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Minimum time required to cover a Binary Array

  • Last Updated : 17 Nov, 2021

Given an integer N which represents the size of a boolean array that is initially filled with 0, and also given an array arr[] of size K consisting of (1-based) indices at which ‘1’ are present in the boolean array. Now, at one unit of time the adjacent cells of the arr[i] in the boolean array become 1 that is 010 becomes 111. Find the minimum time required to convert the whole array into 1s.
Examples: 
 

Input : N = 5, arr[] = {1, 4} 
Output :
Explanation: 
Initially the boolean array is of size 5 filled with 5 zeros. arr[] represents places at which 1 is present in the boolean array. Therefore, the boolean array becomes 10010. 
Now, at time (t) = 1, the 0s at 3rd and 5th position becomes 1 => 10111 and at the same time, 0 at 2nd position becomes 1 => 11111. All the 1s initially increment their adjacent 0s at the same moment of time. So at t=1, the string gets converted to all 1s.
Input : N=7, arr[] = {1, 7} 
Output :
Explanation: 
At time (t) = 1, 1000001 becomes 1100011 
At time (t) = 2, 1100011 becomes 1110111 
At time (t) = 3, 1110111 becomes 1111111 
Hence, minimum time is 3 to change the binary array into 1. 
 

 

Approach: 
To solve the problem mentioned above we have to observe that we need to find the longest segment of zeroes until 1 appears. For example, for a binary number 00010000010000, the longest segment of 0s is from 4th to 10th position. Now, observe that there are 5 0s between the indices which is an odd number. Hence, we can conclude that to cover 5 zeros we need 5/2 + 1 that is 3 units of time because all the other segments will get filled in less than or equal to 3 units of time. 
 

  • If the longest zero segments are odd then we can conclude that x/2 + 1 unit of time is required where x is the number of 0s in the longest segment.
  • If the longest zero segments are even then we can conclude that x/2 units of time are required where x is the number of 0s in the longest segment.

We can calculate the maximum length contiguous segment until 1 occurs in the boolean array and return the answer depends upon whether the length is odd or even.
Below is the implementation of the above approach: 
 

C++




// CPP implementation to find the
// Minimum time required to cover a Binary Array
#include <bits/stdc++.h>
using namespace std;
 
// function to calculate the time
int solve(vector<int> arr, int n)
{
 
    int k = arr.size();
 
    // Map to mark or store the binary values
    bool mp[n + 2];
 
    // Firstly fill the boolean
    // array with all zeroes
    for (int i = 0; i <= n; i++) {
        mp[i] = 0;
    }
 
    // Mark the 1s
    for (int i = 0; i < k; i++) {
        mp[arr[i]] = 1;
    }
 
    // Number of 0s until first '1' occurs
    int leftSegment = arr[0] - 1;
 
    // Maximum Number of 0s in between 2 '1's.
    for (int i = 1; i < k; i++) {
        leftSegment = max(leftSegment, arr[i] - arr[i - 1] - 1);
    }
 
    // Number of 0s from right until first '1' occurs
    int rightSegment = n - arr[k - 1];
 
    // Return maximum from left and right segment
    int maxSegment = max(leftSegment, rightSegment);
 
    int tim;
 
    // check if count is odd
    if (maxSegment & 1)
        tim = (maxSegment / 2) + 1;
 
    // check ifcount is even
    else
        tim = maxSegment / 2;
 
    // return the time
    return tim;
}
 
// driver code
int main()
{
    // initialise N
    int N = 5;
 
    // array initialisation
    vector<int> arr = { 1, 4 };
 
    cout << solve(arr, N);
}

Java




// Java implementation to find the
// Minimum time required to cover a Binary Array
class GFG {
 
    // function to calculate the time
    static int solve(int []arr, int n)
    {
     
        int k = arr.length;
     
        // Map to mark or store the binary values
        int mp[] = new int[n + 2];
     
        // Firstly fill the boolean
        // array with all zeroes
        for (int i = 0; i <= n; i++) {
            mp[i] = 0;
        }
     
        // Mark the 1s
        for (int i = 0; i < k; i++) {
            mp[arr[i]] = 1;
        }
     
        // Number of 0s until first '1' occurs
        int leftSegment = arr[0] - 1;
     
        // Maximum Number of 0s in between 2 '1's.
        for (int i = 1; i < k; i++) {
            leftSegment = Math.max(leftSegment, arr[i] - arr[i - 1] - 1);
        }
     
        // Number of 0s from right until first '1' occurs
        int rightSegment = n - arr[k - 1];
     
        // Return maximum from left and right segment
        int maxSegment = Math.max(leftSegment, rightSegment);
     
        int tim;
     
        // check if count is odd
        if ((maxSegment & 1) == 1)
            tim = (maxSegment / 2) + 1;
     
        // check ifcount is even
        else
            tim = maxSegment / 2;
     
