Modify array by replacing every array element with minimum possible value of arr[j] + |j – i|

Given an array arr[] of size N, the task is to find a value for each index such that the value at index i is arr[j] + |j – i| where 1 ? j ? N, the task is to find the minimum value for each index from 1 to N.

Example:

Input: N = 5, arr[] = {1, 4, 2, 5, 3}
Output: {1, 2, 2, 3, 3}
Explanation: 
arr[0] = arr[0] + |0-0| = 1
arr[1] = arr[0] + |0-1| = 2
arr[2] = arr[2] + |2-2| = 2
arr[3] = arr[2] + |2-3| = 3
arr[4] = arr[4] + |4-4| = 3
The output array will give minimum value at every i^th position.

Input: N = 4, arr[] = {1, 2, 3, 4}
Output: {1, 2, 3, 4} 

Naive Approach: The idea is to use two nested for loops for traversing the array and for each i^th index, find and print the minimum value of arr[j] + |i-j|. 

Implementation :

C++

#include <bits/stdc++.h>
using namespace std;

// function that prints the minimum possible
// value of arr[j] + |j - i|
void minAtEachIndex(int n, int arr[]) {
    // loop to modify every element
    for(int i = 0; i < n; i++) {
        // to get the minimum value for current ith element
        int minValue = INT_MAX;
        // to find the minimum value in range [1,N-1]
        for(int j = 0; j < n; j++) {
            // compute value and update minimum value
            int val = arr[j] + abs(i-j);
            if(val < minValue) {
                minValue = val;
            }
        }
        // print minimum value
        cout << minValue << " ";
    }
    return;
}

//Driver code
int main() {
    int arr[] = {1, 4, 2, 5, 3};
    int n = sizeof(arr) / sizeof(arr[0]);
    // function call
    minAtEachIndex(n, arr);
    return 0;
}

// this code is contributed by bhardwajji

Java

import java.util.*;

public class Main {
// function that prints the minimum possible
// value of arr[j] + |j - i|
public static void minAtEachIndex(int n, int[] arr) {
// loop to modify every element
for (int i = 0; i < n; i++) {
// to get the minimum value for current ith element
int minValue = Integer.MAX_VALUE;
// to find the minimum value in range [1,N-1]
for (int j = 0; j < n; j++) {
// compute value and update minimum value
int val = arr[j] + Math.abs(i - j);
if (val < minValue) {
minValue = val;
}
}
// print minimum value
System.out.print(minValue + " ");
}
return;
}
// Driver code
public static void main(String[] args) {
    int arr[] = {1, 4, 2, 5, 3};
    int n = arr.length;
    // function call
    minAtEachIndex(n, arr);
}
}

Python3

import sys

# function that prints the minimum possible
# value of arr[j] + |j - i|
def minAtEachIndex(n, arr):

    # loop to modify every element
    for i in range(n):

        # to get the minimum value for current ith element
        minValue = sys.maxsize

        # to find the minimum value in range [1,N-1]
        for j in range(n):

            # compute value and update minimum value
            val = arr[j] + abs(i-j)
            if val < minValue:
                minValue = val

        # print minimum value
        print(minValue, end=' ')
    return

# Driver code
if __name__ == '__main__':
    arr = [1, 4, 2, 5, 3]
    n = len(arr)

    # function call
    minAtEachIndex(n, arr)

C#

using System;

class Program {
    // function that prints the minimum possible
    // value of arr[j] + |j - i|
    static void minAtEachIndex(int n, int[] arr) {
        // loop to modify every element
        for(int i = 0; i < n; i++) {
            // to get the minimum value for current ith element
            int minValue = int.MaxValue;
            // to find the minimum value in range [1,N-1]
            for(int j = 0; j < n; j++) {
                // compute value and update minimum value
                int val = arr[j] + Math.Abs(i-j);
                if(val < minValue) {
                    minValue = val;
                }
            }
            // print minimum value
            Console.Write(minValue + " ");
        }
    }

    //Driver code
    static void Main() {
        int[] arr = {1, 4, 2, 5, 3};
        int n = arr.Length;
        // function call
        minAtEachIndex(n, arr);
    }
}

