Solution Explanation for Similar RGB Color

The problem asks to find the shorthand RGB color that is most similar to a given full RGB color. A shorthand RGB color is represented as "#XYZ", where X, Y, and Z are hexadecimal digits representing the red, green, and blue components, respectively. The full RGB color is "#ABCDEF", where AB, CD, and EF are the hexadecimal representations of the red, green, and blue components.

The similarity between two colors is defined as -(AB - UV)² - (CD - WX)² - (EF - YZ)², where "#ABCDEF" and "#UVWXYZ" are the two colors. We need to find the shorthand color that maximizes this similarity.

Approach:

The key observation is that to find the most similar shorthand color, we can independently round each component (R, G, B) of the full color to the nearest multiple of 17 (because each component in the shorthand is represented by a single hex digit, which ranges from 00 to ff, in steps of 17).

1. Extract Components: The input color string is parsed to extract the red, green, and blue components.

2. Rounding to Nearest Multiple of 17: Each component (a hex string like "09", "f1", "66") is converted to its decimal equivalent using int(x, 16). Then we divide by 17. If the remainder is greater than 8, we round up (add 1 to the quotient). Otherwise, we keep the quotient. This quotient is then multiplied by 17 to get the nearest multiple of 17, which is converted back to a two-digit hexadecimal string using '{:02x}'.format(17 * y) (Python), String.format("%02x", 17 * q) (Java), or fmt.Sprintf("%02x", 17*q) (Go).

3. Construct Shorthand Color: The rounded components are concatenated to form the shorthand color string with a "#" prefix.

Time and Space Complexity Analysis:

• Time Complexity: O(1). The algorithm involves a constant number of operations regardless of the input size (the input is always a 7-character string).
• Space Complexity: O(1). The algorithm uses a constant amount of extra space to store the intermediate variables.

Code Examples (with explanations):

The provided code examples in Python, Java, and Go all follow the same approach described above. The f() function in each language encapsulates the logic of rounding to the nearest multiple of 17. The main part of the code then extracts the components, calls f() on each, and constructs the final result.

Example (Python):

def similarRGB(color: str) -> str:
    def f(x):  # Helper function for rounding
        y, z = divmod(int(x, 16), 17) # Convert hex to decimal, then divide by 17. y is quotient, z is remainder
        if z > 8:  # Round up if remainder > 8
            y += 1
        return '{:02x}'.format(17 * y) # Convert back to 2-digit hex
 
    a, b, c = color[1:3], color[3:5], color[5:7] # Extract RGB components
    return f'#{f(a)}{f(b)}{f(c)}' # Construct shorthand color

The other languages (Java and Go) implement the same logic with their respective syntax and standard libraries. The core idea remains consistent: extract components, round to the nearest multiple of 17, and construct the shorthand color.