Problem:
Score : 600 points

Problem Statement

There are N dots in a two-dimensional plane. The coordinates of the i-th dot are (x_i, y_i).

We will repeat the following operation as long as possible:

Choose four integers a, b, c, d (a \neq c, b \neq d) such that there are dots at exactly three of the positions (a, b), (a, d), (c, b) and (c, d), and add a dot at the remaining position.

We can prove that we can only do this operation a finite number of times. Find the maximum number of times we can do the operation.

Constraints

1 \leq N \leq 10^5

1 \leq x_i, y_i \leq 10^5

If i \neq j, x_i \neq x_j or y_i \neq y_j.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the maximum number of times we can do the operation.

Sample Input 1

3
1 1
5 1
5 5

Sample Output 1

1

By choosing a = 1, b = 1, c = 5, d = 5, we can add a dot at (1, 5). We cannot do the operation any more, so the maximum number of operations is 1.

Sample Input 2

2
10 10
20 20

Sample Output 2

0

There are only two dots, so we cannot do the operation at all.

Sample Input 3

9
1 1
2 1
3 1
4 1
5 1
1 2
1 3
1 4
1 5

Sample Output 3

16

We can do the operation for all choices of the form a = 1, b = 1, c = i, d = j (2 \leq i,j \leq 5), and no more. Thus, the maximum number of operations is 16.

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

local n = io.read("*n")

local parent = {}
for i = 1, 2 * n do parent[i] = i end

local function uf_findroot(idx)
  local idx_update = idx
  while parent[idx] ~= idx do
    idx = parent[idx]
  end
  while parent[idx_update] ~= idx do
    parent[idx_update], idx_update = idx, parent[idx_update]
  end
  return idx
end

local x, y = {}, {}
local xinvmap, yinvmap = {}, {}
local xary, yary = {}, {}
for i = 1, n do
  x[i], y[i] = io.read("*n", "*n")
  if not xinvmap[x[i]] then
    table.insert(xary, x[i])
    xinvmap[x[i]] = #xary
  end
  if not yinvmap[y[i]] then
    table.insert(yary, y[i])
    yinvmap[y[i]] = n + #yary
  end
end

for i = 1, n do
  local xi, yi = x[i], y[i]
  local idx1, idx2 = xinvmap[xi], yinvmap[yi]
  local p1, p2 = uf_findroot(idx1), uf_findroot(idx2)
  parent[p2], parent[idx2] = p1, p1
end
local groups = {}
for i = 1, 2 * n do
  local p = uf_findroot(i)
  if not groups[p] then
    groups[p] = {}
    groups[p].xcnt = 0
    groups[p].ycnt = 0
    groups[p].pointcnt = 0
  end
  if i <= n then
    groups[p].xcnt = groups[p].xcnt + 1
  else
    groups[p].ycnt = groups[p].ycnt + 1
  end
end
for i = 1, n do
  local xi, yi = x[i], y[i]
  local p = uf_findroot(xinvmap[xi])
  groups[p].pointcnt = groups[p].pointcnt + 1
end
local ret = 0
for k, v in pairs(groups) do
  if 1 < v.xcnt and 1 < v.ycnt then
    ret = ret + v.xcnt * v.ycnt - v.pointcnt
  end
end
print(ret)