Unbounded Binary Search Example (Find the point where a monotonically increasing function becomes positive first time)

Given a function 'int f(unsigned int x)' which takes a non-negative integer 'x' as input and returns an integer as output. The function is monotonically increasing with respect to the value of x, i.e., the value of f(x+1) is greater than f(x) for every input x. Find the value 'n' where f() becomes positive for the first time. Since f() is monotonically increasing, values of f(n+1), f(n+2),... must be positive and values of f(n-2), f(n-3), … must be negative. 
Find n in O(logn) time, you may assume that f(x) can be evaluated in O(1) time for any input x. 

A simple solution is to start from i equals to 0 and one by one calculate the value of f(i) for 1, 2, 3, 4 … etc until we find a positive f(i). This works but takes O(n) time.
Can we apply Binary Search to find n in O(Logn) time? We can't directly apply Binary Search as we don't have an upper limit or high index. The idea is to do repeated doubling until we find a positive value, i.e., check values of f() for following values until f(i) becomes positive.

  f(0) 
  f(1)
  f(2)
  f(4)
  f(8)
  f(16)
  f(32)
  ....
  ....
  f(high)
Let 'high' be the value of i when f() becomes positive for first time.

Can we apply Binary Search to find n after finding 'high'? We can apply Binary Search now, we can use 'high/2' as low and 'high' as high indexes in binary search. The result n must lie between 'high/2' and 'high'.
The number of steps for finding 'high' is O(Logn). So we can find 'high' in O(Logn) time. What about the time taken by Binary Search between high/2 and high? The value of 'high' must be less than 2*n. The number of elements between high/2 and high must be O(n). Therefore, the time complexity of Binary Search is O(Logn) and the overall time complexity is 2*O(Logn) which is O(Logn). 
 

// C++ code for binary search
#include<bits/stdc++.h> 
using namespace std;

int binarySearch(int low, int high); // prototype 

// Let's take an example function 
// as f(x) = x^2 - 10*x - 20 Note that
// f(x) can be any monotonically increasing function 
int f(int x) { return (x*x - 10*x - 20); } 

// Returns the value x where above
// function f() becomes positive 
// first time. 
int findFirstPositive() 
{ 
    // When first value itself is positive 
    if (f(0) > 0) 
        return 0; 

    // Find 'high' for binary search by repeated doubling 
    int i = 1; 
    while (f(i) <= 0) 
        i = i*2; 

    // Call binary search 
    return binarySearch(i/2, i); 
} 

// Searches first positive value
// of f(i) where low <= i <= high 
int binarySearch(int low, int high) 
{ 
    if (high >= low) 
    { 
        int mid = low + (high - low)/2; /* mid = (low + high)/2 */

        // If f(mid) is greater than 0 and
        // one of the following two 
        // conditions is true: 
        // a) mid is equal to low 
        // b) f(mid-1) is negative 
        if (f(mid) > 0 && (mid == low || f(mid-1) <= 0)) 
            return mid; 

        // If f(mid) is smaller than or equal to 0 
        if (f(mid) <= 0) 
            return binarySearch((mid + 1), high); 
        else // f(mid) > 0 
            return binarySearch(low, (mid -1)); 
    } 

    /* Return -1 if there is no 
    positive value in given range */
    return -1; 
} 

/* Driver code */
int main() 
{ 
    cout<<"The value n where f() becomes" <<
        "positive first is "<< findFirstPositive(); 
    return 0; 
}

// This code is contributed by rathbhupendra

#include <stdio.h>
int binarySearch(int low, int high); // prototype

// Let's take an example function as f(x) = x^2 - 10*x - 20
// Note that f(x) can be any monotonically increasing function
int f(int x) { return (x*x - 10*x - 20); }

// Returns the value x where above function f() becomes positive
// first time.
int findFirstPositive()
{
    // When first value itself is positive
    if (f(0) > 0)
        return 0;

    // Find 'high' for binary search by repeated doubling
    int i = 1;
    while (f(i) <= 0)
        i = i*2;

    //  Call binary search
    return binarySearch(i/2, i);
}

// Searches first positive value of f(i) where low <= i <= high
int binarySearch(int low, int high)
{
    if (high >= low)
    {
        int mid = low + (high - low)/2; /* mid = (low + high)/2 */

