Find position of an element in a sorted array of infinite numbers

2.7

Suppose you have a sorted array of infinite numbers, how would you search an element in the array?

Source: Amazon Interview Experience.

Since array is sorted, the first thing clicks into mind is binary search, but the problem here is that we don’t know size of array.
If the array is infinite, that means we don’t have proper bounds to apply binary search. So in order to find position of key, first we find bounds and then apply binary search algorithm.

Let low be pointing to 1st element and high pointing to 2nd element of array, Now compare key with high index element,
->if it is greater than high index element then copy high index in low index and double the high index.
->if it is smaller, then apply binary search on high and low indices found.

Below are implementations of above algorithm

C++

// C++ program to demonstrate working of an algorithm that finds
// an element in an array of infinite size
#include<iostream>
using namespace std;

// Simple binary search algorithm
int binarySearch(int arr[], int l, int r, int x)
{
    if (r>=l)
    {
        int mid = l + (r - l)/2;
        if (arr[mid] == x)
            return mid;
        if (arr[mid] > x)
            return binarySearch(arr, l, mid-1, x);
        return binarySearch(arr, mid+1, r, x);
    }
    return -1;
}

// function takes an infinite size array and a key to be
//  searched and returns its position if found else -1.
// We don't know size of arr[] and we can assume size to be
// infinite in this function.
// NOTE THAT THIS FUNCTION ASSUMES arr[] TO BE OF INFINITE SIZE
// THEREFORE, THERE IS NO INDEX OUT OF BOUND CHECKING
int findPos(int arr[], int key)
{
    int l = 0, h = 1;
    int val = arr[0];

    // Find h to do binary search
    while (val < key)
    {
        l = h;        // store previous high
        h = 2*h;      // double high index
        val = arr[h]; // update new val
    }

    // at this point we have updated low and
    // high indices, Thus use binary search 
    // between them
    return binarySearch(arr, l, h, key);
}

// Driver program
int main()
{
    int arr[] = {3, 5, 7, 9, 10, 90, 100, 130, 
                               140, 160, 170};
    int ans = findPos(arr, 10);
    if (ans==-1)
        cout << "Element not found";
    else
        cout << "Element found at index " << ans;
    return 0;
}

Java

// Java program to demonstrate working of 
// an algorithm that finds an element in an 
// array of infinite size

class Test
{
	// Simple binary search algorithm
	static int binarySearch(int arr[], int l, int r, int x)
	{
	    if (r>=l)
	    {
	        int mid = l + (r - l)/2;
	        if (arr[mid] == x)
	            return mid;
	        if (arr[mid] > x)
	            return binarySearch(arr, l, mid-1, x);
	        return binarySearch(arr, mid+1, r, x);
	    }
	    return -1;
	}
	
	// Method takes an infinite size array and a key to be
	//  searched and returns its position if found else -1.
	// We don't know size of arr[] and we can assume size to be
	// infinite in this function.
	// NOTE THAT THIS FUNCTION ASSUMES arr[] TO BE OF INFINITE SIZE
	// THEREFORE, THERE IS NO INDEX OUT OF BOUND CHECKING
	static int findPos(int arr[],int key)	
	{
		int l = 0, h = 1;
		int val = arr[0];
 
		// Find h to do binary search
		while (val < key)
		{
			l = h;        // store previous high
			h = 2*h;      // double high index
			val = arr[h]; // update new val
		}
 
		// at this point we have updated low
		//  and high indices, thus use binary 
		// search between them
		return binarySearch(arr, l, h, key);
	}

	// Driver method to test the above function
	public static void main(String[] args) 
	{
	    int arr[] = new int[]{3, 5, 7, 9, 10, 90, 
                             100, 130, 140, 160, 170};
		int ans = findPos(arr,10);
		
		if (ans==-1)
	        System.out.println("Element not found");
	    else
	        System.out.println("Element found at index " + ans);
    }
}

Python

# Python Program to demonstrate working of an algorithm that finds
# an element in an array of infinite size

# Binary search algorithm implementation
def binary_search(arr,l,r,x):

    if r >= l:
        mid = l+(r-l)/2

        if arr[mid] == x:
            return mid

        if arr[mid] > x:
            return binary_search(arr,l,mid-1,x)

        return binary_search(arr,mid+1,r,x)

    return -1

# function takes an infinite size array and a key to be
# searched and returns its position if found else -1.
# We don't know size of a[] and we can assume size to be
# infinite in this function.
# NOTE THAT THIS FUNCTION ASSUMES a[] TO BE OF INFINITE SIZE
# THEREFORE, THERE IS NO INDEX OUT OF BOUND CHECKING
def findPos(a, key):

    l, h, val = 0, 1, arr[0]

    # Find h to do binary search
    while val < key:
        l = h            #store previous high
        h = 2*h          #double high index
        val = arr[h]       #update new val

    # at this point we have updated low and high indices,
    # thus use binary search between them
    return binary_search(a, l, h, key)

# Driver function
arr = [3, 5, 7, 9, 10, 90, 100, 130, 140, 160, 170]
ans = findPos(arr,10)
if ans == -1:
    print "Element not found"
else:
    print"Element found at index",ans

# This code is contributed by __Devesh Agrawal__

Output:

Element found at index 4

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).

This article is contributed by Gaurav Sharma. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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