# Computing the largest empty rectangle

- First Online:

DOI: 10.1007/3-540-12920-0_4

- Cite this paper as:
- Chazelle B., Drysdale R.L., Lee D.T. (1984) Computing the largest empty rectangle. In: Fontet M., Mehlhorn K. (eds) STACS 84. STACS 1984. Lecture Notes in Computer Science, vol 166. Springer, Berlin, Heidelberg

## Abstract

We consider the following problem: Given a rectangle containing N points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result[13] takes O(N^{2}) worst-case and O(Nlog^{2}N) expected time. This paper presents an O(N log^{3}N) time, O(N log N) space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Strong and Bentley[1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree.

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