Minimum Number of Arrows to Burst Balloons

Minimum Number of Arrows to Burst Balloons

1. What is this problem about?

In the Minimum Number of Arrows to Burst Balloons interview question, you are presented with a 2D plane where balloons are represented as horizontal intervals [start, end]. You can shoot arrows vertically from the x-axis. An arrow shot at position x will burst all balloons whose intervals include x. Your goal is to find the absolute minimum number of arrows needed to pop every single balloon in the set.

2. Why is this asked in interviews?

This is a classic "Interval Overlap" problem, making it a favorite for Tier-1 companies like Apple, Microsoft, and Goldman Sachs. It tests a candidate's ability to handle ranges and identify greedy optimizations. Interviewers want to see if you can recognize that the problem isn't about the balloons themselves, but about finding the most "crowded" points on the x-axis where a single arrow can provide maximum value.

3. Algorithmic pattern used

This problem follows the Array, Sorting, Greedy interview pattern. The key is to sort the balloons based on their ending coordinates. By doing this, you ensure that you are always looking at the earliest possible point where a balloon must be popped. You then greedily place an arrow at the end of the current balloon, which gives it the best chance of overlapping with subsequent balloons that start before or at that same coordinate.

4. Example explanation

Consider balloons at: [[10, 16], [2, 8], [1, 6], [7, 12]]. Sort by end: [[1, 6], [2, 8], [7, 12], [10, 16]].

1. Look at first balloon [1, 6]. Shoot arrow at x=6. (1 arrow)
2. Next balloon [2, 8] also covers x=6. It bursts.
3. Next balloon [7, 12] starts at 7, which is after 6. We need a new arrow. Shoot at x=12. (2 arrows)
4. Next balloon [10, 16] covers x=12. It bursts. Total arrows: 2.

5. Common mistakes candidates make

A very common mistake is sorting by the start coordinate instead of the end. While it's possible to solve it that way, the logic becomes significantly more complex. Another error is not handling the boundary conditions correctly—specifically, whether an arrow at the exact edge of a balloon counts as a burst (it usually does). Using incorrect data types for large coordinates can also lead to issues in some environments.

6. Interview preparation tip

Interval problems are extremely common. Whenever you see a task involving "overlapping ranges" or "scheduling," your first instinct should be to try sorting the data. Practice articulating why sorting by the end point is "greedily optimal"—it’s because it leaves the most room for future intervals to be included in the same group.