Partitioning with Two Lines in the Plane

The following problem was raised in a paper by Dan Willard [6]: Given n points (ai, bi) (i = 1,. . . , n), find two straight lines that partition the plane in such a way that each (closed) quadrant contains at least /n/4] points. Cole, Sharir, and Yap [l] propose an algorithm which is based on the scheme for applying parallel algorithms in the design of serial ones that is presented in [3]. Their algorithm runs in O(n log’n) but the authors argue that an 0( n log n) algorithm is possible if the guiding routine is a sorting network of depth n with bounded degree. Here we show that the ideas presented in [2,4] apply to the two-line partitioning problem as well, so we obtain here an O(n) algorithm for the latter.