3027. Find the Number of Ways to Place People II

Problem Description

In this problem, we're given a 2D array points representing coordinates of certain points on a plane. Each point is meant to be occupied by one individual, and there are two particular individuals of interest: Alice and Bob. The task is to figure out how many ways Alice can be positioned at a point that will serve as the upper left corner of a rectangular fence, and Bob can be positioned at a point that will serve as the lower right corner of the same fence, without any other person being inside or on the boundary of this fence.

The essence of this problem revolves around the concept that we can only place Alice and Bob in such a way that Alice is above and to the left of Bob (based on their x and y coordinates). Thus, no individual should have coordinates such that they fall within or on the boundary of the rectangle defined by Alice's and Bob's locations.

To summarize, the challenge is to find all pairs (Alice, Bob) that allow the fence to be built without enclosing or including any other individuals on its boundary or within its area.

Intuition

Approaching this problem begins with the understanding that if we have Alice’s position fixed as the upper left corner, then Bob must be positioned diagonally across from her at the lower right corner. This means Bob’s x-coordinate must be greater than Alice’s x-coordinate, and his y-coordinate must be less than Alice’s y-coordinate, to form a valid rectilinear fence.

The intuition behind the solution hinges on efficiently checking potential Bob points for every potential Alice point. We must avoid counting any point that would be inside or on the proposed fence boundary.

Sorting points by their x-coordinate distinctly helps in the forward search since it naturally orders Bobs to the right of Alice. The y-coordinates are sorted inversely to deal with ensuring that Alice's y-coordinate is more than Bob's. After sorting, iterate through each Alice candidate while keeping track of the maximal y-coordinate observed for Bob that is less than or equal to Alice’s y-coordinate—the idea being that we don't want any point that sits on or inside the imaginary fence.

For each potential Alice, we search for potential Bobs. If we find a Bob with a y-coordinate less than Alice's but greater than any we've previously encountered, we know we've found a pair that can build a fence without including another individual, so we increment our pairs count. The process is repeated until all viable Alice and Bob pairs are accounted for.

Learn more about Math and Sorting patterns.

Solution Approach

To implement the solution, the following steps are taken:

1. Sort the Points: Firstly, we sort the points array. This is done based on the x-coordinates in ascending order so that potential Bobs are to the right of Alice. For points with the same x-coordinate, we sort based on the y-coordinate in descending order so that potential Alices are above Bobs. This is accomplished with the line:

   1points.sort(key=lambda x: (x[0], -x[1]))

   This step is crucial as it aligns all points to a common reference that respects the direction constraints (right and down) mentioned in the problem statement.

2. Initialize Counter: We have an ans variable set to 0, which will keep track of the number of valid pairs (Alice, Bob) found.

3. Iterate Through Points: We iterate through the sorted points list, treating each point as a potential Alice's position.

4. Track Maximum y-coordinate: As we iterate through potential Bob positions (to the right of Alice), we maintain a max_y variable set to negative infinity to track the highest y-coordinate of points encountered that could serve as a Bob's position and still not invalidate the fence.

5. Check and Update Fence Conditions: For each potential Bob, we check if any previous point's y-coordinate (greater than max_y but less than or equal to current Alice's y-coordinate) can be used as Bob's position. This is done with the following check:

   1if max_y < y2 <= y1:
   2    max_y = y2
   3    ans += 1

   This checks whether the current point can be a Bob without any other point on the fence. If it can, we update max_y to this new value and increment ans.

6. Return the Count: After iterating over all points, the ans value will contain the total number of valid (Alice, Bob) pairs that can be placed such that no other point will be inside or on the fence. This value is returned as the final answer:

   1return ans

Algorithm and Data Structures: The solution employs a simple sorting algorithm along with a linear iteration and tracking of maximum y-values. There was no need for complex data structures as the sorted list itself was sufficient to find all possible pairs.

Patterns Used: What stands out is the use of the two-pointer approach, in a way, where we keep a moving pointer for Alice and another moving pointer for Bob, within the sorted array. This pattern is efficient because it omits the need for nested iterations over the entire set of points for every Alice, reducing the potential time complexity.

Example Walkthrough

Let's use a small example to illustrate the solution approach.

Consider the following points representing coordinates on a plane: points = [[1, 3], [3, 0], [3, 4], [2, 2]]

Following the steps of the solution:

1. Sort the Points: After sorting based on the given key, we have the points array sorted as [[1, 3], [2, 2], [3, 4], [3, 0]]. Notice how the points are sorted left to right first, and then for the same x-coordinate, they are sorted from top to bottom.

2. Initialize Counter: We set ans = 0, ready to count valid pairs.

3. Iterate Through Points: We treat each point as potential Alice's position and move from left to right, top to bottom.

4. Track Maximum y-coordinate: We start with max_y = -inf as we haven't encountered any Bobs yet.

5. Check and Update Fence Conditions:

   • Firstly, we consider point [1, 3] as a potential Alice. The max_y is still -inf.
   • We move to the next point [2, 2]. Could this be Bob for Alice [1, 3]? Yes, because max_y < 2 <= 3, so we update max_y = 2, and increment ans to 1.
   • Then, we consider point [3, 4] - this cannot be Bob for the current Alice because its y-coordinate is not less than Alice's.
   • Lastly, we check point [3, 0]. This could potentially serve as Bob to Alice [1, 3], as it satisfies the condition max_y < 0 <= 3. Therefore, we update the max_y to 0, and increment ans to 2.

   At this moment, if we try to find a Bob for Alice [2, 2], no point to its right has a lesser y-coordinate. Similarly, points [3, 4] and [3, 0] serve as potential Alice positions but have no valid Bobs to the right.

6. Return the Count: Since we only found valid (Alice, Bob) configurations with the initial Alice [1, 3], the final ans is 2.

The result tells us there are two distinct ways to position Alice and Bob for the creation of a fence that does not enclose or include others on its boundary. The pairs found were ([1, 3], [3, 0]) and ([1, 3], [2, 2]).

Solution Implementation

Time and Space Complexity

The time complexity of the code provided is not O(1). It first sorts the list, which has O(n log n) complexity where n is the length of the points list. Then it uses a nested loop to iterate through the points list, leading to a worst-case scenario of O(n^2) complexity since for each point, it potentially compares it to all other points. Therefore, the overall time complexity is O(n^2) due to the nested loop after sorting.

The space complexity is O(1) (ignoring the space taken up by the input and the sort implementation), as the code only uses a fixed amount of additional space – a few variables like ans and max_y, which do not scale with the size of the input.

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