        // return the time
        return tim;
    }
     
    // driver code
    public static void main (String[] args)
    {
        // initialise N
        int N = 5;
     
        // array initialisation
        int arr[] = { 1, 4 };
     
        System.out.println(solve(arr, N));
    }
}
 
// This code is contributed by AnkitRai01

Python3




# Python3 implementation to find the
# Minimum time required to cover a Binary Array
 
# function to calculate the time
def solve(arr, n) :
 
    k = len(arr)
 
    # Map to mark or store the binary values
    # Firstly fill the boolean
    # array with all zeroes
    mp = [False for i in range(n + 2)]
 
    # Mark the 1s
    for i in range(k) :
        mp[arr[i]] = True
 
    # Number of 0s until first '1' occurs
    leftSegment = arr[0] - 1
 
    # Maximum Number of 0s in between 2 '1's.
    for i in range(1,k) :
        leftSegment = max(leftSegment, arr[i] - arr[i - 1] - 1)
 
    # Number of 0s from right until first '1' occurs
    rightSegment = n - arr[k - 1]
 
    # Return maximum from left and right segment
    maxSegment = max(leftSegment, rightSegment);
 
    tim = 0
 
    # check if count is odd
    if (maxSegment & 1) :
        tim = (maxSegment // 2) + 1
 
    # check ifcount is even
    else :
        tim = maxSegment // 2
 
    # return the time
    return tim
 
# Driver code
# initialise N
N = 5
 
# array initialisation
arr = [ 1, 4 ]
 
print(solve(arr, N))
 
# This code is contributed by Sanjit_Prasad

C#




// C# implementation to find the
// Minimum time required to cover
// a Binary Array
using System;
 
class GFG{
 
// Function to calculate the time
static int solve(int[] arr, int n)
{
    int k = arr.Length;
 
    // Map to mark or store the binary values
    int[] mp = new int[n + 2];
 
    // Firstly fill the boolean
    // array with all zeroes
    for(int i = 0; i <= n; i++)
    {
        mp[i] = 0;
    }
 
    // Mark the 1s
    for(int i = 0; i < k; i++)
    {
        mp[arr[i]] = 1;
    }
 
    // Number of 0s until first '1' occurs
    int leftSegment = arr[0] - 1;
 
    // Maximum Number of 0s in between 2 '1's.
    for(int i = 1; i < k; i++)
    {
        leftSegment = Math.Max(leftSegment,
                               arr[i] -
                               arr[i - 1] - 1);
    }
 
    // Number of 0s from right until first '1' occurs
    int rightSegment = n - arr[k - 1];
 
    // Return maximum from left and right segment
    int maxSegment = Math.Max(leftSegment,
                              rightSegment);
 
    int tim;
 
    // Check if count is odd
    if ((maxSegment & 1) == 1)
        tim = (maxSegment / 2) + 1;
 
    // Check ifcount is even
    else
        tim = maxSegment / 2;
 
    // Return the time
    return tim;
}
 
// Driver code
public static void Main ()
{
     
    // Initialise N
    int N = 5;
 
    // Array initialisation
    int[] arr = { 1, 4 };
 
    Console.Write(solve(arr, N));
}
}
 
// This code is contributed by chitranayal

Javascript




<script>
 
// JavaScript implementation to find the
// Minimum time required to cover a Binary Array
 
    // function to calculate the time
    function solve(arr, n)
    {
       
        let k = arr.length;
       
        // Map to mark or store the binary values
        let mp = Array.from({length: n+2}, (_, i) => 0);
       
        // Firstly fill the boolean
        // array with all zeroes
        for (let i = 0; i <= n; i++) {
            mp[i] = 0;
        }
       
        // Mark the 1s
        for (let i = 0; i < k; i++) {
            mp[arr[i]] = 1;
        }
       
        // Number of 0s until first '1' occurs
        let leftSegment = arr[0] - 1;
       
        // Maximum Number of 0s in between 2 '1's.
        for (let i = 1; i < k; i++) {
            leftSegment = Math.max(leftSegment, arr[i] -
            arr[i - 1] - 1);
        }
       
        // Number of 0s from right until first '1' occurs
        let rightSegment = n - arr[k - 1];
       
        // Return maximum from left and right segment
        let maxSegment = Math.max(leftSegment, rightSegment);
       
        let tim;
       
        // check if count is odd
        if ((maxSegment & 1) == 1)
            tim = (maxSegment / 2) + 1;
       
        // check ifcount is even
        else
            tim = maxSegment / 2;
       
        // return the time
        return tim;
    }
// Driver Code
     
    // initialise N
        let N = 5;
       
        // array initialisation
        let arr = [ 1, 4 ];
       
        document.write(solve(arr, N));
  
</script>
Output: 
1

 

Time Complexity: O(N)

Auxiliary Space: O(N)


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