Problem Description

In this problem, we are given a list of points on a 2D plane, where each point is represented as a pair of integers (x, y). These pairs are the coordinates of the points. We are also given an integer k. The task is to count how many unique pairs of points (i, j) have a specific distance between them, where the distance is defined as (x1 XOR x2) + (y1 XOR y2) and XOR stands for the bitwise XOR operation. A key point to note is that a valid pair (i, j) must have the indices i less than j to ensure pairs are not counted multiple times and to maintain uniqueness.

Intuition

The intuition behind the solution emerges from understanding how XOR operation works and then optimizing the brute force approach of checking each possible pair of points to see if they meet our distance criteria.

Here are the steps to understand and arrive at the solution:

1. Understanding XOR: The XOR operation has a crucial property: if a XOR b = c, then a = b XOR c and b = a XOR c. This property will help us quickly find potential pairs that can have the distance k between them.

2. Iterating Efficiently: A naive approach would involve checking every possible pair of points, which would be inefficient with large datasets. We can optimize this by smartly iterating over possible values of one coordinate, reducing the search space.

3. Counter Dictionary: We use a Counter (dictionary subclass in Python that helps keep track of element counts) to keep track of how many points with certain coordinates we already have seen. The key of the counter will be the coordinate pair, and the value is the number of times we have seen that specific pair.

4. Calculating Potential Matches: For each new point (x2, y2) we're inspecting, we iterate through possible values of a (from 0 to k) and calculate a corresponding b such that a + b = k. With these values of a and b, we can determine what the coordinate (x1, y1) of a potential matching point would need to be using the XOR property mentioned earlier.

5. Counting Valid Pairs: For each of these potential matching coordinates (x1, y1), we use our counter to see if we have previously come across such a point. If we have, we know that the current point (x2, y2) and the counted (x1, y1) would form a valid pair, so we add the count from the counter to our answer.

6. Updating the Counter: After running through all possible a and corresponding b, we add the current point (x2, y2) to our counter, indicating that we have now seen this particular point.

7. Return the Answer: Once we have iterated over all points and calculated the valid pairs, we return the total count.

Through this method, we can efficiently and systematically count all the pairs that have the XOR-based distance of k, without having to evaluate each pair of points individually.

Solution Approach

The solution uses a blend of counting methodology with smart iteration based on the properties of the XOR operation to count the number of valid point pairs that are at a distance k from each other.

Here is a step-by-step explanation of the algorithm:

1. Initialization:

   • A Counter object cnt is initialized to keep track of how many times a particular point (x, y) is seen. In Python, a Counter is a dictionary that stores elements as keys and their counts as values.
   • An integer ans is initialized to 0, which will accumulate the number of valid pairs that meet the distance criteria.

2. Iterate Over Points:

   • We iterate over each point (x2, y2) in the coordinates array using a for-loop. This loop is responsible for looking at each point and determining how many points we've seen so far can form a valid pair with it at distance k.

3. Determining Potential Pairs Based on Distance k:

   • Inside the first loop, another for-loop runs through integer values from 0 to k inclusive. With each iteration, we take a as the current value of the loop, and we compute b such that b = k - a.
   • We then use the XOR property to find what x1 and y1 would be if x2 XOR x1 = a and y2 XOR y1 = b. This gives us the coordinates of a point that could be at the required distance k from (x2, y2).

4. Counting and Summing Valid Pairs:

   • Using the calculated (x1, y1), we check the cnt dictionary to see if we already came across this point in our previous iterations. If so, the value from the dictionary for the key (x1, y1) is added to ans. This step counts how many times we have a point that can pair with our current point to create a distance of k.

5. Updating the Counter:

   • After inspecting all possible matching points for (x2, y2), we update the cnt dictionary by incrementing the count of the current point (x2, y2). This means that now (x2, y2) can be considered as a potential match for future points.

6. Returning the Result:

   • Once all points have been accounted for, and all valid pairs have been counted, the final value of ans, which represents the number of valid pairs with the specified distance, is returned.

The algorithm efficiently uses a single pass through the points while leveraging the constant-time lookup feature of dictionaries in Python to keep the overall performance manageable. Moreover, the approach takes full advantage of the XOR operation’s properties, avoiding an exhaustive enumeration of all possible point pairs, which could otherwise become computationally expensive.

Example Walkthrough

Let's take an example with the following points and k = 5:

points = [(0, 0), (1, 4), (4, 1), (5, 0)]

Now, let's walk through the solution approach step by step for k = 5:

1. Initialization:

   • cnt = Counter() is initialized to keep track of how many times a particular point is seen.
   • ans = 0 is initialized to accumulate the number of valid pairs.

2. Iterate Over Points:

   • First we consider the point (0, 0). There are no previous points to compare it with, so we simply move on after adding it to the cnt.

3. Determining Potential Pairs Based on Distance k:

   • Examining the next point (1, 4). We iterate for a from 0 to 5 (our k value).
     • When a = 0: Then b = 5, we look for a point with coordinates such that 0 XOR x1 = 0 and 4 XOR y1 = 5. This gives us x1 = 0 and y1 = 1. But we have not seen (0, 1) before, so there is no match.
     • When a = 1: Then b = 4, we seek a point where 1 XOR x1 = 1 and 4 XOR y1 = 4. We get x1 = 0 and y1 = 0 which we have seen, so we add 1 to ans.
     • We continue this process for other values of a until 5 but find no more matches.
   • cnt[(1,4)] += 1

4. Next Points:

   • For (4, 1), repeating the process for values of a:
     • When a = 0: Then b = 5, and we look for 4 XOR x1 = 0 & 1 XOR y1 = 5 which yields x1 = 4, y1 = 4. No such point exists yet.
     • When a = 1: Then b = 4, and 4 XOR x1 = 1 & 1 XOR y1 = 4 which yields x1 = 5, y1 = 5. No match.
     • And so on, until a match is found for a = 4: Then b = 1. Here, x1 = 0 & y1 = 0 is a match, so we add 1 to ans.
   • cnt[(4, 1)] += 1

5. Last Point:

   • For (5, 0), following the same method for different values of a, we find a match for a = 1 & b = 4, which corresponds to the point (4, 4). It's not in cnt, so no match is added to ans.
   • cnt[(5, 0)] += 1

6. Returning the Result:

   • Having processed all points, we find that ans = 2. There are two unique pairs of points with XOR-based distance 5: (0, 0) & (1, 4), (0, 0) & (4, 1).

The algorithm allowed us to figure out the number of valid point pairs with the desired property by efficiently iterating through the list of points and using the Counter to avoid unnecessary pair checks.

Solution Implementation

Time and Space Complexity

Time Complexity

The given Python code has a nested loop where the outer loop goes through each coordinate in the list coordinates, and for each of these coordinates, the inner loop iterates k + 1 times. If n is the number of coordinates, the total number of iterations of the inner loop across all executions of the outer loop is n * (k + 1). Since the other operations inside the inner loop, including dictionary access and updates, take constant time, the time complexity is O(n * k).

Space Complexity

The space complexity of the code is determined by the additional space required for the Counter object cnt. In the worst-case scenario, if all coordinates are unique, the counter will have an entry for each coordinate in the list. Therefore, if there are n coordinates, the space complexity will be O(n) for storing all the unique coordinates in cnt.

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