Three-Dimensional Layers of Maxima

Abstract

We present an O(n log n)-time algorithm to solve the three-dimensional layers-of-maxima problem. This is an improvement over the prior O(n log n log log n)-time solution. A previous claimed O(n log n)-time solution due to Atallah et al.[2] has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph.Key Words. Fractional cascading, Layers of maxima, Planar point location. Introduction.A point p ∈ d dominates another point q ∈ d if each coordinate of p exceeds that of q. Given a set S of n points in d , the maximum points are those that are not dominated by any point in S. The maxima set problem, of finding all the maximum points in S, is a classic problem in computational geometry, dating back to the early days of the discipline [11]. Interestingly, the algorithm presented in the original paper by Kung et al. [11] is still the most efficient known method for solving this problem; it runs in O(n log n) time when d = 2 or 3 and O(n log d−2 n) time when d ≥ 4. This problem has subsequently been studied in many other contexts, including solutions for parallel computation models [9], for point sets subject to insertions and deletions [10], and for moving points [7].The layers-of-maxima problem iterates this discovery: after finding the maximum points, remove them from S and find the maximum points in the remaining set, iterating until S becomes empty. The iteration index in which a point is a maximum is defined to be its layer. More formally, for p ∈ S, layer( p) = 1 if p is a maximum point; otherwise, layer( p) = 1 + max{layer(q) : q dominates p}. The layers-of-maxima problem, which is related to the convex layers problem [4], is to determine layer( p) for each p ∈ S, given S.With some effort [1], the three-dimensional layers-of-maxima problem can be solved in time O(n log n log log n) using techniques from dynamic fractional cascading [13]. We sketch this result in Section 2. Atallah et al. [2] claim an O(n log n)-time algorithm, but their presentation appears to have several problems. A simple, linear-time reduction from sorting gives an (n log n)-time lower bound in the comparison model.

In-Place Algorithms for Computing (Layers of) Maxima

“…We conclude that the only three possible outcomes are Outcomes (3), (5), and (6), i.e., pq|r, prq, and p|rq. As we will explain in the following, these configurations can be distinguished solely by the < x -order of p, r, and q: If the correct outcome is prq, all three points lie on the same layer, and since a layer is monotone in decreasing y-direction and in increasing x-direction, it follows that in this case the x-order of the three points has to be prq.…”

“…For constant dimensionality d ≥ 4, their divide-and-conquer approach yields an algorithm with O(n log d−2 n) running time [3,17], and Matoušek [20] gave an O(n 2.688 ) algorithm for the case d = n. The problem has also been studied for dynamically changing point sets in two dimensions [16] and under assumptions about the distribution of the input points in higher dimensions [4,14]. Buchsbaum and Goodrich [5] presented an O(n log n) algorithm for computing the layers of maxima for point sets in three dimensions. Their approach is based on the plane-sweeping paradigm and relies on dynamic fractional cascading to maintain a point-location structure for dynamically changing two-dimensional layers of maxima.…”

“…Given two points p and q, the point p is said to dominate the point q iff the coordinates of p are larger than the coordinates of q in all dimensions. A point p is said to be a maximal point (or: a maximum) of P iff it is not dominated by any other point in P. The union MAX(P) of all points in P that are maximal is called the set of maxima of P. This notion can be extended in a natural way to compute layers of maxima [5]. After MAX(P) has been identified, the computation is repeated for P := P \ MAX(P), i.e., the next layer of maxima is computed.…”

Optimal dynamic vertical ray shooting in rectilinear planar subdivisions

“…Buchsbaum and Goodrich [5] present an algorithm to solve the three-dimensional layers-of-maxima problem. We use our data structure to implement their algorithm in linear space as follows.…”