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Mastering String Multiplication: A Guide to Multiplying String-Represented Integers

By Talent Navigator

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Published May 28, 2025

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4 min read

https://www.youtube.com/watch?v=aDapLH1SjLk&ab_channel=TalentNavigator

Understanding how to multiply string-represented integers is essential for tackling a common coding challenge, especially in interview settings such as LeetCode. This article will detail a step-by-step approach to effectively multiply two numbers represented as strings, all while adhering to the restriction of avoiding direct integer conversion.

The Problem Statement

Given two non-negative integers as strings, num1 and num2, the objective is to return their product, also as a string. The challenge lies in the restriction that neither built-in big libraries nor direct conversions to integers can be employed. This forces us to think creatively about how multiplication is fundamentally conducted.

Understanding the Constraints

Before delving into the solution, let’s outline important constraints and considerations:

• Input Types: Inputs will be provided as strings.
• No Direct Conversion: We cannot convert the strings to integers to perform operations directly.
• Output Requirements: The result must also be returned as a string.

Edge Case:

• If either num1 or num2 is "0", the product is "0". Therefore, we can return "0" immediately in this scenario to avoid unnecessary calculations.

Approach to Solve the Problem

To achieve our goal, let’s break down our approach into a structured algorithm:

Step 1: Initialize Variables

• Storage for Result: First, we need storage to hold the result of our multiplication. Since the maximum possible length of the product of two numbers (with m and n as their lengths) can be m + n, we can create an array (or list) with this length initialized to zero.

Step 2: Multiply Digit by Digit

• Utilize a nested loop where each digit in num1 is multiplied by each digit in num2 in a reversed manner (from the least significant digit to the most significant).

Step 3: Compute and Store the Result

• For each digit multiplication, the result will fit into our previously initialized array. If the product of digits is two digits (e.g., 18 when multiplying 3 and 6), we should:
  1. Store the units in the current position (using modulo operation).
  2. Carry over the tens to the next position in the result array (using integer division).

Step 4: Convert Back to String

• After populating our results array, we need to handle leading zeros and convert the array contents back to a string format.

Step 5: Implementation Details

Python code-

def multiply(num1: str, num2: str) -> str:
    if num1 == "0" or num2 == "0":
        return "0"

    m, n = len(num1), len(num2)
    result = [0] * (m + n)

    for i in range(m - 1, -1, -1):
        for j in range(n - 1, -1, -1):
            mul = (ord(num1[i]) - ord('0')) * (ord(num2[j]) - ord('0'))
            p1, p2 = i + j, i + j + 1
            total = mul + result[p2]
            result[p2] = total % 10
            result[p1] += total // 10

    result_str = ''.join(map(str, result)).lstrip('0')
    return result_str if result_str else "0"

Explanation of the Implementation

• Output Handling: After performing all the multiplications and accumulating any carries, we convert the result list back to a string. To prevent leading zeros, we utilize lstrip('0').
• Character to Integer Conversion: The character-to-integer conversion is achieved by subtracting '0' from the character, thus converting the character to its corresponding integer value.
• Efficiency: This algorithm runs in O(m * n) time, where m is the length of num1 and n is the length of num2, making it efficient given the constraints.

Conclusion

Multiplying strings that represent large integers is a common programming challenge that tests one's understanding of fundamental mathematical operations without relying on built-in functionalities. By following this structured approach, programmers can effectively tackle this challenge while gaining deeper insights into string manipulation and number representation in programming.