An Analysis of a Recursive and an Iterative Algorithm for Generating Permutations Modified for Travelling Salesman Problem

Abstract

This paper presents the results of a comparative analysis between a recursive and an iterative algorithm when generating permutation. A number of studies discussing the problem and some methods dealing with its solution are analyzed. Recursion and iteration are approaches used in computer programs to implement different algorithms. An iterative approach is the repeated execution of the same source code until a certain end condition is met. On the other hand, a recursive approach uses a recursive function that repeatedly calls itself. This function contains a source code that must be executed repeatedly. Both algorithms presented in this paper can be used to generate permutations of an n element set. The algorithms are modified so that they can be used to solve the Travelling Salesman Problem (TSP) with a small number of vertices. Several publications that discuss the TSP and some approaches to its solution are also presented. The methodology and the conditions for conducting the experiments are described in details. The obtained results have been analyzed; they show that for the same conditions the iterative algorithm works from of 2 3 to 2 4 times faster than the recursive algorithm in all the tested input data. Several approaches to optimize the two algorithms in terms of the number of permutations tested when searching a minimal Hamiltonian cycle are presented.

Different Applications of the Genetic Mutation Operator for Symetric Travelling Salesman Problem

“…This problem is solved by repeating vertices being replaced by missing ones. After generating new solutions, their scores are calculated using the CalculateScore function (lines [25][26]. The complexity of this method is Θ(m 2 ), where m is the number of vertices in the graph.…”

“…After the process of generating new solutions (lines [11][12][13][14][15][16][17] is completed, it is checked whether there are same solutions among them. When two identical solutions are found, the second one mutates (lines [18][19][20][21][22][23][24][25][26][27][28][29]. This leads to an increase in the computational complexity of the whole procedure, which is already cubic and depends both on the number of vertices in the graph and the number of individuals in the population, along with the number of reproductions generated.…”

“…One major class of graph theory problems -NP-hard, can be solved well by some approximate algorithms such as GAs [20]- [24]. Finding an exact solution to these problems (with a large input size) can take a long time [25]. The approaches based on the backtracking method yield the best results but only for problems with small input size.…”

An analysis between exact and approximate algorithms for the k-center problem in graphs

“…If a connected undirected graph G=(V, E) is given, it is necessary to find a vertex v ∈ V such that the distance from it to the farthest vertex (for example u) is minimal, i.e., distance(v, u)=minimum. In this case, the vertex v is called the center of the graph, and the distance from the vertex u to the vertex v (i.e., distance (v, u)) is called the radius of the graph [1], [29].…”

A Hybrid Water Flow-Like Algorithm and Variable Neighbourhood Search for Traveling Salesman Problem

“…Until today, the search for an effective technique of solving TSP is still a phenomenal effort. In recent years, many metaheuristic approaches were proposed and applied for solving TSP [32]. Some approaches are population-based.…”