# 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

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 ` `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; ` `} `

## 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 `

Output:

```3
```

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