Minimum Number of Lines to Cover Points

This problem asks for the minimum number of straight lines needed to cover all points in a given array. The solution uses a bitmask approach with dynamic programming to efficiently explore all possible combinations of lines.

Approach:

The core idea is to represent the subsets of covered points using a bitmask. Each bit in the bitmask corresponds to a point; a set bit indicates the point is covered. The dynamic programming approach recursively explores all possible subsets, attempting to cover points using the minimum number of lines.

The check function determines if three points are collinear. The dfs function recursively explores possible line combinations:

1. Base Case: If all points are covered (state == (1 << n) - 1), return 0 lines.

2. Memoization: Use a memoization table (f) to store results for previously computed states.

3. Iteration: Iterate through points not yet covered. For each uncovered point i, try to find another point j to form a line.

4. Collinearity Check: Extend this line to include other collinear points k, updating the nxt (next) state to reflect their coverage.

5. Recursive Call: Recursively call dfs with the updated nxt state and add 1 to the line count.

6. Minimum Lines: Keep track of the minimum lines needed across all possibilities.

Time Complexity: O(3n), where n is the number of points. The worst case is when no points are collinear, requiring exhaustive search through all possible subsets.

Space Complexity: O(2n) due to the memoization table, which stores results for all possible bitmask states.

Code Explanations (Python3 Example):

class Solution:
    def minimumLines(self, points: List[List[int]]) -> int:
        def check(i, j, k):  # Check collinearity
            x1, y1 = points[i]
            x2, y2 = points[j]
            x3, y3 = points[k]
            return (x2 - x1) * (y3 - y1) == (x3 - x1) * (y2 - y1)
 
        @cache  # Memoization decorator
        def dfs(state):
            if state == (1 << n) - 1:  # Base case: all points covered
                return 0
            ans = inf  # Initialize minimum lines to infinity
            for i in range(n):
                if not (state >> i & 1):  # Check if point i is not covered
                    for j in range(i + 1, n):
                        nxt = state | 1 << i | 1 << j  # Add points i and j
                        for k in range(j + 1, n):
                            if not (nxt >> k & 1) and check(i, j, k):  #Check collinearity and coverage
                                nxt |= 1 << k  #Add collinear point k
                        ans = min(ans, dfs(nxt) + 1)  #Recursive call, update min lines
                    if i == n - 1: #Handle cases where only one point is left to cover.
                        ans = min(ans, dfs(state | 1 << i) + 1)
            return ans
 
        n = len(points)
        return dfs(0)

The other languages (Java, C++, Go) follow a very similar structure, differing mainly in syntax and the way memoization is handled. The core algorithm remains consistent across all implementations.