#### DMCA

## Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths (1993)

### Cached

### Download Links

- [robotics.stanford.edu]
- [theory.lcs.mit.edu]
- [theory.lcs.mit.edu]
- [people.csail.mit.edu]
- DBLP

### Other Repositories/Bibliography

Citations: | 75 - 0 self |

### Citations

1086 |
A note on two problems in connection with graphs
- Dijkstra
- 1959
(Show Context)
Citation Context ...ap [8] to implement a priority queue; the running time increases to O(m n log n) if a standard heap is used instead. The algorithm operates by running Dijkstra's singlesource shortest paths algorithm =-=[3]-=- in parallel from all nodes in the graph, using information gained at each node to reduce the work done at other nodes. Our algorithm is likely to be fast in practice, because it is known [9, 15] that... |

989 | Matrix multiplication via arithmetic progressions - Coppersmith, Winograd - 1990 |

739 | Fibonacci heaps and their uses in improved network optimization algorithms
- Fredman, Tarjan
- 1987
(Show Context)
Citation Context ...graph. The Hidden Paths Algorithm runs in time Supported by a National Science Foundation Graduate Fellowship. y Supported by NSF PYI grant CCR-8858030-03. O(m n+n 2 log n) if we use a Fibonacci heap =-=[8]-=- to implement a priority queue; the running time increases to O(m n log n) if a standard heap is used instead. The algorithm operates by running Dijkstra's singlesource shortest paths algorithm [3] in... |

667 | Data structures and network algorithms - Tarjan - 1983 |

569 |
Algorithm 97: Shortest Path
- Floyd
- 1962
(Show Context)
Citation Context ... all-pairs shortest path problem for all generalized path weight functions. Previous Work The most widely known algorithms for the all-pairs shortest paths problem are those of Dijkstra [3] and Floyd =-=[5]-=-. Dijkstra's algorithm for the single-source shortest path problem can be run from each vertex (as noted by Johnson [11]), resulting in a running time of \Theta(mn + n 2 log n) if Fibonacci heaps are ... |

384 | Introdution to Automata Theory - Hopcroft, Ullman - 1979 |

312 | Multi-Terminal Network Flows. - Gomory, Hu - 1961 |

221 |
Efficient Algorithms for Shortest Paths in Sparse Networks
- Johnson
- 1977
(Show Context)
Citation Context ...hms for the all-pairs shortest paths problem are those of Dijkstra [3] and Floyd [5]. Dijkstra's algorithm for the single-source shortest path problem can be run from each vertex (as noted by Johnson =-=[11]-=-), resulting in a running time of \Theta(mn + n 2 log n) if Fibonacci heaps are used to implement priority queues. Floyd's algorithm, which can handle negative edge weights, works by dynamic programmi... |

115 | Algorithm 360: Shortest-path forest with topological ordering. - Dial - 1969 |

102 | The shortest-path problem for graphs with random arc-lengths
- Frieze, Grimmett
- 1985
(Show Context)
Citation Context ...lgorithm [3] in parallel from all nodes in the graph, using information gained at each node to reduce the work done at other nodes. Our algorithm is likely to be fast in practice, because it is known =-=[9, 15]-=- that m = O(n log n) with high probability when the input graph is the complete graph with edge weights chosen independently from any of a large class of probability distributions, including the unifo... |

89 | Clique partitions, graph compression and speeding-up algorithms, in
- Feder, Motwani
- 1991
(Show Context)
Citation Context ... bound of Section 3 they have a worst case running time of \Omega\Gamma mn). Fast algorithms exist for special cases of the all-pairs shortest paths problem, for instance when the graph is unweighted =-=[4]-=- or planar [6]. Fredman [7] shows that O(n 5=2 ) comparisons between sums of edge weights suffice to solve the allpairs shortest paths problem. He uses this fact to do preprocessing, producing an algo... |

84 | On the Exponent of the All Pairs Shortest Path Problem, - Alon, Galil, et al. - 1997 |

