Papers › Computing The Packedness of Curves
Computing The Packedness of Curves
Sepideh Aghamolaei, Vahideh Keikha, Mohammad Ghodsi, Ali Mohades
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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))/ϵn³) 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/ϵ³)^(4/3)polylogn/ϵ) time (6+ϵ)-approximation algorithm by providing a (4+ϵ)-approximation O(n(log² n)(log² 1/ϵ)+n/ϵ) time algorithm, and the existing O(n²) time 2-approximation algorithm improving the existing O(n²logn) time 2-approximation algorithm. Our exact c-packedness algorithm takes O(n⁵) time, which is the first exact algorithm for disks. We show using α-fat shapes instead of disks adds a factor α² to the approximation. We also give a data-structure for computing the curve-length inside query disks. It has O(n⁶logn) construction time, uses O(n⁶) space, and has query time O(logn+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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