Quickest Paths: Faster Algorithms and Dynamization

Given a network $N=(V,E,{c},{l})$, where $G=(V,E)$, $|V|=n$ and $|E|=m$, is a directed graph, ${c}(e) \u3e 0$ is the capacity and ${l}(e) \ge 0$ is the lead time (or delay) for each edge $e\in E$, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single--pair quickest path problem runs in time $O(r m+r n \log n)$, where $r$ is the number of distinct capacities of $N$ \cite{ROS}. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to $O(r^{\ast} m+r^{\ast} n \log n)$, where $r^{\ast}$ is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in $N$. For sparse networks, we present an algorithm with time complexity $O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma})$, where $\tilde{\gamma}$ is a topological measure of $N$. Since for sparse networks $\tilde{\gamma}$ ranges from $1$ up to $\Theta(n)$, this constitutes an improvement over the previously known bound of $O(r n \log n)$ in all cases that $\tilde{\gamma}=o(n)$. For planar networks, the complexity becomes $O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma})$. Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

The Quick Time Dependent Quickest Flow Problem: A Lesson in Zero-Sum Cycles

A quick solution technique for the integral time-dependent quickest flow problem with no waiting is presented. The proposed technique is based on the successive shortest path approach and modifies an existing algorithm to improve its average performance. At each iteration, a reoptimization procedure is employed to determine the augmenting path given updates to the residual graph. The residual graph, by construction, almost always contains zero-sum cycles when employed in this context. These zero-sum cycles pose a unique problem for the reoptimization technique. A heuristic that can be embedded in the reoptimization algorithm to provide path solutions in the presence of zero-sum cycles has been proposed. In the computational experiments, the heuristic provided an optimal solution nearly 100% of the times. Further, a modified implementation of an existing path-finding algorithm has been used to solve the time-dependent quickest flow problem with source waiting

Quickest paths: faster algorithms and dynamization

Given a network $N=(V,E,{c},{l})$, where $G=(V,E)$, $|V|=n$ and $|E|=m$, is a directed graph, ${c}(e) > 0$ is the capacity and ${l}(e) \ge 0$ is the lead time (or delay) for each edge $e\in E$, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single pair quickest path problem runs in time $O(r m+r n \log n)$, where $r$ is the number of distinct capacities of $N$. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to $O(r^{\ast} m+r^{\ast} n \log n)$, where $r^{\ast}$ is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in $N$. For sparse networks, we present an algorithm with time complexity $O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma})$, where $\tilde{\gamma}$ is a topological measure of $N$. Since for sparse networks $\tilde{\gamma}$ ranges from $1$ up to $\Theta(n)$, this constitutes an improvement over the previously known bound of $O(r n \log n)$ in all cases that $\tilde{\gamma}=o(n)$. For planar networks, the complexity becomes $O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma})$. Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

Safety Aware Vehicle Routing Algorithm, A Weighted Sum Approach

Driving is an essential part of work life for many people. Although driving can be enjoyable and pleasant, it can also be stressful and dangerous. Many people around the world are killed or seriously injured while driving. According to the World Health Organization (WHO), about 1.25 million people die each year as a result of road traffic crashes. Road traffic injuries are also the leading cause of death among young people. To prevent traffic injuries, governments must address road safety issues, an endeavor that requires involvement from multiple sectors (transport, police, health, education). Effective intervention should include designing safer infrastructure and incorporating road safety features into land-use and transport planning. The aim of this research is to design an algorithm to help drivers find the safest path between two locations. Such an algorithm can be used to find the safest path for a school bus travelling between bus stops, a heavy truck carrying inflammable materials, poison gas, or explosive cargo, or any driver who wants to avoid roads with higher numbers of accidents. In these applications, a path is safe if the danger factor on either side of the path is no more than a given upper bound. Since travel time is another important consideration for all drivers, the suggested algorithm utilizes traffic data to consider travel time when searching for the safest route. The key achievements of the work presented in this thesis are summarized as follows. Defining the Safest and Quickest Path Problem (SQPP), in which the goal is to find a short and low-risk path between two locations in a road network at a given point of time. Current methods for representing road networks, travel times and safety level were investigated. Two approaches to defining road safety level were identified, and some methods in each approach were presented. An intensive review of traffic routing algorithms was conducted to identify the most well-known algorithms. An empirical study was also conducted to evaluate the performance of some routing algorithms, using metrics such as scalability and computation time. This research approaches the SQPP problem as a bi-objective Shortest Path Problem (SPP), for which the proposed Safety Aware Algorithm (SAA) aims to output one quickest and safest route. The experiments using this algorithm demonstrate its efficacy and practical applicability

Use Quickest Path to Evaluate the Performance for An E-Commerce Network

In the E-Commerce network, it is an important issue to reduce the transmission time. The quickest path problem is to find the path in the network to send a given amount of data from the source to the sink with minimum transmission time. This problem traditionally assumed that the capacity of each arc in the network is deterministic. However, the capacity of each arc is stochastic due to failure, maintenance, etc. in many real-life networks. This paper proposes a simple algorithm to evaluate the probability that d units of data can be sent through the E-Commerce network within T units of time. Such a probability is a performance index for E-Commerce networks

Delaying a convoy

This thesis studies "the convoy-path interdiction problem" (CPIP) in which an interdictor uses limited resources to attack and disrupt road segments ("arcs") or road intersections ("nodes") in a road network in order to delay an adversary's convoy from reaching its destination. The convoy will move between a known origin node a and destination node b using a "quickest path." We first show how to compute, using an A* search, the convoy's quickest path under the assumptions that the convoy may occupy several arcs simultaneously, each arc may have a different speed limit, and the convoy maintains constant inter-vehicle spacing. The basic model assumes that the convoy moves in a single lane of traffic; an extension handles arcs that may have multiple lanes. Using that algorithm as a subroutine, a decomposition algorithm solves the optimal interdiction problem. Interdiction of a node or arc makes that node or arc impassable. Computational results are presented on grid networks with up to 629 nodes and 2452 arcs with varying levels of interdiction resource. Using Xpress-MP optimization software and a 2 GHz Pentium IV computer, the largest network problem solves in no more than 360 seconds given that at most 4 arcs can be interdicted.http://archive.org/details/delayingconvoy109451754Approved for public release; distribution is unlimited