Adaptive Techniques to find Optimal Planar Boxes

Given a set P of n planar points, two axes and a realvalued score function f() on subsets of P , the Optimal Planar Box problem consists in finding a box (i.e. an axis-aligned rectangle) H maximizing f(H ∩ P ). We consider the case where f() is monotone decomposable, i.e. there exists a composition function g() monotone in its two arguments such that f(A) = g(f(A1), f(A2)) for every subset A ⊆ P and every partition {A1, A2} of A. In this context we propose a solution for the Optimal Planar Box problem which performs in the worst case O(n lg n) score compositions and coordinate comparisons, and much less on other classes of instances defined by various measures of difficulty. A side result of its own interest is a fully dynamic MCS Splay tree data structure supporting insertions and deletions with the dynamic finger property, improving upon previous results [Cortés et al., J.Alg. 2009].