Javascript

// function that prints the minimum possible value of arr[j] + |j - i|
function minAtEachIndex(n, arr) {
  // loop over every element in the array
  for (let i = 0; i < n; i++) {
    // set the minimum value to the maximum safe integer value
    let minValue = Number.MAX_SAFE_INTEGER;
    // loop over every element in the array to find the minimum value
    for (let j = 0; j < n; j++) {
      // compute the value of arr[j] + |j - i|
      let val = arr[j] + Math.abs(i - j);
      // update the minimum value if val is smaller than the current minimum
      if (val < minValue) {
        minValue = val;
      }
    }
    // print the minimum value
    document.write(minValue + " ");
  }
}

// example array
let arr = [1, 4, 2, 5, 3];
// get the length of the array
let n = arr.length;
// call the minAtEachIndex function with the array and its length
minAtEachIndex(n, arr);

Output:

1 2 2 3 3

Time Complexity: O(N²)
Auxiliary Space: O(1)

Efficient Approach: The idea is to use prefix sum technique from both left and right array traversal and find the minimum for each index. Follow the steps below to solve the problem:

1. Take two auxiliary array dp1[] and dp2[] where dp1[] store the answer for the left to right traversal and dp2[] stores the answer for the right to left traversal.
2. Traverse the array arr[] from i = 2 to N-1 and calculate min(arr[i], dp1[i-1] + 1).
3. Traverse the array arr[] form i = N-1 to 1 and calculate min(arr[i], dp2[i+1] + 1).
4. Again traverse the array from 1 to N and print min(dp1[i], dp2[i]) at each iteration.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to find minimum value of
// arr[j] + |j - i| for every array index
void minAtEachIndex(int n, int arr[])
{
    // Stores minimum of a[j] + |i - j|
    // upto position i
    int dp1[n];

    // Stores minimum of a[j] + |i-j|
    // upto position i from the end
    int dp2[n];

    int i;

    dp1[0] = arr[0];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i
    for (i = 1; i < n; i++)
        dp1[i] = min(arr[i], dp1[i - 1] + 1);

    dp2[n - 1] = arr[n - 1];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i from the end
    for (i = n - 2; i >= 0; i--)
        dp2[i] = min(arr[i], dp2[i + 1] + 1);

    vector<int> v;

    // Traversing from [0, N] and storing minimum
    // of a[j] + |i - j| from starting and end
    for (i = 0; i < n; i++)
        v.push_back(min(dp1[i], dp2[i]));

    // Print the required array
    for (auto x : v)
        cout << x << " ";
}

// Driver code
int main()
{

    // Given array arr[]
    int arr[] = { 1, 4, 2, 5, 3 };

    // Size of the array
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function Call
    minAtEachIndex(N, arr);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;
import java.util.ArrayList;
import java.util.List;

class GFG{

// Function to find minimum value of
// arr[j] + |j - i| for every array index
static void minAtEachIndex(int n, int arr[])
{

    // Stores minimum of a[j] + |i - j|
    // upto position i
    int dp1[] = new int[n];

    // Stores minimum of a[j] + |i-j|
    // upto position i from the end
    int dp2[] = new int[n];

    int i;

    dp1[0] = arr[0];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i
    for(i = 1; i < n; i++)
        dp1[i] = Math.min(arr[i], dp1[i - 1] + 1);

    dp2[n - 1] = arr[n - 1];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i from the end
    for(i = n - 2; i >= 0; i--)
        dp2[i] = Math.min(arr[i], dp2[i + 1] + 1);

    ArrayList<Integer> v = new ArrayList<Integer>();

    // Traversing from [0, N] and storing minimum
    // of a[j] + |i - j| from starting and end
    for(i = 0; i < n; i++)
        v.add(Math.min(dp1[i], dp2[i]));

    // Print the required array
    for(int x : v)
        System.out.print(x + " ");
}

// Driver code
public static void main(String[] args)
{

    // Given array arr[]
    int arr[] = { 1, 4, 2, 5, 3 };

    // Size of the array
    int N = arr.length;

