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On the Complexity of TimeDependent Shortest Paths
"... We investigate the complexity of shortest paths in timedependent graphs, in which the costs of edges vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions are (polynomialsize) piecewise line ..."
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We investigate the complexity of shortest paths in timedependent graphs, in which the costs of edges vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions are (polynomialsize) piecewise linear, the shortest path from s to d can change Θ(log n) n times, settling a severalyearold conjecture of Dean [Technical Reports, 1999, 2004]. We also show that the complexity is polynomial if the slopes of the linear function come from a restricted class, present an outputsensitive algorithm for the general case, and describe a scheme for a (1 + ɛ)approximation of the travel time function in nearquadratic space. Finally, despite the fact that the arrival time function may have superpolynomial complexity, we show that a minimum delay path for any departure time interval can be computed in polynomial time. 1
Online Computation of Fastest Path in TimeDependent Spatial Networks ⋆
"... Abstract. The problem of pointtopoint fastest path computation in static spatial networks is extensively studied with many precomputation techniques proposed to speedup the computation. Most of the existing approaches make the simplifying assumption that traveltimes of the network edges are cons ..."
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Abstract. The problem of pointtopoint fastest path computation in static spatial networks is extensively studied with many precomputation techniques proposed to speedup the computation. Most of the existing approaches make the simplifying assumption that traveltimes of the network edges are constant. However, with realworld spatial networks the edge traveltimes are timedependent, where the arrivaltime to an edge determines the actual traveltime on the edge. In this paper, we study the online computation of fastest path in timedependent spatial networks and present a technique which speedsup the path computation. We show that our fastest path computation based on a bidirectional timedependent A * search significantly improves the computation time and storage complexity. With extensive experiments using real datasets (including a variety of large spatial networks with real traffic data) we demonstrate the efficacy of our proposed techniques for online fastest path computation. 1
Algorithmica DOI 10.1007/s0045301297147 On the Complexity of TimeDependent Shortest Paths
, 2012
"... Abstract We investigate the complexity of shortest paths in timedependent graphs where the costs of edges (that is, edge travel times) vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions a ..."
Abstract
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Abstract We investigate the complexity of shortest paths in timedependent graphs where the costs of edges (that is, edge travel times) vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions are (polynomialsize) piecewise linear, the shortest path from s to d can change n Θ(log n) times, settling a severalyearold conjecture of Dean (Technical Reports, 1999, 2004). However, despite the fact that the arrival time function may have superpolynomial complexity, we show that a minimum delay path for any departure time interval can be computed in polynomial time. We also show that the complexity is polynomial if the slopes of the linear function come from a restricted class and describe an efficient scheme for computing a (1+ɛ)approximation of the travel time function. Keywords Timedependent shortest path · Piecewise linear delay functions · Parametric shortest path · Approximation algorithms