... . In this paper we give three subcubic cost algorithms for the all pairs shortest distance (APSD) and path (APSP) problems. The first is a parallel algorithm that solves the APSD problem for a directed graph with unit edge costs in O(log 2 n) time with O(n ffi p log n) processors where = 2:68 ..."
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Cited by 24 (5 self)
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. In this paper we give three subcubic cost algorithms for the all pairs shortest distance (APSD) and path (APSP) problems. The first is a parallel algorithm that solves the APSD problem for a directed graph with unit edge costs in O(log 2 n) time with O(n ffi p log n) processors where = 2:688 on an EREWPRAM. The second parallel algorithm solves the APSP, and consequently APSD, problem for a directed graph with nonnegative general costs (real numbers) in O(log 2 n) time with o(n 3 ) subcubic cost. Previously this cost was greater than O(n 3 ). Finally we improve with respect to M the complexity O((Mn) ) of a sequential algorithm for a graph with edge costs up to M into O(M 1=3 n (6+!)=3 (log n) 2=3 (log log n) 1=3 ) in the APSD problem. 1 Introduction The all pairs shortest path (APSP) problem is to compute shortest paths between all pairs of vertices of a directed graph with nonnegative real numbers as edge costs. The all pairs shortest distance problem ...
An experimental study of a parallel shortest path algorithm for solving largescale graph instances
 Ninth Workshop on Algorithm Engineering and Experiments (ALENEX 2007)
, 2007
"... We present an experimental study of the single source shortest path problem with nonnegative edge weights (NSSP) on largescale graphs using the $\Delta$stepping parallel algorithm. We report performance results on the Cray MTA2, a multithreaded parallel computer. The MTA2 is a highend shared m ..."
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Cited by 17 (3 self)
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We present an experimental study of the single source shortest path problem with nonnegative edge weights (NSSP) on largescale graphs using the $\Delta$stepping parallel algorithm. We report performance results on the Cray MTA2, a multithreaded parallel computer. The MTA2 is a highend shared memory system offering two unique features that aid the efficient parallel implementation of irregular algorithms: the ability to exploit finegrained parallelism, and lowoverhead synchronization primitives. Our implementation exhibits remarkable parallel speedup when compared with competitive sequential algorithms, for lowdiameter sparse graphs. For instance, $\Delta$stepping on a directed scalefree graph of 100 million vertices and 1 billion edges takes less than ten seconds on 40 processors of the MTA2, with a relative speedup of close to 30. To our knowledge, these are the first performance results of a shortest path problem on realistic graph instances in the order of billions of vertices and edges.
Parallel Shortest Path Algorithms for Solving . . .
, 2006
"... We present an experimental study of the single source shortest path problem with nonnegative edge weights (NSSP) on largescale graphs using the ∆stepping parallel algorithm. We report performance results on the Cray MTA2, a multithreaded parallel computer. The MTA2 is a highend shared memory s ..."
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Cited by 15 (3 self)
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We present an experimental study of the single source shortest path problem with nonnegative edge weights (NSSP) on largescale graphs using the ∆stepping parallel algorithm. We report performance results on the Cray MTA2, a multithreaded parallel computer. The MTA2 is a highend shared memory system offering two unique features that aid the efficient parallel implementation of irregular algorithms: the ability to exploit finegrained parallelism, and lowoverhead synchronization primitives. Our implementation exhibits remarkable parallel speedup when compared with competitive sequential algorithms, for lowdiameter sparse graphs. For instance, ∆stepping on a directed scalefree graph of 100 million vertices and 1 billion edges takes less than ten seconds on 40 processors of the MTA2, with a relative speedup of close to 30. To our knowledge, these are the first performance results of a shortest path problem on realistic graph instances in the order of billions of vertices and edges.
HammockonEars Decomposition: A Technique for the Efficient Parallel Solution of Shortest Paths and Other Problems
 Theoretical Computer Science
, 1996
"... We show how to decompose efficiently in parallel any graph into a number, ~ fl, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. Frederickson and the parallel ear decom ..."
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Cited by 9 (6 self)
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We show how to decompose efficiently in parallel any graph into a number, ~ fl, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. Frederickson and the parallel ear decomposition technique, thus we call it the hammockonears decomposition. We mention that hammockonears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in O(logn log log n) time using O(n + m) CREW PRAM processors, for an nvertex, medge graph or digraph. The hammockonears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of sparse (di)graphs. This class consists of all (di)graphs which have a ~ fl between 1 and \Theta(n...
Buckets strike back: Improved Parallel ShortestPaths
 Proc. 16th Intl. Par. Distr. Process. Symp. (IPDPS
, 2002
"... We study the averagecase complexity of the parallel singlesource shortestpath (SSSP) problem, assuming arbitrary directed graphs with n nodes, m edges, and independent random edge weights uniformly distributed in [0; 1]. We provide a new bucketbased parallel SSSP algorithm that runs in T = O(log ..."
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Cited by 7 (2 self)
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We study the averagecase complexity of the parallel singlesource shortestpath (SSSP) problem, assuming arbitrary directed graphs with n nodes, m edges, and independent random edge weights uniformly distributed in [0; 1]. We provide a new bucketbased parallel SSSP algorithm that runs in T = O(log 2 n min i f2 i L + jV i jg) averagecase time using O(n+m+T ) work on a PRAM where L denotes the maximum shortestpath weight and jV i j is the number of graph vertices with indegree at least 2 i . All previous algorithms either required more time or more work. The minimum performance gain is a logarithmic factor improvement; on certain graph classes, accelerations by factors of more than n 0:4 can be achieved. The algorithm allows adaptation to distributed memory machines, too.
Finding the k Shortest Paths in Parallel
, 2000
"... . A concurrentread exclusivewrite PRAM algorithm is developed to find the k shortest paths between pairs of vertices in an edgeweighted directed graph. Repetitions of vertices along the paths are allowed. The algorithm computes an implicit representation of the k shortest paths to a given destina ..."
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Cited by 4 (0 self)
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. A concurrentread exclusivewrite PRAM algorithm is developed to find the k shortest paths between pairs of vertices in an edgeweighted directed graph. Repetitions of vertices along the paths are allowed. The algorithm computes an implicit representation of the k shortest paths to a given destination vertex from every vertex of a graph with n vertices and m edges, using O(m +nk log 2 k) work and O(log 3 k log # k + log n(log log k +log # n)) time, assuming that a shortest path tree rooted at the destination is precomputed. The paths themselves can be extracted from the implicit representation in O(log k +log n) time, and O(n log n+L) work, where L is the total length of the output. Key Words. Parallel graph algorithms, Data structures, Shortest paths. 1. Introduction. The problem of finding shortest paths in an edgeweighted graph is an important and wellstudied problem in computer science. The more general problem of computing the k shortest paths between vertices of...
Efficient Sequential and Parallel Algorithms for the Negative Cycle Problem
 TO APPEAR IN ISAAC’94
, 1994
"... We present here an algorithm for detecting (and outputting, if exists) a negative cycle in an nvertex planar digraph G with real edge weights. Its running time ranges from O(n) up to O(n 1.5 log n) as a certain topological measure of G varies from 1 up to Θ(n). Moreover, an efficient CREW PRAM imp ..."
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Cited by 4 (3 self)
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We present here an algorithm for detecting (and outputting, if exists) a negative cycle in an nvertex planar digraph G with real edge weights. Its running time ranges from O(n) up to O(n 1.5 log n) as a certain topological measure of G varies from 1 up to Θ(n). Moreover, an efficient CREW PRAM implementation is given. Our algorithm applies also to digraphs whose genus γ is o(n).