Problem
A number of bacteria lie on an infinite grid of cells, each bacterium in its own cell.
Each second, the following transformations occur (all simultaneously):
- If a bacterium has no neighbor to its north and no neighbor to its west, then it will die.
- If a cell has no bacterium in it, but there are bacteria in the neighboring cells to the north and to the west, then a new bacterium will be born in that cell.
Upon examining the grid, you note that there are a positive, finite number of bacteria in one or more rectangular regions of cells.
Determine how many seconds will pass before all the bacteria die.
Here is an example of a grid that starts with 6 cells containing bacteria, and takes 6 seconds for all the bacteria to die. '1's represent cells with bacteria, and '0's represent cells without bacteria.
000010 011100 010000 010000 000000 000000 001110 011000 010000 000000 000000 000110 001100 011000 000000 000000 000010 000110 001100 000000 000000 000000 000010 000110 000000 000000 000000 000000 000010 000000 000000 000000 000000 000000 000000
Input
The input consists of:
One line containing C, the number of test cases. Then for each test case: One line containing R, the number of rectangles of cells that initially contain bacteria. R lines containing four space-separated integers X1 Y1 X2 Y2. This indicates that all the cells with X coordinate between X1 and X2, inclusive, and Y coordinate between Y1 and Y2, inclusive, contain bacteria. The rectangles may overlap.North is in the direction of decreasing Y coordinate.
West is in the direction of decreasing X coordinate.
Output
For each test case, output one line containing "Case #N: T", where N is the case number (starting from 1), and T is the number of seconds until the bacteria all die.
Limits
Time limit: 30 5 seconds per test set.
Memory limit: 1GB.
1 ≤ C ≤ 100.
Small dataset (Test set 1 - Visible; 6 Points)
1 ≤ R ≤ 10
1 ≤ X1 ≤ X2 ≤ 100
1 ≤ Y1 ≤ Y2 ≤ 100
Large dataset (Test set 2 - Hidden; 25 Points)
1 ≤ R ≤ 1000
1 ≤ X1 ≤ X2 ≤ 1000000
1 ≤ Y1 ≤ Y2 ≤ 1000000
The number of cells initially containing bacteria will be at most 1000000.
Sample
1 3 5 1 5 1 2 2 4 2 2 3 2 4
Case #1: 6