Ambiguous Coordinates

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StringsRecursion

We had some 2-dimensional coordinates, like "(1, 3)" or "(2, 0.5)". Then, we removed all commas, decimal points, and spaces and ended up with the string s.

For example, "(1, 3)" becomes s = "(13)" and "(2, 0.5)" becomes s = "(205)".

Return a list of strings representing all possibilities for what our original coordinates could have been.

Our original representation never had extraneous zeroes, so we never started with numbers like "00", "0.0", "0.00", "1.0", "001", "00.01", or any other number that can be represented with fewer digits. Also, a decimal point within a number never occurs without at least one digit occurring before it, so we never started with numbers like ".1".

The final answer list can be returned in any order. All coordinates in the final answer have exactly one space between them (occurring after the comma.)

Example 1:

Input: s = "(123)"
Output: ["(1, 2.3)","(1, 23)","(1.2, 3)","(12, 3)"]

Example 2:

Input: s = "(0123)"
Output: ["(0, 1.23)","(0, 12.3)","(0, 123)","(0.1, 2.3)","(0.1, 23)","(0.12, 3)"]
Explanation: 0.0, 00, 0001 or 00.01 are not allowed.

Example 3:

Input: s = "(00011)"
Output: ["(0, 0.011)","(0.001, 1)"]

Constraints:

4 <= s.length <= 12
s[0] == '(' and s[s.length - 1] == ')'.
The rest of s are digits.

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 maximum length of the input string `s`? Can `s` be empty?
Can the input string `s` contain leading zeros, and if so, how should they be handled when forming the coordinates?
If no valid coordinates can be formed from the input string, what should the return value be? An empty list?
Are there any invalid characters besides digits in the input string `s` that I need to consider?
If multiple valid sets of coordinates are possible, is there any specific ordering required in the output list?

Brute Force Solution

Approach

We're given a string of numbers, and we want to find all possible ways to split it into two numbers that could represent coordinates. The brute force approach is to try every single possible split and check if the resulting numbers are valid coordinates.

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

Start by trying to split the string into two parts: the first part containing just the first digit and the second part containing the rest.
Then try the first part containing the first two digits, and the second part containing the rest, and so on until the first part contains all but the last digit.
For each of these splits, check if the first part can be a valid x-coordinate.
To do that, try placing a decimal point in all possible positions within the digits of the first part (or not at all), and see if the resulting number is a valid number.
A valid number should not have leading zeros unless it is just a single zero, and if it has a decimal point, it should not have trailing zeros.
Do the same for the second part to check if it can be a valid y-coordinate.
If both the first and second parts are valid coordinates, then we have found a possible solution. Save that solution.
Repeat the process of splitting, placing the decimal point, and validating for every possible split of the original string.
In the end, you'll have a collection of all possible solutions, each one representing a possible way to interpret the original string as a set of valid coordinates.

Code Implementation

def ambiguous_coordinates(s):
    s = s[1:-1]
    results = []
    for i in range(1, len(s)): 
        first_part = s[:i]
        second_part = s[i:]

        # Check for valid x and y coordinates
        possible_x = possible_numbers(first_part)
        possible_y = possible_numbers(second_part)

        for x_coordinate in possible_x:
            for y_coordinate in possible_y:
                results.append(f'({x_coordinate}, {y_coordinate})')
    return results

def possible_numbers(s):
    possible_numbers_list = []
    for i in range(len(s) + 1):
        if i == 0:
            possible_numbers_list.append(s)
            continue

        number_with_decimal = s[:i] + '.' + s[i:]
        possible_numbers_list.append(number_with_decimal)

    valid_numbers = []
    for number in possible_numbers_list:
        if is_valid(number):
            valid_numbers.append(number)

    return valid_numbers

def is_valid(s):
    if '.' not in s:
        if s[0] == '0' and len(s) > 1:
            return False

    # No leading zero for integer part
    if '.' in s and s[0] == '0' and s[1] != '.':
        return False

    # No trailing zero for decimal part.
    if '.' in s and s[-1] == '0':
        return False

    # Only a single '0' is ok
    if s == '0':
        return True

    # A number like 0. is invalid
    if '.' in s and len(s) == s.index('.')+1:
        return False

    return True

Big(O) Analysis

Time Complexity

O(n^3) – The algorithm iterates through all possible splits of the input string of length n, which takes O(n) time. For each split, it tries all possible decimal point placements in both the x and y coordinates. Placing the decimal can take up to O(n) time for each coordinate, resulting in O(n) for x and O(n) for y. Thus for both coordinates it takes O(n), so overall it's O(n) * O(n) * O(n). This simplifies to O(n^3).

Space Complexity

O(N^2) – The primary space complexity stems from storing valid coordinate strings. The algorithm iterates through all possible splits of the input string of length N, leading to O(N) possible splits. For each split, it generates possible valid numbers by inserting a decimal point at various locations. In the worst-case, each split can generate O(N) valid coordinate strings (x and y). Since we collect all possible valid coordinates, the space to store them can grow up to O(N * N) = O(N^2) in the worst-case scenario where most generated numbers are valid. Therefore, the overall space complexity is O(N^2).

Optimal Solution

Approach

We are given a string representing coordinates with the form (x, y) and we want to find all valid interpretations by inserting a comma and possibly decimal points. The key idea is to systematically explore all possible placements of the comma and decimal points, ensuring that each resulting number is valid according to the problem's rules.

