Cities in Alberta tend to be laid out as rectangular grids of blocks. Blocks are labeled with coordinates ~0~ to ~R-1~ from north to south and ~0~ to ~C-1~ from west to east.

The quality of living in each particular block has been ranked by a distinct number, called quality rank, between ~1~ and ~R \cdot C~, where ~1~ is the best and ~R \cdot C~ is the worst.

The city planning department wishes to identify a rectangular set of blocks with dimensions ~H~ from north to south and ~W~ from west to east, such that the median quality rank among all blocks in the rectangle is the best. ~H~ and ~W~ are odd numbers not exceeding ~R~ and ~C~ respectively. The median quality rank among an odd number of quality ranks is defined to be the quality rank ~m~ in the set such that the number of quality ranks better than ~m~ equals the number of quality ranks worse than ~m~.

You are to implement a procedure rectangle(R,C,H,W,Q) where ~R~ and ~C~ represent the total size of the city, ~H~ and ~W~ represent the dimensions of the set of blocks, and ~Q~ is an array such that ~Q[a][b]~ is the quality rank for the block labeled ~a~ from north to south and ~b~ from west to east.

Your implementation of rectangle must return a number: the best (numerically smallest) possible median quality rank of an ~H~ by ~W~ rectangle of blocks.

Each test run will only call rectangle once.

Example 1

R=5, C=5, H=3, W=3,
Q=  5 11 12 16 25
   17 18  2  7 10
    4 23 20  3  1
   24 21 19 14  9
    6 22  8 13 15

For this example, the best (numerically smallest) median quality rank of ~9~ is achieved by the middle-right rectangle of ~Q~ shown in bold. That is,

rectangle(R,C,H,W,Q)=9

Example 2

R=2, C=6, H=1, W=5,
Q=  6  1  2 11  7  5
    9  3  4 10 12  8

For this example the correct answer is ~5~.

Subtask 1 [20 points]

Assume ~R~ and ~C~ do not exceed ~30~.

Subtask 2 [20 points]

Assume ~R~ and ~C~ do not exceed ~100~.

Subtask 3 [20 points]

Assume ~R~ and ~C~ do not exceed ~300~.

Subtask 4 [20 points]

Assume ~R~ and ~C~ do not exceed ~1\,000~.

Subtask 5 [20 points]

Assume ~R~ and ~C~ do not exceed ~3\,000~.

Implementation Details

• Implementation folder: /home/ioi2010-contestant/quality/ (prototype: quality.zip)
• To be implemented by contestant: quality.c or quality.cpp or quality.pas
• Contestant interface: quality.h or quality.pas
• Grader interface: none
• Sample grader: grader.c or grader.cpp or grader.pas
• Sample grader input: grader.in.1 grader.in.2 etc.
  Note: The first line of input contains: ~R,C,H,W~ The following lines contain the elements of ~Q~, in row-major order.
• Expected output for sample grader input: grader.expect.1 grader.expect.2 etc.