ALGORITHMS

Floyd's Algorithm

Abstract

In this paper we are going to analyze two algorithms for solving a particular problem. In the first part we are going to analyze Floyd's algorithm which is used for finding the shortest distances between all vertices of a weighted graph with no cycles with negative weights using the method of dynamic programming.

Floyd's Algorithm

Introduction

Pattern matching or pattern based search is a term for symbolic processing methods, which is based on a standard form consisting of discrete structures. They are used for the identification of the subsets of a discrete structure. A discrete structure cosnsists of discrete elements and relations between them (Coquand, pp.71-83, 1992). The examples of such elements are strings and graphs. The pattern itself is also a discrete structure, but by using additional Meta characters we can describe a whole class of discrete structures. In contrast to the pattern recognition interpretation, continuous structures, pattern matching operates from the outset on a symbolic representation (Fenning, pp.84-141, 2000).

The pattern matching plays not only plays an essential role in the search, but also in the pattern and rule based transformation of discrete structures plays a central role. In replacement or transformation systems the process of pattern making could be identified as the first step of the process. In this part the patterns are identified along with the analysis of different parts of the structure (Sestsoft, 1999). In the next step the partial structures then go as a parameter in the transformation function. Examples of such transformations are text substitution in string and graph rewriting systems (Vincent, & Francois, 2007).

Discussion

Floyd's algorithm is an algorithm used for finding the shortest distances between all vertices of a weighted graph with no cycles with negative weights using the method of dynamic programming. This is a basic algorithm for finding the shortest between all the vertices of a weighted graph. This algorithm was simultaneously published in the papers of Robert Floyd and Stephen Uorshella in the year 1962, although in 1959, Bernard Roy posted almost exactly the same algorithm, but it went unnoticed (KLEE, & LARMAN, 1977).

The Floyd-Warshall algorithm takes as input a directed graph and value (V, E) in the form of an adjacency matrix giving the weight of an arc when the value exists and 8 otherwise. The weight of a path between two vertices is the sum of the weights on the arcs constituting the path. The arcs of the graph may have negative weights, but the graph does not possess strictly negative weight cycle. The algorithm calculates, for each pair of vertices, the minimum weight among all paths between two vertices.

Let V = {1, 2, 3, 4, ..., n} the set of vertices of G and i and j are two vertices of V. Consider a path p between i and j of minimum weight whose intermediate vertices are in {1, 2, 3, ..., k}. The Floyd-Warshall algorithm is based on the following observation:

Either p does not borrow the top k

Is p exactly once borrows the top k (because the cycles are nonnegative weight) and p is ...