83 | Linear and Combinatorial Optimization in Ordered Algebraic Structures. - Zimmermann - 1981 |

79 |
New Bounds on the Complexity of the Shortest Path Problem,
- Fredman
- 1976
(Show Context)
Citation Context ...ve a worst case running time of \Omega\Gamma mn). Fast algorithms exist for special cases of the all-pairs shortest paths problem, for instance when the graph is unweighted [4] or planar [6]. Fredman =-=[7]-=- shows that O(n 5=2 ) comparisons between sums of edge weights suffice to solve the allpairs shortest paths problem. He uses this fact to do preprocessing, producing an algorithm that runs in time O(n... |

75 |
A generalization of Dijkstra’s algorithm
- KNUTH
- 1977
(Show Context)
Citation Context ... pair of vertices a path between them of minimum weight. To make this problem tractable, we impose restrictions on the weight function. Generalized weight functions have also been studied by Knuth in =-=[13]-=-. Definition 4.1 Consider a weight function k \Delta k: ffl it is consistent if for all u; v; w, k(v ; w)ksk(v ; 0 w)k implies k(u ; v ; w)ksk(u ; v ; 0 w)k; and similarly for k(u ; v)ksk(u ; 0 v)k. f... |

73 | Scaling Algorithms for the Shortest Paths Problem, - Goldberg - 1995 |

55 |
Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph.
- NAGAMOCHI, IBARAKI
- 1992
(Show Context)
Citation Context ...revealed. The philosophy of the Hidden Paths Algorithm is thus similar to recent algorithms for connectivity, which work by first finding a sparse subgraph (or certificate) with the same connectivity =-=[2, 17]-=-. We have shown a lower bound of \Omega\Gamma mn) on the running time of comparison based algorithms for allpairs shortest paths. It is of particular interest that the construction and verification al... |

50 |
Data Structures and Network Algorithms, volume 44
- Tarjan
- 1983
(Show Context)
Citation Context ...ch assigns to every path a weight equal to the weight of the maximal edge on the path. Solving single source shortest paths under this weight function is referred to as the bottleneck path problem in =-=[20]-=-. An algorithm for a generalized shortest paths problem receives as input the graph and a black box for the weight function. We assume that the black box takes constant time to compute the weight of a... |

46 | A new upper bound on the complexity of the all pairs shortest path problem - Takaoka - 1992 |

44 | On the all-pairs-shortest-path problem - Seidel - 1992 |

39 | N.: Planar graph decomposition and all pairs shortest paths
- Frederickson
- 1991
(Show Context)
Citation Context ...ion 3 they have a worst case running time of \Omega\Gamma mn). Fast algorithms exist for special cases of the all-pairs shortest paths problem, for instance when the graph is unweighted [4] or planar =-=[6]-=-. Fredman [7] shows that O(n 5=2 ) comparisons between sums of edge weights suffice to solve the allpairs shortest paths problem. He uses this fact to do preprocessing, producing an algorithm that run... |

30 |
A new algorithm for finding all shortest paths in a graph of positive arcs in average time O(n2 log2 n).
- Spira
- 1973
(Show Context)
Citation Context ...s, works by dynamic programming and runs in time \Theta(n 3 ). If the edge weights are independently and identically distributed random variables, a variant of Dijkstra's algorithm developed by Spira =-=[18]-=- has an expected running time of O(n 2 log 2 n). Bloniarz [1] provided an algorithm with an expected running time of O(n 2 log n log n). Another algorithm, developed by Frieze and Grimmet [9], achieve... |

29 | Witnesses for boolean matrix multiplication and for shortest paths - Alon, Galil, et al. - 1992 |

27 |
The Effect of Algebraic Structure on the Computational Complexity of Matrix Multiplication
- Kerr
- 1970
(Show Context)
Citation Context ...the computational complexity of the all-pairs shortest paths problem have been proved in some other models. If the permissible operations are addition and minimum in a straight line computation, Kerr =-=[12]-=- shows that any algorithm requires \Omega\Gamma n 3 ) running time. Regarding algebraic decision tree complexity, Spira and Pan [19] show that\Omega\Gamma n 2 ) comparisons between sums of edge weight... |

