Split a Numeric String into Fibonacci Sequence

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Given a numeric string S representing a large number, the task is to form a Fibonacci Sequence of at least length 3 from the given string. If such a split is not possible, print -1.

Examples:

Input: S = “5712”
Output: 5 7 12
Explanation:
Since 5 + 7 = 12, the splits {5}, {7}, {12} forms a Fibonacci sequence.

Input: S = “11235813″
Output: 1 1 2 3 5 8 13

Approach:
To solve the problem, the idea is to use Backtracking to find a sequence that follows the conditions of the Fibonacci Sequence.

Follow the steps below to solve the problem:

Initialize a vector seq[] to store the Fibonacci sequence.
Initialize a variable pos which points to the current index of the string S, initially 0.
Iterate over the indices [pos, length – 1]:
- Add the number S[pos: i] to the Fibonacci sequence seq if the length of seq is less than 2 or the current number is equal to the sum of the last two numbers of seq. Recur for the index i + 1 and proceed.
- If the last added number S[pos: i] does not form a Fibonacci sequence and returns false after recursion, then remove it from the seq.
- Otherwise, end the loop and return true as the Fibonacci sequence is formed.
If pos exceeds the length of S, then:
- If the length of the sequence seq is greater than or equal to 3, then a Fibonacci sequence is found, hence return true.
- Otherwise, the Fibonacci sequence is not possible and hence returns false.
Finally, if the length of seq is greater than or equal to 3, then print the numbers in seq as the required Fibonacci sequence or, otherwise print -1.

Below is the illustration of the recursive structure where only one branch is extended to get the result:

Below is the implementation of the above approach:

C++

// C++ program of the above approach
#include <bits/stdc++.h>
using namespace std;
 
#define LL long long
 
// Function that returns true if
// Fibonacci sequence is found
bool splitIntoFibonacciHelper(int pos,
                              string S,
                              vector<int>& seq)
{
    // Base condition:
    // If pos is equal to length of S
    // and seq length is greater than 3
    if (pos == S.length()
        and (seq.size() >= 3)) {
 
        // Return true
        return true;
    }
 
    // Stores current number
    LL num = 0;
 
    for (int i = pos; i < S.length(); i++) {
 
        // Add current digit to num
        num = num * 10 + (S[i] - '0');
 
        // Avoid integer overflow
        if (num > INT_MAX)
            break;
 
        // Avoid leading zeros
        if (S[pos] == '0' and i > pos)
            break;
 
        // If current number is greater
        // than last two number of seq
        if (seq.size() > 2
            and (num > ((LL)seq.back()
                        + (LL)seq[seq.size()
                                  - 2])))
            break;
 
        // If seq length is less
        // 2 or current number is
        // is equal to the last
        // two of the seq
        if (seq.size() < 2
            or (num == ((LL)seq.back()
                        + (LL)seq[seq.size()
                                  - 2]))) {
 
            // Add to the seq
            seq.push_back(num);
 
            // Recur for i+1
            if (splitIntoFibonacciHelper(i + 1,
                                         S, seq))
                return true;
 
            // Remove last added number
            seq.pop_back();
        }
    }
 
    // If no sequence is found
    return false;
}
 
// Function that prints the Fibonacci
// sequence from the split of string S
void splitIntoFibonacci(string S)
{
    // Initialize a vector to
    // store the sequence
    vector<int> seq;
 
    // Call helper function
    splitIntoFibonacciHelper(0, S,
                             seq);
 
    // If sequence length is
    // greater than 3
    if (seq.size() >= 3) {
 
        // Print the sequence
        for (int i : seq)
            cout << i << " ";
    }
 
    // If no sequence is found
    else {
 
        // Print -1
        cout << -1;
    }
}
 
// Driver Code
int main()
{
    // Given String
    string S = "11235813";
 
    // Function Call
    splitIntoFibonacci(S);
    return 0;
}

Java

// Java program of the above approach 
import java.util.*;
 
class GFG{
 
// Function that returns true if
// Fibonacci sequence is found
static boolean splitIntoFibonacciHelper(int pos,
                                        String S,
                              ArrayList<Long> seq)
{
     
    // Base condition:
    // If pos is equal to length of S
    // and seq length is greater than 3
    if (pos == S.length() && (seq.size() >= 3))
    {
 
