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Binary Search for Rational Numbers without using floating point arithmetic

  • Difficulty Level : Medium
  • Last Updated : 23 Jun, 2021

A rational is represented as p/qb, for example 2/3. Given a sorted array of rational numbers, how to search an element using Binary Search. Use of floating-point arithmetic is not allowed.

Example:  

Input:  arr[] = {1/5, 2/3, 3/2, 13/2}
        x = 3/2
Output: Found at index 2

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To compare two rational numbers p/q and r/s, we can compare p*s with q*r. 

C++




// C++ program for Binary Search for
// Rationalnal Numbers without using
// floating point arithmetic
#include <iostream>
using namespace std;
 
struct Rational
{
    int p;
    int q;
};
 
// Utility function to compare two
// Rationalnal numbers 'a' and 'b'.
// It returns
// 0 --> When 'a' and 'b' are same
// 1 --> When 'a' is greater
//-1 --> When 'b' is greate
int compare(struct Rational a, struct Rational b)
{
     
    // If a/b == c/d then a*d = b*c:
    // method to ignore division
    if (a.p * b.q == a.q * b.p)
        return 0;
    if (a.p * b.q > a.q * b.p)
        return 1;
         
    return -1;
}
 
// Returns index of x in arr[l..r] if
// it is present, else returns -1. It
// mainly uses Binary Search.
int binarySearch(struct Rational arr[], int l, int r,
                 struct Rational x)
{
    if (r >= l)
    {
        int mid = l + (r - l) / 2;
         
        // If the element is present at the middle itself
        if (compare(arr[mid], x) == 0)
            return mid;
         
        // If element is smaller than mid, then it can
        // only be present in left subarray
        if (compare(arr[mid], x) > 0)
            return binarySearch(arr, l, mid - 1, x);
         
        // Else the element can only be present in right
        // subarray
        return binarySearch(arr, mid + 1, r, x);
    }
    return -1;
}
 
// Driver code
int main()
{
    struct Rational arr[] = { { 1, 5 }, { 2, 3 },
                              { 3, 2 }, { 13, 2 } };
    struct Rational x = { 3, 2 };
     
    int n = sizeof(arr) / sizeof(arr[0]);
     
    cout << "Element found at index "
         << binarySearch(arr, 0, n - 1, x);
}
 
// This code is contributed by shivanisinghss2110

C




// C program for Binary Search for Rationalnal Numbers
// without using floating point arithmetic
#include <stdio.h>
 
struct Rational
{
    int p;
    int q;
};
 
// Utility function to compare two Rationalnal numbers
// 'a' and 'b'. It returns
// 0 --> When 'a' and 'b' are same
// 1 --> When 'a' is greater
//-1 --> When 'b' is greate
int compare(struct Rational a, struct Rational b)
{
    // If a/b == c/d  then  a*d = b*c:
    // method to ignore division
    if (a.p * b.q == a.q * b.p)
        return 0;
    if (a.p * b.q > a.q * b.p)
        return 1;
    return -1;
}
 
// Returns index of x in arr[l..r] if it is present, else
// returns -1. It mainly uses Binary Search.
int binarySearch(struct Rational arr[], int l, int r,
                 struct Rational x)
{
   if (r >= l)
   {
        int mid = l + (r - l)/2;
 
        // If the element is present at the middle itself
        if (compare(arr[mid], x) == 0)  return mid;
 
        // If element is smaller than mid, then it can
        // only be present in left subarray
        if (compare(arr[mid], x) > 0)
            return binarySearch(arr, l, mid-1, x);
 
        // Else the element can only be present in right
        // subarray
        return binarySearch(arr, mid+1, r, x);
   }
 
   return -1;
}
 
// Driver method
int main()
{
    struct Rational arr[] = {{1, 5}, {2, 3}, {3, 2}, {13, 2}};
    struct Rational x = {3, 2};
    int n = sizeof(arr)/sizeof(arr[0]);
    printf("Element found at index %d",
            binarySearch(arr, 0, n-1, x));
}

Java




// Java program for Binary Search for Rationalnal Numbers
// without using floating point arithmetic
class GFG
{
 
static class Rational
{
    int p;
    int q;
 
    public Rational(int p, int q)
    {
        this.p = p;
        this.q = q;
    }
     
};
 
// Utility function to compare two Rationalnal numbers
// 'a' and 'b'. It returns
// 0 -. When 'a' and 'b' are same
// 1 -. When 'a' is greater
//-1 -. When 'b' is greate
static int compare(Rational a, Rational b)
{
    // If a/b == c/d then a*d = b*c:
    // method to ignore division
    if (a.p * b.q == a.q * b.p)
        return 0;
    if (a.p * b.q > a.q * b.p)
        return 1;
    return -1;
}
 
// Returns index of x in arr[l..r] if it is present, else
// returns -1. It mainly uses Binary Search.
static int binarySearch(Rational arr[], int l, int r,
                Rational x)
{
     
    if (r >= l)
    {
        int mid = l + (r - l)/2;
 
        // If the element is present at the middle itself
        if (compare(arr[mid], x) == 0) return mid;
 
        // If element is smaller than mid, then it can
        // only be present in left subarray
        if (compare(arr[mid], x) > 0)
            return binarySearch(arr, l, mid - 1, x);
 
        // Else the element can only be present in right
        // subarray
        return binarySearch(arr, mid + 1, r, x);
    }
 
return -1;
}
 
// Driver method
public static void main(String[] args)
{
    Rational arr[] = {new Rational(1, 5),
                        new Rational(2, 3),
                        new Rational(3, 2),
                        new Rational(13, 2)};
    Rational x = new Rational(3, 2);
    int n = arr.length;
    System.out.printf("Element found at index %d",
            binarySearch(arr, 0, n - 1, x));
}
}
 