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

Take the input string, and remove the parentheses at the beginning and the end.
Iterate through all possible positions to place the comma that separates the x and y coordinates.
For each comma position, extract the potential x and y coordinate strings.
For each of the x and y strings, generate all valid number strings by inserting decimal points (or not).
A number string is valid if it adheres to these rules: it cannot have leading zeros unless it is just '0'. It also cannot have trailing zeros after the decimal point.
Combine the valid x and y coordinate strings to form coordinate pairs in the required format (x, y).
Collect all the valid coordinate pairs you have found.
The result is the list of all possible valid coordinate strings.

Code Implementation

def ambiguous_coordinates(input_string):
    input_string = input_string[1:-1]
    number_of_characters = len(input_string)
    result = []

    for i in range(1, number_of_characters):
        x_coordinate_string = input_string[:i]
        y_coordinate_string = input_string[i:]

        possible_x_coordinates = generate_valid_numbers(x_coordinate_string)
        possible_y_coordinates = generate_valid_numbers(y_coordinate_string)

        # Need both coordinates to be valid to create a valid coordinate pair
        if possible_x_coordinates and possible_y_coordinates:
            for x_coordinate in possible_x_coordinates:
                for y_coordinate in possible_y_coordinates:
                    result.append("(" + x_coordinate + ", " + y_coordinate + ")")

    return result

def generate_valid_numbers(number_string):
    number_of_characters = len(number_string)
    result = []

    for i in range(number_of_characters + 1):
        if i == 0:
            # No decimal
            number = number_string
        else:
            # Insert decimal point
            number = number_string[:i] + "." + number_string[i:]

        if is_valid(number):
            result.append(number)

    return result

def is_valid(number_string):
    # Handle leading zero cases (only '0' is valid)
    if number_string.startswith('0') and number_string != '0' and '.' not in number_string:
        return False

    # Handle trailing zero cases after decimal point
    if '.' in number_string:
        integer_part, fractional_part = number_string.split('.')

        #Integer part can't have leading zero except when it's 0
        if integer_part.startswith('0') and integer_part != '0':
            return False

        #Fractional part can't end in zero
        if fractional_part and fractional_part.endswith('0'):
            return False

    return True

Big(O) Analysis

Time Complexity

O(n^3) – Let n be the length of the input string. The primary loop iterates up to n times to determine the position of the comma, splitting the string into two parts (x and y). For each of these two parts, generating valid numbers involves iterating through all possible positions to insert a decimal point. This can take O(n) time for each number. Combining all possible valid x and y numbers requires comparing each valid x against each valid y for each comma position, adding another factor of n in the worst case. Therefore, the overall time complexity is O(n * n * n) = O(n^3).

Space Complexity

O(N^2) – The space complexity is dominated by the storage of valid coordinate pairs. In the worst-case scenario, where nearly every possible combination of x and y coordinates is valid, the number of valid pairs can grow quadratically with the length of the input string N, where N is the length of the input string representing the coordinates. Specifically, we are storing a list of strings, and in the worst case, the number of strings in this list can be O(N^2), with each string taking O(N) space, but overall list size has the largest impact. Therefore, the overall space complexity is O(N^2).

Edge Cases

Case	How to Handle
Empty string input	Return an empty list because an empty string has no coordinates to parse.
String with only parentheses '()'	Return an empty list because there are no numbers to form coordinates.
String with a single digit inside parentheses '(1)'	Return a list with a single coordinate string '(1, )' or '(, 1)', depending on interpretation of single-number coordinate validity.
String with leading/trailing zeros, e.g., '(00, 05)'	Handle the leading/trailing zeros check as part of the valid number generation to filter out invalid coordinates.
String representing very large numbers that could cause integer overflow if converted directly.	Process the numbers as strings to avoid overflow during calculations, only converting to numeric representation for validation when necessary.
String with non-numeric characters (other than parentheses and comma)	Raise an error or return an empty list after validating the input string only contains permitted characters.
String representing floating-point numbers, e.g., '(1.0, 2.5)'	Handle floating-point numbers and accept the string as valid if it follows coordinate formatting rules.
Input string has imbalanced parenthesis i.e. '12, 3)'	Return empty list because it's an invalid formatted string.

Empty string input

How to Handle:

Return an empty list because an empty string has no coordinates to parse.

String with only parentheses '()'

How to Handle:

Return an empty list because there are no numbers to form coordinates.

String with a single digit inside parentheses '(1)'

How to Handle:

Return a list with a single coordinate string '(1, )' or '(, 1)', depending on interpretation of single-number coordinate validity.

String with leading/trailing zeros, e.g., '(00, 05)'

How to Handle:

Handle the leading/trailing zeros check as part of the valid number generation to filter out invalid coordinates.

String representing very large numbers that could cause integer overflow if converted directly.

How to Handle:

Process the numbers as strings to avoid overflow during calculations, only converting to numeric representation for validation when necessary.

String with non-numeric characters (other than parentheses and comma)

How to Handle:

Raise an error or return an empty list after validating the input string only contains permitted characters.

String representing floating-point numbers, e.g., '(1.0, 2.5)'

How to Handle:

Handle floating-point numbers and accept the string as valid if it follows coordinate formatting rules.

Input string has imbalanced parenthesis i.e. '12, 3)'

How to Handle:

Return empty list because it's an invalid formatted string.