Separate Squares II

Separate Squares II

What is this problem about?

The Separate Squares II interview question extends Separate Squares I by finding a horizontal line that minimizes the maximum of the area above and below (or total area difference), but this time with a large number of squares and requiring exact or highly precise computation. The added complexity requires a segment tree or line sweep approach to efficiently compute area below a given y-coordinate across thousands of squares.

Why is this asked in interviews?

Amazon and Google ask this HARD problem because it combines binary search on a continuous domain with efficient area computation using a segment tree or coordinate compression and line sweep. It tests knowledge of computational geometry data structures — specifically the area union problem — applied in geographic information systems, rendering engines, and VLSI design tools where computing the union area under a horizontal line is a performance-critical operation.

Algorithmic pattern used

The pattern is binary search + segment tree with coordinate compression for area under line. Coordinate-compress all unique y-values of square boundaries. For a given query line y, use a sweep-line approach: process horizontal events (square starts and ends) from bottom to top up to y. A segment tree on x-coordinates tracks how many rectangles cover each x-interval, enabling O(log n) total-covered-length queries per event. Binary search on y to minimize the area difference.

Example explanation

Squares at various x/y positions, many overlapping. The line sweep processes square starts (add coverage) and ends (remove coverage) in order of y. At any point, the segment tree gives the total x-length covered by at least one square, which multiplied by the height increment gives the incremental area. Binary search finds the y that equates areas above and below.

Common mistakes candidates make

• Using naive O(n) area computation for each binary search step — with 10^5 squares and 100 binary search iterations, this becomes 10^7 operations per step which may TLE.
• Not handling overlapping squares — naive summation overcounts overlapping areas; the segment tree's coverage count correctly handles this.
• Setting incorrect y-axis binary search bounds when squares have varying y positions.
• Confusing coordinate compression indices with actual coordinates when querying the segment tree.

Interview preparation tip

For the Separate Squares II coding problem, the segment tree line sweep binary search interview pattern is the advanced approach. This is one of the hardest computational geometry problems in interviews — know that the segment tree must maintain both the count of covering rectangles and the total covered length at each node. Google interviewers expect you to outline the segment tree structure before coding. Practice the area-union sweep-line algorithm separately, then combine with binary search for this problem.