Minimize the number of steps required to reach the end of the array | Set 2

Given an integer array arr[] of length N consisting of positive integers, the task is to minimize the number of steps required to reach the arr[N – 1] starting from arr[0]. At a given step if we are at index i we can go to index i – arr[i] or i + arr[i] given we have not visited those indexes before. Also, we cannot go outside the bounds of the array. Print -1 if there is no possible way.

Examples:

Input: arr[] = {1, 1, 1}
Output: 2
The path will be 0 -> 1 -> 2.

Input: arr[] = {2, 1}
Output: -1

Approach: We have already discussed a dynamic programming based approach in this article which has a time complexity of O(n * 2n).
Here we’re going to discuss a BFS based solution:



  1. This problem can be visualized as a directed graph where ith cell is connected with cells i + arr[i] and i – arr[i].
  2. And the graph is un-weighted.

Due to above, BFS can be used to find the shortest path between 0th and the (N – 1)th index. We will use the following algorithm:

  1. Push index 0 in a queue.
  2. Push all the adjacent cells to 0 in the queue.
  3. Repeat the above steps i.e. traverse all the elements in the queue individually again if they have not been visited/traversed before.
  4. Repeat till we don’t reach the index N – 1.
  5. The depth of this traversal will give the minimum steps required to reach the end.

Remember to mark a cell visited after it has been traversed. For this, we will use a boolean array.

Below is the implementation of the above approach:

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the minimum steps
// required to reach the end
// of the given array
int minSteps(int arr[], int n)
{
    // Array to determine whether
    // a cell has been visited before
    bool v[n] = { 0 };
  
    // Queue for bfs
    queue<int> q;
  
    // Push the source i.e. index 0
    q.push(0);
  
    // Variable to store
    // the depth of search
    int depth = 0;
  
    // BFS algorithm
    while (q.size() != 0) {
  
        // Current queue size
        int x = q.size();
        while (x--) {
  
            // Top-most element of queue
            int i = q.front();
            q.pop();
  
            // Base case
            if (v[i])
                continue;
  
            // If we reach the destination
            // i.e. index (n - 1)
            if (i == n - 1)
                return depth;
  
            // Marking the cell visited
            v[i] = 1;
  
            // Pushing the adjacent nodes
            // i.e. indices reachable
            // from the current index
            if (i + arr[i] < n)
                q.push(i + arr[i]);
            if (i - arr[i] >= 0)
                q.push(i - arr[i]);
        }
        depth++;
    }
  
    return -1;
}
  
// Driver code
int main()
{
    int arr[] = { 1, 1, 1, 1, 1, 1 };
    int n = sizeof(arr) / sizeof(int);
  
    cout << minSteps(arr, n);
  
    return 0;
}
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// Java implementation of the approach 
import java.util.*;
  
class GFG 
{
  
// Function to return the minimum steps
// required to reach the end
// of the given array
static int minSteps(int arr[], int n)
{
    // Array to determine whether
    // a cell has been visited before
    boolean[] v = new boolean[n];
  
    // Queue for bfs
    Queue<Integer> q = new LinkedList<>();
  
    // Push the source i.e. index 0
    q.add(0);
  
    // Variable to store
    // the depth of search
    int depth = 0;
  
    // BFS algorithm
    while (q.size() > 0)
    {
  
        // Current queue size
        int x = q.size();
        while (x-- > 0)
        {
  
            // Top-most element of queue
            int i = q.peek();
            q.poll();
  
            // Base case
            if (v[i])
                continue;
  
            // If we reach the destination
            // i.e. index (n - 1)
            if (i == n - 1)
                return depth;
  
            // Marking the cell visited
            v[i] = true;
  
            // Pushing the adjacent nodes
            // i.e. indices reachable
            // from the current index
            if (i + arr[i] < n)
                q.add(i + arr[i]);
            if (i - arr[i] >= 0)
                q.add(i - arr[i]);
        }
        depth++;
    }
  
    return -1;
}
  
// Driver code
public static void main(String[] args) 
{
    int arr[] = { 1, 1, 1, 1, 1, 1 };
    int n = arr.length;
    System.out.println(minSteps(arr, n));
}
}
  
/* This code contributed by PrinciRaj1992 */
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# Python 3 implementation of the approach
  
# Function to return the minimum steps
# required to reach the end
# of the given array
def minSteps(arr,n):
      
    # Array to determine whether
    # a cell has been visited before
    v = [0 for i in range(n)]
  
    # Queue for bfs
    q = []
  
    # Push the source i.e. index 0
    q.append(0)
  
    # Variable to store
    # the depth of search
    depth = 0
  
    # BFS algorithm
    while (len(q) != 0):
        # Current queue size
        x = len(q)
        while (x >= 1):
            # Top-most element of queue
            i = q[0]
            q.remove(i)
              
            x -= 1
  
            # Base case
            if (v[i]):
                continue;
  
            # If we reach the destination
            # i.e. index (n - 1)
            if (i == n - 1):
                return depth
  
            # Marking the cell visited
            v[i] = 1
  
            # Pushing the adjacent nodes
            # i.e. indices reachable
            # from the current index
            if (i + arr[i] < n):
                q.append(i + arr[i])
            if (i - arr[i] >= 0):
                q.append(i - arr[i])
                  
          
        depth += 1
  
    return -1
  
# Driver code
if __name__ == '__main__':
    arr = [1, 1, 1, 1, 1, 1]
    n = len(arr)
  
    print(minSteps(arr, n))
  
# This code is contributed by
# Surendra_Gangwar
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// A C# implementation of the approach 
using System;
using System.Collections.Generic;
  
class GFG 
  
// Function to return the minimum steps 
// required to reach the end 
// of the given array 
static int minSteps(int []arr, int n) 
    // Array to determine whether 
    // a cell has been visited before 
    Boolean[] v = new Boolean[n]; 
  
    // Queue for bfs 
    Queue<int> q = new Queue<int>(); 
  
    // Push the source i.e. index 0 
    q.Enqueue(0); 
  
    // Variable to store 
    // the depth of search 
    int depth = 0; 
  
    // BFS algorithm 
    while (q.Count > 0) 
    
  
        // Current queue size 
        int x = q.Count; 
        while (x-- > 0) 
        
  
            // Top-most element of queue 
            int i = q.Peek(); 
            q.Dequeue(); 
  
            // Base case 
            if (v[i]) 
                continue
  
            // If we reach the destination 
            // i.e. index (n - 1) 
            if (i == n - 1) 
                return depth; 
  
            // Marking the cell visited 
            v[i] = true
  
            // Pushing the adjacent nodes 
            // i.e. indices reachable 
            // from the current index 
            if (i + arr[i] < n) 
                q.Enqueue(i + arr[i]); 
            if (i - arr[i] >= 0) 
                q.Enqueue(i - arr[i]); 
        
        depth++; 
    
  
    return -1; 
  
// Driver code 
public static void Main(String[] args) 
    int []arr = { 1, 1, 1, 1, 1, 1 }; 
    int n = arr.Length; 
    Console.WriteLine(minSteps(arr, n)); 
  
// This code contributed by Rajput-Ji
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Output:
5

Time complexity: O(N)

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