-The Algebraic Path Problem is an important issue with a broad background. Consider the problem to solve the algebraic path problem can be concluded to find the Kleene closure of the adjacency matrix, an algorithm of time complexity 3 ( ) O n to find the transitive closure of matrices over idempotent semiring is constructed, as well as the conditions applicable to it are provided. Compared with the Gauβ-Jordan elimination, this algorithm could extend the applicable range to certain semirings which do not have completeness and closeness, Thus, it has a wide application since it can provide a new method to solve the algebraic path problem over idempotent semirings which do not have completeness and closeness.

Abstract-To judge the loop-nonnegativity of a matrix A over an idempotent semiring and compute the plus-closure of A when it is loop-nonnegative, a Plus_Closure_of_Matrix algorithm of complexity 3 ( ) O n is constructed and proved. As a generalization of Floyd algorithm, Warshall algorithm as well as Gauβ-Jordan Elimination algorithm on idempotent semirings, this algorithm can also be used to solve some Algebraic Path Problems, Shortest Path Problems and the transitive closures of matrices over idempotent semirings even if the idempotent semirings have no completeness and closeness.