Abstract
We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and analyze whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better.
This research has been supported by the Netherlands Organisation for Scientific Research (NWO) under BRICKS/FOCUS grant number 642.065.503, under the project GOGO, and under project no. 639.022.707. M. B. is supported by the German Research Foundation (DFG) under grant number BU 2419/1-1. M. L. is further supported by the U.S. Office of Naval Research under grant N00014-08-1-1015. R. I. S. is also supported by the Netherlands Organisation for Scientific Research (NWO). C. W. is supported by the National Science Foundation grant NSF CCF-0643597.
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Buchin, K. et al. (2010). Median Trajectories. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_40
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DOI: https://doi.org/10.1007/978-3-642-15775-2_40
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