A unifying approach for a class of problems in the computational geometry of polygons (Q1822499)

A generalized problem is defined in terms of functions on sets and illustrated in terms of the computational geometry of simple planar polygons. Although its apparent time complexity is \(O(n^ 2)\), the problem is shown to be solvable for several cases of interest (maximum and minimum distance, intersection detection and reporting) in O(n log n), O(n), or O(log n) time, depending on the nature of a specialized ''selection'' function. (Some of the cases can also be solved by the Voronoi diagram method; but time complexity increases with that approach.) A new use of monotonicity and a new concept of ''locality'' in set mappings contribute significantly to the derivation of the results.