Minimum number of jumps to reach end (O(n) solution)

Software Development
#1

Hello Everyone,

Given an array of integers where each element represents the max number of steps that can be made forward from that element. Write a function to return the minimum number of jumps to reach the end of the array (starting from the first element). If an element is 0, then we cannot move through that element. If we can’t reach the end, return -1.
Examples:

Input: arr[] = {1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9} Output: 3 (1-> 3 → 8 → 9) Explanation: Jump from 1st element to 2nd element as there is only 1 step, now there are three options 5, 8 or 9. If 8 or 9 is chosen then the end node 9 can be reached. So 3 jumps are made. Input: arr[] = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1} Output: 10 Explanation: In every step a jump is needed so the count of jumps is 10.

In this post, its O(n) solution will be discussed.

Implementation:
Variables to be used:

  1. maxReach The variable maxReach stores at all time the maximal reachable index in the array.
  2. step The variable step stores the number of steps we can still take(and is initialized with value at index 0, i.e. initial number of steps)
  3. jump jump stores the amount of jumps necessary to reach that maximal reachable position.

Given array arr = 1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9

  • maxReach = arr[0]; // arr[0] = 1, so the maximum index we can reach at the moment is 1.
    step = arr[0]; // arr[0] = 1, the amount of steps we can still take is also 1.
    jump = 1; // we will always need to take at least one jump.
  • Now, starting iteration from index 1, the above values are updated as follows:
    1. First we test whether we have reached the end of the array, in that case we just need to return the jump variable.

if (i == arr.length - 1) return jump;

  1. Next we update the maxReach. This is equal to the maximum of maxReach and i+arr[i](the number of steps we can take from the current position).

maxReach = Math.max(maxReach, i+arr[i]);

  1. We used up a step to get to the current index, so steps has to be decreased.

step–;

  1. If no more steps are remaining (i.e. steps=0, then we must have used a jump. Therefore increase jump. Since we know that it is possible somehow to reach maxReach, we again initialize the steps to the number of steps to reach maxReach from position i. But before re-initializing step, we also check whether a step is becoming zero or negative. In this case, It is not possible to reach further.

if (step == 0) { jump++; if(i>=maxReach) return -1; step = maxReach - i; }

// C++ program to count Minimum number

// of jumps to reach end

#include <bits/stdc++.h>

using namespace std;

int max( int x, int y)

{

return (x > y) ? x : y;

}

// Returns minimum number of jumps

// to reach arr[n-1] from arr[0]

int minJumps( int arr[], int n)

{

// The number of jumps needed to

// reach the starting index is 0

if (n <= 1)

return 0;

// Return -1 if not possible to jump

if (arr[0] == 0)

return -1;

// initialization

// stores all time the maximal

// reachable index in the array.

int maxReach = arr[0];

// stores the number of steps

// we can still take

int step = arr[0];

// stores the number of jumps

// necessary to reach that maximal

// reachable position.

int jump = 1;

// Start traversing array

int i = 1;

for (i = 1; i < n; i++) {

// Check if we have reached the end of the array

if (i == n - 1)

return jump;

// updating maxReach

maxReach = max(maxReach, i + arr[i]);

// we use a step to get to the current index

step--;

// If no further steps left

if (step == 0) {

// we must have used a jump

jump++;

// Check if the current index/position or lesser index

// is the maximum reach point from the previous indexes

if (i >= maxReach)

return -1;

// re-initialize the steps to the amount

// of steps to reach maxReach from position i.

step = maxReach - i;

}

}

return -1;

}

// Driver program to test above function

int main()

{

int arr[] = { 1, 3, 5, 8, 9, 2, 6, 7, 6, 8, 9 };

int size = sizeof (arr) / sizeof ( int );

// Calling the minJumps function

cout << ( "Minimum number of jumps to reach end is %d " ,

minJumps(arr, size));

return 0;

}

Output

3

Complexity Analysis:

  • Time complexity: O(n).
    Only one traversal of the array is needed.
  • Auxiliary Space: O(1).
    There is no space required.