2152. Minimum Number of Lines to Cover Points

Problem Description

In this problem, we're presented with a set of coordinates, points, where each element points[i] is an array containing the x and y coordinates [x_i, y_i] of a point on a 2D plane. The challenge is to determine the minimum number of straight lines that are necessary to ensure that each point in the array is covered by at least one line.

A key detail to note is that a single straight line can cover multiple points if those points are all collinear, that is, they lie on the same straight line.

Intuition

The intuition behind solving this problem involves a geometric understanding of points and lines. Two points define a unique line. But if we have three or more points, we must check whether they are collinear. If they are, a single line can cover them all.

The solution approach leans on combinatorial optimization and uses depth-first search (DFS) with bitwise state representation and memoization to reduce redundant calculations. We use a state variable to represent the set of points that have already been covered by lines. When all points are covered, the state would equal 2^n - 1, where n is the number of points, as each bit in the state represents whether a corresponding point is covered or not.

To check whether three points are collinear, we use the cross product formula. For points i, j, and k, the intuition is to compare the slopes between points i and j and between i and k. If the slopes are equal, this implies collinearity. Slopes are compared using the cross product to avoid division and possible floating-point inaccuracies.

The dfs function, which employs memoization (as indicated by the @cache decorator), recursively explores all combinations of covering points with the minimum lines. For each unvisited point i, we attempt to draw a line to another unvisited point j and extend this line to cover additional points k if they are collinear with i and j. Whenever we add a line (whether it covers just two points or more), we increment the count of lines.

By exploring all combinations and using memoization to store results of subproblems, we ensure that the final result is the minimum number of lines needed to cover all points.

Learn more about Math, Dynamic Programming, Backtracking and Bitmask patterns.

Solution Approach

The solution implementation utilizes a depth-first search algorithm augmented with memoization (also known as top-down dynamic programming) to explore different combinations efficiently. Here's a breakdown of the key elements of the approach:

• Collinearity Check (check function): The function check(i, j, k) takes three indices corresponding to the points and determines if the points i, j, and k are collinear. This is done by comparing the cross product of vectors ji and ki. The formula ((x2 - x1) * (y3 - y1)) == ((x3 - x1) * (y2 - y1)) checks whether the areas of the triangles formed by the points are the same, confirming collinearity if the area is zero (the points fall in a straight line).

• Depth-First Search (dfs function): The dfs(state) function finds the minimum number of lines needed to cover all points, given the current state. The state variable is a bitmask representing which points have been covered so far, where the ith bit is set if the ith point is covered.

  • The base case is when state == (1 << n) - 1 which means all points are covered, thus zero lines are needed.

  • For each point i that is not covered, the algorithm attempts to draw a new line by combining it with all other points j.

  • If a line connecting points i and j is found, the function looks for any other point k that can be covered with the same line by checking for collinearity.

  • Each time a new line is added, or a point is covered, a new state nxt is created by setting the corresponding bits in state.

  • The minimum number of lines is updated by taking the minimum value between the current ans and the result of dfs(nxt) + 1 (since a new line is added).

• Memoization: The @cache decorator is used on the dfs function to memoize the results of previous state computations. This drastically reduces repetitive calculations and improves execution speed by caching the results of subproblems.

• Bit Manipulation: Bitwise operations on the state variable are used to efficiently manage the set of covered points, check if a point is covered (state >> i & 1), and add points to the set (nxt | 1 << i).

The main function's job is to initiate the dfs with an initial state of 0 (no points covered), and it returns the result of the dfs function as the minimum number of lines needed.

Each of these components works together to create a powerful algorithm capable of solving the problem effectively, navigating the combinatorial complexity by leveraging dynamic programming principles.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Consider a set of points points = [[1,1], [2,2], [3,3], [4,5]]. There are a total of n = 4 points.

Our goal is to find the minimum number of lines that cover all points.

We start with the initial state as 0, indicating no points are covered yet.

1. Starting DFS: We begin with dfs(0).

2. Trying to Cover Points: First, we try to cover point [1,1]. We look at the next point [2,2] and draw a line between them. This line has a slope that both points agree on, and they are collinear.

3. Extending the Line: We then try to include other points on this line. Point [3,3] also lies on this line, as the slope is consistent. However, point [4,5] does not lie on the same line since it will have a different slope when paired with any of the first three points.

4. Updating States: The state after covering points [1,1], [2,2], and [3,3] is now (1 << 0) | (1 << 1) | (1 << 2) which is 111 in binary or 7 in decimal.

5. Covering Remaining Points: Now, dfs(7) is called to cover the remaining points. There's only [4,5] left which is not collinear with the others, and a new line must be drawn. The updated state is 1111 in binary, or 15 in decimal, which means all points are covered.

6. Termination: The base case of dfs is reached since state == (1 << n) - 1, which is 15. So the number of lines needed to cover state 7 is 1.

7. Memoization: The result of dfs(7) is memoized so if we ever encounter this state again, we can return the result immediately without recalculation.

8. Repeating DFS: The process above is repeated for different combinations of starting points and lines drawn. However, in this particular example, after the first TRY, all points are covered in the most optimal way.

Finally, the minimum number of lines needed: So, for the starting state 0, dfs(0) will give us 2, which means 2 lines are needed – one line for points [1,1], [2,2], and [3,3], and another line for point [4,5].

In the actual implementation, the algorithm will explore all possible line combinations, but with memoization, redundant calculations are prevented, leading to an efficient solution.

Solution Implementation

Time and Space Complexity

The given code aims to find the minimum number of lines required to cover all the points provided in a list. It uses a depth-first search approach (dfs) with memoization to reduce redundant calculations.

Time Complexity:

The time complexity of this algorithm can be analyzed as follows:

1. The dfs function is called with different states, representing subsets of points. There are 2^n possible states since each of the n points can either be included or not in a subset.

2. Inside the dfs function, there are two nested loops:

   • The outer loop iterates over n points.
   • The inner loop iterates over n points again.

3. Within the inner loop, for every combination of points (i, j), the code checks for any k point that can be collinear with i and j using the helper function check.

4. The check function performs a constant-time operation to determine if three points are collinear.

Following the above points, the worst-case time complexity would be O(n^3 * 2^n) because:

• For each state, the algorithm iterates over all pairs of points (i, j) which gives O(n^2).
• It further checks for any k that can be on the same line, adding another factor of O(n).
• This is done for all 2^n subsets of points (dfs calls).

Space Complexity:

The space complexity can be considered based on the following:

1. The maximum depth of the recursive dfs is 2^n, as it represents the number of states/subsets.

2. The memoization (@cache) stores results for each state to prevent re-computation, hence it could store up to 2^n entries.

The space complexity therefore is O(2^n) due to the memoization and the recursion stack.

In summary:

• Time Complexity: O(n^3 * 2^n)
• Space Complexity: O(2^n)

Learn more about how to find time and space complexity quickly using problem constraints.