Computing Maximal Layers of Points in Ef(n)

In this paper we present a randomized algorithm for computing the collection of maximal layers for a point set in E (k = f(n)). The input to our algorithm is a point set P = {p1, ..., pn} with pi ∈ E . The proposed algorithm achieves a runtime of O ( kn 2− 1 log k +logk (1+ 2 k+1 ) log n ) when P is a random order and a runtime of O(knk (k−1))/2 log n) for an arbitrary P . Both bounds hold in expectation. Additionally, the run time is bounded by O(kn) in the worst case. This is the first non-trivial algorithm whose run-time remains polynomial whenever f(n) is bounded by some polynomial in n while remaining sub-quadratic in n for constant k. The algorithm is implemented using a new data-structure for storing and answering dominance queries over the set of incomparable points.