# Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms

### Maher Helaoui

#### Abstract

The shortest path problem is one of the classic problems in graph theory. The problem is to provide a solution algorithm returning the optimum route, taking into account a valuation function, between two nodes of a graph G. It is known that the classic shortest path solution is proved if the set of valuation is IR or a subset of IR and the combining operator is the classic sum (+). However, many combinatorial problems can be solved by using shortest path solution but use a set of valuation not a subset of IR and/or a combining operator not equal to the classic sum (+). For this reason, relations between particular valuation structure as the semiring and diod structures with graphs and their combinatorial properties have been presented. On the other hand, if the set of valuation is IR or a subset of IR and the combining operator is the classic sum (+), a longest path between two given nodes s and t in a weighted graph G is the same thing as a shortest path in a graph -G derived from G by changing every weight to its negation. In this paper, in order to give a general model that can be used for any valuation structure we propose to model both the valuations of a graph G and the combining operator by a valuation structure S. We discuss the equivalence between longest path and shortest path problem given a valuation structure S. And we present a generalization of the shortest path algorithms according to the properties of the graph G and the valuation structure S.

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#### Paper Citation

#### in Harvard Style

Helaoui M. (2017). **Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms** . In *Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-218-9, pages 306-313. DOI: 10.5220/0006145303060313

#### in Bibtex Style

@conference{icores17,

author={Maher Helaoui},

title={Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms},

booktitle={Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2017},

pages={306-313},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006145303060313},

isbn={978-989-758-218-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 6th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - Extended Shortest Path Problem - Generalized Dijkstra-Moore and Bellman-Ford Algorithms

SN - 978-989-758-218-9

AU - Helaoui M.

PY - 2017

SP - 306

EP - 313

DO - 10.5220/0006145303060313