Matrix algorithm for determination of the elementary paths and elementary circuits using exotic semirings

Idempotent mathematics is based on replacing the usual arithmetic operations with a new set of basic operations, that is on replacing numerical fields by idempotent semirings. Exotic semirings such as the max-plus algebra Rmax or concatenation semiring P(Σ) have been introduced in connection with various fields: graph theory, Markov decision processes, language theory, discrete event systems theory, see [4], [2], [6], [1]. In this paper we will have to consider various semirings, and will universally use the notation ⊕,⊗, ε, e with a context dependent manning (e.g. ⊕ := max in Rmax but ⊕ := ∪ in P(Σ), ε := −∞ in Rmax but ε := ∅ in P(Σ )). In many fields of applications, the graphs are used widely for modelling of practical problems. This paper will focus on two algebraic path problems, namely: the elementary path problem (EPP) and the elementary circuit problem (ECP). For a directed graph G = (V,E) (|V | = n) and u, v ∈ V , (EPP) and (ECP) are formulated as follows: • (EPP) enumerate the elementary paths from u to v of length k (1 ≤ k ≤ n− 1); • (ECP) enumerate the elementary circuits starting in u of length k (1 ≤ k ≤ n). For to solve the above problems we give a method based on a n × n matrix with entries in the semiring of distinguished languages. These algebraic path problems are applied into large domains: combinatorial optimization, traffic control, Internet routing etc. The paper is organized as follows. In Section 2 we construct a special idempotent semiring denoted by P∗(Σ∗dw) and named the semiring of distinguished languages. The semiring of matrices with entries in P∗(Σ∗dw) is presented in Section 3. This algebraic tool is used to establishing of a one-to-one correspondence between the distinguished words and the elementary paths in a directed graph. In Section 4 we give an algorithm to determine all the elementary paths and elementary circuits in a directed graph. This new practical algorithm is based on the latin composition of distinguished languages.