Aritmética tropical y programación dinámica

Abstract

This monograph paper examines the application of the Floyd-Warshall algorithm in the context of tropical arithmetic, adjusting its structure to address optimization problems in weighted graphs using tropical sum and product. Through the construction of a tropical adjacency matrix and its boosting, the minimum weights representing the optimal alignment costs are determined. In the first phase, the tropical Floyd-Warshall algorithm is used to solve optimization problems in dynamic programming, particularly in path minimization, which is applied to the computation of optimal routes between nodes, highlighting its potential in various planning areas. Next, a biological sequence alignment problem is modeled, representing it in an alignment graph, where the lowest weight paths that reflect the optimal alignment are identified, validating the effectiveness of the tropical model in sequence-related problems. This approach provides a robust framework for addressing combinatorial optimization problems.