omputing homotopic line simplification

rticle history: eceived 3 December 2011 eceived in revised form 29 October 2012 ccepted 7 February 2014 vailable online xxxx ommunicated by P. Agarwal eywords: omputational geometry implification omotopy ath ine urve hain 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 (n6m2) + T F (n) time, where T F (n) is the time to compute all shortcuts pi p j 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 pi p j of P is homotopic to the sub-path pi, . . . , p j . 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 (m log(n + m) + n log n log(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 T F (n). © 2014 Published by Elsevier B.V.