Studi Perbandingan Kinerja Algoritma Dijkstra dan Shortest Path Faster Algorithm Sebagai Metode Penyelesaian Minimum Steiner Tree pada Studi Kasus E-Olymp 1445 Road Network

Steiner Tree berperan besar dalam permodelan VLSI chip, penyelesaian permasalahan routing, dan networking. Permasalahan yang timbul selanjutnya adalah bagaimana cara mengkomputasi Minimum Steiner Tree dengan efisien. Topik Tugas Akhir ini mengulas dua algoritma yang digunakan untuk menyelesaikan permasalahan Minimum Steiner Tree dengan efisien, yaitu menggunakan Dijkstra dan Shortest Path Faster Algorithm. Melalui pengujian dan studi kasus, didapat hasil bahwa Shortest Path Faster Algorithm memiliki kinerja yang lebih baik dari Dijkstra untuk menyelesaikan permasalahan Minimum Steiner Tree. ================================================================================================================================ Steiner Tree plays a major role in modeling VLSI chips, solving routing, and networking problems. The next problem that arises is how to efficiently compute the Minimum Steiner Tree. This Final Project Topic reviews two algorithms that are used to efficiently solve the Minimum Steiner Tree problem, using Dijkstra and Shortest Path Faster Algorithm. Through testing and case studies, the results show that the Shortest Path Faster Algorithm has better performance than Dijkstra to solve the Minimum Steiner Tree problem

The Steiner Ratio for the Obstacle-Avoiding Steiner Tree Problem

This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P, called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles

Новый подход к проблеме Гильберта-Поллака

Предлагается новый подход к обоснованию справедливости гипотезы Гильберта-Поллака, которая высказанаоб оценке отношения длины дерева Штейнера к длине минимального остовного дерева на множестве точек плоскости. При введении подходящих параметров задача сводится к задаче нелинейного программирования.Пропонується новий підхід до обгрунтування справедливості гіпотези Гільберта-Поллака, яка висловлена про оцінку відношення довжини мінімального дерева Штейнера до довжини мінімального остовного дерева на множині точок площини. Шляхом введення підходящих параметрів задача зводиться до задачі нелінійного програмування.New approach to justification of Hilbert-Pollak hypothesis concerning the estimate of ratio of minimum Steiner tree length to the minimal skeleton tree length for a set of points on the plane. The problem is reduced to a problem of nonlinear programming by introducing proper new parameters

Reconfiguration of Minimum Steiner Trees via Vertex Exchanges

In this paper, we study the problem of deciding if there is a transformation between two given minimum Steiner trees of an unweighted graph such that each transformation step respects a prescribed reconfiguration rule and results in another minimum Steiner tree of the graph. We consider two reconfiguration rules, both of which exchange a single vertex at a time, and generalize the known reconfiguration problem for shortest paths in an unweighted graph. This generalization implies that our problems under both reconfiguration rules are PSPACE-complete for bipartite graphs. We thus study the problems with respect to graph classes, and give some boundaries between the polynomial-time solvable and PSPACE-complete cases