        // If f(mid) is greater than 0 and one of the following two
        // conditions is true:
        // a) mid is equal to low
        // b) f(mid-1) is negative
        if (f(mid) > 0 && (mid == low || f(mid-1) <= 0))
            return mid;

        // If f(mid) is smaller than or equal to 0
        if (f(mid) <= 0)
            return binarySearch((mid + 1), high);
        else // f(mid) > 0
            return binarySearch(low, (mid -1));
    }

    /* Return -1 if there is no positive value in given range */
    return -1;
}

/* Driver program to check above functions */
int main()
{
    printf("The value n where f() becomes positive first is %d",
           findFirstPositive());
    return 0;
} 

// Java program for Binary Search
import java.util.*;

class Binary
{
    public static int f(int x) 
    { return (x*x - 10*x - 20); }

    // Returns the value x where above 
    // function f() becomes positive
    // first time.
    public static int findFirstPositive()
    {
        // When first value itself is positive
        if (f(0) > 0)
            return 0;

        // Find 'high' for binary search
        // by repeated doubling
        int i = 1;
        while (f(i) <= 0)
            i = i * 2;

        // Call binary search
        return binarySearch(i / 2, i);
    }

    // Searches first positive value of
    // f(i) where low <= i <= high
    public static int binarySearch(int low, int high)
    {
        if (high >= low)
        {   
            /* mid = (low + high)/2 */
            int mid = low + (high - low)/2; 

            // If f(mid) is greater than 0 and
            // one of the following two
            // conditions is true:
            // a) mid is equal to low
            // b) f(mid-1) is negative
            if (f(mid) > 0 && (mid == low || f(mid-1) <= 0))
                return mid;

            // If f(mid) is smaller than or equal to 0
            if (f(mid) <= 0)
                return binarySearch((mid + 1), high);
            else // f(mid) > 0
                return binarySearch(low, (mid -1));
        }

        /* Return -1 if there is no positive
        value in given range */
        return -1;
    }
    
    // driver code
    public static void main(String[] args)
    {
        System.out.print ("The value n where f() "+
                         "becomes positive first is "+
                         findFirstPositive());
    }
}

// This code is contributed by rishabh_jain

# Python3 program for Unbound Binary search.

# Let's take an example function as 
# f(x) = x^2 - 10*x - 20
# Note that f(x) can be any monotonically 
# increasing function
def f(x): 
    return (x * x - 10 * x - 20)

# Returns the value x where above function
# f() becomes positive first time.
def findFirstPositive() :
    
    # When first value itself is positive
    if (f(0) > 0):
        return 0

    # Find 'high' for binary search
    # by repeated doubling
    i = 1
    while (f(i) <= 0) :
        i = i * 2

    # Call binary search
    return binarySearch(i/2, i)

# Searches first positive value of 
# f(i) where low <= i <= high
def binarySearch(low, high):
    if (high >= low) :
        
        # mid = (low + high)/2
        mid = low + (high - low)/2;  

        # If f(mid) is greater than 0 
        # and one of the following two
        # conditions is true:
        # a) mid is equal to low
        # b) f(mid-1) is negative
        if (f(mid) > 0 and (mid == low or f(mid-1) <= 0)) :
            return mid;

        # If f(mid) is smaller than or equal to 0
        if (f(mid) <= 0) :
            return binarySearch((mid + 1), high)
        else : # f(mid) > 0
            return binarySearch(low, (mid -1))
    
    # Return -1 if there is no positive
    # value in given range 
    return -1;

# Driver Code
print ("The value n where f() becomes "+
      "positive first is ", findFirstPositive());

# This code is contributed by rishabh_jain

// C# program for Binary Search
using System;

class Binary
{
    public static int f(int x) 
    { 
        return (x*x - 10*x - 20); 
    }

    // Returns the value x where above 
    // function f() becomes positive
    // first time.
    public static int findFirstPositive()
    {
        // When first value itself is positive
        if (f(0) > 0)
            return 0;

        // Find 'high' for binary search
        // by repeated doubling
        int i = 1;
        while (f(i) <= 0)
            i = i * 2;

