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Ambiguous Coordinates

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.

Solution Explanation:

This problem asks to reconstruct all possible coordinate pairs from a string that has had commas, decimal points, and spaces removed. The solution uses backtracking to explore all possible ways to insert decimal points and split the string into two coordinates.

Approach:

The core idea is to break down the problem into two parts: finding all valid representations for the x-coordinate and finding all valid representations for the y-coordinate. We can then combine these possibilities to get all possible coordinate pairs.

1. f(i, j) function (Helper Function):

This function takes a substring of s from index i to j (exclusive of j) and returns all possible valid representations of a coordinate from that substring. A representation is considered valid if:

  • It doesn't start with a leading zero unless it's just "0". (e.g., "01" is invalid, "0" and "10" are valid).
  • It doesn't end with a trailing zero (e.g., "10" is invalid, "1" is valid).

The function iterates through all possible positions to insert a decimal point. For each position, it checks for validity and adds the resulting string to the res list if it's valid.

2. Main Logic:

The main part of the solution iterates through all possible split points within the string s (excluding the parentheses). For each split point i, it calls f(1, i) to get all possible x-coordinates and f(i, n - 1) to get all possible y-coordinates (where n is the length of the original string). It then combines these possibilities to form coordinate pairs and adds them to the ans list.

Time Complexity Analysis:

The time complexity is dominated by the nested loops in the main part of the solution. The outer loop iterates n-3 times (where n is the length of s), and the inner loops iterate through the results of f(1, i) and f(i, n-1). In the worst case, f(i, j) can return j-i strings. So, the complexity is approximately O(n * (n/2)^2), which simplifies to O(n^3). However, it is closer to O(n^2) in practice, because the average number of possible decimal placements in a substring is less than the substring length.

Space Complexity Analysis:

The space complexity is determined by the size of the ans list, which in the worst case can contain a large number of coordinate pairs. The space used by the f function is relatively small and doesn't dominate the space complexity. Therefore, the space complexity can also be considered approximately O(n^2) in the worst case but closer to linear in practice.

Code Explanation (Python):

The Python code directly implements this approach. The f function is neatly encapsulated for code readability and modularity. The main part then uses list comprehensions for a concise way to generate the coordinate pairs. The other language implementations follow a very similar structure.

The provided solutions in different languages highlight the core algorithm's adaptability across various programming paradigms. The approach remains consistent: split, validate, and combine.