Farmer John's Cheese Block
Farmer John has a block of cheese in the shape of a cube. It lies on the 3-dimensional coordinate plane, extending from (0,0,0) to (N,N,N) (2≤N≤1000). Farmer John will perform a series of Q (1≤Q≤2⋅105) update operations to his cheese block.
For each update operation, FJ will carve out the 1 by 1 by 1 block of cheese extending from integer coordinates (x,y,z) to (x+1,y+1,z+1), where 0≤x,y,z<N. It is guaranteed that there will exist a 1 by 1 by 1 block of cheese at the location FJ carves. Since FJ is playing Moocraft, gravity does not cause parts of the cheese to fall if cheese below is carved.
After each update, output the number of distinct configurations that FJ can stick a 1 by 1 by N brick in the cheese block such that no part of the brick overlaps with any remaining cheese. Every vertex of the brick must have integer coordinates in the range [0,N] for all three axes. FJ may rotate the brick however he wants.
After the first three updates, the 1×2×1 brick spanning [0,1]×[0,2]×[0,1] does not overlap with the remaining cheese, so it contributes toward the answer.
SCORING:
- Inputs 2-4: N≤10 and Q≤1000
- Inputs 5-7: N≤100 and Q≤1000
- Inputs 8-16: No additional constraints
Problem credits: Chongtian Ma, Alex Liang
Source: USA Computing Olympiad.
Input Format
The first line contains N and Q.
The following Q lines contain x, y, and z, the coordinates to be carved.
Output Format
After each update operation, output an integer, the number of configurations.