Problem:
Score : 500 points

Problem Statement

We have a chocolate bar partitioned into H horizontal rows and W vertical columns of squares.

The square (i, j) at the i-th row from the top and the j-th column from the left is dark if S_{i,j} is 0, and white if S_{i,j} is 1.

We will cut the bar some number of times to divide it into some number of blocks. In each cut, we cut the whole bar by a line running along some boundaries of squares from end to end of the bar.

How many times do we need to cut the bar so that every block after the cuts has K or less white squares?

Constraints

1 \leq H \leq 10

1 \leq W \leq 1000

1 \leq K \leq H \times W

S_{i,j} is 0 or 1.

Input

Input is given from Standard Input in the following format:

H W K
S_{1,1}S_{1,2}...S_{1,W}
:
S_{H,1}S_{H,2}...S_{H,W}

Output

Print the number of minimum times the bar needs to be cut so that every block after the cuts has K or less white squares.

Sample Input 1

3 5 4
11100
10001
00111

Sample Output 1

2

For example, cutting between the 1-st and 2-nd rows and between the 3-rd and 4-th columns - as shown in the figure to the left - works.

Note that we cannot cut the bar in the ways shown in the two figures to the right.

Sample Input 2

3 5 8
11100
10001
00111

Sample Output 2

0

No cut is needed.

Sample Input 3

4 10 4
1110010010
1000101110
0011101001
1101000111

Sample Output 3

3

Solution in LuaJIT (2.0.4) (judge verdict: Accepted)

local mmi, mma = math.min, math.max
local bls, brs = bit.lshift, bit.rshift
local h, w, k = io.read("*n", "*n", "*n", "*l")
local map = {}
for i = 1, h do
  local s = io.read()
  for j = 1, w do
    local ss = s:sub(j, j)
    map[(i - 1) * w + j] = ss == "1"
  end
end
local ans = false
local tot = 2^(h - 1)
for it = 0, tot - 1 do
  local tmpi = it
  local box = {1}
  local pos = 1
  for j = 2, h do
    if it % 2 == 1 then
      pos = pos + 1
    end
    box[j] = pos
    it = brs(it, 1)
  end
  local bsize = pos
  local ret = bsize - 1
  -- print(tmpi, ret)
  local bcnt = {}
  for i = 1, bsize do
    bcnt[i] = 0
  end
  local valid = true
  local start_w = 1
  for j = 1, w do
    for i = 1, h do
      local idx = (i - 1) * w + j
      if map[idx] then
        bcnt[box[i]] = bcnt[box[i]] + 1
      end
    end
    local over = false
    for ib = 1, bsize do
      if k < bcnt[ib] then
        over = true
        break
      end
    end
    if over then
      if start_w == j then
        valid = false
        break
      end
      start_w = j
      ret = ret + 1
      for ib = 1, bsize do
        bcnt[ib] = 0
      end
      for i = 1, h do
        local idx = (i - 1) * w + j
        if map[idx] then
          bcnt[box[i]] = bcnt[box[i]] + 1
        end
      end
    end
  end
  if valid then
    if not ans then ans = ret
    else ans = mmi(ans, ret)
    end
  end
end
print(ans)