Geometric Approach to Optimal Path Problem with Uncertain Arc Lengths

Atefeh Hasan-Zadeh

doi:10.14419/jacst.v9i1.30674

Abstract

In this paper, the problem of finding the shortest paths, one of the most important problems in science and technology has been geometrically studied. Shortest path algorithm has been generalized to the shortest cycles in each homotopy class on a surface with arbitrary topology, using the universal covering space notion in the algebraic topology. Then, a general algorithm has been presented to compute the shortest cycles (geometrically rather than combinatorial) in each homotopy class. The algorithm can handle surface meshes with the desired topology, with or without boundary. It also provides a fundamental framework for other algorithms based on universal coverage space due to the capacity and flexibility of the framework.Â

Highlights

The shortest problem is finding a path with minimal distance, time, or cost between the source node and the destination node
The theory of uncertainty has provided a new approach to addressing non-deterministic factors in programming problems
As mentioned about homotopy paths, without losing the whole, it can be studied a restricted version of the above problem: Restricted shortest cycle problem: With a surface mesh X with an arbitrary topology, it should be found in each homotopy class the shortest cycle passing through a given point x0 X

Summary

Introduction

The shortest problem is finding a path with minimal distance, time, or cost between the source node and the destination node. In the 1950s and 1960s, effective algorithms were proposed or developed by Bellman [1], Dijkstra [2], Dreyfus [3] and Floyd [4], which made the short-haul problem a central element of a network. Since Dubois and Prade [9] proposed the Short Fuzzy Path (FSPP) problem in 1980, the fuzzy theory has begun to attract networked researchers. Some researchers, such as Klein [10], Ji and Iwamura [11] have worked extensively in this area. The concepts of the time-dependent shortest paths in large graphs and the shortest path in an uncertain network is taken into consideration. Algebraic concepts has been explained and the shortest path problem can be solved geometrically rather than combinatorial on a surface with arbitrary topology with or without boundary

Literature Review

Methodology

Results

Conclusions