Minimum number of Fibonacci jumps to reach end

Given an array of 0s and 1s, If any particular index i has value 1 then it is a safe index and if the value is 0 then it is an unsafe index. A man is standing at index -1(source) can only land on a safe index and he has to reach the Nth index (last position). At each jump, the man can only travel a distance equal to any Fibonacci Number. You have to minimize the number of steps, provided man can jump only in forward direction.

Note: First few Fibonacci numbers are – 0, 1, 1, 2, 3, 5, 8, 13, 21….

Examples:



Input: arr[]= {0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0}
Output: 3

The person will jump from:
1) index -1 to index 4 (a distance of jump = 5)
2) index 4 to index 6 (a distance of jump = 2)
3) index 6 to the destination (a distance of jump = 5)

Input: arr[]= {0, 0}
Output: 1

The person will jump from:
1) index -1 to destination (a distance of jump = 3)

Approach:

  • First, we will create an array fib[] for storing fibonacci numbers.
  • Then, we will create a DP array and initialize it with INT_MAX and the 0th index DP[0] = 0 and will move in the same way as Coin Change Problem with minimum number of coins.
  • The DP definition and recurrence is as followed: 
     
 DP[i] = min( DP[i], 1 + DP[i-fib[j]] )

where i is the current index and j denotes
the jth fibonacci number of which the
jump is possible
  • Here DP[i] denotes the minimum steps required to reach index i considering all Fibonacci numbers.

Below is the implementation of the approach:

C++

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// A Dynamic Programming based
// C++ program to find minimum
// number of jumps to reach
// Destination
#include <bits/stdc++.h>
using namespace std;
#define MAX 1e9
  
// Function that returns the min
// number of jump to reach the
// destination
int minJumps(int arr[], int N)
{
    // We consider only those Fibonacci
    // numbers which are less than n,
    // where we can consider fib[30]
    // to be the upper bound as this
    // will cross 10^5
    int fib[30];
    fib[0] = 0;
    fib[1] = 1;
    for (int i = 2; i < 30; i++)
        fib[i] = fib[i - 1] + fib[i - 2];
  
    // DP[i] will be storing the minimum
    // number of jumps required for
    // the position i. So DP[N+1] will
    // have the result we consider 0
    // as source and N+1 as the destination
    int DP[N + 2];
  
    // Base case (Steps to reach source is)
    DP[0] = 0;
  
    // Initialize all table values as Infinite
    for (int i = 1; i <= N + 1; i++)
        DP[i] = MAX;
  
    // Compute minimum jumps required
    // considering each Fibonacci
    // numbers one by one.
  
    // Go through each positions
    // till destination.
    for (int i = 1; i <= N + 1; i++) {
  
        // Calculate the minimum of that
        // position if all the Fibonacci
        // numbers are considered
        for (int j = 1; j < 30; j++) {
  
            // If the position is safe
            // or the position is the
            // destination then only we
            // calculate the minimum
            // otherwise the cost is
            // MAX as default
            if ((arr[i - 1] == 1
                 || i == N + 1)
                && i - fib[j] >= 0)
                DP[i] = min(DP[i],
                            1 + DP[i - fib[j]]);
        }
    }
  
    // -1 denotes if there is
    // no path possible
    if (DP[N + 1] != MAX)
        return DP[N + 1];
    else
        return -1;
}
  
// Driver program to test above function
int main()
{
    int arr[] = { 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    cout << minJumps(arr, n);
    return 0;
}

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Java














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// A Dynamic Programming based  


// Java program to find minimum  


// number of jumps to reach  


// Destination 


import java.util.*;  


import java.lang.*;  


  


class GFG  


{ 


      


// Function that returns the min  


// number of jump to reach the  


// destination 


public static int minJumps(int arr[], int N)  


{ 


    int MAX = 1000000; 


      


    // We consider only those Fibonacci  


    // numbers which are less than n,  


    // where we can consider fib[30]  


    // to be the upper bound as this  


    // will cross 10^5  


    int[] fib = new int[30];  


    fib[0] = 0;  


    fib[1] = 1; 


      


    for (int i = 2; i < 30; i++)  


    fib[i] = fib[i - 1] + fib[i - 2];  


      


    // DP[i] will be storing the minimum  


    // number of jumps required for  


    // the position i. So DP[N+1] will  


    // have the result we consider 0  


    // as source and N+1 as the destination  


    int[] DP = new int[N + 2];  


      


    // Base case (Steps to reach source is)  


    DP[0] = 0;  


  


    // Initialize all table values as Infinite  


    for (int i = 1; i <= N + 1; i++)  


        DP[i] = MAX;  


  


    // Compute minimum jumps required  


    // considering each Fibonacci  


    // numbers one by one.  