27 | Linear verification for spanning trees
- Komlós
- 1985
(Show Context)
Citation Context ...ification algorithms have the same worst case complexity. Compare this to the situation for the minimum spanning tree problem, where there is a linear-time algorithm to verify a minimum spanning tree =-=[14]-=-, although no algorithm is known that finds one in linear-time. The comparison based lower bound shows that any improvement in the worst case complexity of shortest paths algorithms, such as Fredman's... |

25 | On shortest paths in graphs with random weights. - Hassin, Zemel - 1985 |

22 |
Algorithms for parallel k-vertex connectivity and sparse certificates.
- CHERIYAN, THURIMELLA
- 1991
(Show Context)
Citation Context ...revealed. The philosophy of the Hidden Paths Algorithm is thus similar to recent algorithms for connectivity, which work by first finding a sparse subgraph (or certificate) with the same connectivity =-=[2, 17]-=-. We have shown a lower bound of \Omega\Gamma mn) on the running time of comparison based algorithms for allpairs shortest paths. It is of particular interest that the construction and verification al... |

17 |
A shortest-path algorithm with expected time O(n2 log n log∗ n).
- Bloniarz
- 1983
(Show Context)
Citation Context ...robability when the input graph is the complete graph with edge weights chosen independently from any of a large class of probability distributions, including the uniform distribution on the interval =-=[0; 1]-=-. Our second contribution is a new lower bound, given in Section 3. Most algorithms for the allpairs shortest path problem use the edge weight function only in comparing the weights of two paths in th... |

12 |
A bidirectional shortest-path algorithm with good average-case behavior. Algorithmica
- Luby, Ragde
- 1989
(Show Context)
Citation Context ...lgorithm [3] in parallel from all nodes in the graph, using information gained at each node to reduce the work done at other nodes. Our algorithm is likely to be fast in practice, because it is known =-=[9, 15]-=- that m = O(n log n) with high probability when the input graph is the complete graph with edge weights chosen independently from any of a large class of probability distributions, including the unifo... |

9 | Efficient algorithms for path problems with general cost criteria - Lengauer, Theune - 1991 |

8 |
Mixed-approach algorithms for transitive closure
- Jakobsson
- 1991
(Show Context)
Citation Context ...n algorithm similar to the Hidden Paths Algorithm, with the same time bound, has been developed independently by McGeoch [16]. A variant of our algorithm has been developed independently by Jakobsson =-=[10]-=- as a transitive closure algorithm. Both these algorithms require data structures which are more complex than those used by the Hidden Paths Algorithm. Lower bounds on the computational complexity of ... |

6 |
On and updating shortest paths and spanning trees
- Spira, Pan
- 1973
(Show Context)
Citation Context ...tions are addition and minimum in a straight line computation, Kerr [12] shows that any algorithm requires \Omega\Gamma n 3 ) running time. Regarding algebraic decision tree complexity, Spira and Pan =-=[19]-=- show that\Omega\Gamma n 2 ) comparisons between sums of edge weights are necessary to solve the single-source shortest paths problem. 2 Finding Shortest Paths In this section, we describe an algorith... |

3 |
A new all-pairs shortest-path algorithm
- McGeoch
- 1991
(Show Context)
Citation Context ...t allow such comparisons in order to improve on the \Omega\Gamma n 3 ) bound. An algorithm similar to the Hidden Paths Algorithm, with the same time bound, has been developed independently by McGeoch =-=[16]-=-. A variant of our algorithm has been developed independently by Jakobsson [10] as a transitive closure algorithm. Both these algorithms require data structures which are more complex than those used ... |

3 | Tarjan, "Faster scaling algorithms for network problems - Gabow, E - 1989 |

1 | Minimum paths in directed graphs", Operations Research Quarterly - Frieze - 1977 |