Find the index of first 1 in an infinite sorted array of 0s and 1s

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Given an infinite sorted array consisting 0s and 1s. The problem is to find the index of first ‘1’ in that array. As the array is infinite, therefore it is guaranteed that number ‘1’ will be present in the array.

Examples:

Input : arr[] = {0, 0, 1, 1, 1, 1} 
Output : 2

Input : arr[] = {1, 1, 1, 1,, 1, 1}
Output : 0

Approach: The problem is closely related to the problem of finding position of an element in a sorted array of infinite numbers. As the array is infinte, therefore we do not know the upper and lower bounds between which we have to find the occurrence of first ‘1’. Below is an algorithm to find the upper and lower bounds.
Algorithm:

posOfFirstOne(arr)
    Declare l = 0, h = 1
    while arr[h] == 0
        l = h
    h = 2*h;
    return indexOfFirstOne(arr, l, h)
}

Here h and l are the required upper and lower bounds. indexOfFirstOne(arr, l, h) is used to find the index of occurrence of first ‘1’ between these two bounds.

// C++ implementation to find the index of first 1
// in an infinite sorted array of 0's and 1's
#include <bits/stdc++.h>
using namespace std;
// function to find the index of first '1'
// binary search technique is applied
int indexOfFirstOne(int arr[], int low, int high)
{
    int mid;
    while (low <= high) {
        mid = (low + high) / 2;
        // if true, then 'mid' is the index of first '1'
        if (arr[mid] == 1 &&
            (mid == 0 || arr[mid - 1] == 0))
            break;
        // first '1' lies to the left of 'mid'
        else if (arr[mid] == 1)
            high = mid - 1;
        // first '1' lies to the right of 'mid'
        else
            low = mid + 1;
    }
    // required index
    return mid;
}
// function to find the index of first 1 in
// an infinite sorted array of 0's and 1's
int posOfFirstOne(int arr[])
{
    // find the upper and lower bounds between
    // which the first '1' would be present
    int l = 0, h = 1;
    // as the array is being considered infinite
    // therefore 'h' index will always exist in
    // the array
    while (arr[h] == 0) {
        // lower bound
        l = h;
        // upper bound
        h = 2 * h;
    }
    // required index of first '1'
    return indexOfFirstOne(arr, l, h);
}
// Driver program to test above
int main()
{
    int arr[] = { 0, 0, 1, 1, 1, 1 };
    cout << "Index = "
         << posOfFirstOne(arr);
    return 0;
}

Output:

Index = 2

Let p be the position of element to be searched. Number of steps for finding high index ‘h’ is O(Log p). The value of ‘h’ must be less than 2*p. The number of elements between h/2 and h must be O(p). Therefore, time complexity of Binary Search step is also O(Log p) and overall time complexity is 2*O(Log p) which is O(Log p).

Disclaimer: This does not belong to TechCodeBit, its an article taken from the below
source and credits.
source and credits: http://www.geeksforgeeks.org
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rakesh

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