Given an array **arr[]** of positive integers, the task is to count the maximum possible number of pairs **(arr[i], arr[j])** such that **arr[i] + arr[j]** is a **power of 2**.

**Note:** One element can be used at most once to form a pair.

**Examples:**

Input:arr[] = {3, 11, 14, 5, 13}

Output:2

All valid pairs are (13, 3) and (11, 5) both sum up to 16 which is a power of 2.

We could have used (3, 5) but by doing so maximum of 1 pair could only be formed.

Therefore, (3, 5) is not optimal.

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

Output:1

1 and 3 can be paired to form 4, which is a power of 2.

A **simple solution **is to consider every pair and check if sum of this pair is a power of 2 or not. Time Complexity of this solution is O(n * n)

An **Efficient Approach:** is to find the largest element from the array say **X** then find the largest element from the rest of the array elements **Y** such that **Y ≤ X** and **X + Y** is a **power of 2**. This is an optimal selection of pair because even if **Y** makes a valid pair with some other element say **Z** then **Z** will be left to pair with an element other than **Y** (if possible) to maximize the number of valid pairs.

## C++

`// C++ implementation of above approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to return the count of valid pairs ` `int` `countPairs(` `int` `a[], ` `int` `n) ` `{ ` ` ` `// Storing occurrences of each element ` ` ` `unordered_map<` `int` `, ` `int` `> mp; ` ` ` `for` `(` `int` `i = 0; i < n; i++) ` ` ` `mp[a[i]]++; ` ` ` ` ` `// Sort the array in deceasing order ` ` ` `sort(a, a + n, greater<` `int` `>()); ` ` ` ` ` `// Start taking largest element each time ` ` ` `int` `count = 0; ` ` ` `for` `(` `int` `i = 0; i < n; i++) { ` ` ` ` ` `// If element has already been paired ` ` ` `if` `(mp[a[i]] < 1) ` ` ` `continue` `; ` ` ` ` ` `// Find the number which is greater than ` ` ` `// a[i] and power of two ` ` ` `int` `cur = 1; ` ` ` `while` `(cur <= a[i]) ` ` ` `cur <<= 1; ` ` ` ` ` `// If there is a number which adds up with a[i] ` ` ` `// to form a power of two ` ` ` `if` `(mp[cur - a[i]]) { ` ` ` ` ` `// Edge case when a[i] and crr - a[i] is same ` ` ` `// and we have only one occurrence of a[i] then ` ` ` `// it cannot be paired ` ` ` `if` `(cur - a[i] == a[i] and mp[a[i]] == 1) ` ` ` `continue` `; ` ` ` ` ` `count++; ` ` ` ` ` `// Remove already paired elements ` ` ` `mp[cur - a[i]]--; ` ` ` `mp[a[i]]--; ` ` ` `} ` ` ` `} ` ` ` ` ` `// Return the count ` ` ` `return` `count; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `a[] = { 3, 11, 14, 5, 13 }; ` ` ` `int` `n = ` `sizeof` `(a) / ` `sizeof` `(a[0]); ` ` ` `cout << countPairs(a, n); ` ` ` `return` `0; ` `} ` |

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## Python3

`# Python3 implementation of above approach ` ` ` `# Function to return the count ` `# of valid pairs ` `def` `countPairs(a, n) : ` ` ` ` ` `# Storing occurrences of each element ` ` ` `mp ` `=` `dict` `.fromkeys(a, ` `0` `) ` ` ` `for` `i ` `in` `range` `(n) : ` ` ` `mp[a[i]] ` `+` `=` `1` ` ` ` ` `# Sort the array in deceasing order ` ` ` `a.sort(reverse ` `=` `True` `) ` ` ` ` ` `# Start taking largest element ` ` ` `# each time ` ` ` `count ` `=` `0` ` ` `for` `i ` `in` `range` `(n) : ` ` ` ` ` `# If element has already been paired ` ` ` `if` `(mp[a[i]] < ` `1` `) : ` ` ` `continue` ` ` ` ` `# Find the number which is greater ` ` ` `# than a[i] and power of two ` ` ` `cur ` `=` `1` ` ` `while` `(cur <` `=` `a[i]) : ` ` ` `cur ` `=` `cur << ` `1` ` ` ` ` `# If there is a number which adds ` ` ` `# up with a[i] to form a power of two ` ` ` `if` `(cur ` `-` `a[i] ` `in` `mp.keys()) : ` ` ` ` ` `# Edge case when a[i] and crr - a[i] ` ` ` `# is same and we have only one occurrence ` ` ` `# of a[i] then it cannot be paired ` ` ` `if` `(cur ` `-` `a[i] ` `=` `=` `a[i] ` `and` `mp[a[i]] ` `=` `=` `1` `) : ` ` ` `continue` ` ` ` ` `count ` `+` `=` `1` ` ` ` ` `# Remove already paired elements ` ` ` `mp[cur ` `-` `a[i]] ` `-` `=` `1` ` ` `mp[a[i]] ` `-` `=` `1` ` ` ` ` `# Return the count ` ` ` `return` `count ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `a ` `=` `[ ` `3` `, ` `11` `, ` `14` `, ` `5` `, ` `13` `] ` ` ` `n ` `=` `len` `(a) ` ` ` `print` `(countPairs(a, n)) ` ` ` `# This code is contributed by Ryuga ` |

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**Output:**

2

Note that the below operation in above code can be done in O(1) time using the last approach discussed in Smallest power of 2 greater than or equal to n

`// Find the number which is greater than ` `// a[i] and power of two ` `int` `cur = 1; ` `while` `(cur <= a[i]) ` ` ` `cur <<= 1; ` |

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After optimizing above expression, time complexity of this solution becomes O(n Log n)

## Recommended Posts:

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- Number of special pairs possible from the given two numbers

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