Count number of ways to jump to reach end

3.3

Given an array of numbers where each element represents the max number of jumps that can be made forward from that element. For each array element, count number of ways jumps can be made from that element to reach the end of the array. If an element is 0, then move cannot be made through that element. The element that cannot reach to the end should have a count “-1”.

Examples:

Input : {3, 2, 0, 1}
Output : 2 1 -1 0
For 3 number of steps or jumps that 
can be taken are 1, 2 or 3. The different ways are:
3 -> 2 -> 1
3 -> 1

For 2 number of steps or jumps that 
can be taken are 1, or 2. The different ways are:
2 -> 1

For 0 number of steps or jumps that 
can be taken are 0. 
One cannot move forward from this point.

For 1 number of steps or jumps that 
can be taken are 1. But the element is at
the end so no jump is required.

Input : {1, 3, 5, 8, 9, 1, 0, 7, 6, 8, 9}
Output : 52 52 28 16 8 -1 -1 4 2 1 0

This problem is a variation of Minimum number of jumps to reach end(Method 3). Here we need to count all ways to reach end from every cell.

The solution is a modified version of the solution to the problem of Minimum number of jumps to reach end(Method 3).
This problem aims to count the different ways to jump from each element so as to reach to the end. For each element the count is being calculated by adding the counts of all those forward elements that can reach to the end and to which the current element could reach + 1(if the element can directly reach to the end).

Algorithm:

countWays(arr, n)
    Initialize array count_jump[n] = {0}

    count_jump[n-1] = 0
    for i = n-2 to 0
        if arr[i] >= (n-i-1)
         count_jump[i]++
        for j=i+1; j < n-1 && j <= arr[i]+i; i++
          if count_jump[j] != -1
             count_jump[i] += count_jump[j]
        if count_jump[i] == 0
         count_jump[i] = -1

    for i = 0 to n-1
        print count_jump[i]
// C++ implementation to count number
// of ways to jump to reach end
#include <bits/stdc++.h>
using namespace std;

// function to count ways to jump to
// reach end for each array element
void countWaysToJump(int arr[], int n)
{
    // count_jump[i] store number of ways
    // arr[i] can reach to the end
    int count_jump[n];
    memset(count_jump, 0, sizeof(count_jump));

    // Last element does not require to jump.
    // Count ways to jump for remaining
    // elements
    for (int i=n-2; i>=0; i--)
    {
        // if the element can directly
        // jump to the end
        if (arr[i] >= n - i - 1)
            count_jump[i]++;

        // add the count of all the elements
        // that can reach to end and arr[i] can
        // reach to them
        for (int j=i+1; j < n-1 && j <= arr[i] + i; j++)

            // if element can reach to end then add
            // its count to count_jump[i]
            if (count_jump[j] != -1)
                 count_jump[i] += count_jump[j];

        // if arr[i] cannot reach to the end
        if (count_jump[i] == 0)
            count_jump[i] = -1;
    }

    // print count_jump for each
    // array element
    for (int ai=0; i<n; i++)
        cout << count_jump[i] << " ";
}

// Driver program to test above
int main()
{
    int arr[] = {1, 3, 5, 8, 9, 1, 0, 7, 6, 8, 9};
    int n = sizeof(arr) / sizeof(arr[0]);
    countWaysToJump(arr, n);
    return 0;
}

Output:

52 52 28 16 8 -1 -1 4 2 1 0

Time Complexity: O(n2) in worst case.

This article is contributed by Ayush Jauhari. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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