  


    // Go through each positions  


    // till destination.  


    for (int i = 1; i <= N + 1; i++) 


    {  


  


        // Calculate the minimum of that  


        // position if all the Fibonacci  


        // numbers are considered  


        for (int j = 1; j < 30; j++)  


        {  


  


            // If the position is safe  


            // or the position is the  


            // destination then only we  


            // calculate the minimum  


            // otherwise the cost is  


            // MAX as default  


            if ((i == N + 1 ||  


                 arr[i - 1] == 1) && 


                 i - fib[j] >= 0)  


                DP[i] = Math.min(DP[i], 1 + 


                                 DP[i - fib[j]]);  


        }  


    }  


  


    // -1 denotes if there is  


    // no path possible  


    if (DP[N + 1] != MAX)  


        return DP[N + 1];  


    else


        return -1;  


}  


  


// Driver Code  


public static void main (String[] args)  


{  


    int[] arr = new int[]{ 0, 0, 0, 1, 1, 0, 


                              1, 0, 0, 0, 0 };  


    int n = 11;  


    int ans = minJumps(arr, n);  


    System.out.println(ans);  


} 


} 


  


// This code is contributed by Mehul 



















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Python3














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# A Dynamic Programming based Python3 program  


# to find minimum number of jumps to reach 


# Destination 


MAX = 1e9


  


# Function that returns the min number  


# of jump to reach the destination 


def minJumps(arr, N): 


      


    # We consider only those Fibonacci 


    # numbers which are less than n, 


    # where we can consider fib[30] 


    # to be the upper bound as this 


    # will cross 10^5 


    fib = [0 for i in range(30)] 


    fib[0] = 0


    fib[1] = 1


    for i in range(2, 30): 


        fib[i] = fib[i - 1] + fib[i - 2] 


  


    # DP[i] will be storing the minimum 


    # number of jumps required for 


    # the position i. So DP[N+1] will 


    # have the result we consider 0 


    # as source and N+1 as the destination 


    DP = [0 for i in range(N + 2)] 


  


    # Base case (Steps to reach source is) 


    DP[0] = 0


  


    # Initialize all table values as Infinite 


    for i in range(1, N + 2): 


        DP[i] = MAX


  


    # Compute minimum jumps required 


    # considering each Fibonacci 


    # numbers one by one. 


  


    # Go through each positions 


    # till destination. 


    for i in range(1, N + 2): 


  


        # Calculate the minimum of that 


        # position if all the Fibonacci 


        # numbers are considered 


        for j in range(1, 30): 


  


            # If the position is safe or the  


            # position is the destination then  


            # only we calculate the minimum 


            # otherwise the cost is MAX as default 


            if ((arr[i - 1] == 1 or i == N + 1) and 


                                    i - fib[j] >= 0): 


                DP[i] = min(DP[i], 1 + DP[i - fib[j]]) 


  


    # -1 denotes if there is 


    # no path possible 


    if (DP[N + 1] != MAX): 


        return DP[N + 1] 


    else: 


        return -1


  


# Driver Code 


arr = [0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0] 


n = len(arr) 


  


print(minJumps(arr, n - 1)) 


  


# This code is contributed by Mohit Kumar 



















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C#














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// A Dynamic Programming based  


// C# program to find minimum  


// number of jumps to reach  


// Destination 


using System; 


using System.Collections.Generic; 


  


class GFG  


{ 


      


// Function that returns the min  


// number of jump to reach the  


// destination 


public static int minJumps(int []arr, int N)  


{ 


    int MAX = 1000000; 


      


    // We consider only those Fibonacci  


    // numbers which are less than n,  


    // where we can consider fib[30]  


    // to be the upper bound as this  


    // will cross 10^5  


    int[] fib = new int[30];  


    fib[0] = 0;  


    fib[1] = 1; 


      


    for (int i = 2; i < 30; i++)  


    fib[i] = fib[i - 1] + fib[i - 2];  


      


    // DP[i] will be storing the minimum  


    // number of jumps required for  


    // the position i. So DP[N+1] will  


    // have the result we consider 0  


    // as source and N+1 as the destination  


    int[] DP = new int[N + 2];  


      


    // Base case (Steps to reach source is)  


    DP[0] = 0;  


  


    // Initialize all table values as Infinite  


    for (int i = 1; i <= N + 1; i++)  


        DP[i] = MAX;  


  


    // Compute minimum jumps required  


    // considering each Fibonacci  


    // numbers one by one.  


  


    // Go through each positions  


    // till destination.  


    for (int i = 1; i <= N + 1; i++) 


    {  


  


        // Calculate the minimum of that  


        // position if all the Fibonacci  


        // numbers are considered  


        for (int j = 1; j < 30; j++)  


        {  


  


            // If the position is safe  


            // or the position is the  


            // destination then only we  


            // calculate the minimum  


            // otherwise the cost is  


            // MAX as default  


            if ((i == N + 1 ||  


                arr[i - 1] == 1) && 


                i - fib[j] >= 0)  


                DP[i] = Math.Min(DP[i], 1 + 


                                 DP[i - fib[j]]);  


        }  


    }  


  


    // -1 denotes if there is  


    // no path possible  


    if (DP[N + 1] != MAX)  


        return DP[N + 1];  


    else


        return -1;  


}  


  


// Driver Code  


public static void Main (String[] args)  


{  


    int[] arr = new int[]{ 0, 0, 0, 1, 1, 0, 


                              1, 0, 0, 0, 0 };  


    int n = 11;  


    int ans = minJumps(arr, n);  


    Console.WriteLine(ans);  


} 


} 


  


// This code is contributed by Princi Singh 



















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Output:


3





Time Complexity: O(Nlog(N))





    

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