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Complex Number Multiplication

Medium

Asked by:

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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].
imaginary is the imaginary part and is an integer in the range [-100, 100].
i² == -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 + i2 + 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 + i2 - 2 * i = -2i, and you need convert it to the form of 0+-2i.

Constraints:

num1 and num2 are valid complex numbers.

Try coding it in LeetCode

Complex Number Multiplication

Medium

Asked by:

6 views

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].
imaginary is the imaginary part and is an integer in the range [-100, 100].
i² == -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 + i2 + 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 + i2 - 2 * i = -2i, and you need convert it to the form of 0+-2i.

Constraints:

num1 and num2 are valid complex numbers.

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the format of the input strings representing the complex numbers? Specifically, are they always in the form 'a+bi' and will 'a' and 'b' always be integers?
What is the range of possible values for the real and imaginary parts (a and b) of the complex numbers? Should I be concerned about potential integer overflow during multiplication?
Should the returned string be in the exact format 'c+di', including the '+' sign even if 'd' is negative or zero, and are there any restrictions on whitespace?
Are the input strings guaranteed to be valid complex numbers in the specified format? If not, how should I handle invalid input?
Can the real or imaginary part of either complex number be zero?

Brute Force Solution

Approach

We're given two complex numbers as text, and we need to multiply them together and return the result, also as text. The brute force approach involves directly applying the distributive property of multiplication, similar to how you'd multiply two binomials in algebra.

Here's how the algorithm would work step-by-step:

First, separate each complex number into its real and imaginary parts.
Then, treat the multiplication like multiplying two expressions: (a + bi) * (c + di).
Multiply each part of the first complex number by each part of the second complex number: a*c, a*di, bi*c, and bi*di.
Combine the real parts (a*c and bi*di) and simplify, remembering that i squared is -1.
Combine the imaginary parts (a*di and bi*c).
Finally, put the real and imaginary parts together in the standard complex number format: real + imaginary i.

Code Implementation

def complexNumberMultiply(num1: str, num2: str) -> str:

    # Split the complex numbers into real and imaginary parts
    real_part_num1, imaginary_part_num1 = map(int, num1[:-1].split('+'))
    real_part_num2, imaginary_part_num2 = map(int, num2[:-1].split('+'))

    # Calculate the real part of the result.
    # Remember i*i = -1, so (b*i)*(d*i) = -b*d, which reduces the real part
    real_part = (real_part_num1 * real_part_num2) - \
                (imaginary_part_num1 * imaginary_part_num2)

    # Calculate the imaginary part of the result
    imaginary_part = (real_part_num1 * imaginary_part_num2) + \
                     (imaginary_part_num1 * real_part_num2)

    # Format the result as a string
    return str(real_part) + '+' + str(imaginary_part) + 'i'

Big(O) Analysis

Time Complexity

O(1) – The algorithm involves extracting real and imaginary parts from two complex numbers represented as strings, and then performing a fixed number of arithmetic operations (multiplications and additions) to compute the real and imaginary parts of the resulting complex number. These operations take constant time. The length of the input string doesn't affect the number of these operations. Therefore, the time complexity is O(1).

Space Complexity

O(1) – The described algorithm primarily uses a fixed number of variables to store the real and imaginary parts of the complex numbers during multiplication (a, b, c, d, ac, adi, bic, bi_di, real_result, imaginary_result). The amount of extra space required does not scale with the input string size N (the length of the complex number strings). Therefore, the auxiliary space complexity is constant.

Optimal Solution

Approach

The most efficient way to multiply complex numbers presented as strings involves directly applying the distributive property of multiplication. This method avoids unnecessary string manipulations and directly computes the real and imaginary components of the result. It treats the complex numbers as algebraic expressions and multiplies them out accordingly.

Here's how the algorithm would work step-by-step:

Recognize that each complex number string is in the form 'a+bi', where 'a' is the real part and 'b' is the imaginary part.
Extract the real and imaginary parts from both complex number strings. Think of these as four individual numbers that you'll use in the next step.
Multiply the complex numbers using the distributive property, just like you would with algebraic expressions: (a + bi) * (c + di) = ac + adi + bci + bdi*i. Remember that i*i (i squared) is -1.
Simplify the result by combining the real parts (ac - bd) and the imaginary parts (ad + bc).
Form the result string using the simplified real and imaginary parts in the format 'real+imaginaryi'.
Return the resulting complex number string.

