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We study the Flight Planning Problem for a single aircraft, which deals with finding a path of minimal travel time in an airway network. Flight time along arcs is affected by wind speed and direction, which are functions of time. We consider three variants of the problem, which can be modeled as, respectively, a classical shortest path problem in a metric space, a time-dependent shortest path problem with piecewise linear travel time functions, and a time-dependent shortest path problem with piecewise differentiable travel time functions. The shortest path problem and its time-dependent variant have been extensively studied, in particular, for road networks. Airway networks, however, have different characteristics: the average node degree is higher and shortest paths usually have only few arcs. We propose A* algorithms for each of the problem variants. In particular, for the third problem, we introduce an application-specific "super-optimal wind" potential function that overestimates optimal wind conditions on each arc, and establish a linear error bound. We compare the performance of our methods with the standard Dijkstra algorithm and the Contraction Hierarchies (CHs) algorithm. Our computational results on real world instances show that CHs do not perform as well as on road networks. On the other hand, A* guided by our potentials yields very good results. In particular, for the case of piecewise linear travel time functions, we achieve query times about 15 times shorter than CHs.

We study the Flight Planning Problem for a single aircraft, where we look for a minimum cost path in the airway network, a directed graph. Arc evaluation, such as weather computation, is computationally expensive due to non-linear functions, but required for exactness. We propose several pruning methods to thin out the search space for Dijkstra's algorithm before the query commences. We do so by using innate problem characteristics such as an aircraft's tank capacity, lower and upper bounds on the total costs, and in particular, we present a method to reduce the search space even in the presence of regional crossing costs.
We test all pruning methods on real-world instances, and show that incorporating crossing costs into the pruning process can reduce the number of nodes by 90\% in our setting.

Planning efficient routes fast becomes ever more important,
especially in the context of aircraft trajectories. As time-dependent wind
conditions factor into the shortest path query, we use an artificial wind
vector called Super-Optimal Wind as a means of creating a suitable
potential function for the A* algorithm, thus speeding up the query.
We assess the quality of Super-Optimal Wind both theoretically and
computationally, and use Super-Optimal Wind in a real-world instance.