# Pair of fibonacci numbers with a given sum and minimum absolute difference

Given an integer N(less than 10^6), the task is to find a pair of Fibonacci numbers whose sum is equal to the given N and the absolute difference between the chosen pair is minimum.
Print -1 if there is no solution.

Examples:

Input: N = 199
Output: 55 144
Explanation
199 can be represented as sum 55 and 144 which has the minimum difference.

Input: N = 1830
Output: 233 1597
Explanation
1830 can be represented as sum 233 and 1597 which has the minimum difference.

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

Approach: The idea is to use hashing to precompute and store the Fibonacci nodes up to the maximum value to make checking easy and efficient (in O(1) time).

After precomputing the hash:

1. Start a loop from (N / 2) to 1 (to minimize the absolute difference) and check whether the loop counter ‘i’ and ‘N – i’ are both Fibonacci.
2. If they are Fibonacci, then we will print them and break out of the loop.
3. If the number N cannot be represented as the sum of two Fibonacci numbers, then we will print -1.

Below is the implementation of the above approach:

## C++

 `// C++ program to find the pair of ` `// Fibonacci numbers with a given sum ` `// and minimum absolute difference ` ` `  `#include ` `using` `namespace` `std; ` `#define MAX 1000005 ` ` `  `// Hash to store all the ` `// Fibonacci numbers ` `set<``int``> fib; ` ` `  `// Function to generate fibonacci Series ` `// and create hash table ` `// to check Fibonacci numbers ` `void` `fibonacci() ` `{ ` `    ``// Adding the first two Fibonacci ` `    ``// numbers into the Hash set ` `    ``int` `prev = 0, curr = 1, len = 2; ` `    ``fib.insert(prev); ` `    ``fib.insert(curr); ` ` `  `    ``// Computing the remaining Fibonacci ` `    ``// numbers based on the previous ` `    ``// two numbers ` `    ``while` `(len <= MAX) { ` `        ``int` `temp = curr + prev; ` `        ``fib.insert(temp); ` `        ``prev = curr; ` `        ``curr = temp; ` `        ``len++; ` `    ``} ` `} ` ` `  `// Function to find the pair of ` `// Fibonacci numbers with the given ` `// sum and minimum absolute difference ` `void` `findFibonacci(``int` `N) ` `{ ` ` `  `    ``// Start from N/2 such that the ` `    ``// difference between i and ` `    ``// N - i will be minimum ` `    ``for` `(``int` `i = N / 2; i > 1; i--) { ` ` `  `        ``// If both 'i' and 'sum - i' are ` `        ``// fibonacci numbers then print ` `        ``// them and break the loop ` `        ``if` `(fib.find(i) != fib.end() ` `            ``&& fib.find(N - i) != fib.end()) { ` ` `  `            ``cout << i << ``" "` `<< (N - i) << endl; ` `            ``return``; ` `        ``} ` `    ``} ` ` `  `    ``// If there is no Fibonacci pair ` `    ``// possible ` `    ``cout << ``"-1"` `<< endl; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``// Generate the Fibonacci numbers ` `    ``fibonacci(); ` ` `  `    ``int` `sum = 199; ` ` `  `    ``// Find the Fibonacci pair ` `    ``findFibonacci(sum); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to find the pair of ` `// Fibonacci numbers with a given sum ` `// and minimum absolute difference ` `import` `java.util.*; ` `  `  `class` `GFG{ ` `    ``// Hash to store all the ` `    ``// Fibonacci numbers ` `    ``static` `int` `MAX=``1000005``; ` `    ``static`  `Set fib=``new` `HashSet(); ` `     `  `    ``// Function to generate fibonacci Series ` `    ``// and create hash table ` `    ``// to check Fibonacci numbers ` `    ``static` `void` `fibonacci() ` `    ``{ ` `        ``// Adding the first two Fibonacci ` `        ``// numbers into the Hash set ` `        ``int` `prev = ``0``, curr = ``1``, len = ``2``; ` `        ``fib.add(prev); ` `        ``fib.add(curr); ` `     `  `        ``// Computing the remaining Fibonacci ` `        ``// numbers based on the previous ` `        ``// two numbers ` `        ``while` `(len <= MAX) { ` `            ``int` `temp = curr + prev; ` `            ``fib.add(temp); ` `            ``prev = curr; ` `            ``curr = temp; ` `            ``len++; ` `        ``} ` `    ``} ` `     `  `    ``// Function to find the pair of ` `    ``// Fibonacci numbers with the given ` `    ``// sum and minimum absolute difference ` `    ``static` `void` `findFibonacci(``int` `N) ` `    ``{ ` `     `  `        ``// Start from N/2 such that the ` `        ``// difference between i and ` `        ``// N - i will be minimum ` `        ``for` `(``int` `i = N / ``2``; i > ``1``; i--) { ` `     `  `            ``// If both 'i' and 'sum - i' are ` `            ``// fibonacci numbers then print ` `            ``// them and break the loop ` `            ``if` `(fib.contains(i)  && fib.contains(N - i)) { ` `     `  `                ``System.out.println(i+``" "``+(N - i)); ` `                ``return``; ` `            ``} ` `        ``} ` `     `  `        ``// If there is no Fibonacci pair ` `        ``// possible ` `        ``System.out.println(``"-1"``); ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``// Generate the Fibonacci numbers ` `        ``fibonacci(); ` `     `  `        ``int` `sum = ``199``; ` `     `  `        ``// Find the Fibonacci pair ` `        ``findFibonacci(sum); ` `     `  `         `  `    ``} ` `} ` ` `  ` `  `// This code is contributed by AbhiThakur `

