Lecture Notes: 2D Maxima

In this lecture, we will discuss the maxima problem defined as follows. Let (x1, y1) and (x2, y2) be two different points in R. We say that the former dominates the latter if x1 ≥ x2 and y1 ≥ y2 (note that the two equalities cannot hold simultaneously because these are two different points). Let P be a set of n points in R. A point p ∈ P is a maximal point of P if p is not dominated by any point in P . We want to design an algorithm to report all the maximal points of P efficiently. In the example of Figure 1, points 1, 2, 6, and 8 should be reported. We will assume that P is in general position (in particular, no two points have the same x-coordinate, or the same y-coordinate). It is not hard to observe that the maximal points must form a “staircase”, namely, if we walk along them in ascending order of x-coordinate, their y-coordinates must be descending.