An Efficient Algorithm for the Fast Delivery Problem

Abstract

We study a problem where k autonomous mobile agents are initially located on distinct nodes of a weighted graph (with n nodes and m edges). Each autonomous mobile agent has a predefined velocity and is only allowed to move along the edges of the graph. We are interested in delivering a package, initially positioned in a source node s, to a destination node y. The delivery is achieved by the collective effort of the autonomous mobile agents, which can carry and exchange the package among them. The objective is to compute a delivery schedule that minimizes the delivery time of the package. In this paper, we propose an O(kn log n + km) time algorithm for this problem. This improves the previous state-of-the-art O(k 2 m + kn 2 + APSP) time algorithm for this problem, where APSP stands for the running-time of an algorithm for the All-Pairs Shortest Paths problem.

A Robust Graph Theoretic Solution of Routing in Intelligent Networks

“…The solution in [16] suggests that these algorithms can be implemented in all kind of networks with complexity of Dijkstra O ðV 2 + EÞ, Bellman O ðVÞ 3 , and Floyd O ðVEÞ. The paper [17] suggests a solution with O ðkV log ðkVÞ + kEÞ complexity for a fast delivery problem. The complexity of Bellman is O ðVÞ 3 , and the complexity of Dijkstra is O ðVÞ 2 if the network is fully connected or near to the fully connected [18].…”