        // Return true
        return true;
    }
 
    // Stores current number
    long num = 0;
 
    for(int i = pos; i < S.length(); i++)
    {
         
        // Add current digit to num
        num = num * 10 + (S.charAt(i) - '0');
 
        // Avoid integer overflow
        if (num > Integer.MAX_VALUE)
            break;
 
        // Avoid leading zeros
        if (S.charAt(pos) == '0' && i > pos)
            break;
 
        // If current number is greater
        // than last two number of seq
        if (seq.size() > 2 && 
           (num > ((long)seq.get(seq.size() - 1) +
                   (long)seq.get(seq.size() - 2))))
            break;
 
        // If seq length is less
        // 2 or current number is
        // is equal to the last
        // two of the seq
        if (seq.size() < 2 ||
            (num == ((long)seq.get(seq.size() - 1) +
                     (long)seq.get(seq.size() - 2))))
        {
             
            // Add to the seq
            seq.add(num);
 
            // Recur for i+1
            if (splitIntoFibonacciHelper(i + 1,
                                         S, seq))
                return true;
 
            // Remove last added number
            seq.remove(seq.size() - 1);
        }
    }
     
    // If no sequence is found
    return false;
}
 
// Function that prints the Fibonacci
// sequence from the split of string S
static void splitIntoFibonacci(String S)
{
     
    // Initialize a vector to
    // store the sequence
    ArrayList<Long> seq = new ArrayList<>();
 
    // Call helper function
    splitIntoFibonacciHelper(0, S, seq);
 
    // If sequence length is
    // greater than 3
    if (seq.size() >= 3) 
    {
         
        // Print the sequence
        for (int i = 0; i < seq.size(); i++)
            System.out.print(seq.get(i) + " ");
    }
 
    // If no sequence is found
    else
    {
 
        // Print -1
        System.out.print("-1");
    }
}
 
// Driver code
public static void main(String[] args)
{
     
    // Given String
    String S = "11235813";
     
    // Function Call
    splitIntoFibonacci(S);
}
}
 
// This code is contributed by offbeat

Python3

# Python3 program of the above approach
import sys
 
# Function that returns true if
# Fibonacci sequence is found
def splitIntoFibonacciHelper(pos, S, seq):
 
    # Base condition:
    # If pos is equal to length of S
    # and seq length is greater than 3
    if (pos == len(S) and (len(seq) >= 3)):
  
        # Return true
        return True
  
    # Stores current number
    num = 0
     
    for i in range(pos, len(S)):
  
        # Add current digit to num
        num = num * 10 + (ord(S[i]) - ord('0'))
  
        # Avoid integer overflow
        if (num > sys.maxsize):
            break
  
        # Avoid leading zeros
        if (ord(S[pos]) == ord('0') and i > pos):
            break
  
        # If current number is greater
        # than last two number of seq
        if (len(seq) > 2 and
                (num > (seq[-1] +
                        seq[len(seq) - 2]))):
            break
  
        # If seq length is less
        # 2 or current number is
        # is equal to the last
        # two of the seq
        if (len(seq) < 2 or
           (num == (seq[-1] +
                    seq[len(seq) - 2]))):
  
            # Add to the seq
            seq.append(num)
  
            # Recur for i+1
            if (splitIntoFibonacciHelper(
                i + 1, S, seq)):
                return True
  
            # Remove last added number
            seq.pop()
  
    # If no sequence is found
    return False
 
# Function that prints the Fibonacci
# sequence from the split of string S
def splitIntoFibonacci(S):
     
    # Initialize a vector to
    # store the sequence
    seq = []
  
    # Call helper function
    splitIntoFibonacciHelper(0, S, seq)
  
    # If sequence length is
    # greater than 3
    if (len(seq) >= 3):
  
        # Print the sequence
        for i in seq:
            print(i, end = ' ')
             
    # If no sequence is found
    else:
  
        # Print -1
        print(-1, end = '')
         
# Driver Code
if __name__=='__main__':
     
    # Given String
    S = "11235813"
  
    # Function Call
    splitIntoFibonacci(S)
 
# This code is contributed by pratham76

C#

// C# program of the above approach 
using System;
using System.Collections; 
using System.Collections.Generic; 
 
class GFG{
     
// Function that returns true if
// Fibonacci sequence is found
static bool splitIntoFibonacciHelper(int pos,
                                     string S,
                                ArrayList seq)
{
     
