Number Line Obstacles: Check Whether a Centered Block Can Be Built

You are working on an infinite integer number line that starts with no obstacles. Process a sequence of operations of two types:

• [1, x] builds an obstacle at coordinate x. It is guaranteed that coordinate x does not already hold an obstacle when the operation is performed.
• [2, x, size] checks whether a block centered on x and extending size - 1 units in each direction could be built, that is, whether none of the coordinates from x - (size - 1) to x + (size - 1) holds an obstacle. For example, for x = 0 and size = 3, it checks coordinates -2 through 2. The check produces "1" if the block could be built and "0" otherwise. It does not build the block, so it never changes the number line.

Return the outputs of all type-2 operations, in the order the operations appear, concatenated into one string.

Function Signature

def check_blocks(operations: list[list[int]]) -> str:

Rules

• Operations are processed in the given order. A type-2 operation sees exactly the obstacles built by the type-1 operations that come before it.
• The checked range is inclusive at both ends and contains 2 * size - 1 coordinates. With size = 1, only x itself is checked.
• Each type-2 operation contributes exactly one character to the result. If there are no type-2 operations, return the empty string "".

Constraints

• 1 <= len(operations) <= 10^5
• Each operation is either [1, x] or [2, x, size].
• -10^9 <= x <= 10^9
• 1 <= size <= 10^9
• No two type-1 operations build at the same coordinate.
• The checked endpoints x - (size - 1) and x + (size - 1) lie between -1999999999 and 1999999999, so they fit in a 32-bit signed integer.

Examples

Example 1

Input: operations = [[1, 2], [2, 0, 3], [2, 0, 2], [1, -5], [2, -3, 3], [2, -3, 2]]
Output: "0101"

An obstacle is built at 2. The check at 0 with size 3 covers -2 to 2 and hits the obstacle at 2, giving "0". With size 2 it covers -1 to 1, which is clear, giving "1". After an obstacle is built at -5, the check at -3 with size 3 covers -5 to -1 and gives "0", while size 2 covers -4 to -2 and gives "1".

Example 2

Input: operations = [[2, 7, 1], [1, 7], [2, 7, 1], [2, 10, 3], [2, 10, 4]]
Output: "1010"

The first check covers only 7, which is still empty. After the obstacle at 7 is built, the same check gives "0". The check at 10 with size 3 covers 8 to 12 and is clear; with size 4 it covers 7 to 13, which contains the obstacle.

Example 3

Input: operations = [[1, 1000000000], [2, -1000000000, 1000000000], [2, 1, 1000000000]]
Output: "10"

The first check covers -1999999999 to -1, which does not contain 1000000000. The second covers -999999998 to 1000000000, and its right end is the obstacle.