## Parallel Shortest Path Algorithms for Solving . . . (2006)

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Citations: | 9 - 3 self |

### BibTeX

@MISC{Madduri06parallelshortest,

author = {Kamesh Madduri and David A. Bader and Jonathan W. Berry and Joseph R. Crobak},

title = {Parallel Shortest Path Algorithms for Solving . . . },

year = {2006}

}

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### Abstract

We present an experimental study of the single source shortest path problem with non-negative edge weights (NSSP) on large-scale graphs using the ∆-stepping parallel algorithm. We report performance results on the Cray MTA-2, a multithreaded parallel computer. The MTA-2 is a high-end shared memory system offering two unique features that aid the efficient parallel implementation of irregular algorithms: the ability to exploit fine-grained parallelism, and low-overhead synchronization primitives. Our implementation exhibits remarkable parallel speedup when compared with competitive sequential algorithms, for low-diameter sparse graphs. For instance, ∆-stepping on a directed scale-free graph of 100 million vertices and 1 billion edges takes less than ten seconds on 40 processors of the MTA-2, 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.

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Citation Context ...oretic analysis metrics such as betweenness centrality [27, 9, 41, 43, 34] are based on shortest path algorithms. Our parallel implementations target the scale-free graph family, a well-studied model =-=[7, 24, 12, 53, 52]-=- for real-world large-scale graphs that captures features such as a low diameter, heavy-tailed degree distributions modeled by power laws, and self similarity. We also conduct an experimental study of... |

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Citation Context ...CS Implementation Challenge [18] on Shortest Paths. Sequential algorithms for the single source shortest path problem with non-negative edge weights (NSSP) are studied extensively, both theoretically =-=[22, 20, 25, 26, 56, 58, 35, 32, 48]-=- and experimentally [21, 30, 29, 15, 61, 31]. Let m and n denote the number of edges and vertices in the graph respectively. Nearly all NSSP algorithms are based on the classical Dijkstra’s [22] algor... |

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Citation Context ...oretic analysis metrics such as betweenness centrality [27, 9, 41, 43, 34] are based on shortest path algorithms. Our parallel implementations target the scale-free graph family, a well-studied model =-=[7, 24, 12, 53, 52]-=- for real-world large-scale graphs that captures features such as a low diameter, heavy-tailed degree distributions modeled by power laws, and self similarity. We also conduct an experimental study of... |

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Citation Context ...oretic analysis metrics such as betweenness centrality [27, 9, 41, 43, 34] are based on shortest path algorithms. Our parallel implementations target the scale-free graph family, a well-studied model =-=[7, 24, 12, 53, 52]-=- for real-world large-scale graphs that captures features such as a low diameter, heavy-tailed degree distributions modeled by power laws, and self similarity. We also conduct an experimental study of... |

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Citation Context ...CS Implementation Challenge [18] on Shortest Paths. Sequential algorithms for the single source shortest path problem with non-negative edge weights (NSSP) are studied extensively, both theoretically =-=[22, 20, 25, 26, 56, 58, 35, 32, 48]-=- and experimentally [21, 30, 29, 15, 61, 31]. Let m and n denote the number of edges and vertices in the graph respectively. Nearly all NSSP algorithms are based on the classical Dijkstra’s [22] algor... |

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Citation Context ...ial optimization problems, shortest path algorithms are finding increasing relevance in the domain of complex network analysis. Popular graph theoretic analysis metrics such as betweenness centrality =-=[27, 9, 41, 43, 34]-=- are based on shortest path algorithms. Our parallel implementations target the scale-free graph family, a well-studied model [7, 24, 12, 53, 52] for real-world large-scale graphs that captures featur... |

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Citation Context ...ential algorithms for the single source shortest path problem with non-negative edge weights (NSSP) are studied extensively, both theoretically [22, 20, 25, 26, 56, 58, 35, 32, 48] and experimentally =-=[21, 30, 29, 15, 61, 31]-=-. Let m and n denote the number of edges and vertices in the graph respectively. Nearly all NSSP algorithms are based on the classical Dijkstra’s [22] algorithm. Using Fibonacci heaps [25], Dijkstra’s... |

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Citation Context ...r parallel NSSP algorithms have been reviewed in detail by Meyer and Sanders [48, 51]. There are no known PRAM algorithms that run in sub-linear time and O(m + n log n) work. Parallel priority queues =-=[23, 11]-=- for implementing Dijkstra’s algorithm have been developed, but these linear work algorithms have a worst-case time bound of Ω(n), as they only perform edge relaxations in parallel. Several matrix-mul... |

