In this paper we consider the classical min--\# curve simplification problem in three different variants. Let \delta>0, P be  a polygonal curve  with n vertices in \mathbb{R^d, and D(\cdot,\cdot) be a distance measure. We aim to simplify P by }  another polygonal curve P' with minimum number of vertices satisfying D(P,P' ) \leq \delta. We obtain three main results for this problem: (1) An O(n^4)-time algorithm when D(P,P') is the Fr\'echet distance and vertices in P' are selected from a subsequence of vertices in P. (2) An   \emph{  NP-hardness result for the case that D(P ,P') is the directed Hausdorff distance from P' to P and the vertices of P' can lie anywhere on P while respecting the order of edges along P. (3) For any \epsilon>0, an O^*(n^2\log n \log \log n)-time algorithm that computes P'whose vertices can lie anywhere in the space and whose Fr\'echet distance to P is at most (1+\epsilon)\delta with at most 2m+1 links, where m is the number of links in the optimal simplified curve and O^* hides polynomial factors of 1/\epsilon .
Due to its many applications,curve simplification is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve P with n vertices ^d, and D(\cdot,\cdot) be a distance measure. We aim to simplify P by } , the goal is to find another polygonal curve P' with a smaller number of vertices such that P' is sufficiently similar to P. Quality guarantees of a simplification are usually given in alocal sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work, we aim to provide a systematic overview of curve simplification problems underglobal distance measures that bound the distance between P and P'. We consider six different curve distance measures: three variants of theHausdorff distance and three variants of the\emph{Fr\'echet distance. And we study different restrictions on the choice of vertices for P'. We provide polynomial-time algorithms for some variants of the global curve simplification problem and show NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area .
R
In this paper we consider the classical min--\# curve simplification problem in three different variants. Let \delta>0, P be
Due to its many applications,
A
curve simplification
A
is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given
A
P
D
in \mathbb{R
A
, the goal is to find
R
minimum
a smaller
R
satisfying D(P,P' ) \leq \delta. We obtain three main results for this problem: (1) An O(n^4)-time algorithm when D(P,P') is the Fr\'echet distance and vertices in P' are selected from a subsequence of vertices in P. (2) An
such that P' is sufficiently similar to P. Quality guarantees of a simplification are usually given in a
A
local
A
sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work, we aim to provide a systematic overview of curve simplification problems under
A
global
A
distance measures that bound the distance between P and P'. We consider six different curve distance measures: three variants of the
A
Hausdorff
A
distance and three variants of the
A
Fr\'echet
A
distance. And we study different restrictions on the choice of vertices for P'. We provide polynomial-time algorithms for some variants of the global curve simplification problem and show
R
result for the case that D(P ,P') is the directed Hausdorff distance from P' to P and the vertices of P' can lie anywhere on P while respecting the order of edges along P. (3) For any \epsilon>0, an O^*(n^2\log n \log \log n)-time algorithm that computes P'whose vertices can lie anywhere in the space and whose Fr\'echet distance to P is at most (1+\epsilon)\delta with at most 2m+1 links, where m is the number of links in the optimal simplified curve and O^* hides polynomial factors of 1/\epsilon
for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area