## Buckets strike back: Improved Parallel Shortest-Paths (2002)

Venue: | Proc. 16th Intl. Par. Distr. Process. Symp. (IPDPS |

Citations: | 6 - 2 self |

### BibTeX

@INPROCEEDINGS{Meyer02bucketsstrike,

author = {Ulrich Meyer},

title = {Buckets strike back: Improved Parallel Shortest-Paths},

booktitle = {Proc. 16th Intl. Par. Distr. Process. Symp. (IPDPS},

year = {2002},

pages = {1--8},

publisher = {IEEE Computer Society}

}

### OpenURL

### Abstract

We study the average-case complexity of the parallel single-source shortest-path (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 bucket-based parallel SSSP algorithm that runs in T = O(log 2 n min i f2 i L + jV i jg) average-case time using O(n+m+T ) work on a PRAM where L denotes the maximum shortest-path weight and jV i j is the number of graph vertices with in-degree 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.

### Citations

2275 | Emergence of scaling in random networks
- Barabási, Albert
- 1999
(Show Context)
Citation Context ....5 Performance Gain on Power Law Graphs. Many sparse massive graphs such as the WWW graph and telephone call graphs share universal characteristics which can be described by the so-called "power =-=law" [2, 4, 2-=-1]: the number of nodes, y, of a given in-degree x is proportional to x for some constants> 0. For most massive graphs, 2s 4: independently, Kumar et. al. [21] and Babarasi et. al. [4] reporteds 2:1 f... |

1843 | Random Graphs
- Bollobás
- 2001
(Show Context)
Citation Context ...ohen [9] and Shi and Spencer [28]. Most of these algorithms can be modified to run on the weakest PRAM model without concurrent read/write capability. Parallel shortest path problems on random graphs =-=[6]-=- where each of the n 2 possible edges is present with a certain probability have been studied intensively [8, 10, 15, 17, 25, 26, 27]. Under the assumption of independent random edge weights uniformly... |

1544 | A note on two problems in connexion with graphs
- Dijkstra
- 1959
(Show Context)
Citation Context ... facilitate easy exposition we focus on the PRAM model and only sketch how our SSSP algorithm can be converted to DMMs. 1.1 Previous Work. The classical sequential SSSP result is Dijkstra's algorithm =-=[11-=-]; implemented with Fibonacci heaps it solves SSSP on arbitrary directed graphs with nonnegative edge weights in O(n log n + m) time. A number of faster algorithms have been developed on the more pow... |

1466 |
Network flows: Theory, Algorithm, and Applications
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context ...rage-case complexity of the parallel single-source shortest-path (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 bucket-based parallel SSSP algorithm that runs in T = O(log 2 n min i f2 i L + jV i jg) average-case time using O(n+m+T ) work on a PRAM where L denotes the maximum shortest-pat... |

1157 |
A bridging model for parallel computation
- Valiant
- 1990
(Show Context)
Citation Context ...rimental parallel machines like the SB-PRAM [13], it is valuable to highlight the main ideas of a parallel algorithm without tedious details caused by a particular architecture. Other models like BSP =-=[30]-=- view a parallel computer as a collection of sequential processors, each one having its own local memory, so called distributed memory machines (DMMs). The PUs are interconnected by a network that all... |

658 |
An Introduction to Parallel Algorithms
- Jájá
- 1992
(Show Context)
Citation Context ... the Hungarian Academy of Sciences, Center of Excellence, MTA SZTAKI, Budapest. has finite, nonnegative derivative at 0. This is also true for our algorithm. The parallel random access machine (PRAM) =-=[19]-=- is one of the most widely studied abstract models of a parallel computer. A PRAM consists of P independent processors (processing units, PUs) and a shared memory, which these processors can synchrono... |

350 | A random graph model for massive graphs
- Aiello, Chung, et al.
- 2000
(Show Context)
Citation Context ....5 Performance Gain on Power Law Graphs. Many sparse massive graphs such as the WWW graph and telephone call graphs share universal characteristics which can be described by the so-called "power =-=law" [2, 4, 2-=-1]: the number of nodes, y, of a given in-degree x is proportional to x for some constants> 0. For most massive graphs, 2s 4: independently, Kumar et. al. [21] and Babarasi et. al. [4] reporteds 2:1 f... |

304 | Trawling the Web for Emerging Cyber Communities
- Kumar, Raghavan, et al.
- 1999
(Show Context)
Citation Context ....5 Performance Gain on Power Law Graphs. Many sparse massive graphs such as the WWW graph and telephone call graphs share universal characteristics which can be described by the so-called "power =-=law" [2, 4, 2-=-1]: the number of nodes, y, of a given in-degree x is proportional to x for some constants> 0. For most massive graphs, 2s 4: independently, Kumar et. al. [21] and Babarasi et. al. [4] reporteds 2:1 f... |