## C#

 `// C# program to find the pair of ` `// Fibonacci numbers with a given sum ` `// and minimum absolute difference ` `using` `System; ` `using` `System.Collections.Generic; ` ` `  `class` `GFG{ ` `    ``// Hash to store all the ` `    ``// Fibonacci numbers ` `    ``static` `int` `MAX=100005; ` `    ``static`  `HashSet<``int``> fib=``new` `HashSet<``int``>(); ` `      `  `    ``// Function to generate fibonacci Series ` `    ``// and create hash table ` `    ``// to check Fibonacci numbers ` `    ``static` `void` `fibonacci() ` `    ``{ ` `        ``// Adding the first two Fibonacci ` `        ``// numbers into the Hash set ` `        ``int` `prev = 0, curr = 1, len = 2; ` `        ``fib.Add(prev); ` `        ``fib.Add(curr); ` `      `  `        ``// Computing the remaining Fibonacci ` `        ``// numbers based on the previous ` `        ``// two numbers ` `        ``while` `(len <= MAX) { ` `            ``int` `temp = curr + prev; ` `            ``fib.Add(temp); ` `            ``prev = curr; ` `            ``curr = temp; ` `            ``len++; ` `        ``} ` `    ``} ` `      `  `    ``// Function to find the pair of ` `    ``// Fibonacci numbers with the given ` `    ``// sum and minimum absolute difference ` `    ``static` `void` `findFibonacci(``int` `N) ` `    ``{ ` `      `  `        ``// Start from N/2 such that the ` `        ``// difference between i and ` `        ``// N - i will be minimum ` `        ``for` `(``int` `i = N / 2; i > 1; i--) { ` `      `  `            ``// If both 'i' and 'sum - i' are ` `            ``// fibonacci numbers then print ` `            ``// them and break the loop ` `            ``if` `(fib.Contains(i)  && fib.Contains(N - i)) { ` `      `  `                ``Console.WriteLine(i+``" "``+(N - i)); ` `                ``return``; ` `            ``} ` `        ``} ` `      `  `        ``// If there is no Fibonacci pair ` `        ``// possible ` `        ``Console.WriteLine(``"-1"``); ` `    ``} ` `      `  `    ``// Driver code ` `    ``public` `static` `void` `Main(String []args) ` `    ``{ ` `        ``// Generate the Fibonacci numbers ` `        ``fibonacci(); ` `      `  `        ``int` `sum = 199; ` `      `  `        ``// Find the Fibonacci pair ` `        ``findFibonacci(sum); ` `    ``} ` `} ` ` `  `// This code is contributed by sapnasingh4991 `

Output:

```55 144
```

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