Maintaining Contour Trees of Dynamic Terrains
We study the problem of maintaining the contour tree T of a terrain σ, represented as a triangulated xy-monotone surface, as the heights of its vertices vary continuously with time. We characterize the combinatorial changes in T and how they relate to topological changes in. We present a kinetic data structure (KDS) for maintaining T efficiently. It maintains certificates that fail, i.e., an event occurs, only when the heights of two adjacent vertices become equal or two saddle vertices appear on the same contour. Assuming that the heights of two vertices of κ become equal only O(1) times and these instances can be computed in O(1) time, the KDS processes O(κ + n) events, where n is the number of vertices in σ and κ; is the number of events at which the combinatorial structure of T changes, and processes each event in O(log n) time. The KDS can be extended to maintain an augmented contour tree and a join/split tree.
Agarwal, PK; Mølhave, T; Revsbæk, M; Safa, I; Wang, Y; Yang, J
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