Abstract
In this paper, we present a new algorithm of the time-dependent shortest path problem with time windows. Give a directed graph , where V is a set of nodes, E is a set of edges with a non-negative transit-time function . For each node , a time window within which the node may be visited and , is non-negative of the service and leaving time of the node. A source node s, a destination node d and a departure time t0, the time-dependent shortest path problem with time windows asks to find an s, d-path that leaves a source node s at a departure time t0; and minimizes the total arrival time at a destination node d. This formulation generalizes the classical shortest path problem in which ce are constants. Our algorithm of the time windows gave the generalization of the ALT algorithm and A* algorithm for the classical problem according to Goldberg and Harrelson [1], Dreyfus [2] and Hart et al. [3].
Highlights
The shortest path problem on graphs is a problem with many real-life applications such as: route planning in an internet, car navigation system, traffic simulation or logistic optimization
We have found the first algorithm for the time-dependent shortest path problem with time windows that speeds up the calculation using preprocessing and we have observed that it is several time faster than the generalized Dijkstra’s algorithm
We show by the induction that, every active node ν must get the optimal distance label, i.e., the earliest arrival time at node ν for leaving s at time t0
Summary
The shortest path problem on graphs is a problem with many real-life applications such as: route planning in an internet, car navigation system, traffic simulation or logistic optimization. Dreyfus [2] suggested a polynomial-time straightforward generalization of the Dijkstra’s algorithm He did not notice that it works correctly only for instances satisfying the First-In First-Out (FIFO) property, i.e., for any edg= e e (v, w) ∈ E and tv ≤ tw , it holds that tv + ce (tv ) ≤ tw + ce (tw ).
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