Généralisation de l'algorithme de Warshall

Warshall’s algorithm—survey and applications

“…Lines 5 and 6 in the Warshall's algorithm presented in Listing 1 can be expressed as ← + • using the operations defined above. If instead of the operations + and • we use two operations ⊕ and ⊙ from a semiring, a generalized Warshall's algorithm results [11]:…”

“…Choosing for ⊕ the min operation (minimum of two real numbers), and for ⊙ the real addition (+), we obtain the well-known Floyd-Warshall algorithm as a special case of the generalized Warshall's algorithm [5,11,12] :…”

“…Warshall's algorithm [14] with its generalization [11] is widely used in graph theory [1, 3-5, 8-10, 12] and in various fields of sciences (e.g. fuzzy [13] and quantum [6] theory, Kleene algebra [7]).…”

Linguistic characterization of preference relations as a basis for choice in social systems

“…The well-known Warshall algorithm for the computation of the transitive closure of a nonfuzzy relation may readily be extended to the computation of the right-hand member of (2.8) [9], [10]. A fuzzy relation, R, is reflexive if…”

Dioïds and semirings: Links to fuzzy sets and other applications

“…In the last two or three decades however, the range of available mathematical models has been considerably enlarged as a result of the development of new modelling paradigms such as Fuzzy Set theory [29,31,60,84,85] and also due to the needs for new problem-solving techniques on graphs [1,48,74,75,78,79,86,87] and new Operations Research applications [3,8,17,37,80,88]. From these recent evolutions, the basic algebraic structures referred to as dioïds and semirings have emerged, somewhat systematically (often under different names), in modelling and solving an extremely rich variety of problems.…”