Veronika Horvat (2014) *Semirings and the shortest path problem*. EngD thesis.

## Abstract

The thesis presents the problem of finding the shortest path in a directed graph with the Bellman's algorithm. This algorithm a version of Bellman equation with the novelty that the we work with matrices and operations over tropical semiring. Given a weighted directed graph, its arc prices are written in a matrix A. The matrix has a quasi-inverse A^* over tropical semiring, which is used to compute the minimal solution of the system of equations expressed from the Bellman equation. The solution of the algorithm is a vector of prices that represents the prices of the shortest paths to all nodes in the graph.

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