Block Placement Queries

Block Placement Queries

What is this problem about?

Block Placement Queries is a complex "Hard" level problem involving dynamic updates and range queries on a 1D coordinate system. You start with an empty infinite line and receive two types of queries:

1. Place an obstacle at a given coordinate.
2. Check if it's possible to place a block of a specific length between the start (0) and a target coordinate, given the existing obstacles. The challenge lies in handling these queries efficiently as the number of obstacles grows.

Why is this asked in interviews?

This problem is asked by top-tier companies like Uber, Visa, and Meta to evaluate your mastery of advanced data structures. It tests whether you can go beyond basic arrays and trees to use Segment Trees or Binary Indexed Trees (BIT). It requires a deep understanding of how to manage intervals and how to find the "maximum gap" between points in a dynamic environment.

Algorithmic pattern used

This problem is a classic application of the Segment Tree or Segment Tree with Lazy Propagation pattern. Each leaf in the segment tree represents a gap between two obstacles. The internal nodes store the maximum gap within their range. Alternatively, you can use a Balanced BST (like a std::set in C++) to keep track of sorted obstacles and a Segment Tree to store the distances between adjacent obstacles.

Example explanation

1. Initial state: No obstacles. Max gap in range [0, 10] is 10.
2. Query: Place obstacle at 4. Now we have two gaps: [0, 4] (size 4) and [4, infinity].
3. Query: Place obstacle at 7. Now we have gaps: [0, 4] (size 4), [4, 7] (size 3), and [7, infinity].
4. Query: Can we place a block of size 5 in the range [0, 8]?
   • Looking at the obstacles in [0, 8]: {4, 7}.
   • Gaps are 4 and 3. Max gap is 4.
   • 4 < 5, so the answer is No.

Common mistakes candidates make

The most common mistake is using a naive list to store obstacles and sorting it for every query, which results in $O(Q imes N \log N)$ or $O(Q imes N)$ complexity. Another mistake is failing to handle the "boundary" cases, such as the gap between the last obstacle and the target coordinate. Finally, many struggle with the implementation details of a Segment Tree, which is quite complex to write from scratch under interview pressure.

Interview preparation tip

Practice implementing a Segment Tree for range maximum queries. Once you're comfortable with the basics, try adapting it to handle dynamic point updates. This is a recurring theme in advanced competitive programming and "Hard" level interviews.