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Subtraction of two large numbers using 9’s complement

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  • Last Updated : 15 Nov, 2022
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Given two strings str1 and str2 of given lengths N and M respectively, each representing a large number, the task is to subtract one from the other using 9’s complement.
Examples:

Input: N = 17, str1 = “12345678987654321”, M = 11, str2 = “22324324343” 
Output: 12345656663329978
Input: N = 20, Str1 = “12345334233242431433”, M = 20, Str2 = “12345334233242431432” 
Output: 1

Approach: 
The basic idea is similar to Subtraction of two numbers using 2’s complement

Subtraction of given strings can be written as 
Str1 – Str2 = Str1 + (- Str2) = Str1 + (9’s complement of Str2) 

Follow the steps below to solve the problem: 
 

  • Compare the lengths of the two strings and store the smaller of the two in str2.
  • Calculate 9’s complement of str2.
  • Now, add 9’s complement of str2 to str1.
  • If any carry is generated, insert at the beginning of str1.
  • If no carry is generated, then the complement of str1 is the final answer.

Below is the implementation of the above approach:

C++




// C++ Program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return sum of two
// large numbers given as strings
string sumBig(string a, string b)
{
 
    // Compare their lengths
    if (a.length() > b.length())
        swap(a, b);
 
    // Stores the result
    string str = "";
 
    // Store the respective lengths
    int n1 = a.length(),
        n2 = b.length();
 
    int diff = n2 - n1;
 
    // Initialize carry
    int carry = 0;
 
    // Traverse from end of both strings
    for (int i = n1 - 1; i >= 0; i--) {
        // Compute sum of
        // current digits and carry
        int sum = ((a[i] - '0')
                   + (b[i + diff] - '0')
                   + carry);
 
        // Store the result
        str.push_back(sum % 10 + '0');
 
        // Update carry
        carry = sum / 10;
    }
 
    // Add remaining digits of str2[]
    for (int i = n2 - n1 - 1; i >= 0; i--) {
 
        int sum = ((b[i] - '0') + carry);
 
        str.push_back(sum % 10 + '0');
        carry = sum / 10;
    }
 
    // Add remaining carry
    if (carry)
        str.push_back(carry + '0');
 
    // Reverse resultant string
    reverse(str.begin(), str.end());
 
    return str;
}
 
// Function return 9's
// complement of given number
string complement9(string v)
{
    // Stores the complement
    string complement = "";
 
    for (int i = 0; i < v.size(); i++) {
 
        // Subtract every bit from 9
        complement += '9' - v[i] + '0';
    }
 
    // Return the result
    return complement;
}
 
// Function returns subtraction
// of two given numbers as strings
string subtract(string a, string b)
{
 
    // If second string is larger
    if (a.length() < b.length())
        swap(a, b);
 
    // Calculate respective lengths
    int l1 = a.length(),
        l2 = b.length();
 
    // If lengths are equal
    int diffLen = l1 - l2;
 
    for (int i = 0; i < diffLen; i++) {
 
        // Insert 0's to the beginning
        // of b to make both the lengths equal
        b = "0" + b;
    }
 
    // Add (complement of B) and A
    string c = sumBig(a, complement9(b));
 
    // If length of new string is greater
    // than length of first string,
    // than carry is generated
    if (c.length() > a.length()) {
        string::iterator it;
 
        // bit1 is the carry bit
        char bit1 = c[0];
        string bit = { bit1 };
        it = c.begin();
        c.erase(it);
        c = sumBig(c, bit);
        return c;
    }
 
    // If both lengths are equal
    else {
        return complement9(c);
    }
}
 
// Driver Code
int main()
{
 
    string str1 = "12345678987654321";
    string str2 = "22324324343";
 
    cout << subtract(str1, str2) << endl;
 
    return 0;
}

Java




import java.io.*;
import java.lang.*;
import java.util.*;
public class Main
{
 
  // Function to return sum of two
  // large numbers given as strings
  static String sumBig(String a, String b)
  {
 
    // Compare their lengths
    if (a.length() > b.length()) {
      String c = a;
      a = b;
      b = c;
    }
 
    // Stores the result
    String str = "";
 
    // Store the respective lengths
    int n1 = a.length(), n2 = b.length();
 
    int diff = n2 - n1;
 
    // Initialize carry
    int carry = 0;
 
    // Traverse from end of both strings
    for (int i = n1 - 1; i >= 0; i--) {
      // Compute sum of
      // current digits and carry
      int sum
        = ((a.charAt(i) - '0')
           + (b.charAt(i + diff) - '0') + carry);
 
      // Store the result
      str += (char)(sum % 10 + '0');
 
      // Update carry
      carry = sum / 10;
    }
 
    // Add remaining digits of str2[]
    for (int i = n2 - n1 - 1; i >= 0; i--) {
 
      int sum = ((b.charAt(i) - '0') + carry);
      str += (char)(sum % 10 + '0');
      carry = sum / 10;
    }
 
    // Add remaining carry
    if (carry != 0)
      str += (char)(carry + '0');
 
