Optimal time-convex hull under the Lp metrics

Authors:
Bang-Sin Dai;Mong-Jen Kao;D. T. Lee
Affiliations:
Dep. of Computer Sci. and Infor. Engineering, National Taiwan Uni., Taiwan;Research Center for Infor. Tech. Innovation, Academia Sinica, Taiwan;Research Center for Infor. Tech. Innovation, Academia Sinica, Taiwan,Dep. of Computer Sci. and Engineering, National Chung-Hsing Uni., Taiwan,Dep. of Computer Sci. and Infor. Engineering, National ...
Venue:
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
Year:
2013

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Abstract

We consider the problem of computing the time-convex hull of a point set under the general Lp metric in the presence of a straight-line highway in the plane. The traveling speed along the highway is assumed to be faster than that off the highway, and the shortest time-path between a distant pair may involve traveling along the highway. The time-convex hull TCH(P) of a point set P is the smallest set containing both P and all shortest time-paths between any two points in TCH(P). In this paper we give an algorithm that computes the time-convex hull under the Lp metric in optimal $\mathcal{O}(n\log n)$ time for a given set of n points and a real number p with 1≤p≤∞.