43. Multiply Strings - Detailed Explanation

Problem Statement

You are given two non-negative numbers, num1 and num2, represented as strings. Your task is to return the product of these numbers, also represented as a string. You are not allowed to use any built-in library for handling large integers or convert the inputs directly to integers.

Example 1
Input: num1 = "2", num2 = "3"
Output: "6"
Explanation: 2 multiplied by 3 equals 6.

Example 2
Input: num1 = "123", num2 = "456"
Output: "56088"
Explanation: Multiplying 123 by 456 gives 56088.

Example 3
Input: num1 = "0", num2 = "52"
Output: "0"
Explanation: Any number multiplied by 0 is 0.

Constraints:

• Both num1 and num2 consist only of digits (0-9).

• They do not contain any leading zeros, except the number "0" itself.

• (1 \leq \text{num1.length}, \text{num2.length} \leq 200)

Hints

1. Simulate School Multiplication:
   Imagine multiplying the numbers as you would do on paper. Multiply each digit of one number by every digit of the other, taking care of the position of each digit.

2. Use an Array for Intermediate Results:
   Since the maximum length of the product of two numbers of length (m) and (n) is (m+n), create an array of size (m+n) to store the intermediate sums and manage the carry appropriately.

Approach 1: Simulation of Multiplication (School Algorithm)

Explanation

This approach simulates the traditional multiplication algorithm taught in school:

1. Initialize:
   Let (m) and (n) be the lengths of num1 and num2 respectively. Create an array result of size (m+n) initialized with zeros.

2. Multiply Digit by Digit:
   For each digit in num1 (starting from the rightmost digit), multiply it with every digit in num2 (also starting from the rightmost digit). The product of the digits at positions (i) and (j) will contribute to the positions (i+j) and (i+j+1) in the result array.

3. Manage Carry:
   Add the multiplication result to the current position in the result array and manage the carry by updating the next left position.

4. Convert to String:
   After processing all digits, skip any leading zeros in the result array and convert the remaining digits into a string.

Step-by-Step Walkthrough

• Step 1:
  Let num1 = "123" and num2 = "456". Initialize an array of size (3 + 3 = 6):
  result = [0, 0, 0, 0, 0, 0].

• Step 2:
  Start with the rightmost digit of num1 ('3') and multiply it with each digit of num2. For instance, multiplying '3' and '6' gives 18. Add 18 to the appropriate position in result and update the carry.

• Step 3:
  Continue this process for each digit, updating the array with sums and carries.

• Step 4:
  After processing, if the array is [0, 5, 6, 0, 8, 8], remove any leading zeros and convert it to "56088".

Complexity Analysis

• Time Complexity: (O(m \times n)), where (m) is the length of num1 and (n) is the length of num2.

• Space Complexity: (O(m + n)), used by the result array.

Approach 2: Alternative Brute Force (Not Recommended)

Explanation

A naive approach might involve converting the strings to integers, multiplying them, and converting the result back to a string. However, this approach is not allowed by the problem constraints since you must not use any built-in library for handling big integers or perform direct conversions.

Code Implementations

Python Code