// This code is contributed by Rajput-Ji

Python3




# Python3 program for Binary Search
# for Rationalnal Numbers without
# using floating point arithmetic
class Rational:
     
    def __init__(self, a = 0, b = 0):
         
        self.p = a
        self.q = b
 
# Utility function to compare two
# Rationalnal numbers 'a' and 'b'.
# It returns
# 0 --> When 'a' and 'b' are same
# 1 --> When 'a' is greater
#-1 --> When 'b' is greate
def compare(a: Rational, b: Rational) -> int:
 
    # If a/b == c/d  then  a*d = b*c:
    # method to ignore division
    if (a.p * b.q == a.q * b.p):
        return 0
    if (a.p * b.q > a.q * b.p):
        return 1
         
    return -1
 
# Returns index of x in arr[l..r] if
# it is present, else returns -1. It
# mainly uses Binary Search.
def binarySearch(arr: list, l: int,
                   r: int, x: Rational) -> int:
 
    if (r >= l):
 
        mid = l + (r - l) // 2
 
        # If the element is present at the
        # middle itself
        if (compare(arr[mid], x) == 0):
            return mid
 
        # If element is smaller than mid, then
        # it can only be present in left subarray
        if (compare(arr[mid], x) > 0):
            return binarySearch(arr, l, mid - 1, x)
 
        # Else the element can only be present
        # in right subarray
        return binarySearch(arr, mid + 1, r, x)
 
    return -1
 
# Driver code
if __name__ == "__main__":
     
    arr = [ Rational(1, 5), Rational(2, 3),
            Rational(3, 2), Rational(13, 2) ]
    x = Rational(3, 2)
    n = len(arr)
     
    print("Element found at index %d" % (
        binarySearch(arr, 0, n - 1, x)))
 
# This code is contributed by sanjeev2552

C#




// C# program for Binary Search for Rationalnal Numbers
// without using floating point arithmetic
using System;
 
class GFG
{
 
class Rational
{
    public int p;
    public int q;
 
    public Rational(int p, int q)
    {
        this.p = p;
        this.q = q;
    }
     
};
 
// Utility function to compare two Rationalnal numbers
// 'a' and 'b'. It returns
// 0 -. When 'a' and 'b' are same
// 1 -. When 'a' is greater
//-1 -. When 'b' is greate
static int compare(Rational a, Rational b)
{
    // If a/b == c/d then a*d = b*c:
    // method to ignore division
    if (a.p * b.q == a.q * b.p)
        return 0;
    if (a.p * b.q > a.q * b.p)
        return 1;
    return -1;
}
 
// Returns index of x in arr[l..r] if it is present, else
// returns -1. It mainly uses Binary Search.
static int binarySearch(Rational []arr, int l, int r,
                Rational x)
{
    if (r >= l)
    {
        int mid = l + (r - l)/2;
 
        // If the element is present at the middle itself
        if (compare(arr[mid], x) == 0) return mid;
 
        // If element is smaller than mid, then it can
        // only be present in left subarray
        if (compare(arr[mid], x) > 0)
            return binarySearch(arr, l, mid - 1, x);
 
        // Else the element can only be present in right
        // subarray
        return binarySearch(arr, mid + 1, r, x);
    }
return -1;
}
 
// Driver method
public static void Main(String[] args)
{
    Rational []arr = {new Rational(1, 5),
                        new Rational(2, 3),
                        new Rational(3, 2),
                        new Rational(13, 2)};
    Rational x = new Rational(3, 2);
    int n = arr.Length;
    Console.Write("Element found at index {0}",
            binarySearch(arr, 0, n - 1, x));
}
}
 
// This code is contributed by 29AjayKumar

Javascript




<script>
// Javascript program for Binary Search for Rationalnal Numbers
// without using floating point arithmetic
class Rational
{
    constructor(p, q)
    {
        this.p = p;
        this.q = q;
    }
}
 
// Utility function to compare two Rationalnal numbers
// 'a' and 'b'. It returns
// 0 -. When 'a' and 'b' are same
// 1 -. When 'a' is greater
//-1 -. When 'b' is greate
function compare(a,b)
{
    // If a/b == c/d then a*d = b*c:
    // method to ignore division
    if (a.p * b.q == a.q * b.p)
        return 0;
    if (a.p * b.q > a.q * b.p)
        return 1;
    return -1;
}
 
// Returns index of x in arr[l..r] if it is present, else
// returns -1. It mainly uses Binary Search.
function binarySearch(arr,l,r,x)
{
    if (r >= l)
    {
        let mid = l + Math.floor((r - l)/2);
  
        // If the element is present at the middle itself
        if (compare(arr[mid], x) == 0) return mid;
  
        // If element is smaller than mid, then it can
        // only be present in left subarray
        if (compare(arr[mid], x) > 0)
            return binarySearch(arr, l, mid - 1, x);
  
        // Else the element can only be present in right
        // subarray
        return binarySearch(arr, mid + 1, r, x);
    }
  
return -1;
}
 
// Driver method
let arr=[new Rational(1, 5),
                        new Rational(2, 3),
                        new Rational(3, 2),
                        new Rational(13, 2)];
let x = new Rational(3, 2);
let n = arr.length;
document.write("Element found at index ",
            binarySearch(arr, 0, n - 1, x));
 
// This code is contributed by rag2127
</script>

Output: 

Element found at index 2

Time Complexity: O(log n)

Auxiliary Space: O(1)

Thanks to Utkarsh Trivedi for suggesting above solution.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above 

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