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Place the Vertices Anywhere on the Curve and Simplify

Rudi Ali Gholami

Fundamenta Informaticae

2021

Vol. 180, nr 3

275--287

A polygonal curve is simplified to reduce its number of vertices, while maintaining similarity to its original shape. Numerous results have been published for vertex-restricted simplification, in which the vertices of the simplified curve are a subset of the vertices of the input curve. In curve-restricted simplification, i.e. when the vertices of the simplified curve are allowed to be placed on the edges of the input curve, the number of vertices may be much more reduced. In this paper, we present algorithms for computing curve-restricted simplifications of polygonal curves under the local Hausdorff distance measure.

Approximate Hotspots of Orthogonal Trajectories

Rudi Ali Gholami

Fundamenta Informaticae

2019

Vol. 167, nr 4

271--285

We study the problem of finding hotspots, i.e. regions, in which a moving entity spends a significant amount of time, for polygonal trajectories. The fastest exact algorithm, due to Gudmundsson, van Kreveld, and Staals (2013) finds an axis-parallel square hotspot of fixed side length in O(n2) for a trajectory with n edges. Limiting ourselves to the case in which the entity moves in a direction parallel either to the x or to the y-axis, we present an approximation algorithm with the time complexity O(n log3 n) and approximation factor 1/2.