        // Call binary search
        return binarySearch(i / 2, i);
    }

    // Searches first positive value of
    // f(i) where low <= i <= high
    public static int binarySearch(int low, int high)
    {
        if (high >= low)
        { 
            /* mid = (low + high)/2 */
            int mid = low + (high - low)/2; 

            // If f(mid) is greater than 0 and
            // one of the following two
            // conditions is true:
            // a) mid is equal to low
            // b) f(mid-1) is negative
            if (f(mid) > 0 && (mid == low ||
                             f(mid-1) <= 0))
                return mid;

            // If f(mid) is smaller than or equal to 0
            if (f(mid) <= 0)
                return binarySearch((mid + 1), high);
            else 
            
                // f(mid) > 0
                return binarySearch(low, (mid -1));
        }

        /* Return -1 if there is no positive
        value in given range */
        return -1;
    }
    
    // Driver code
    public static void Main()
    {
       Console.Write ("The value n where f() " +
                      "becomes positive first is " +
                       findFirstPositive());
    }
}

// This code is contributed by nitin mittal

<?php
// PHP program for Binary Search

// Let's take an example function 
// as f(x) = x^2 - 10*x - 20
// Note that f(x) can be any 
// monotonically increasing function
function f($x) 
{ 
    return ($x * $x - 10 * $x - 20); 
}

// Returns the value x where above 
// function f() becomes positive 
// first time.
function findFirstPositive()
{
    // When first value 
    // itself is positive
    if (f(0) > 0)
        return 0;

    // Find 'high' for binary 
    // search by repeated doubling
    $i = 1;
    while (f($i) <= 0)
        $i = $i * 2;

    // Call binary search
    return binarySearch(intval($i / 2), $i);
}

// Searches first positive value
// of f(i) where low <= i <= high
function binarySearch($low, $high)
{
    if ($high >= $low)
    {
        /* mid = (low + high)/2 */
        $mid = $low + intval(($high - 
                              $low) / 2); 

        // If f(mid) is greater than 0 
        // and one of the following two
        // conditions is true:
        // a) mid is equal to low
        // b) f(mid-1) is negative
        if (f($mid) > 0 && ($mid == $low || 
                          f($mid - 1) <= 0))
            return $mid;

        // If f(mid) is smaller 
        // than or equal to 0
        if (f($mid) <= 0)
            return binarySearch(($mid + 1), $high);
        else // f(mid) > 0
            return binarySearch($low, ($mid - 1));
    }

    /* Return -1 if there is no 
    positive value in given range */
    return -1;
}

// Driver Code
echo "The value n where f() becomes ". 
                 "positive first is ". 
                 findFirstPositive() ;

// This code is contributed by Sam007
?>

<script>
    // Javascript program for Binary Search
    
    function f(x) 
    { 
        return (x*x - 10*x - 20); 
    }
  
    // Returns the value x where above 
    // function f() becomes positive
    // first time.
    function findFirstPositive()
    {
        // When first value itself is positive
        if (f(0) > 0)
            return 0;
  
        // Find 'high' for binary search
        // by repeated doubling
        let i = 1;
        while (f(i) <= 0)
            i = i * 2;
  
        // Call binary search
        return binarySearch(parseInt(i / 2, 10), i);
    }
  
    // Searches first positive value of
    // f(i) where low <= i <= high
    function binarySearch(low, high)
    {
        if (high >= low)
        { 
            /* mid = (low + high)/2 */
            let mid = low + parseInt((high - low)/2, 10); 
  
            // If f(mid) is greater than 0 and
            // one of the following two
            // conditions is true:
            // a) mid is equal to low
            // b) f(mid-1) is negative
            if (f(mid) > 0 && (mid == low ||
                             f(mid-1) <= 0))
                return mid;
  
            // If f(mid) is smaller than or equal to 0
            if (f(mid) <= 0)
                return binarySearch((mid + 1), high);
            else 
              
                // f(mid) > 0
                return binarySearch(low, (mid -1));
        }
  
        /* Return -1 if there is no positive
        value in given range */
        return -1;
    }
    
    document.write ("The value n where f() " +
                      "becomes positive first is " +
                       findFirstPositive());

</script>

Output : 

The value n where f() becomes positive first is 12

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