43. Multiply Strings
Description
Given two non-negative integers num1
and num2
represented as strings, return the product of num1
and num2
, also represented as a string.
Note: You must not use any built-in BigInteger library or convert the inputs to integer directly.
Example 1:
Input: num1 = "2", num2 = "3" Output: "6"
Example 2:
Input: num1 = "123", num2 = "456" Output: "56088"
Constraints:
1 <= num1.length, num2.length <= 200
num1
andnum2
consist of digits only.- Both
num1
andnum2
do not contain any leading zero, except the number0
itself.
Solutions
Solution 1: Simulating Mathematical Multiplication
Assume the lengths of $num1$ and $num2$ are $m$ and $n$ respectively, then the length of their product can be at most $m + n$.
The proof is as follows:
- If $num1$ and $num2$ both take the minimum value, then their product is ${10}^{m - 1} \times {10}^{n - 1} = {10}^{m + n - 2}$, with a length of $m + n - 1$.
- If $num1$ and $num2$ both take the maximum value, then their product is $({10}^m - 1) \times ({10}^n - 1) = {10}^{m + n} - {10}^m - {10}^n + 1$, with a length of $m + n$.
Therefore, we can apply for an array of length $m + n$ to store each digit of the product.
From the least significant digit to the most significant digit, we calculate each digit of the product in turn, and finally convert the array into a string.
Note to check whether the most significant digit is $0$, if it is, remove it.
The time complexity is $O(m \times n)$, and the space complexity is $O(m + n)$. Here, $m$ and $n$ are the lengths of $num1$ and $num2$ respectively.
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