A Pathfinding Problem for Fork-Join Directed Acyclic Graphs with Unknown Edge Length

Kunihiko Hiraishi

doi:10.3390/a14120367

Abstract

In a previous paper by the author, a pathfinding problem for directed trees is studied under the following situation: each edge has a nonnegative integer length, but the length is unknown in advance and should be found by a procedure whose computational cost becomes exponentially larger as the length increases. In this paper, the same problem is studied for a more general class of graphs called fork-join directed acyclic graphs. The problem for the new class of graphs contains the previous one. In addition, the optimality criterion used in this paper is stronger than that in the previous paper and is more appropriate for real applications.

Highlights

In a previous paper by the author [1], a pathfinding problem for directed trees is studied under the following situation: each edge has a nonnegative integer length, but the length is unknown in advance and should be found by a procedure whose computational cost becomes exponentially larger as the length increases
We have studied a pathfinding problem of fork-join directed acyclic graphs (FJ-DAGs) with unknown edge length and have proposed a strategy that gives a shortest path
The strategy is optimal in the sense that the characteristic vector consisting of the number of fail oracle calls for each length is minimized with respect to a lexicographical order, provided that there are no waste fail calls

Summary

Introduction

In a previous paper by the author [1], a pathfinding problem for directed trees is studied under the following situation: each edge has a nonnegative integer length, but the length is unknown in advance and should be found by a procedure whose computational cost becomes exponentially larger as the length increases Such a situation arises in an operation synthesis problem for reconfigurable cloud computing systems [2,3]. Online shortest path problems for graphs with uncertainty are studied [11], and there are many applications such as route finding in transit networks [12]. In these problems, the weight of each edge can change in an arbitrary way.

Problem Formulation

Estimate and Characteristic Vector

Canonical Estimate

A Solution to PFJUEL

Conclusions