Problem Description

You have been given an array named nums with a size n, which includes distinct integers ranging from 1 to n. You are also given a positive integer k. Your task is to count how many non-empty subarrays of nums have a median that is equal to k.

Remember, the median of an array is defined as the middle element once the array is sorted in ascending order. For an array with an even number of elements, there isn’t a single middle element, so in that case, you take the left middle element as the median.

A subarray is any sequence of consecutive elements from the array. It can be as short as one element, or as long as the entire array.

To illustrate, consider the array [2, 3, 1, 4]. After sorting, this array becomes [1, 2, 3, 4]. The median is the left middle element, which is 2 for this array. For the array [8, 4, 3, 5, 1], after sorting it becomes [1, 3, 4, 5, 8]. The median for this array is 4 since there is an odd number of elements.

Your objective is to find and return the number of these subarrays in which the median is exactly k.

Intuition

Let's try to decode the provided solution to this problem. The main idea here is to use the index of k in nums as an anchor point. Since the problem tells us nums consists of distinct integers from 1 to n, we know k will appear exactly once in nums.

We can search for the contiguous subarrays where k is the median. If k is the median of a subarray, for the subarray to remain valid when adding elements to the left or right, we must maintain a balance or near-balance in the number of elements that are less than k and the number of elements greater than k.

While iterating through the elements, we keep track of this balance with a counter x, which we increment if we add an element greater than k to our subarray, and decrement if the element is less than k. The counter value represents the difference between the count of elements greater than k and less than k to the right of k.

The solution approach calculates this balance for the elements to the right and left of k separately, then combines these results. For the elements on the right of k, we just need to keep track of instances where this balance is 0 or 1. For the elements on the left of k, we also have to consider the complementary counts from the right side to ensure that k remains the median when combining subarrays from both sides.

Essentially, the algorithm loops twice: once over the elements to the right of k's index and once over the elements to the left. In each loop, it maintains a count (cnt) of occurrences of each balance value. This count is key to finding the number of valid subarrays when we combine elements from both sides. The final answer is the sum of all valid subarray counts.

Learn more about Prefix Sum patterns.

Solution Approach

The implementation of the solution revolves around a two-part process. In the first part, we find the index i of the integer k within the array nums using nums.index(k). This is crucial as we will compute the balance of larger and smaller elements in the subarrays with respect to k.

To assist in counting, a Counter object named cnt is instantiated. This data structure is a dictionary subclass from Python's collections module that counts hashable objects.

The core part of the algorithm operates on the premise that when we add an element greater than k, it could potentially disrupt the balance required for k to be the median. Thus, every time we encounter such an element, we increase our balance counter x. Conversely, an element less than k contributes toward maintaining the median, and thus we decrease the counter x by one.

Analyzing the Right of k

We create a subarray of nums starting from the element just after k with nums[i + 1 :]. With a for-loop over this subarray, for every element v that is greater than k, we increment x and for every element less than k, we decrement x. The conditions 0 <= x <= 1 ensure that the median of the new subarray including v remains k (as k should not be overtaken by a higher number of elements that are either greater or smaller). For each balance situation satisfying this condition, we increase the solution count ans. The cnt counter keeps track of how many times each x value occurred.

Analyzing the Left of k

Once the right side is processed, we reset x to 0 and run a second for-loop, iterating over the elements to the left of k. The loop decrements the index j each time, moving leftwards. For each element nums[j], x is adjusted in the same way (increment for elements greater than k, decrement for elements less than k).

For each balance value x, two additions are made to the ans:

• 0 <= x <= 1 is again checked, and if true, ans is incremented.
• The corresponding counts for -x and -x + 1 in the cnt Counter (the balance counts from the right side) are added to ans.

These additions account for the subarrays extending both left and right from element k, whose medians are still k.

Finally, the ans which holds the count of all qualifying subarrays is returned.

This solution leverages the balance theory, counter collections, indexing, and iteration in reverse to efficiently find all the subarrays with k as their median.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Consider the array nums = [3, 1, 4, 2] with n = 4 and let's find the number of subarrays where the median equals k = 2.

First, we need to find the index of k in nums; k is at index i = 3.

Now, let's analyze the elements to the right of k, which in this case, are none because k is at the last index. So, we move on to the next part of the process.

Next, we will analyze the elements to the left of k. We iterate from index i - 1 to 0 (that is, from right to left). We initialize our balance counter x to 0 and a Counter to keep track of the occurrences of x.

• At index i - 1 (index 2), the value is 4 (greater than k). We increment x to 1 (x = x + 1). There are no elements to the right, so we don't update our answer yet.
• At index i - 2 (index 1), the value is 1 (less than k). We decrement x to 0 (x = x - 1). If we were to choose the subarray [3, 1, 4, 2] up to now, the sorted version would be [1, 2, 3, 4], and k would be the median. We increment our answer ans by 1.
• At index i - 3 (index 0), the value is 3 (greater than k). We increment x to 1 again. The subarray [1, 4, 2] has k as the median after sorting to [1, 2, 4]. ans is incremented by 1.

Finishing this process, we have found 2 subarrays where k is the median: [3, 1, 4, 2] and [1, 4, 2]. Therefore, by following the steps outlined in the solution approach, we would return 2 as the final answer.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be analyzed in the following steps:

1. Finding the index of k in nums using nums.index(k) – This operation is O(n) where n is the number of elements in nums.

2. Iterating over the elements to the right of k, incrementing x, and updating the ans and the Counter for each element – The iteration is done once for each element to the right of k, which is O(n) in the worst case (when k is the first element).

3. Iterating over the elements to the left of k, incrementing x, and updating ans based on the current x value and the count from the Counter – Similar to the step above, this is also O(n) in the worst case (when k is the last element).

Given that these steps are performed sequentially, we add the complexities resulting in a total time complexity of O(n) + O(n) + O(n) = O(n).

Space Complexity

The space complexity of the code can be analyzed by considering the additional data structures used:

1. cnt - A Counter object which in the worst case stores counts for each possible distinct value of x. Since x represents a running difference between the count of values greater than k and less than k, x can be at most n in absolute value in the worst case, leading to 2n + 1 different possible values (including zero). Thus, the Counter can have at most O(n) elements.

Hence, the space complexity of the code is O(n).

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