Complex Number Multiplication

Medium

Amazon

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Strings

A complex number can be represented as a string on the form "real+imaginaryi" where:

real is the real part and is an integer in the range [-100, 100]. For example, in the string "1+2i", the real part is 1.
imaginary is the imaginary part and is an integer in the range [-100, 100]. For example, in the string "1+2i", the imaginary part is 2.
i^2 == -1.

Given two complex numbers num1 and num2 as strings, return a string of the complex number that represents their multiplications.

Example 1:

Input: num1 = "1+1i", num2 = "1+1i"
Output: "0+2i"
Explanation: (1 + i) * (1 + i) = 1 + i^2 + 2 * i = 2i, and you need convert it to the form of 0+2i.

Example 2:

Input: num1 = "1+-1i", num2 = "1+-1i"
Output: "0+-2i"
Explanation: (1 - i) * (1 - i) = 1 + i^2 - 2 * i = -2i, and you need convert it to the form of 0+-2i.

Write a function to perform this complex number multiplication, considering the constraints and edge cases, and aiming for an optimal solution.

Complex Number Multiplication

Medium

Amazon

1 view

Topics:

Strings

A complex number can be represented as a string on the form "real+imaginaryi" where:

real is the real part and is an integer in the range [-100, 100]. For example, in the string "1+2i", the real part is 1.
imaginary is the imaginary part and is an integer in the range [-100, 100]. For example, in the string "1+2i", the imaginary part is 2.
i^2 == -1.

Given two complex numbers num1 and num2 as strings, return a string of the complex number that represents their multiplications.

Example 1:

Input: num1 = "1+1i", num2 = "1+1i"
Output: "0+2i"
Explanation: (1 + i) * (1 + i) = 1 + i^2 + 2 * i = 2i, and you need convert it to the form of 0+2i.

Example 2:

Input: num1 = "1+-1i", num2 = "1+-1i"
Output: "0+-2i"
Explanation: (1 - i) * (1 - i) = 1 + i^2 - 2 * i = -2i, and you need convert it to the form of 0+-2i.

Write a function to perform this complex number multiplication, considering the constraints and edge cases, and aiming for an optimal solution.

Solution

Naive Solution

The most straightforward way to solve this problem is to parse the input strings to extract the real and imaginary parts, perform the complex number multiplication, and then format the result back into the required string format.

Steps:

Parse Input: Split each input string into its real and imaginary parts.
Multiply: Perform complex number multiplication: (a + bi) * (c + di) = (ac - bd) + (ad + bc)i
Format Output: Construct the output string in the format "real+imaginaryi".

Code (Python):

def complexNumberMultiply(num1: str, num2: str) -> str:
    def parse_complex(num: str) -> tuple[int, int]:
        real, imaginary = num.split('+')
        imaginary = imaginary[:-1]
        return int(real), int(imaginary)

    real1, imaginary1 = parse_complex(num1)
    real2, imaginary2 = parse_complex(num2)

    real_part = (real1 * real2) - (imaginary1 * imaginary2)
    imaginary_part = (real1 * imaginary2) + (real2 * imaginary1)

    return f"{real_part}+{imaginary_part}i"

Explanation:

The parse_complex function extracts the real and imaginary parts from the input string.
The real and imaginary parts of the product are calculated using the formula for complex number multiplication.
The result is formatted into the required string format.

Big O Analysis

Time Complexity: O(1) - The operations performed are constant time operations.
Space Complexity: O(1) - Only a constant amount of extra space is used.

Optimal Solution

The provided naive solution is already optimal as the complex number multiplication takes constant time. No other approach could theoretically provide better time complexity.

Edge Cases:

Input strings are guaranteed to be in the correct format, so no validation is strictly necessary.
The ranges of the real and imaginary parts are bounded [-100, 100], which means the product will not cause any integer overflow problems.

Solution

Naive Solution

Steps:

Parse Input: Split each input string into its real and imaginary parts.
Multiply: Perform complex number multiplication: (a + bi) * (c + di) = (ac - bd) + (ad + bc)i
Format Output: Construct the output string in the format "real+imaginaryi".

Code (Python):

def complexNumberMultiply(num1: str, num2: str) -> str:
    def parse_complex(num: str) -> tuple[int, int]:
        real, imaginary = num.split('+')
        imaginary = imaginary[:-1]
        return int(real), int(imaginary)

    real1, imaginary1 = parse_complex(num1)
    real2, imaginary2 = parse_complex(num2)

    real_part = (real1 * real2) - (imaginary1 * imaginary2)
    imaginary_part = (real1 * imaginary2) + (real2 * imaginary1)

    return f"{real_part}+{imaginary_part}i"

Explanation:

The parse_complex function extracts the real and imaginary parts from the input string.
The real and imaginary parts of the product are calculated using the formula for complex number multiplication.
The result is formatted into the required string format.

Big O Analysis

Time Complexity: O(1) - The operations performed are constant time operations.
Space Complexity: O(1) - Only a constant amount of extra space is used.

Optimal Solution

The provided naive solution is already optimal as the complex number multiplication takes constant time. No other approach could theoretically provide better time complexity.

Edge Cases:

Input strings are guaranteed to be in the correct format, so no validation is strictly necessary.
The ranges of the real and imaginary parts are bounded [-100, 100], which means the product will not cause any integer overflow problems.