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- Type of Document: Article
- DOI: 10.1016/j.comgeo.2014.02.002
- Abstract:
- In this paper, we study a variant of the well-known line-simplification problem. For this problem, we are given a polygonal path P=p1, p2,...,pn and a set S of m point obstacles in the plane, with the goal being to determine an optimal homotopic simplification of P. This means finding a minimum subsequence Q=q1,q2,..., qk (q1=p1 and qk=pn) of P that approximates P within a given error ε that is also homotopic to P. In this context, the error is defined under a distance function that can be a Hausdorff or Fréchet distance function, sometimes referred to as the error measure. In this paper, we present the first polynomial-time algorithm that computes an optimal homotopic simplification of P in O(n6 m2)+TF(n) time, where TF(n) is the time to compute all shortcuts pipj with errors of at most ε under the error measure F. Moreover, we define a new concept of strongly homotopic simplification where every link qlql+1 of Q corresponding to the shortcut pipj of P is homotopic to the sub-path pi,...,pj. We present a method that in O(n(m+n)log(n+m)) time identifies all such shortcuts. If P is x-monotone, we show that this problem can be solved in O(mlog(n+m)+nlognlog(n+m)+k) time, where k is the number of such shortcuts. We can use Imai and Iri's framework [24] to obtain the simplification at the additional cost of TF(n)
- Keywords:
- Computational geometry ; Homotopy ; Path Simplification ; Algorithms ; Optimization ; Additional costs ; Distance functions ; Error measures ; Homotopies ; Line simplification ; Polygonal path ; Polynomial-time algorithms
- Source: Computational Geometry: Theory and Applications ; Vol. 47, issue. 7 , 2014 , pp. 728-739 ; ISSN: 09257721
- URL: http://www.sciencedirect.com/science/article/pii/S0925772114000352