Under the promise that the convex hull vertices form a subsequence of the input, the hull can be computed in O(n sqrt(log n)) deterministic time or O(n log^ε n) expected time, and the promise is tight because even one out-of-order hull point forces an Omega(n log n) lower bound.
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Quadtrees and related structures are 2-Presortable, admitting expected O(n sqrt(log n)) algorithms given presorts along both axes.
Data structures achieve O(log* n) per operation for tree-sum queries on decremental forests using micro-macro decomposition, plus a universally optimal algorithm in the group model.
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Computing Planar Convex Hulls with a Promise
Under the promise that the convex hull vertices form a subsequence of the input, the hull can be computed in O(n sqrt(log n)) deterministic time or O(n log^ε n) expected time, and the promise is tight because even one out-of-order hull point forces an Omega(n log n) lower bound.
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The Presort Hierarchy for Geometric Problems
Quadtrees and related structures are 2-Presortable, admitting expected O(n sqrt(log n)) algorithms given presorts along both axes.
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Fast decremental tree sums in forests
Data structures achieve O(log* n) per operation for tree-sum queries on decremental forests using micro-macro decomposition, plus a universally optimal algorithm in the group model.