    // Function Call
    minAtEachIndex(N, arr);
}
}

// This code is contributed by sanjoy_62

Python3

# Python3 program for the above approach

# Function to find minimum value of
# arr[j] + |j - i| for every array index
def minAtEachIndex(n, arr):

    # Stores minimum of a[j] + |i - j|
    # upto position i
    dp1 = [0] * n

    # Stores minimum of a[j] + |i-j|
    # upto position i from the end
    dp2 = [0] * n

    i = 0

    dp1[0] = arr[0]

    # Traversing and storing minimum
    # of a[j]+|i-j| upto i
    for i in range(1, n):
        dp1[i] = min(arr[i], dp1[i - 1] + 1)

    dp2[n - 1] = arr[n - 1]

    # Traversing and storing minimum
    # of a[j]+|i-j| upto i from the end
    for i in range(n - 2, -1, -1):
        dp2[i] = min(arr[i], dp2[i + 1] + 1)

    v = []

    # Traversing from [0, N] and storing minimum
    # of a[j] + |i - j| from starting and end
    for i in range(0, n):
        v.append(min(dp1[i], dp2[i]))

    # Print the required array
    for x in v:
        print(x, end = " ")

# Driver code
if __name__ == '__main__':

    # Given array arr
    arr = [ 1, 4, 2, 5, 3 ]

    # Size of the array
    N = len(arr)

    # Function Call
    minAtEachIndex(N, arr)

# This code is contributed by shikhasingrajput

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to find minimum value of
// arr[j] + |j - i| for every array index
static void minAtEachIndex(int n, int []arr)
{

    // Stores minimum of a[j] + |i - j|
    // upto position i
    int []dp1 = new int[n];

    // Stores minimum of a[j] + |i-j|
    // upto position i from the end
    int []dp2 = new int[n];
    int i;
    dp1[0] = arr[0];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i
    for(i = 1; i < n; i++)
        dp1[i] = Math.Min(arr[i], dp1[i - 1] + 1);
    dp2[n - 1] = arr[n - 1];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i from the end
    for(i = n - 2; i >= 0; i--)
        dp2[i] = Math.Min(arr[i], dp2[i + 1] + 1);
    List<int> v = new List<int>();

    // Traversing from [0, N] and storing minimum
    // of a[j] + |i - j| from starting and end
    for(i = 0; i < n; i++)
        v.Add(Math.Min(dp1[i], dp2[i]));

    // Print the required array
    foreach(int x in v)
        Console.Write(x + " ");
}

// Driver code
public static void Main(String[] args)
{

    // Given array []arr
    int []arr = { 1, 4, 2, 5, 3 };

    // Size of the array
    int N = arr.Length;

    // Function Call
    minAtEachIndex(N, arr);
}
}

// This code is contributed by shikhasingrajput

Javascript

<script>

// JavaScript program for the above approach

// Function to find minimum value of
// arr[j] + |j - i| for every array index
function minAtEachIndex(n, arr)
{
    // Stores minimum of a[j] + |i - j|
    // upto position i
    var dp1 = Array(n);

    // Stores minimum of a[j] + |i-j|
    // upto position i from the end
    var dp2 = Array(n);

    var i;

    dp1[0] = arr[0];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i
    for (i = 1; i < n; i++)
        dp1[i] = Math.min(arr[i], dp1[i - 1] + 1);

    dp2[n - 1] = arr[n - 1];

    // Traversing and storing minimum
    // of a[j]+|i-j| upto i from the end
    for (i = n - 2; i >= 0; i--)
        dp2[i] = Math.min(arr[i], dp2[i + 1] + 1);

    var v = [];

    // Traversing from [0, N] and storing minimum
    // of a[j] + |i - j| from starting and end
    for (i = 0; i < n; i++)
        v.push(Math.min(dp1[i], dp2[i]));

    // Print the required array
    v.forEach(x => {
        document.write(x + " ");
    });

}

// Driver code

// Given array arr[]
var arr = [1, 4, 2, 5, 3];

// Size of the array
var N = arr.length;

// Function Call
minAtEachIndex(N, arr);

</script>

Output:

1 2 2 3 3

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