Below the Diagonal (Codeforces 266C)
Tags: Math, constructive algorithms

You are given a square matrix consisting of n rows and n columns. We assume that the rows are numbered from 1 to n from top to bottom and the columns are numbered from 1 to n from left to right. Some cells (n - 1 cells in total) of the the matrix are filled with ones, the remaining cells are filled with zeros. We can apply the following operations to the matrix:  Swap i-th and j-th rows of the matrix;  Swap i-th and j-th columns of the matrix. You are asked to transform the matrix into a special form using these operations. In that special form all the ones must be in the cells that lie below the main diagonal. Cell of the matrix, which is located on the intersection of the i-th row and of the j-th column, lies below the main diagonal if i > j.

Input: The first line contains an integer n (2 ≤ n ≤ 1000) — the number of rows and columns. Then follow n - 1 lines that contain one's positions, one per line. Each position is described by two integers xk, yk (1 ≤ xk, yk ≤ n), separated by a space. A pair (xk, yk) means that the cell, which is located on the intersection of the xk-th row and of the yk-th column, contains one.It is guaranteed that all positions are distinct.

Output: Print the description of your actions. These actions should transform the matrix to the described special form.In the first line you should print a non-negative integer m (m ≤ 105) — the number of actions. In each of the next m lines print three space-separated integers t, i, j (1 ≤ t ≤ 2, 1 ≤ i, j ≤ n, i ≠ j), where t = 1 if you want to swap rows, t = 2 if you want to swap columns, and i and j denote the numbers of rows or columns respectively.Please note, that you do not need to minimize the number of operations, but their number should not exceed 105. If there are several solutions, you may print any of them.


Sample Inputs:
2
1 2

3
3 1
1 3

3
2 1
3 2

Sample Outputs:
2
2 1 2
1 1 2

3
2 2 3
1 1 3
1 1 2

0

Solution (Python):
n = int(input())
row = list(range(n + 1))
rinv = row[:]
col = row[:]
cinv = row[:]
ans = []
for i in range(1, n):
    (r, c) = map(int, input().split())
    if col[c] < row[r] and row[r] <= i + 1:
        continue
    if row[r] != i + 1:
        ans.append((1, row[r], i + 1))
        rinv[row[r]] = rinv[i + 1]
        (row[r], row[rinv[i + 1]]) = (row[rinv[i + 1]], row[r])
        rinv[i + 1] = r
    if col[c] > i:
        ans.append((2, col[c], i))
        cinv[col[c]] = cinv[i]
        (col[c], col[cinv[i]]) = (col[cinv[i]], col[c])
        cinv[i] = c
print(len(ans))
for a in ans:
    print(*a)

Solution (Python):
from collections import defaultdict
n = int(input())
arr = [[0] * n for i in range(n)]
cells_col = defaultdict(set)
cells_row = defaultdict(set)
for _ in range(n - 1):
    (i, j) = [int(x) for x in input().split()]
    arr[i - 1][j - 1] = 1
    cells_col[j - 1].add(i - 1)
    cells_row[i - 1].add(j - 1)
ans = []
for i in range(n - 1, -1, -1):
    if i in cells_col:
        for j in range(i - 1, -1, -1):
            if j not in cells_col:
                cells_col[j] = cells_col[i]
                del cells_col[i]
                for row in cells_col[j]:
                    cells_row[row].remove(i)
                    cells_row[row].add(j)
                ans.append((2, i + 1, j + 1))
                break
    if i not in cells_row:
        for j in range(i - 1, -1, -1):
            if j in cells_row:
                cells_row[i] = cells_row[j]
                del cells_row[j]
                for col in cells_row[i]:
                    cells_col[col].remove(j)
                    cells_col[col].add(i)
                ans.append((1, i + 1, j + 1))
                break
    if i in cells_row:
        for col in cells_row[i]:
            cells_col[col].remove(i)
            if not cells_col[col]:
                del cells_col[col]
        del cells_row[i]
print(len(ans))
for i in range(len(ans)):
    print(ans[i][0], ans[i][1], ans[i][2])

Solution (Python):
n = int(input())
P = [list(map(int, input().split())) for _ in range(n - 1)]
col = set()
row = set()
sCol = {}
sRow = {}
for (x, y) in P:
    row.add(x)
    col.add(y)
n_col = len(col)
n_row = len(row)
(cur_col, cur_row) = (1, n)
used_col = {}
used_row = {}
for c in sorted(col):
    if c > n_col:
        while cur_col in used_col:
            cur_col += 1
        used_col[cur_col] = 1
        sCol[c] = cur_col
    else:
        used_col[c] = 1
for r in sorted(row, reverse=True):
    if r < n - n_row + 1:
        while cur_row in used_row:
            cur_row -= 1
        used_row[cur_row] = 1
        sRow[r] = cur_row
    else:
        used_row[r] = 1
for (i, [x, y]) in enumerate(P):
    if x in sRow:
        P[i][0] = sRow[x]
    if y in sCol:
        P[i][1] = sCol[y]
minCol = [float('inf') for _ in range(n_col + 1)]
minCol[0] = 0
for (x, y) in P:
    if x < minCol[y]:
        minCol[y] = x

def sort(minCol):
    swap = []
    pos = {}
    for (col, val) in enumerate(minCol):
        if col == 0:
            continue
        if val not in pos:
            pos[val] = set()
        pos[val].add(col)
    sortCol = sorted(minCol)
    for (i, val) in enumerate(sortCol):
        if i == 0:
            continue
        if val == minCol[i]:
            pos[val].remove(i)
        else:
            col_ = next(iter(pos[val]))
            pos[val].remove(col_)
            pos[minCol[i]].remove(i)
            pos[minCol[i]].add(col_)
            minCol[col_] = minCol[i]
            minCol[i] = val
            swap.append([2, i, col_])
    return swap
swap = sort(minCol)
ans = [[1, x, y] for (x, y) in sRow.items()] + [[2, x, y] for (x, y) in sCol.items()] + swap
print(len(ans))
print('\n'.join([' '.join([str(x) for x in line]) for line in ans]))