Code Implementation

def complexNumberMultiply(num1, num2):
    real_part_1, imaginary_part_1 = map(int, num1[:-1].split('+'))
    real_part_2, imaginary_part_2 = map(int, num2[:-1].split('+'))

    # Calculate real part of result
    real_part_result = (real_part_1 * real_part_2) - \
                         (imaginary_part_1 * imaginary_part_2)

    # Calculate imaginary part of result
    imaginary_part_result = (real_part_1 * imaginary_part_2) + \
                            (imaginary_part_1 * real_part_2)

    # Construct the final result string
    return str(real_part_result) + '+' + str(imaginary_part_result) + 'i'

Big(O) Analysis

Time Complexity

O(1) – The algorithm involves extracting real and imaginary parts from two fixed-size strings. The arithmetic operations (multiplication, addition, subtraction) performed on these extracted numbers take constant time. The formation of the result string also takes constant time. Therefore, the overall time complexity is O(1), constant time, regardless of the input.

Space Complexity

O(1) – The algorithm extracts real and imaginary parts from the input strings, storing them as four integer variables. It then performs arithmetic operations and constructs a new string. The space used for these variables (the four integers and potentially a few more for intermediate calculations, plus the final string which is of constant bounded size), does not scale with the size of the input strings; it remains constant regardless of the length of the complex number strings. Therefore, the auxiliary space complexity is O(1).

Edge Cases

Case	How to Handle
Null or empty input strings	Return a default value like '0+0i' if either input is null or empty to avoid NullPointerExceptions or incorrect parsing.
Input strings not in the specified 'a+bi' format	Throw an IllegalArgumentException or return a default value if the input strings do not match the expected format.
Integer overflow during multiplication of real and imaginary parts	Use long data type to store intermediate multiplication results before converting to Integer, to avoid possible overflow issues.
Real and imaginary parts are very large positive or negative numbers	Using long to avoid overflow during intermediate calculation will handle very large positive or negative numbers.
Real or imaginary parts are zero	The multiplication formula will correctly handle cases where real or imaginary parts are zero, resulting in either zero or correct real/imaginary parts.
Input strings with leading or trailing whitespace	Trim whitespace from the input strings before parsing to ensure accurate extraction of real and imaginary parts.
Input strings with incorrect signs (e.g., 'a-bi' instead of 'a+-bi')	The parsing logic should correctly handle both positive and negative signs for the imaginary part, regardless of their position.
Large number of consecutive calculations	The solution has a constant time complexity and constant space complexity, so consecutive calculation will perform well.

Null or empty input strings

How to Handle:

Return a default value like '0+0i' if either input is null or empty to avoid NullPointerExceptions or incorrect parsing.

Input strings not in the specified 'a+bi' format

How to Handle:

Throw an IllegalArgumentException or return a default value if the input strings do not match the expected format.

Integer overflow during multiplication of real and imaginary parts

How to Handle:

Use long data type to store intermediate multiplication results before converting to Integer, to avoid possible overflow issues.

Real and imaginary parts are very large positive or negative numbers

How to Handle:

Using long to avoid overflow during intermediate calculation will handle very large positive or negative numbers.

Real or imaginary parts are zero

How to Handle:

The multiplication formula will correctly handle cases where real or imaginary parts are zero, resulting in either zero or correct real/imaginary parts.

Input strings with leading or trailing whitespace

How to Handle:

Trim whitespace from the input strings before parsing to ensure accurate extraction of real and imaginary parts.

Input strings with incorrect signs (e.g., 'a-bi' instead of 'a+-bi')

How to Handle:

The parsing logic should correctly handle both positive and negative signs for the imaginary part, regardless of their position.

Large number of consecutive calculations

How to Handle:

The solution has a constant time complexity and constant space complexity, so consecutive calculation will perform well.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the format of the input strings representing the complex numbers? Specifically, are they always in the form 'a+bi' and will 'a' and 'b' always be integers?
What is the range of possible values for the real and imaginary parts (a and b) of the complex numbers? Should I be concerned about potential integer overflow during multiplication?
Should the returned string be in the exact format 'c+di', including the '+' sign even if 'd' is negative or zero, and are there any restrictions on whitespace?
Are the input strings guaranteed to be valid complex numbers in the specified format? If not, how should I handle invalid input?
Can the real or imaginary part of either complex number be zero?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

First, separate each complex number into its real and imaginary parts.
Then, treat the multiplication like multiplying two expressions: (a + bi) * (c + di).
Multiply each part of the first complex number by each part of the second complex number: a*c, a*di, bi*c, and bi*di.
Combine the real parts (a*c and bi*di) and simplify, remembering that i squared is -1.
Combine the imaginary parts (a*di and bi*c).
Finally, put the real and imaginary parts together in the standard complex number format: real + imaginary i.

