# Implementing upper_bound() and lower_bound() for Ordered Set in C++

Prerequisites: Ordered Set and GNU C++ PBDS

Given an ordered set set and a key K, the task is to find the upper bound and lower bound of the element K in the set in C++. If the element is not present or either of the bounds could not be calculated, then print -1.

Ordered set is a policy based data structure in g++ that keeps the unique elements in sorted order. It performs all the operations as performed by the set data structure in STL in log(n) complexity. Apart from that, it performs two additional operations also in log(n) complexity like:
1. order_of_key (K): Number of items strictly smaller than K.
2. find_by_order(K): Kth element in a set (counting from zero).

Examples:

Input: set[] = {10, 20, 30, 40, 50, 60}, K = 30
Output:
Lower Bound of 30: 30
Upper Bound of 30: 40
Explanation:
The lower bound for element 30 is 30 located at position 2
The upper bound for element 30 is 40 located at position 3

Input: set[] = {10, 20, 30, 40, 50, 60}, K = 60
Output:
Lower Bound of 60: -1
Upper Bound of 60: -1

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

Approach:

• upper_bound(): The upper_bound(key) function returns the element which is just greater than key passed in parameter.
• lower_bound(): The lower_bound(key) function returns the element which is equivalent to key passed in the parameter. If the key is not present in the ordered set, then the function should return the element which is greater than the parameter.
• In order to implement the upper_bound and lower_bound functions, the index of the element if present in the ordered set passed as the parameter is found using order_of_key() function.
• Now, both the lower bound and the upper bound can be found by simply comparing the elements present at this index.

Below is the implementation of lower_bound() and upper_bound():

 `// C++ program to implement the ` `// lower_bound() and upper_bound() ` `// using Ordered Set ` ` `  `#include ` `#include ` `#include ` `using` `namespace` `__gnu_pbds; ` `using` `namespace` `std; ` ` `  `// Ordered Set Tree ` `typedef` `tree<``int``, null_type, ` `             ``less<``int``>, ` `             ``rb_tree_tag, ` `             ``tree_order_statistics_node_update> ` `    ``ordered_set; ` ` `  `ordered_set set1; ` ` `  `// Function that returns the lower bound ` `// of the element ` `int` `lower_bound(``int` `x) ` `{ ` `    ``// Finding the position of the element ` `    ``int` `pos = set1.order_of_key(x); ` ` `  `    ``// If the element is not present in the set ` `    ``if` `(pos == set1.size()) { ` `        ``return` `-1; ` `    ``} ` ` `  `    ``// Finding the element at the position ` `    ``else` `{ ` `        ``int` `element ` `            ``= *(set1.find_by_order(pos)); ` ` `  `        ``return` `element; ` `    ``} ` `} ` ` `  `// Function that returns the upper bound ` `// of the element ` `int` `upper_bound(``int` `x) ` `{ ` `    ``// Finding the position of the element ` `    ``int` `pos = set1.order_of_key(x + 1); ` ` `  `    ``// If the element is not present ` `    ``if` `(pos == set1.size()) { ` `        ``return` `-1; ` `    ``} ` ` `  `    ``// Finding the element at the position ` `    ``else` `{ ` `        ``int` `element ` `            ``= *(set1.find_by_order(pos)); ` ` `  `        ``return` `element; ` `    ``} ` `} ` ` `  `// Function to print Upper ` `// and Lower bound of K ` `// in Ordered Set ` `void` `printBound(``int` `K) ` `{ ` ` `  `    ``cout << ``"Lower Bound of "` `         ``<< K << ``": "` `         ``<< lower_bound(K) ` `         ``<< endl; ` `    ``cout << ``"Upper Bound of "` `         ``<< K << ``": "` `         ``<< upper_bound(K) ` `         ``<< endl; ` `} ` ` `  `// Driver's Code ` `int` `main() ` `{ ` `    ``set1.insert(10); ` `    ``set1.insert(20); ` `    ``set1.insert(30); ` `    ``set1.insert(40); ` `    ``set1.insert(50); ` ` `  `    ``int` `K = 30; ` `    ``printBound(K); ` ` `  `    ``K = 60; ` `    ``printBound(K); ` ` `  `    ``return` `0; ` `} `

Output:

```Lower Bound of 30: 30
Upper Bound of 30: 40
Lower Bound of 60: -1
Upper Bound of 60: -1
```

Time Complexity: O(log N)

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