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Citation Context ...ial optimization problems, shortest path algorithms are finding increasing relevance in the domain of complex network analysis. Popular graph theoretic analysis metrics such as betweenness centrality =-=[27, 9, 41, 43, 34]-=- are based on shortest path algorithms. Our parallel implementations target the scale-free graph family, a well-studied model [7, 24, 12, 53, 52] for real-world large-scale graphs that captures featur... |

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Citation Context ...ential algorithms for the single source shortest path problem with non-negative edge weights (NSSP) are studied extensively, both theoretically [22, 20, 25, 26, 56, 58, 35, 32, 48] and experimentally =-=[21, 30, 29, 15, 61, 31]-=-. Let m and n denote the number of edges and vertices in the graph respectively. Nearly all NSSP algorithms are based on the classical Dijkstra’s [22] algorithm. Using Fibonacci heaps [25], Dijkstra’s... |

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Citation Context ...ential algorithms for the single source shortest path problem with non-negative edge weights (NSSP) are studied extensively, both theoretically [22, 20, 25, 26, 56, 58, 35, 32, 48] and experimentally =-=[21, 30, 29, 15, 61, 31]-=-. Let m and n denote the number of edges and vertices in the graph respectively. Nearly all NSSP algorithms are based on the classical Dijkstra’s [22] algorithm. Using Fibonacci heaps [25], Dijkstra’s... |

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Citation Context ...or implementations of key combinatorial optimization and graph theoretic problems such as list ranking [3], connected components and subgraph isomorphism [8], Breadth-First Search and st-connectivity =-=[5]-=-. Our main contributions in this paper are as follows: • We conduct an experimental study of parallel shortest path algorithms designed for shared memory architectures. Prior studies have predominantl... |

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Citation Context ...or Δstepping. The literature contains few experimental studies on parallel NSSP algorithms [37, 54, 39, 59]. Prior implementation results on distributed memory machines resorted to graph partitioning =-=[14, 1, 33]-=-, and then running a sequential NSSP algorithm on the sub-graph. Heuristics are used for load balancing and termination detection [38, 40]. The implemen2tations perform well for certain graph familie... |

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Citation Context ...algorithm have been developed, but these linear work algorithms have a worst-case time bound of Ω(n), as they only perform edge relaxations in parallel. Several matrix-multiplication based algorithms =-=[36, 28]-=-, proposed for the parallel All-Pairs Shortest Paths (APSP), involve running time and efficiency tradeoffs. Parallel approximate NSSP algorithms [42, 16, 57] based on the randomized BreadthFirst searc... |

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Citation Context ...rmance on cache-based systems. On distributed memory clusters, few parallel graph algorithms outperform the best sequential implementations due to long memory latencies and high synchronization costs =-=[4, 3]-=-. Parallel shared memory systems are a more supportive platform. They offer higher memory bandwidth and lower latency than clusters, and the global shared memory greatly improves developer productivit... |

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Citation Context ...or Δstepping. The literature contains few experimental studies on parallel NSSP algorithms [37, 54, 39, 59]. Prior implementation results on distributed memory machines resorted to graph partitioning =-=[14, 1, 33]-=-, and then running a sequential NSSP algorithm on the sub-graph. Heuristics are used for load balancing and termination detection [38, 40]. The implemen2tations perform well for certain graph familie... |

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Citation Context ...ge time for uniformly distributed edge weights. In this paper, we primarily focus on parallel implementations of NSSP. Prior parallel NSSP algorithms have been reviewed in detail by Meyer and Sanders =-=[48, 51]-=-. There are no known PRAM algorithms that run in sub-linear time and O(m + n log n) work. Parallel priority queues [23, 11] for implementing Dijkstra’s algorithm have been developed, but these linear ... |

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Citation Context ... Shortest Paths (APSP), involve running time and efficiency tradeoffs. Parallel approximate NSSP algorithms [42, 16, 57] based on the randomized BreadthFirst search algorithm of Ullman and Yannakakis =-=[60]-=- run in sub-linear time. However, it is not known how to use the Ullman-Yannakakis randomized approach for exact NSSP computations in sub-linear time. Meyer and Sanders give the Δ-stepping [51] NSSP a... |