Code Implementation

def complexNumberMultiply(num1: str, num2: str) -> str:

    # Split the complex numbers into real and imaginary parts
    real_part_num1, imaginary_part_num1 = map(int, num1[:-1].split('+'))
    real_part_num2, imaginary_part_num2 = map(int, num2[:-1].split('+'))

    # Calculate the real part of the result.
    # Remember i*i = -1, so (b*i)*(d*i) = -b*d, which reduces the real part
    real_part = (real_part_num1 * real_part_num2) - \
                (imaginary_part_num1 * imaginary_part_num2)

    # Calculate the imaginary part of the result
    imaginary_part = (real_part_num1 * imaginary_part_num2) + \
                     (imaginary_part_num1 * real_part_num2)

    # Format the result as a string
    return str(real_part) + '+' + str(imaginary_part) + 'i'

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

Recognize that each complex number string is in the form 'a+bi', where 'a' is the real part and 'b' is the imaginary part.
Extract the real and imaginary parts from both complex number strings. Think of these as four individual numbers that you'll use in the next step.
Multiply the complex numbers using the distributive property, just like you would with algebraic expressions: (a + bi) * (c + di) = ac + adi + bci + bdi*i. Remember that i*i (i squared) is -1.
Simplify the result by combining the real parts (ac - bd) and the imaginary parts (ad + bc).
Form the result string using the simplified real and imaginary parts in the format 'real+imaginaryi'.
Return the resulting complex number string.

Code Implementation

def complexNumberMultiply(num1, num2):
    real_part_1, imaginary_part_1 = map(int, num1[:-1].split('+'))
    real_part_2, imaginary_part_2 = map(int, num2[:-1].split('+'))

    # Calculate real part of result
    real_part_result = (real_part_1 * real_part_2) - \
                         (imaginary_part_1 * imaginary_part_2)

    # Calculate imaginary part of result
    imaginary_part_result = (real_part_1 * imaginary_part_2) + \
                            (imaginary_part_1 * real_part_2)

    # Construct the final result string
    return str(real_part_result) + '+' + str(imaginary_part_result) + 'i'

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input strings	Return a default value like '0+0i' if either input is null or empty to avoid NullPointerExceptions or incorrect parsing.
Input strings not in the specified 'a+bi' format	Throw an IllegalArgumentException or return a default value if the input strings do not match the expected format.
Integer overflow during multiplication of real and imaginary parts	Use long data type to store intermediate multiplication results before converting to Integer, to avoid possible overflow issues.
Real and imaginary parts are very large positive or negative numbers	Using long to avoid overflow during intermediate calculation will handle very large positive or negative numbers.
Real or imaginary parts are zero	The multiplication formula will correctly handle cases where real or imaginary parts are zero, resulting in either zero or correct real/imaginary parts.
Input strings with leading or trailing whitespace	Trim whitespace from the input strings before parsing to ensure accurate extraction of real and imaginary parts.
Input strings with incorrect signs (e.g., 'a-bi' instead of 'a+-bi')	The parsing logic should correctly handle both positive and negative signs for the imaginary part, regardless of their position.
Large number of consecutive calculations	The solution has a constant time complexity and constant space complexity, so consecutive calculation will perform well.

Null or empty input strings

How to Handle:

Return a default value like '0+0i' if either input is null or empty to avoid NullPointerExceptions or incorrect parsing.

Input strings not in the specified 'a+bi' format

How to Handle:

Throw an IllegalArgumentException or return a default value if the input strings do not match the expected format.

Integer overflow during multiplication of real and imaginary parts

How to Handle:

Use long data type to store intermediate multiplication results before converting to Integer, to avoid possible overflow issues.

Real and imaginary parts are very large positive or negative numbers

How to Handle:

Using long to avoid overflow during intermediate calculation will handle very large positive or negative numbers.

Real or imaginary parts are zero

How to Handle:

The multiplication formula will correctly handle cases where real or imaginary parts are zero, resulting in either zero or correct real/imaginary parts.

Input strings with leading or trailing whitespace

How to Handle:

Trim whitespace from the input strings before parsing to ensure accurate extraction of real and imaginary parts.

Input strings with incorrect signs (e.g., 'a-bi' instead of 'a+-bi')

How to Handle:

The parsing logic should correctly handle both positive and negative signs for the imaginary part, regardless of their position.

Large number of consecutive calculations

How to Handle:

The solution has a constant time complexity and constant space complexity, so consecutive calculation will perform well.