### Citations

10603 | Introduction to Algorithms
- Cormen, Leiverson, et al.
- 2009
(Show Context)
Citation Context ...network G amounts to checking for a negative cost cycle in the residual graph R f . Negative cycles in a weighted graph can be detected in O(n 3 ) time, where n is the number of vertices in the graph =-=[1]-=-. Theorem 2.2 There is an O(n 3 ) verification algorithm for the minimum-cost flow verification problem. 2.1.3 Verifiers for Matching A matching M of a graph G = (V; E) is a subset of edges with the p... |

1708 |
Combinatorial optimization: algorithms and complexity. Prentice-Hall, Inc., Upper Saddle River,
- Papadimitriou, Steiglitz
- 1982
(Show Context)
Citation Context ...?0 cost(v; w)f(v; w): We define the residual graph R f for a flow f exactly as we did in the section 2.2.1, with the extension that cost(v; w) is the same on R f as on G. We state the following lemma =-=[11]-=- without proof. Lemma 2.2 An s-t flow f in a network is an optimal min-cost flow if and only if there are no negative cost cycles in the residual graph R f . From the above theorem, verifying the opti... |

667 |
Data structures and network algorithms
- Tarjan
- 1983
(Show Context)
Citation Context ...e maximum-flow problem is that of finding a feasible flow f of maximum value v. This problem has a rich and elegant theory and many applications both in operations research and in combinatorics [13], =-=[12]-=-. We need to define a few terms. A residual capacity for a flow f is the function on vertex pairs given by res(v; w) = c(v; w) \Gamma f(v; w). The residual graph R f for a flow f is the graph with ver... |

570 |
Algorithm 97: Shortest Path
- Floyd
- 1962
(Show Context)
Citation Context ...sum of the weights of its edges. The length of a path is the number of edges in the path. The most widely known algorithms for the all-pairs shortest paths problem are those of Dijkstra [9] and Floyd =-=[10]-=-. Dijkstra's algorithm has a running time of \Theta(mn + n 2 log n) when implemented with a heap. Floyd's algorithm runs in \Theta(n 3 ) time. Bellman and Ford [1] also developed an algorithm which ru... |

361 | Self-testing/correcting with applications to numerical problems.
- Blum, Luby, et al.
- 1990
(Show Context)
Citation Context ...doesn't prove the correctness of the executable code of the algorithm verified. The executable code may still be faulty because of errors in the compilation process, hardware faults, etc. Blum et al. =-=[3]-=- introduced the theory of self-testing/correcting, where a self-tester for a program A estimates the probability that A(x) 6= f(x) for a random input x and a selfcorrectorsfor A computes f(x) correctl... |

100 |
The complexity of facets (and some facets of complexity
- PAPADIMITRIOU, YANNAKAKIS
- 1982
(Show Context)
Citation Context ...f and only if k is the size of the largest clique in the graph G. Definition 3.1 A language L is in the class DP if and only if there are two languages L 1 2 NP and L 2 2 coNP such that L = L 1 "=-= L 2 [4], [5]. Obviously, NP-=- ` DP and all DP-complete languages are NP-hard. V(H) is the "exact cost" version of the NP-complete optimization problem N H . The "exact cost" versions of many of the NP-complete... |

75 | Finding the hidden path: Time bounds for all-pairs shortest paths.
- Karger, Koller, et al.
- 1993
(Show Context)
Citation Context ...3 ) time and handles graphs with negative weights also. Many algorithms for the shortest-paths problem use edge weights only to compute and compare the weights of paths. We therefore define a version =-=[8]-=- of the decision tree model that captures this behavior. Definition 2.3 A path-comparison-based algorithm A solving the all-pairs shortest paths problem accepts as input a graph G and a weight functio... |

64 |
The complexity of facets resolved.
- Papadimitriou, Wolfe
- 1988
(Show Context)
Citation Context ... only if k is the size of the largest clique in the graph G. Definition 3.1 A language L is in the class DP if and only if there are two languages L 1 2 NP and L 2 2 coNP such that L = L 1 " L 2 =-=[4], [5]. Obviously, NP ` DP-=- and all DP-complete languages are NP-hard. V(H) is the "exact cost" version of the NP-complete optimization problem N H . The "exact cost" versions of many of the NP-complete opti... |

39 | A Mathematical Theory of Self-Checking, Self-Testing, and SelfCorrecting Programs
- Rubinfeld
- 1990
(Show Context)
Citation Context ... comparisons. We present a path-comparison-based verification algorithm for the all-pairs shortest paths problem which runs in O(mn) time. This is an improvement over the program checker of Rubinfeld =-=[2]-=-, which runs in O(n 3 ) time. There are graphs with \Theta(n 2 ) pairs of vertices whose connecting paths have lengths \Theta(n) each. Hence, we cannot verify all-pairs shortest paths in O(mn) time if... |

27 |
Using Certification Trails to Achieve Software Fault Tolerance
- Sullivan, Masson
- 1990
(Show Context)
Citation Context ...n of existence of polynomial verification algorithms becomes more interesting when we allow an algorithm to leave behind a trail of data, polynomial in length of the input, called certification trail =-=[6]-=-, [7], in addition to its normal output. A result verification algorithm might use this additional information cleverly to verify the result in much less time. The rest of the article is organized as ... |

23 | Certification trails for data structures
- Sullivan, Masson
- 1991
(Show Context)
Citation Context ...existence of polynomial verification algorithms becomes more interesting when we allow an algorithm to leave behind a trail of data, polynomial in length of the input, called certification trail [6], =-=[7]-=-, in addition to its normal output. A result verification algorithm might use this additional information cleverly to verify the result in much less time. The rest of the article is organized as follo... |

11 |
A note on two problems in connextion with graphs
- Dijkstra
- 1959
(Show Context)
Citation Context ...efined as the sum of the weights of its edges. The length of a path is the number of edges in the path. The most widely known algorithms for the all-pairs shortest paths problem are those of Dijkstra =-=[9]-=- and Floyd [10]. Dijkstra's algorithm has a running time of \Theta(mn + n 2 log n) when implemented with a heap. Floyd's algorithm runs in \Theta(n 3 ) time. Bellman and Ford [1] also developed an alg... |

3 |
Maximal Weighted Matching on General Graphs
- Galil, Micali, et al.
- 1982
(Show Context)
Citation Context ...e general weighted matching problem suffices to check for correctness of the matching presented. The best known algorithm for solving the weighted nonbipartite matching is by Galil, Micali, and Gabow =-=[14]-=- which runs in O(nm log n) time. Each result verification stage of the algorithm runs in O(m log n) time. Theorem 2.3 The unweighted general matching problem has a result verification algorithm which ... |