Abstract
In this paper, we study the time-dependent shortest paths problem for two types of time-dependent FIFO networks. First, we consider networks where the availability of links, given by a set of disjoint time intervals for each link, changes over time. Here, each interval is assigned a non-negative real value which represents the travel time on the link during the corresponding interval. The resulting shortest path problem is the time-dependent shortest path problem for availability intervals (\(\mathcal{TDSP}_{\mathrm{int}}\)), which asks to compute all shortest paths to any (or all) destination node(s) d for all possible start times at a given source node s. Second, we study time-dependent networks where the cost of using a link is given by a non-decreasing piece-wise linear function of a real-valued argument. Here, each piece-wise linear function represents the travel time on the link based on the time when the link is used. The resulting shortest paths problem is the time-dependent shortest path problem for piece-wise linear functions (\(\mathcal{TDSP}_{\mathrm{lin}}\)) which asks to compute, for a given source node s and destination d, the shortest paths from s to d, for all possible starting times.
We present an algorithm for the \(\mathcal{TDSP}_{\mathrm{lin}}\) problem that runs in time O((F d +γ)(|E|+|V|log |V|)) where F d is the output size (i.e., number of linear pieces needed to represent the earliest arrival time function to d) and γ is the input size (i.e., number of linear pieces needed to represent the local earliest arrival time functions for all links in the network). We then solve the \(\mathcal{TDSP}_{\mathrm{int}}\) problem in O(λ(|E|+|V|log |V|)) time by reducing it to an instance of the \(\mathcal{TDSP}_{\mathrm{lin}}\) problem. Here, λ denotes the total number of availability intervals in the entire network. Both methods improve significantly on the previously known algorithms.
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Research supported by NSERC, SUN Microsystems of Canada and HPCVL.
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Dehne, F., Omran, M.T. & Sack, JR. Shortest Paths in Time-Dependent FIFO Networks. Algorithmica 62, 416–435 (2012). https://doi.org/10.1007/s00453-010-9461-6
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DOI: https://doi.org/10.1007/s00453-010-9461-6