The frechet distance revisited and extended
Proceedings of the twenty-seventh annual symposium on Computational geometry
A generalization of the convex kakeya problem
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Scandinavian thins on top of cake: on the smallest one-size-fits-all box
FUN'12 Proceedings of the 6th international conference on Fun with Algorithms
Bundling three convex polygons to minimize area or perimeter
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
The fréchet distance revisited and extended
ACM Transactions on Algorithms (TALG)
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Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕ P of P that minimizes the convex hull of ϕ P∪Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕ P and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact near-linear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1+ε)-approximation in time O(ε −1/2log n+ε −3/2log a (1/ε)) if the two sets are convex polygons with n vertices in total, where a∈{0,1,2} depending on the version of the problem.