[Paper Review] Maximum Area Rectangle Separating Red and Blue Points

Bogdan Armaselu, Ovidiu Daescu|arXiv (Cornell University)|Jun 10, 2017

Computational Geometry and Mesh Generation18 references3 citations

TL;DR

This paper presents an optimal O(m log m + n) time, O(m + n) space algorithm for computing the maximum area axis-aligned rectangle that contains all red points and the minimum number of blue points. The approach uses a staircase-based method to efficiently explore candidate rectangles, and the algorithm is proven optimal in the decision model via a reduction from the 1D-Furthest-Adjacent-Pair problem.

ABSTRACT

Given a set R of n red points and a set B of m blue points, we study the problem of finding a rectangle that contains all the red points, the minimum number of blue points and has the largest area. We call such rectangle a maximum separating rectangle. We address the planar, axis-aligned (2D) version, and present an O(mlogm+n) time, O(m+n) space algorithm. The running time reduces to O(m + n) if the points are pre-sorted by one of the coordinates. We further prove that our algorithm is optimal in the decision model of computation.

Motivation & Objective

• To solve the problem of finding an axis-aligned rectangle that contains all red points and the fewest blue points while maximizing area.
• To develop an efficient algorithm for this separation problem with optimal time complexity.
• To prove the optimality of the algorithm in the decision model of computation.
• To address applications in tumor cell separation, urban planning, and integrated circuit defect detection.
• To establish a lower bound for the problem, showing that Ω(m log m + n) time is necessary in the worst case.

Proposed method

• Construct the minimum enclosing rectangle S_min of all red points as a starting constraint.
• Use a staircase-based approach to systematically explore candidate rectangles defined by pairs of blue points.
• For each candidate rectangle, verify it contains all red points and no additional blue points beyond the minimum required.
• Sort blue points by x-coordinate to enable efficient candidate enumeration and area computation.
• Leverage monotonicity properties of the area function to avoid unnecessary checks and ensure optimality.
• Reduce the 1D-Furthest-Adjacent-Pair problem to prove the Ω(m log m + n) lower bound, establishing algorithmic optimality.

Experimental results

Research questions

• RQ1 What is the minimum number of blue points that must be included in an axis-aligned rectangle containing all red points, while maximizing the area?
• RQ2 Can an optimal solution be computed in time better than O(n log²n) for this separating rectangle problem?
• RQ3 Is the proposed algorithm optimal in terms of time complexity, and what is the theoretical lower bound for solving this problem?
• RQ4 How can the problem be reduced to a known hard problem to establish a lower bound?
• RQ5 Can the algorithm be adapted to dynamic or query-based settings where red points are provided at query time?

Key findings

• The proposed algorithm runs in O(m log m + n) time and O(m + n) space, with O(m + n) time if blue points are pre-sorted by x-coordinate.
• The algorithm is proven optimal in the decision model of computation, as Ω(m log m + n) time is required in the worst case.
• The optimality is established via a reduction from the 1D-Furthest-Adjacent-Pair problem, which has a known Ω(m log m) lower bound.
• The maximum area separating rectangle is found by exploring candidate rectangles bounded by pairs of blue points with specific geometric constraints.
• The area function is monotonically increasing with respect to the horizontal span, enabling efficient pruning of suboptimal candidates.
• The method ensures that the resulting rectangle contains all red points and the minimum number of blue points, while maximizing area.