Heuristics for optimum binary search trees and minimum weight triangulation problems

AbstractIn this paper we establish new bounds on the problem of constructing optimum binary search trees with zero-key access probabilities (with applications e.g. to point location problems). We present a linear-time heuristic for constructing such search trees so that their cost is within a factor of 1 + ε from the optimum cost, where ε is an arbitrary small positive constant. Furthermore, by using an interesting amortization argument, we give a simple and practical, linear-time implementation of a known greedy heuristics for such trees.The above results are obtained in a more general setting, namely in the context of minimum length triangulations of so-called semi-circular polygons. They are carried over to binary search trees by proving a duality between optimum (m − 1)-way search trees and minimum weight partitions of infinitely-flat semi-circular polygons into m-gons. With this duality we can also obtain better heuristics for minimum length partitions of polygons by using known algorithms for optimum search trees

Soluciones aproximadas para el problema de Triangulación de Peso Mínimo utilizando ACO

Muchos problemas de optimización en configuraciones geométricas son NP-duros por lo que interesa obtener soluciones aproximadas. En este trabajo proponemos la utilización de una técnica metaheurística, Optimización basada en Colonias de Hormigas (Ant Colony Optimization - ACO) para la resolución aproximada del problema de Triangulación de Peso Mínimo (Minimum Weight Triangulation - MWT) para un conjunto de puntos en el plano. Además presentamos los resultados obtenidos de la evaluación experimental realizada, mostrando el rendimiento del algoritmo ACO propuesto.Presentado en el X Workshop Agentes y Sistemas InteligentesRed de Universidades con Carreras en Informática (RedUNCI

Soluciones aproximadas para el problema de Triangulación de Peso Mínimo utilizando ACO

Muchos problemas de optimización en configuraciones geométricas son NP-duros por lo que interesa obtener soluciones aproximadas. En este trabajo proponemos la utilización de una técnica metaheurística, Optimización basada en Colonias de Hormigas (Ant Colony Optimization - ACO) para la resolución aproximada del problema de Triangulación de Peso Mínimo (Minimum Weight Triangulation - MWT) para un conjunto de puntos en el plano. Además presentamos los resultados obtenidos de la evaluación experimental realizada, mostrando el rendimiento del algoritmo ACO propuesto.Presentado en el X Workshop Agentes y Sistemas InteligentesRed de Universidades con Carreras en Informática (RedUNCI

Optimal area triangulation

Given a set of points in the Euclidean plane, we are interested in its triangulations, i.e., the maximal sets of non-overlapping triangles with vertices in the given points whose union is the convex hull of the point set. With respect to the area of the triangles in a triangulation, several optimality criteria can be considered. We study two of them. The MaxMin area triangulation is the triangulation of the point set that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. In the case when the point set is in a convex position, we present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in $O(n^2log{n})$ time and $O(n^2)$ space. These algorithms are based on dynamic programming. They use a number of geometric properties that are established within this work, and a variety of data structures specific to the problems. Further, we study polynomial time computable approximations to the optimal area triangulations of general point sets. We present geometric properties, based on angular constraints and perfect matchings, and use them to evaluate the approximation factor and to achieve triangulations with good practical quality compared to the optimal ones. These results open new direction in the research on optimal triangulations and set the stage for further investigations on optimization of area