# Largest subset of rectangles such that no rectangle fit in any other rectangle

Given **height** and **width** of **N** rectangles. The task is to find the size of the largest subset such that no pair of rectangles fit within each other. **Note** that if **H1 ≤ H2** and **W1 ≤ W2** then rectangle 1 fits inside rectangle 2.

**Examples:**

Input:arr[] = {{1, 3}, {2, 2}, {1, 3}}

Output:2

The required sub-set is {{1, 3}, {2, 2}}

{1, 3} is included only once as it can fit in {1, 3}

Input:arr[] = {{1, 5}, {2, 4}, {1, 1}, {3, 3}}

Output:3

**Approach:** The above problem can be solved using Dynamic Programming and sorting. Initially, we can sort the N pairs on the basis of heights. A recursive function can be written where there will be two states.

The first state being, if the present rectangle does not fit in the previous rectangle or the vice versa, then we call for the next state with the present rectangle being the previous rectangle and moving to the next rectangle.

dp[present][previous] = max(dp[present][previous], 1 + dp[present + 1][present])

If it does fit in, we call the next state with the previous rectangle and moving to the next rectangle.

dp[present][previous] = max(dp[present][previous], dp[present + 1][previous])

Memoization can be further used to avoid repetitively the same states being called.

Below is the implementation of the above approach:

## C++

`// C++ implementation of the approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` `#define N 10 ` `int` `dp[N][N]; ` ` ` `// Recursive function to get the largest subset ` `int` `findLongest(pair<` `int` `, ` `int` `> a[], ` `int` `n, ` ` ` `int` `present, ` `int` `previous) ` `{ ` ` ` `// Base case when it exceeds ` ` ` `if` `(present == n) { ` ` ` `return` `0; ` ` ` `} ` ` ` ` ` `// If the state has been visited previously ` ` ` `else` `if` `(previous != -1) { ` ` ` `if` `(dp[present][previous] != -1) ` ` ` `return` `dp[present][previous]; ` ` ` `} ` ` ` ` ` `// Initialize ` ` ` `int` `ans = 0; ` ` ` ` ` `// No elements in subset yet ` ` ` `if` `(previous == -1) { ` ` ` ` ` `// First state which includes current index ` ` ` `ans = 1 + findLongest(a, n, ` ` ` `present + 1, present); ` ` ` ` ` `// Second state which does not include current index ` ` ` `ans = max(ans, findLongest(a, n, ` ` ` `present + 1, previous)); ` ` ` `} ` ` ` `else` `{ ` ` ` `int` `h1 = a[previous].first; ` ` ` `int` `h2 = a[present].first; ` ` ` `int` `w1 = a[previous].second; ` ` ` `int` `w2 = a[present].second; ` ` ` ` ` `// If the rectangle fits in, then do not include ` ` ` `// the current index in subset ` ` ` `if` `((h1 <= h2 && w1 <= w2)) { ` ` ` `ans = max(ans, findLongest(a, n, ` ` ` `present + 1, previous)); ` ` ` `} ` ` ` `else` `{ ` ` ` ` ` `// First state which includes current index ` ` ` `ans = 1 + findLongest(a, n, ` ` ` `present + 1, present); ` ` ` ` ` `// Second state which does not include current index ` ` ` `ans = max(ans, findLongest(a, n, ` ` ` `present + 1, previous)); ` ` ` `} ` ` ` `} ` ` ` ` ` `return` `dp[present][previous] = ans; ` `} ` ` ` `// Function to get the largest subset ` `int` `getLongest(pair<` `int` `, ` `int` `> a[], ` `int` `n) ` `{ ` ` ` `// Initialize the DP table with -1 ` ` ` `memset` `(dp, -1, ` `sizeof` `dp); ` ` ` ` ` `// Sort the array ` ` ` `sort(a, a + n); ` ` ` ` ` `// Get the answer ` ` ` `int` `ans = findLongest(a, n, 0, -1); ` ` ` `return` `ans; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` ` ` `// (height, width) pairs ` ` ` `pair<` `int` `, ` `int` `> a[] = { { 1, 5 }, ` ` ` `{ 2, 4 }, ` ` ` `{ 1, 1 }, ` ` ` `{ 3, 3 } }; ` ` ` `int` `n = ` `sizeof` `(a) / ` `sizeof` `(a[0]); ` ` ` ` ` `cout << getLongest(a, n); ` ` ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Python3

# Python3 implementation of the approach

# Recursive function to get the

# largest subset

def findLongest(a, n, present, previous):

# Base case when it exceeds

if present == n:

return 0

# If the state has been visited

# previously

elif previous != -1:

if dp[present][previous] != -1:

return dp[present][previous]

# Initialize

ans = 0

# No elements in subset yet

if previous == -1:

# First state which includes

# current index

ans = 1 + findLongest(a, n, present + 1,

present)

# Second state which does not

# include current index

ans = max(ans, findLongest(a, n, present + 1,

previous))

else:

h1 = a[previous][0]

h2 = a[present][0]

w1 = a[previous][1]

w2 = a[present][1]

# If the rectangle fits in, then do

# not include the current index in subset

if h1 <= h2 and w1 <= w2:
ans = max(ans, findLongest(a, n, present + 1,
previous))
else:
# First state which includes
# current index
ans = 1 + findLongest(a, n, present + 1,
present)
# Second state which does not
# include current index
ans = max(ans, findLongest(a, n, present + 1,
previous))
dp[present][previous] = ans
return ans
# Function to get the largest subset
def getLongest(a, n):
# Sort the array
a.sort()
# Get the answer
ans = findLongest(a, n, 0, -1)
return ans
# Driver code
if __name__ == "__main__":
# (height, width) pairs
a = [[1, 5], [2, 4], [1, 1], [3, 3]]
N = 10
# Initialize the DP table with -1
dp = [[-1 for i in range(N)]
for j in range(N)]
n = len(a)
print(getLongest(a, n))
# This code is contributed
# by Rituraj Jain
[tabbyending]

**Output:**

3

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