Computing homotopic line simplification

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 = p 1 , p 2 , … , p n 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 = q 1 , q 2 , … , q k ( q 1 = p 1 and q k = p n ) of P that approximates P within a given error e that is also homotopic to P . In this context, the error is defined under a distance function that can be a Hausdorff or Frechet 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 ( n 6 m 2 ) + T F ( n ) time, where T F ( n ) is the time to compute all shortcuts p i p j with errors of at most e under the error measure F. Moreover, we define a new concept of strongly homotopic simplification where every link q l q l + 1 of Q corresponding to the shortcut p i p j of P is homotopic to the sub-path p i , … , 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 ) .