Discrete Dynamic Shortest Path Problems in Transportation Applications: Complexity and Algorithms with Optimal Run Time

Abstract

A solution is provided for what appears to be a 30-year-old problem dealing with the discovery of the most efficient algorithms possible to compute all-to-one shortest paths in discrete dynamic networks. This problem lies at the heart of efficient solution approaches to dynamic network models that arise in dynamic transportation systems, such as intelligent transportation systems (ITS) applications. The all-to-one dynamic shortest paths problem and the one-to-all fastest paths problems are studied. Early results are revisited and new properties are established. The complexity of these problems is established, and solution algorithms optimal for run time are developed. A new and simple solution algorithm is proposed for all-to-one, all departure time intervals, shortest paths problems. It is proved, theoretically, that the new solution algorithm has an optimal run time complexity that equals the complexity of the problem. Computer implementations and experimental evaluations of various solution algorithms support the theoretical findings and demonstrate the efficiency of the proposed solution algorithm. The findings should be of major benefit to research and development activities in the field of dynamic management, in particular real-time management, and to control of large-scale ITSs.The shortest paths problem in networks has been the subject of extensive research for many years (1). The analysis of transportation networks is one of the many application areas in which the computation of shortest paths is one of the most fundamental problems. The majority of published research on shortest paths algorithms, however, dealt with static networks that have fixed topology and fixed link costs.Interest in the concept of dynamic management of transportation systems has been increasing. New advances have brought renewed interest in the study of shortest paths problems with a new twist: Link costs generally depend on the entry time of a link. This results in a new family of shortest paths problems known as dynamic, or time-dependent, shortest paths problems.Chabini (2) distinguishes various types of dynamic shortest path problems depending on the following: (a) fastest versus minimum cost (or shortest) path problems; (b) discrete versus continuous representation of time; (c) first-in-first-out (FIFO) networks versus non-FIFO networks, in which one can depart later at the beginning of one or more arcs and arrive earlier at their end; (d) waiting is allowed versus waiting is not allowed at nodes; (e) types of shortest path questions asked: one-to-all for a given departure time or all departure times, and all-to-one for all departure times; and ( f ) integer versus real valued link travel times and link travel costs.In fastest path problems, the cost of a link is the travel time of that link. In minimum cost paths problems, link costs can be of general form. Although the fastest paths problem is a particular case of the minimum cost paths problem, the distinction between the two is particularly important for the design of ef...

A Fictitious Play Approach to Large-Scale Optimization

“…However, a crucial distinguishing characteristic of this version of the problem is the dynamic, time-dependent, nature of the congestion on the links in the network. The problem of finding a system-optimal assignment of routes in a dynamic network is the subject of a great deal of research in the field of Intelligent Transportation Systems; in particular, several versions of a pertinent subproblem of finding a shortest path in a dynamic network has been successfully addressed (see, for example, Smith, 1993 andChabini, 1998). Specifically, for given routing decisions of all other vehicles in the network, the dynamic (i.e., time-dependent) shortest path for one vehicle can be efficiently computed if the network conditions are such that a single vehicle will have little effect on congestion, and if some mild assumptions of the assignment mappings are satisfied.…”

Snapshot Centrality Indices in Dynamic FIFO Networks

“…The fire is then assumed to start at two random locations. Figure 5 shows the variation of the dynamic snapshot betweenness for nodes (8,6) and (8,7). Node (8,6) has the largest snapshot betweenness values for most of the time and then decreases sharply to 0.…”

“…Node (8,6) has the largest snapshot betweenness values for most of the time and then decreases sharply to 0. On the other hand node (8,7) had smaller values initially, but then increases to a constant value over time (because the fire does not spread to this area of the network within the time window).…”

“…Dynamic shortest paths more recently became very popular with the emergence of Intelligent or Dynamic Transportation Systems e.g. Chabini [7,8], Ahuja [3] or Demetrescu [11]. There are two well established approaches to this problem, which are quite different in essence.…”

“…Previous work on dynamic shortest paths depended on continuous cost functions, and on the time interval being [0, ∞) generating algorithms that are not efficient. In that research, the dynamic costs are considered as integers in order to have the continuous case equations with t ∈ [0, ∞) translated into similar equations in the discrete case with the time t ∈ {0, 1, 2, ...} [8,10]. In this section we introduce an accurate method to transform the equations in the continuous space into equations in the discrete space.…”