    // Reverse resultant string
    String rev = "";
 
    for (int i = str.length() - 1; i >= 0; i--)
      rev = rev + str.charAt(i);
 
    return rev;
  }
  // Function return 9's
  // complement of given number
  static String complement9(String v)
  {
    // Stores the complement
    String complement = "";
 
    for (int i = 0; i < v.length(); i++) {
 
      // Subtract every bit from 9
      int x = '9' - v.charAt(i) + '0';
      complement += (char)x;
    }
    // Return the result
    return complement;
  }
 
  // Function returns subtraction
  // of two given numbers as strings
  static String subtract(String a, String b)
  {
 
    // If second string is larger
    if (a.length() < b.length()) {
      String c = a;
      a = b;
      b = c;
    }
 
    // Calculate respective lengths
    int l1 = a.length(), l2 = b.length();
 
    // If lengths are equal
    int diffLen = l1 - l2;
 
    for (int i = 0; i < diffLen; i++) {
 
      // Insert 0's to the beginning
      // of b to make both the lengths equal
      b = "0" + b;
    }
 
    // Add (complement of B) and A
    String c = sumBig(a, complement9(b));
 
    // If length of new string is greater
    // than length of first string,
    // than carry is generated
    if (c.length() > a.length()) {
 
      // bit1 is the carry bit
      char bit1 = c.charAt(0);
      String bit = "";
      bit += bit1;
      StringBuilder sb = new StringBuilder(c);
      sb.deleteCharAt(0);
      c = sb.toString();
      c = sumBig(c, bit);
      return c;
    }
 
    // If both lengths are equal
    else {
      return complement9(c);
    }
  }
  public static void main(String[] args)
  {
    String str1 = "12345678987654321";
    String str2 = "22324324343";
 
    System.out.println(subtract(str1, str2));
  }
}
 
// This code is contributed by garg28harsh.

Python3




# Python3 Program to implement
# the above approach
 
# Function to return sum of two
# large numbers given as strings
def sumBig(a, b):
 
    # Compare their lengths
    if (len(a) > len(b)):
     
        temp = a;
        a = b;
        b = temp;
     
     
    # Stores the result
    str1 = "";
 
    # Store the respective lengths
    n1 = len(a)
    n2 = len(b);
 
    diff = n2 - n1;
 
    # Initialize carry
    carry = 0;
 
    # Traverse from end of both strings
    for i in range(n1 - 1, -1, -1):
        # Compute sum of
        # current digits and carry
        sums = ( int(a[i]) + int(b[i + diff]) + carry);
 
        # Store the result
        str1 += str((sums % 10));
 
        # Update carry
        carry = int(sums / 10);
     
 
    # Add remaining digits of str2[]
    for i in range(n2 - n1 - 1, -1, -1):
 
        sums = (int(b[i]) + carry);
 
        str1 += str((sums % 10));
        carry = int(sums/ 10);
     
    # Add remaining carry
    if (carry > 0):
        str1 += str(carry)
 
    # Reverse resultant string
    str1 = str1[::-1]
 
    return str1;
 
# Function return 9's
# complement of given number
def complement9( v):
 
    # Stores the complement
    complement = "";
     
    for i in range(len(v)):
 
        # Subtract every bit from 9
        complement += str(9 - int(v[i]));
     
    # Return the result
    return complement;
 
# Function returns subtraction
# of two given numbers as strings
def subtract(a, b):
 
    # If second string is larger
    if (len(a) < len(b)):
     
        temp = a;
        a = b;
        b = temp;
     
    # Calculate respective lengths
    l1 = len(a)
    l2 = len(b);
 
    # If lengths are equal
    diffLen = l1 - l2;
     
    for i in range(diffLen):
 
        # Insert 0's to the beginning
        # of b to make both the lengths equal
        b = "0" + b;
     
    # Add (complement of B) and A
    c = sumBig(a, complement9(b));
 
    # If length of new string is greater
    # than length of first string,
    # than carry is generated
    if (len(c) > len(a)):
 
        # bit1 is the carry bit
        bit1 = c[0];
        bit = [ bit1 ];
        c = c[1::];
        c = sumBig(c, bit);
        return c;
     
    # If both lengths are equal
    else :
        return complement9(c);
     
# Driver Code
str1 = "12345678987654321";
str2 = "22324324343";
 
print(subtract(str1, str2));
 
# This code is contributed by phasing17.

C#




// C# Program to implement
// the above approach
using System;
using System.Linq;
using System.Collections.Generic;
 
class GFG
{
  // Function to return sum of two
  // large numbers given as strings
  static string sumBig(string a, string b)
  {
 
    // Compare their lengths
    if (a.Length > b.Length)
    {var temp = b; b = a; a = temp; };
 
    // Stores the result
    string str = "";
 
    // Store the respective lengths
    int n1 = a.Length,
    n2 = b.Length;
 
    int diff = n2 - n1;
 
    // Initialize carry
    int carry = 0;
 
    // Traverse from end of both strings
    for (int i = n1 - 1; i >= 0; i--) {
      // Compute sum of
      // current digits and carry
      int sum = ((a[i] - '0')
                 + (b[i + diff] - '0')
                 + carry);
 
      // Store the result
      str += (char)(sum % 10 + '0');
 