96 |
Undirected single-source shortest paths with positive integer weights in linear time
- Thorup
- 1999
(Show Context)
Citation Context ...SSSP on arbitrary directed graphs with nonnegative edge weights in O(n log n + m) time. A number of faster algorithms have been developed on the more powerful RAM (random access machine) model, see [=-=29]-=- for an overview. In particular, Thorup [29] has given the first O(n + m) worst-case time RAM algorithm for undirected graphs with integer or float edge weights. The average-case analysis of shortest-... |

77 |
Relaxed heaps: An alternative to Fibonacci heaps with applications to parallel computation
- Driscoll, Gabow, et al.
- 1988
(Show Context)
Citation Context ...o parallel O(n log n +m) work PRAM SSSP algorithm with worst-case sublinear running time for arbitrary digraphs with nonnegative edge weights. The O(n log n + m) work solution by Driscoll et. al. [1=-=-=-2] has 1 running time O(n log n). An O(n) time algorithm requiringsO(m log n) work was presented by Brodal et. al. [7]. All faster known algorithms require more work, e.g., the approach by Han et. a... |

39 |
Algorithmic theory of random graphs. Random Structures and Algorithms
- Frieze, McDiarmid
- 1997
(Show Context)
Citation Context ...abbreviated dist(v); the weight of a path is the sum of the weights of its edges. Assuming independent random edge weights is a standard setting for the average-case analysis of graph algorithms; see =-=[14]-=- for many examples. The uniform edge weight distribution is mostly chosen in order to keep the proofs simple. Frequently, the obtained results also hold asymptotically in the more general situation of... |

37 | The Diameter of Random Massive Graphs
- Lu
- 2000
(Show Context)
Citation Context ...meters are usually very small; on random graph classes -- with O(n) edges -- that are widely considered to be appropriate models of real massive graphs like the WWW it turns out that L = O(log n) whp [22]. For such graphs the number of nodes having in-degree at least d is approximately given by (n P xd x ), which for constants 2 and arbitrary d 1 can be bounded by (n R 1 d x dx) = (... |

34 | A simple shortest path algorithm with linear average time
- Goldberg
- 2001
(Show Context)
Citation Context ...(APSP) problem for the complete graph with random edge weights. Recently, the first linear O(n + m) average-case time algorithms for arbitrary directed graphs with random edge weights have been given =-=[16, 2-=-4]. So far there is no parallel O(n log n +m) work PRAM SSSP algorithm with worst-case sublinear running time for arbitrary digraphs with nonnegative edge weights. The O(n log n + m) work solution b... |

29 |
Fast parallel space allocation, estimation and integer sorting
- Bast, Hagerup
- 1995
(Show Context)
Citation Context ... emanating from nodes in R is built. An immediate parallel relaxation of the set Req might cause conflicts, therefore Req is first grouped by target nodes (using semi-sorting with small hashed values =-=[5, 26]-=-), and then the strictest relaxation request for each target node (group) is selected. The selected requests are grouped once more by target buckets and finally each group is appended in parallel afte... |

28 | Shortest-Paths on arbitrary directed graphs in linear Average-Case time
- Meyer
- 2001
(Show Context)
Citation Context ...(APSP) problem for the complete graph with random edge weights. Recently, the first linear O(n + m) average-case time algorithms for arbitrary directed graphs with random edge weights have been given =-=[16, 2-=-4]. So far there is no parallel O(n log n +m) work PRAM SSSP algorithm with worst-case sublinear running time for arbitrary digraphs with nonnegative edge weights. The O(n log n + m) work solution b... |

27 | Parallelizing Dijkstra’s shortest path algorithm
- Crauser, Mehlhorn, et al.
- 1998
(Show Context)
Citation Context ...without concurrent read/write capability. Parallel shortest path problems on random graphs [6] where each of the n 2 possible edges is present with a certain probability have been studied intensively =-=[8, 10, 15, 17, 25, 26, 27]-=-. Under the assumption of independent random edge weights uniformly distributed in the interval [0; 1] the fastest work-efficient parallel SSSP algorithm for random graphs [25, 26] requires O(log 2 n)... |

25 | Practical parallel algorithms for personalized communication and integer sorting
- Bader, Helman, et al.
- 1996
(Show Context)
Citation Context ...that store outgoing edges of v. Figure 3 depicts the role of spreading and grouping for load balancing. The grouping steps of the algorithm can be implemented by DMM integer-sorting algorithms, e.g., =-=[3]-=-. The choice of the sorting algorithm determines how many supersteps are needed to implement a phase of the PRAM algorithm, and hence how many processors can be reasonably used. 3.5 Performance Gain o... |

25 | Efficient Parallel Algorithms for Computing All
- Han, Pan, et al.
- 1992
(Show Context)
Citation Context ...s 1 running time O(n log n). An O(n) time algorithm requiringsO(m log n) work was presented by Brodal et. al. [7]. All faster known algorithms require more work, e.g., the approach by Han et. al. [18] needs O(log 2 n) time and O(n 3 (log log n= log n) 1=3 ) work. The algorithm of Klein and Subramanian [20] takes O( p n log L log n log n) time and O( p n m log L log n) work where L ... |