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Citation Context ...rmance on cache-based systems. On distributed memory clusters, few parallel graph algorithms outperform the best sequential implementations due to long memory latencies and high synchronization costs =-=[4, 3]-=-. Parallel shared memory systems are a more supportive platform. They offer higher memory bandwidth and lower latency than clusters, and the global shared memory greatly improves developer productivit... |

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Citation Context ...r parallel NSSP algorithms have been reviewed in detail by Meyer and Sanders [48, 51]. There are no known PRAM algorithms that run in sub-linear time and O(m + n log n) work. Parallel priority queues =-=[23, 11]-=- for implementing Dijkstra’s algorithm have been developed, but these linear work algorithms have a worst-case time bound of Ω(n), as they only perform edge relaxations in parallel. Several matrix-mul... |

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Citation Context ...Several matrix-multiplication based algorithms [36, 28], proposed for the parallel All-Pairs Shortest Paths (APSP), involve running time and efficiency tradeoffs. Parallel approximate NSSP algorithms =-=[42, 16, 57]-=- based on the randomized BreadthFirst search algorithm of Ullman and Yannakakis [60] run in sub-linear time. However, it is not known how to use the Ullman-Yannakakis randomized approach for exact NSS... |

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Citation Context ...s in sub-linear time with linear average case work. Several theoretical improvements [50, 46, 47] are given for Δstepping. The literature contains few experimental studies on parallel NSSP algorithms =-=[37, 54, 39, 59]-=-. Prior implementation results on distributed memory machines resorted to graph partitioning [14, 1, 33], and then running a sequential NSSP algorithm on the sub-graph. Heuristics are used for load ba... |

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Citation Context ...or Δstepping. The literature contains few experimental studies on parallel NSSP algorithms [37, 54, 39, 59]. Prior implementation results on distributed memory machines resorted to graph partitioning =-=[14, 1, 33]-=-, and then running a sequential NSSP algorithm on the sub-graph. Heuristics are used for load balancing and termination detection [38, 40]. The implemen2tations perform well for certain graph familie... |

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Citation Context ... which could be executed in parallel. For random graphs with uniformly distributed edge weights, this algorithm runs in sub-linear time with linear average case work. Several theoretical improvements =-=[50, 46, 47]-=- are given for Δstepping. The literature contains few experimental studies on parallel NSSP algorithms [37, 54, 39, 59]. Prior implementation results on distributed memory machines resorted to graph p... |

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Citation Context ...Several matrix-multiplication based algorithms [36, 28], proposed for the parallel All-Pairs Shortest Paths (APSP), involve running time and efficiency tradeoffs. Parallel approximate NSSP algorithms =-=[42, 16, 57]-=- based on the randomized BreadthFirst search algorithm of Ullman and Yannakakis [60] run in sub-linear time. However, it is not known how to use the Ullman-Yannakakis randomized approach for exact NSS... |

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Citation Context ...s in sub-linear time with linear average case work. Several theoretical improvements [50, 46, 47] are given for Δstepping. The literature contains few experimental studies on parallel NSSP algorithms =-=[37, 54, 39, 59]-=-. Prior implementation results on distributed memory machines resorted to graph partitioning [14, 1, 33], and then running a sequential NSSP algorithm on the sub-graph. Heuristics are used for load ba... |

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Citation Context ... Intel Xeon machine with 6GB memory, 1MB cache and running RedHat Enterprise Linux 4 (linux kernel 2.6.9). We compare the sequential performance of our implementation with the DIMACS reference solver =-=[19]-=-. Both the codes are compiled with the Intel C compiler (icc) Version 9.0, with the flags -O3. The source code is freely available online [44]. 4.2 Problem Instances We evaluate the sequential and par... |

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Citation Context ... which could be executed in parallel. For random graphs with uniformly distributed edge weights, this algorithm runs in sub-linear time with linear average case work. Several theoretical improvements =-=[50, 46, 47]-=- are given for Δstepping. The literature contains few experimental studies on parallel NSSP algorithms [37, 54, 39, 59]. Prior implementation results on distributed memory machines resorted to graph p... |

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Citation Context ...s in sub-linear time with linear average case work. Several theoretical improvements [50, 46, 47] are given for Δstepping. The literature contains few experimental studies on parallel NSSP algorithms =-=[37, 54, 39, 59]-=-. Prior implementation results on distributed memory machines resorted to graph partitioning [14, 1, 33], and then running a sequential NSSP algorithm on the sub-graph. Heuristics are used for load ba... |