Streaming Algorithms for Line Simplification

Abstract

We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p 0 , p 1 , p 2 , . . . in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k Comput Geom (2010) 43: 497-515 points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k 2 ) additional storage.

Efficient Observer-Dependent Simplification in Polygonal Domains

“…Then, the error of the resulting simplification with 2k points is not bigger than (2 + ) times the error of the optimal offline simplification with k points. This method is based on the general algorithm proposed in [1].…”

“…Proof We put vertices p j ∈ C 1 into a range searching data structure so that the following query can be answered efficiently: For a vertex p i ∈ C 1 find all vertices p j ∈ C 1 …”

A Near-Linear Time Guaranteed Algorithm for Digital Curve Simplification under the Fréchet Distance

“…Moreover, even if the theoretical complexity of the algorithm is in O(n log(n)), experiments show a linear behavior in practice. 1] with α(0) = 0, α(1) = 1, β(0) = 0, β(1) = 1, the Fréchet distance δ F (f, g) between two curves f and g is defined as…”

“…Instead, we use an approximation of the Fréchet distance proposed in [1]. More precisely, they show that error(i, j) can be upper and lower bounded by functions of two values, namely ω(i, j) and b(i, j).…”

“…b(i, j) is the length of the longest backpath in the direction − − → p i p j (see Figure 4). We have the following property [1], which leads to Algorithm 1 below. The Fréchet error of a shortcut p i p j satisfies:…”