15 | A parallel priority queue with constant time operations
- Brodal, TrÃd’ff, et al.
- 1998
(Show Context)
Citation Context ...nnegative edge weights. The O(n log n + m) work solution by Driscoll et. al. [12] has 1 running time O(n log n). An O(n) time algorithm requiringsO(m log n) work was presented by Brodal et. al. [7]. All faster known algorithms require more work, e.g., the approach by Han et. al. [18] needs O(log 2 n) time and O(n 3 (log log n= log n) 1=3 ) work. The algorithm of Klein and Subramanian [20] tak... |

15 | A randomized parallel algorithm for single-source shortestpaths
- Klein, Subramanian
- 1992
(Show Context)
Citation Context ... al. [7]. All faster known algorithms require more work, e.g., the approach by Han et. al. [18] needs O(log 2 n) time and O(n 3 (log log n= log n) 1=3 ) work. The algorithm of Klein and Subramanian [20] takes O( p n log L log n log n) time and O( p n m log L log n) work where L is the maximum shortest-path weight, i.e. L = max v2G;dist(v) dist(v). Similar results have been obtained by ... |

11 | Parallel shortest path for arbitrary graphs
- Sanders
- 2000
(Show Context)
Citation Context ... emanating from nodes in R is built. An immediate parallel relaxation of the set Req might cause conflicts, therefore Req is first grouped by target nodes (using semi-sorting with small hashed values =-=[5, 26]-=-), and then the strictest relaxation request for each target node (group) is selected. The selected requests are grouped once more by target buckets and finally each group is appended in parallel afte... |

9 |
Time-work tradeoffs of the single-source shortest paths problem
- Shi, Spencer
- 1999
(Show Context)
Citation Context ... log n) time and O( p n m log L log n) work where L is the maximum shortest-path weight, i.e. L = max v2G;dist(v) dist(v). Similar results have been obtained by Cohen [9] and Shi and Spencer [28]. Most of these algorithms can be modified to run on the weakest PRAM model without concurrent read/write capability. Parallel shortest path problems on random graphs [6] where each of the n 2 possibl... |

8 |
Expected parallel time and sequential space complexity of graph and digraph problems
- Reif, Spirakis
- 1992
(Show Context)
Citation Context ...without concurrent read/write capability. Parallel shortest path problems on random graphs [6] where each of the n 2 possible edges is present with a certain probability have been studied intensively =-=[8, 10, 15, 17, 25, 26, 27]-=-. Under the assumption of independent random edge weights uniformly distributed in the interval [0; 1] the fastest work-efficient parallel SSSP algorithm for random graphs [25, 26] requires O(log 2 n)... |

7 | HPP: A High{Performance PRAM
- Formella, Keller, et al.
- 1996
(Show Context)
Citation Context ...limited number of available PUs) and work (the total number of operations needed). Even though the strict PRAM model is only implemented on a number of experimental parallel machines like the SB-PRAM =-=[13]-=-, it is valuable to highlight the main ideas of a parallel algorithm without tedious details caused by a particular architecture. Other models like BSP [30] view a parallel computer as a collection of... |

7 |
A parallel algorithm for all-pairs shortest paths in a random graph
- Frieze, Rudolph
- 1985
(Show Context)
Citation Context ...without concurrent read/write capability. Parallel shortest path problems on random graphs [6] where each of the n 2 possible edges is present with a certain probability have been studied intensively =-=[8, 10, 15, 17, 25, 26, 27]-=-. Under the assumption of independent random edge weights uniformly distributed in the interval [0; 1] the fastest work-efficient parallel SSSP algorithm for random graphs [25, 26] requires O(log 2 n)... |

5 |
Using selective path-doubling for parallel shortest-path computation
- Cohen
- 1997
(Show Context)
Citation Context ... O( p n log L log n log n) time and O( p n m log L log n) work where L is the maximum shortest-path weight, i.e. L = max v2G;dist(v) dist(v). Similar results have been obtained by Cohen [9] and Shi and Spencer [28]. Most of these algorithms can be modified to run on the weakest PRAM model without concurrent read/write capability. Parallel shortest path problems on random graphs [6] wher... |

5 |
stepping: A parallel shortest path algorithm
- Meyer, Sanders
- 1998
(Show Context)
Citation Context |

4 | Randomized parallel algorithms
- Clementi, Rolim, et al.
- 1996
(Show Context)
Citation Context |

4 |
are better than buckets: parallel shortest paths on unbalanced graphs
- Heaps
- 2000
(Show Context)
Citation Context ...raphs with large maximum degree d, the algorithms of [25, 26] perform poorly: d) time is needed. If the number of high-degree nodes is rather small, then the running time can be considerably improved =-=[2-=-3]: let jV i j denote the number of graph vertices with in-degree at least 2 i then SSSP can be solved in TH = O(log 3 n min i f2 i L+ jV i jg) time on average. However, the algorithm needs non-line... |

2 |
A sharper analysis of a parallel algorithm for the all pairs shortest path problem
- Gu, Takaoka
- 1990
(Show Context)
Citation Context |