Computing The Packedness of Curves
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A polygonal curve P with n vertices is c-packed, if the sum of the lengths of the parts of the edges of the curve that are inside any disk of radius r is at most cr, for any r>0. Similarly, the concept of c-packedness can be defined for any scaling of a given shape. Assuming L is the diameter of P and δ is the minimum distance between points on disjoint edges of P, we show the approximation factor of the existing O(log (L/δ)/ϵn^3) time algorithm is 1+ϵ-approximation algorithm. The massively parallel versions of these algorithms run in O(log (L/δ)) rounds. We improve the existing O((n/ϵ^3)^4/3n/ϵ) time (6+ϵ)-approximation algorithm by providing a (4+ϵ)-approximation O(n(log^2 n)(log^2 1/ϵ)+n/ϵ) time algorithm, and the existing O(n^2) time 2-approximation algorithm improving the existing O(n^2log n) time 2-approximation algorithm. Our exact c-packedness algorithm takes O(n^5) time, which is the first exact algorithm for disks. We show using α-fat shapes instead of disks adds a factor α^2 to the approximation. We also give a data-structure for computing the curve-length inside query disks. It has O(n^6log n) construction time, uses O(n^6) space, and has query time O(log n+k), where k is the number of intersected segments with the query shape. We also give a massively parallel algorithm for relative c-packedness with O(1) rounds.
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