## Optimal Traversal of Directed Hypergraphs (1992)

Citations: | 21 - 2 self |

### BibTeX

@TECHREPORT{Ausiello92optimaltraversal,

author = {Giorgio Ausiello and Roberto Giaccio and Giuseppe F. Italiano and Umberto Nanni},

title = {Optimal Traversal of Directed Hypergraphs},

institution = {},

year = {1992}

}

### Years of Citing Articles

### OpenURL

### Abstract

A directed hypergraph is defined by a set of nodes and a set of hyperarcs, each connecting a set of source nodes to a single target node. Directed hypergraphs are used in several contexts to model different combinatorial structures, such as functional dependencies [Ull82], Horn clauses in propositional calculus [AI91], AND-OR graphs [Nil82], Petri nets [Pet62]. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using hyperarcs. Unlike paths in graphs, hyperpaths are suitable of different definitions of measure, corresponding to different concepts arising in various applications. In this paper we consider the problem of finding minimal hyperpaths according to several measures. We show that some of these problems are, not surprisingly, NP-hard. However, if the measure function on hyperpaths matches certain conditions (which we define as value-based measure functions) , the problem turns out to be solvable in polynomial time. We...

### Citations

10885 |
Computer and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...f size k or less. Then (P 1 ), (P 2 ), and (P 3 ) are NP-complete. Proof. We consider the three problems separately. Consider first problem (P 1 ). We use a reduction from Minimum Cover (in short MC) =-=[GJ79]-=-. Let A = fa 1 ; a 2 ; : : : ; an g be a set of elements and S = fS 1 ; S 2 ; : : : ; Sm g be a family of subsets of A such that S i=1;2;:::;m S i = A. We recall that, given A; S and an integer k, MC ... |

821 |
A note on two problems in connection with graphs”. Numerische Mathematik, 1:260–271
- Dijkstra
- 1959
(Show Context)
Citation Context ...sed as an argument to the procedure Scan. All the optimal hyperpaths from T are found in nondecreasing order. Algorithm Distance generalizes to directed hypergraphs Dijkstra's shortest path algorithm =-=[Dij59]-=-. Algorithm Distance operates in conjunction with procedure Scan. The algorithm uses a priority queue PQ where a node x is enqueued by procedure Scan (line 6) as soon as it is visited (that is when a ... |

572 |
Fibonacci heaps and their uses in improved network optimization algorithms
- Fredman, Tarjan
- 1987
(Show Context)
Citation Context ...t most n insertions and n deletions of the minimum key are performed in the priority queue PQ, plus at most h priority decreases. The claimed bound derives from the implementation of PQ as an F -heap =-=[FT87]-=-, provided that for any compound node z, the function /(DIST[z i 1 ]; DIST[z i 2 ]; : : : ; DIST[z i q ]) is computable in O(q) time. 2 5 Dynamic Maintenance of Value-Based Measures In many applicatio... |

556 |
Principles of Artificial Intelligence
- Nilsson
- 1980
(Show Context)
Citation Context ...e. Directed hypergraphs are used in several contexts to model different combinatorial structures, such as functional dependencies [Ull82], Horn clauses in propositional calculus [AI91], AND-OR graphs =-=[Nil82]-=-, Petri nets [Pet62]. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using hyperarcs. Unlike paths in graphs, hyperpaths are suitable of differen... |

270 |
Principle of Database Systems
- Ullman
- 1982
(Show Context)
Citation Context ...yperarcs, each connecting a set of source nodes to a single target node. Directed hypergraphs are used in several contexts to model different combinatorial structures, such as functional dependencies =-=[Ull82]-=-, Horn clauses in propositional calculus [AI91], AND-OR graphs [Nil82], Petri nets [Pet62]. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using ... |

177 |
An algorithm for the organization of information
- Adel'son-Vel'skii, Landis
- 1962
(Show Context)
Citation Context ...tions of hyperarcs in the structure is arbitrary. In such a situation an efficient dynamic data structure for compound nodes has to be maintained. The compound nodes will be maintained in an AVL tree =-=[AVL62]-=- referred to as T c (while N c denotes the set of compound nodes). This will allow one to efficiently check whether the source set of the hyperarc to be introduced corresponds to a compound node alrea... |

115 | An incremental algorithm for a generalization of the shortestpath problem
- Ramalingam, Reps
- 1996
(Show Context)
Citation Context ...of superior function has been introduced by Knuth [Knu77] in the context of a generalization of Dijkstra's algorithm to context-free grammars, and has been slightly generalized by Ramalingam and Reps =-=[RR92]-=-, who also provide dynamic algorithms. Value-based measures are a proper subclass of superior functions. A first difference is that in the grammar problem each production may be associated to a differ... |