    // Base condition:
    // If pos is equal to length of S
    // and seq length is greater than 3
    if (pos == S.Length && (seq.Count >= 3))
    {
 
        // Return true
        return true;
    }
 
    // Stores current number
    long num = 0;
 
    for(int i = pos; i < S.Length; i++)
    {
         
        // Add current digit to num
        num = num * 10 + (S[i] - '0');
 
        // Avoid integer overflow
        if (num > Int64.MaxValue)
            break;
     
        // Avoid leading zeros
        if (S[pos] == '0' && i > pos)
            break;
 
        // If current number is greater
        // than last two number of seq
        if (seq.Count> 2 && 
           (num > ((long)seq[seq.Count - 1] +
                   (long)seq[seq.Count - 2])))
            break;
 
        // If seq length is less
        // 2 or current number is
        // is equal to the last
        // two of the seq
        if (seq.Count < 2 ||
           (num == ((long)seq[seq.Count - 1] +
                    (long)seq[seq.Count - 2])))
        {
             
            // Add to the seq
            seq.Add(num);
 
            // Recur for i+1
            if (splitIntoFibonacciHelper(i + 1,
                                         S, seq))
                return true;
 
            // Remove last added number
            seq.Remove(seq.Count - 1);
        }
    }
     
    // If no sequence is found
    return false;
}
 
// Function that prints the Fibonacci
// sequence from the split of string S
static void splitIntoFibonacci(string S)
{
     
    // Initialize a vector to
    // store the sequence
    ArrayList seq = new ArrayList();
 
    // Call helper function
    splitIntoFibonacciHelper(0, S, seq);
 
    // If sequence length is
    // greater than 3
    if (seq.Count >= 3) 
    {
         
        // Print the sequence
        for(int i = 0; i < seq.Count; i++)
            Console.Write(seq[i] + " ");
    }
 
    // If no sequence is found
    else
    {
         
        // Print -1
        Console.Write("-1");
    }
}
 
// Driver Code
public static void Main(string[] args)
{
     
    // Given String
    string S = "11235813";
     
    // Function call
    splitIntoFibonacci(S);
}
}
 
// This code is contributed by rutvik_56

Javascript

<script>
    // Javascript program of the above approach
     
    // Function that returns true if
    // Fibonacci sequence is found
    function splitIntoFibonacciHelper(pos, S, seq)
    {
 
        // Base condition:
        // If pos is equal to length of S
        // and seq length is greater than 3
        if (pos == S.length && (seq.length >= 3))
        {
 
            // Return true
            return true;
        }
 
        // Stores current number
        let num = 0;
 
        for(let i = pos; i < S.length; i++)
        {
 
            // Add current digit to num
            num = num * 10 + (S[i] - '0');
 
            // Avoid integer overflow
            if (num > Number.MAX_VALUE)
                break;
 
            // Avoid leading zeros
            if (S[pos] == '0' && i > pos)
                break;
 
            // If current number is greater
            // than last two number of seq
            if (seq.length> 2 &&
               (num > (seq[seq.length - 1] +
                       seq[seq.length - 2])))
                break;
 
            // If seq length is less
            // 2 or current number is
            // is equal to the last
            // two of the seq
            if (seq.length < 2 ||
               (num == (seq[seq.length - 1] +
                        seq[seq.length - 2])))
            {
 
                // Add to the seq
                seq.push(num);
 
                // Recur for i+1
                if (splitIntoFibonacciHelper(i + 1, S, seq))
                    return true;
 
                // Remove last added number
                seq.pop();
            }
        }
 
        // If no sequence is found
        return false;
    }
 
    // Function that prints the Fibonacci
    // sequence from the split of string S
    function splitIntoFibonacci(S)
    {
 
        // Initialize a vector to
        // store the sequence
        let seq = [];
 
        // Call helper function
        splitIntoFibonacciHelper(0, S, seq);
 
        // If sequence length is
        // greater than 3
        if (seq.length >= 3)
        {
 
            // Print the sequence
            for(let i = 0; i < seq.length; i++)
                document.write(seq[i] + " ");
        }
 
        // If no sequence is found
        else
        {
 
            // Print -1
            document.write("-1");
        }
    }
     
    // Given String
    let S = "11235813";
      
    // Function call
    splitIntoFibonacci(S);
 
// This code is contributed by suresh07.
</script>

Output:

1 1 2 3 5 8 13

Time Complexity: O(N²)
Space Complexity: O(N)

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