# Number of pairs whose sum is a power of 2

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 â‰¤ 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.

**Implementation:**

## 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 decreasing 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;` `}` |

## Java

`// Java implementation of above approach` `import` `java.util.TreeMap;` `class` `Count` `{` ` ` `// Function to return the count of valid pairs` ` ` `static` `int` `countPairs(` `int` `[] a, ` `int` `n)` ` ` `{` ` ` `// To keep the element in sorted order` ` ` `TreeMap<Integer, Integer> map = ` `new` `TreeMap<>();` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++)` ` ` `{` ` ` `map.put(a[i], ` `1` `);` ` ` `}` ` ` ` ` `// Start taking largest element each time` ` ` `int` `count = ` `0` `;` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++)` ` ` `{` ` ` `// If element has already been paired` ` ` `if` `(map.get(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` `(map.containsKey(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] && map.get(a[i]) == ` `1` `)` ` ` `continue` `;` ` ` `count++;` ` ` `// Remove already paired elements` ` ` `map.put(cur - a[i], map.get(cur - a[i]) - ` `1` `);` ` ` `map.put(a[i], map.get(a[i]) - ` `1` `);` ` ` `}` ` ` `}` ` ` `// Return the count` ` ` `return` `count;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{` ` ` `int` `[] a = { ` `3` `, ` `11` `, ` `14` `, ` `5` `, ` `13` `};` ` ` `int` `n = a.length;` ` ` `System.out.println(countPairs(a, n));` ` ` `}` `}` `// This code is contributed by Vivekkumar Singh` |

## 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 decreasing 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` |

## C#

`// C# implementation of above approach` `using` `System;` `using` `System.Collections.Generic; ` `class` `GFG` `{` ` ` `// Function to return the count of valid pairs` ` ` `static` `int` `countPairs(` `int` `[] a, ` `int` `n)` ` ` `{` ` ` `// To keep the element in sorted order` ` ` `Dictionary<` `int` `, ` ` ` `int` `> map = ` `new` `Dictionary<` `int` `,` ` ` `int` `>();` ` ` `for` `(` `int` `i = 0; i < n; i++)` ` ` `{` ` ` `if` `(!map.ContainsKey(a[i]))` ` ` `map.Add(a[i], 1);` ` ` `}` ` ` ` ` `// Start taking largest element each time` ` ` `int` `count = 0;` ` ` `for` `(` `int` `i = 0; i < n; i++)` ` ` `{` ` ` `// If element has already been paired` ` ` `if` `(map[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` `(map.ContainsKey(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] && map[a[i]] == 1)` ` ` `continue` `;` ` ` `count++;` ` ` `// Remove already paired elements` ` ` `map[cur - a[i]] = map[cur - a[i]] - 1;` ` ` `map[a[i]] = map[a[i]] - 1;` ` ` `}` ` ` `}` ` ` ` ` `// Return the count` ` ` `return` `count;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `Main(String[] args) ` ` ` `{` ` ` `int` `[] a = { 3, 11, 14, 5, 13 };` ` ` `int` `n = a.Length;` ` ` `Console.WriteLine(countPairs(a, n));` ` ` `}` `}` `// This code is contributed by Princi Singh` |

## Javascript

`<script>` `// JavaScript Program to implement` `// the above approach` ` ` `// Function to return the count of valid pairs` ` ` `function` `countPairs(a, n)` ` ` `{` ` ` `// To keep the element in sorted order` ` ` `let map = ` `new` `Map();` ` ` `for` `(let i = 0; i < n; i++)` ` ` `{` ` ` `map.set(a[i], 1);` ` ` `}` ` ` ` ` `// Start taking largest element each time` ` ` `let count = 0;` ` ` `for` `(let i = 0; i < n; i++)` ` ` `{` ` ` `// If element has already been paired` ` ` `if` `(map.get(a[i]) < 1)` ` ` `continue` `;` ` ` `// Find the number which is greater than` ` ` `// a[i] and power of two` ` ` `let 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` `(map.has(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] && map.get(a[i]) == 1)` ` ` `continue` `;` ` ` `count++;` ` ` `// Remove already paired elements` ` ` `map.set(cur - a[i], map.get(cur - a[i]) - 1);` ` ` `map.set(a[i], map.get(a[i]) - 1);` ` ` `}` ` ` `}` ` ` `// Return the count` ` ` `return` `count;` ` ` `}` `// Driver Code` ` ` `let a = [ 3, 11, 14, 5, 13 ];` ` ` `let n = a.length;` ` ` `document.write(countPairs(a, n));` `</script>` |

**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

## C

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

## Javascript

`// JavaScript program to find the ` `// number which is greater than` `// a[i] and power of two` `var` `cur = 1;` `while` `(cur <= a[i])` ` ` `cur <<= 1;` `//This code is contributed by phasing17` |

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