100 | Directed hypergraphs and applications
- Gallo, Longo, et al.
- 1993
(Show Context)
Citation Context ... ! q, where p 1 ; : : : ; p k and q are propositional symbols), implications in problem solving (where hypergraphs can be used as an alternative to and-or graphs), Datalog [GR90], Operations Research =-=[GLNP93], Petri Ne-=-ts [AFN92]. Work partially supported by EU ESPRIT Long Term Research Project ALCOM-IT under contract no.20244. y Dipartimento di Informatica e Sistemistica, Universit`a di Roma "La Sapienza"... |

65 | Incremental Algorithms for Minimal Length Paths - Ausiello, Italiano, et al. - 1990 |

64 |
A Generalization of Dijkstra’s Algorithm
- Knuth
- 1977
(Show Context)
Citation Context ...sure function are given. The achieved time bound is O(jHj + n log n) for a hypergraph H with n nodes and an overall description of size jHj. A notion of superior function has been introduced by Knuth =-=[Knu77]-=- in the context of a generalization of Dijkstra's algorithm to context-free grammars, and has been slightly generalized by Ramalingam and Reps [RR92], who also provide dynamic algorithms. Value-based ... |

33 | Minimal representation of directed hypergraphs - Ausiello, D’Atri, et al. - 1986 |

24 |
Graph algorithms for functional dependency manipulation
- Ausiello, D’Atri, et al.
- 1983
(Show Context)
Citation Context ...ost over the sequence is O(x + log h), wheresx is the average size of a hyperarc. 1 Introduction A directed hypergraph is a generalization of the concept of directed graph. It was first introduced in =-=[ADS83]-=- to represent functional dependencies in relational data base schemata. While directed graphs are normally used for representing one-to-one functional relations over finite sets, in several areas of c... |

19 | Hyperpaths and shortest hyperpaths - Nguyen, Pallottino - 1989 |

18 |
On-line algorithms for polynomially solvable satisfiability problems
- Ausiello, Italiano
- 1991
(Show Context)
Citation Context ...to a single target node. Directed hypergraphs are used in several contexts to model different combinatorial structures, such as functional dependencies [Ull82], Horn clauses in propositional calculus =-=[AI91]-=-, AND-OR graphs [Nil82], Petri nets [Pet62]. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using hyperarcs. Unlike paths in graphs, hyperpaths a... |

18 |
Dynamic maintenance of directed hypergraphs
- Ausiello, Italiano, et al.
- 1990
(Show Context)
Citation Context ...t the notion of transitive closure of a hypergraph has been defined in [ADS83] and efficient algorithms for transitive closure maintenance under dynamic insertion of hyperarcs have been introduced in =-=[AIN90]-=-. The problem of defining cost measures in hyperpaths is more delicate. In fact, hyperpaths can be measured and compared according to a much broader range of measures than simple graphs. Indeed we sho... |

13 |
A hypergraph approach to logical inference for datalog formulae, working paper
- Gallo, Rago
- 1990
(Show Context)
Citation Context ...orn clause p 1 ; : : : ; p k ! q, where p 1 ; : : : ; p k and q are propositional symbols), implications in problem solving (where hypergraphs can be used as an alternative to and-or graphs), Datalog =-=[GR90]-=-, Operations Research [GLNP93], Petri Nets [AFN92]. Work partially supported by EU ESPRIT Long Term Research Project ALCOM-IT under contract no.20244. y Dipartimento di Informatica e Sistemistica, Uni... |

11 | On line maintenance of minimal directed hypergraphs - Italiano, Nanni - 1989 |

6 | Strongly equivalent directed hypergraphs, in: Analysis and Design of Algorithms for combinatorial problems - Ausiello, D’Atri, et al. - 1985 |

6 |
Linear time algorithms for liveness and boundedness in conflict-free petri nets
- Alimonti, Feuerstein, et al.
- 1992
(Show Context)
Citation Context ... : ; p k and q are propositional symbols), implications in problem solving (where hypergraphs can be used as an alternative to and-or graphs), Datalog [GR90], Operations Research [GLNP93], Petri Nets =-=[AFN92]. Work par-=-tially supported by EU ESPRIT Long Term Research Project ALCOM-IT under contract no.20244. y Dipartimento di Informatica e Sistemistica, Universit`a di Roma "La Sapienza", Via Salaria 113 - ... |

5 |
Communication with Automata, Supplement 1 to
- Petri, A
- 1962
(Show Context)
Citation Context ...phs are used in several contexts to model different combinatorial structures, such as functional dependencies [Ull82], Horn clauses in propositional calculus [AI91], AND-OR graphs [Nil82], Petri nets =-=[Pet62]-=-. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using hyperarcs. Unlike paths in graphs, hyperpaths are suitable of different definitions of mea... |

4 |
On-line algorithms for satisfiability formulae with uncertainty
- Ausiello, Giaccio
- 1997
(Show Context)
Citation Context ... existence of a hyperpath in a directed hypergraph representing the whole Horn formula. More in general we might consider a knowledge-based system, based on Horn calculus, with fuzzy production rules =-=[AG97]-=-. If each rule is given a confidence value representing its reliability, it is interesting to consider various problems, such as the degree of consistency of a set of rules, or the most reliable proof... |