      // Update carry
      carry = sum / 10;
    }
 
    // Add remaining digits of str2[]
    for (int i = n2 - n1 - 1; i >= 0; i--) {
 
      int sum = ((b[i] - '0') + carry);
 
      str+= (char)(sum % 10 + '0');
      carry = sum / 10;
    }
 
    // Add remaining carry
    if (carry != 0)
      str += (char)(carry + '0');
 
    // Reverse resultant string
    char[] strc = str.ToCharArray();
    Array.Reverse(strc);
    return new string(strc);
  }
 
  // Function return 9's
  // complement of given number
  static string complement9(string v)
  {
    // Stores the complement
    string complement = "";
 
    for (int i = 0; i < v.Length; i++) {
 
      // Subtract every bit from 9
      complement += (char)('9' - v[i] + '0');
    }
 
    // Return the result
    return complement;
  }
 
  // Function returns subtraction
  // of two given numbers as strings
  static string subtract(string a, string b)
  {
 
    // If second string is larger
    if (a.Length < b.Length)
    {var temp = b; b = a; a = temp; };
 
    // Calculate respective lengths
    int l1 = a.Length,
    l2 = b.Length;
 
    // If lengths are equal
    int diffLen = l1 - l2;
 
    for (int i = 0; i < diffLen; i++) {
 
      // Insert 0's to the beginning
      // of b to make both the lengths equal
      b = "0" + b;
    }
 
    // Add (complement of B) and A
    string c = sumBig(a, complement9(b));
 
    // If length of new string is greater
    // than length of first string,
    // than carry is generated
    if (c.Length > a.Length) {
      List<char> c1 = c.ToList();
 
 
      // bit1 is the carry bit
      char bit1 = c[0];
      string bit = "" + bit1;
 
      c1.RemoveAt(0);
      c = new string(c1.ToArray());
 
      c = sumBig(c, bit);
      return c;
    }
 
    // If both lengths are equal
    else {
      return complement9(c);
    }
  }
 
  // Driver Code
  public static void Main(string[] args)
  {
 
    string str1 = "12345678987654321";
    string str2 = "22324324343";
 
    Console.WriteLine(subtract(str1, str2));
  }
}
 
// This code is contributed by phasing17.

Javascript




// JS Program to implement
// the above approach
 
// Function to return sum of two
// large numbers given as strings
function sumBig(a, b)
{
 
    // Compare their lengths
    if (a.length > b.length)
    {
        let temp = a;
        a = b;
        b = temp;
    }
     
    // Stores the result
    let str = "";
 
    // Store the respective lengths
    let n1 = a.length,
        n2 = b.length;
 
    let diff = n2 - n1;
 
    // Initialize carry
    let carry = 0;
 
    // Traverse from end of both strings
    for (let i = n1 - 1; i >= 0; i--) {
        // Compute sum of
        // current digits and carry
        let sum = ( parseInt(a[i])
                   + parseInt(b[i + diff])
                   + carry);
 
        // Store the result
        str += ((sum % 10).toString());
 
        // Update carry
        carry = Math.floor(sum / 10);
    }
 
    // Add remaining digits of str2[]
    for (var i = n2 - n1 - 1; i >= 0; i--) {
 
        var sum = (parseInt(b[i]) + carry);
 
        str += ((sum % 10).toString());
        carry = Math.floor(sum / 10);
    }
 
    // Add remaining carry
    if (carry > 0)
        str += (carry.toString());
 
    // Reverse resultant string
    str = str.split("").reverse().join("")
 
    return str;
}
 
// Function return 9's
// complement of given number
function complement9( v)
{
    // Stores the complement
    let complement = "";
 
    for (var i = 0; i < v.length; i++) {
 
        // Subtract every bit from 9
        complement += (9 - parseInt(v[i])).toString();
    }
 
    // Return the result
    return complement;
}
 
// Function returns subtraction
// of two given numbers as strings
function subtract(a, b)
{
 
    // If second string is larger
    if (a.length < b.length)
    {
        let temp = a;
        a = b;
        b = temp;
    }
 
    // Calculate respective lengths
    let l1 = a.length,
        l2 = b.length;
 
    // If lengths are equal
    let diffLen = l1 - l2;
 
    for (let i = 0; i < diffLen; i++) {
 
        // Insert 0's to the beginning
        // of b to make both the lengths equal
        b = "0" + b;
    }
 
    // Add (complement of B) and A
    let c = sumBig(a, complement9(b));
 
    // If length of new string is greater
    // than length of first string,
    // than carry is generated
    if (c.length > a.length) {
 
        // bit1 is the carry bit
        let bit1 = c[0];
        let bit = [ bit1 ];
        c = c.substr(1);
        c = sumBig(c, bit);
        return c;
    }
 
    // If both lengths are equal
    else {
        return complement9(c);
    }
}
 
// Driver Code
let str1 = "12345678987654321";
let str2 = "22324324343";
 
console.log(subtract(str1, str2));
 
// This code is contributed by phasing17.

Output: 

12345656663329978

 

Time Complexity: O(max(N, M)) 
Auxiliary Space: O(max(N, M))
 


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