Block Placement Queries

This problem requires tracking obstacles and verifying if a block can fit at a given position. By using binary search over the valid answer space combined with segment trees or binary indexed trees, you can efficiently process placement queries. The challenge is handling both obstacle additions and block placement checks while maintaining fast updates.

Problem Statement

You are given an infinite number line starting at 0 extending positively. You receive a sequence of queries in a 2D array. Queries are either type 1, adding an obstacle at a specific position x, or type 2, checking if a block of a given size sz can be placed starting at position x without overlapping obstacles.

Return a boolean array results where results[i] is true if the block in the ith type 2 query can be placed, and false otherwise. Each query must be processed sequentially, and type 1 updates affect subsequent type 2 queries. The goal is to efficiently handle these queries using array structures and binary search over possible positions.

Examples

Example 1

Input: queries = [[1,2],[2,3,3],[2,3,1],[2,2,2]]

Output: [false,true,true]

For query 0, place an obstacle at x = 2 . A block of size at most 2 can be placed before x = 3 .

Example 2

Input: queries = [[1,7],[2,7,6],[1,2],[2,7,5],[2,7,6]]

Output: [true,true,false]

Constraints

• 1 <= queries.length <= 15 * 104
• 2 <= queries[i].length <= 3
• 1 <= queries[i][0] <= 2
• 1 <= x, sz <= min(5 * 104, 3 * queries.length)
• The input is generated such that for queries of type 1, no obstacle exists at distance x when the query is asked.
• The input is generated such that there is at least one query of type 2.

Solution Approach

Track obstacles using efficient data structures

Use a segment tree or binary indexed tree to track the nearest obstacle positions. Each type 1 query updates the data structure with a new obstacle, and type 2 queries check the distance to the next obstacle using the structure.

Binary search over valid placement

For type 2 queries, perform binary search on the valid range starting from x to determine if a block of size sz can fit before encountering the next obstacle. This ensures fast checks even when the number line is conceptually infinite.

Sequential query processing

Process queries in order to ensure each obstacle placement affects subsequent checks. This pattern is essential because type 2 queries depend on all previous type 1 updates, enforcing the correct obstacle constraints.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the number of queries and the efficiency of the segment tree or binary indexed tree updates, generally O(Q log Q) where Q is the number of queries. Space complexity is O(Q) to store obstacles and support fast queries over the number line.

What Interviewers Usually Probe

• Asks if you can handle an infinite number line efficiently without simulating every position.
• Hints at using segment trees or binary indexed trees to manage obstacles.
• Checks if you are applying binary search to find the valid placement range instead of naive iteration.

Common Pitfalls or Variants

Common pitfalls

• Attempting to check block placement by iterating over each number line position, leading to TLE.
• Not updating obstacle structures before type 2 queries, causing incorrect placement results.
• Ignoring the sequential effect of type 1 queries on type 2 queries, leading to false positives.

Follow-up variants

• Queries on a finite number line with wrap-around constraints, still requiring binary search for placement.
• Blocks with variable sizes that can be rotated, changing the binary search logic slightly.
• Tracking multiple independent number lines with simultaneous obstacle updates, needing separate trees per line.

FAQ

What is the main pattern used in Block Placement Queries?

The key pattern is binary search over the valid answer space, combined with a data structure to track obstacles efficiently.

Can type 2 queries affect type 1 updates?

No, type 2 queries only check placement; they do not modify obstacles. Only type 1 queries update the obstacle data structure.

Why use a segment tree or BIT instead of a simple array?

A segment tree or BIT allows fast updates and queries for the next obstacle position without scanning the entire number line, crucial for performance.

How do we handle an infinite number line?

We do not simulate the entire line. Instead, we track only positions with obstacles and use binary search to determine valid placement ranges.

What are common mistakes when solving this problem?

Naively iterating over positions, ignoring the sequential impact of obstacles, or failing to use efficient structures for fast queries are the main pitfalls.