Problem Description

In this problem, we are provided with an integer array nums, and two positive integers m and k. Our task is to find the maximum sum of all almost unique subarrays of length k from nums. An almost unique subarray is defined as a subarray that contains at least m distinct elements. A subarray refers to a contiguous sequence of elements within the array that is non-empty. If there is no subarray that meets the criteria, then the function should return 0.

Intuition

To tackle this problem, the intuition is to use a sliding window approach combined with a Counter (dictionary in Python) to keep track of the frequency of each element within the window. Here is the thinking process:

1. We start by creating a window of size k at the beginning of the array and use a Counter to track the elements' frequencies within this window.

2. We calculate the sum of the elements in the initial window as a possible candidate for the maximum sum.

3. If the number of distinct elements in this window is at least m, we consider the sum of this window as the initial answer.

4. We then slide the window by one position to the right, updating the Counter and sum accordingly by adding the new element to the window and removing the leftmost element that goes out of the window.

5. After updating the Counter and sum for the new window, we check if the window still contains at least m distinct elements. If it does, we compare the current sum with the previous maximum sum and update the maximum sum if the current sum is higher.

6. This process continues for each possible subarray of size k until we reach the end of the array.

7. The highest sum found during this sliding window process is returned as the result. If no such subarray is found throughout the process, the return value will be 0.

By utilizing a sliding window and a Counter, we maintain an efficient check over the uniqueness of the elements and the sum of the elements in the subarray, thus arriving at the desired solution in a linear time complexity relative to the size of the input array.

Learn more about Sliding Window patterns.

Solution Approach

The solution uses a sliding window approach to efficiently calculate the sum of each k-length subarray while tracking the distinct elements within that subarray. Here's a step-by-step walkthrough of the implementation:

1. Initialize the Counter:

   • Create a Counter that will hold the frequency of each element in the current window.
   • Initialize the sum, s, of the first window by adding up the first k elements of nums.

2. Initial Check:

   • Check if the initial window (first k elements) is almost unique, meaning it has at least m distinct elements. If it is, store this sum as ans.

3. Sliding the Window:

   • Loop through the array starting from the (k+1)-th element to the end.
   • For each step in the loop:
     • Increment the count of the new element that enters the window into the Counter.
     • Decrement the count of the element that is leaving the window.
     • Update the sum, s, by subtracting the value of the element that leaves the window and adding the value of the element that enters the window.

4. Update the Counter and Sum:

   • If the count of the element that leaves the window drops to zero, remove it from the Counter since it's no longer part of the current window. This step is crucial as it ensures the Counter only contains elements present in the window.

5. Update the Maximum Sum:

   • After updating the Counter for the new window, check again if we have at least m distinct elements.
   • If the current subarray satisfies the almost unique condition, compare the current sum with the previously stored maximum sum, ans, and update ans if the current sum is greater.

6. Return the Result:

   • After processing all possible windows, return the maximum sum found, stored in ans. If no valid window was found, ans will be 0, which is the correct return value in that case.

This algorithm efficiently computes the maximum sum of almost unique subarrays with a time complexity of O(n), where n is the length of nums, by performing constant work for each element in nums. The use of the sliding window pattern combined with a Counter for frequency tracking is what makes this approach effective.

Example Walkthrough

Let's consider an example to illustrate the solution approach. Suppose we have the array nums = [4, 3, 2, 3, 5, 4, 1], and we want to find the maximum sum of almost unique subarrays of length k = 3 with at least m = 2 distinct elements.

1. Initialize the Counter:

   • Create a Counter and initally it would be empty as no window is considered yet.
   • We start with the first window nums[0:3] which is [4, 3, 2]. The sum s is 4 + 3 + 2 = 9.

2. Initial Check:

   • The initial window [4, 3, 2] has exactly 3 distinct elements, which is more than m = 2. Therefore, we initially store this sum as ans = 9.

3. Sliding the Window:

   • Move to the next window by sliding one position to the right. Now we consider nums[1:4], which is [3, 2, 3].
   • Update the Counter by decrementing the count of 4 (as it leaves the window) and incrementing the count of the new element 3 (added again but already present).

4. Update the Counter and Sum:

   • Remove 4 from the Counter if its count drops to 0 (in this case, it does).
   • The sum of the new window [3, 2, 3] is now 3 + 2 + 3 = 8. We update s accordingly.

5. Update the Maximum Sum:

   • The new window [3, 2, 3] has only 2 distinct elements which meets the almost unique condition of at least m = 2.
   • We compare the sum 8 with the previous maximum ans = 9. Since 8 is less than 9, we do not update ans.

6. Sliding Further:

   • Repeat the process for the next window [2, 3, 5]. Remove 3 from the counter, add 5, and calculate the new sum s = 2 + 3 + 5 = 10.
   • Check if the new sum 10 is greater than ans = 9. It is, so we update ans to 10.

7. Continue:

   • Continue with the above steps until we have slid over the entire array with the window.
   • For nums[3:6] -> [3, 5, 4], the sum is 12 and it's almost unique, update ans.
   • For nums[4:7] -> [5, 4, 1], the sum is 10 but ans is still greater, so no change.

8. Return the Result:

   • After sliding through all windows, the highest sum we found was 12 for the subarray [3, 5, 4]. Therefore, we return ans = 12.
   • If we had not found any almost unique subarrays, we would return 0.

Through this example, we've demonstrated how the sliding window approach combined with a Counter can be used to efficiently solve this problem. The key is to keep updating the sum and distinct element count for each window and maintaining the maximum sum encountered that meets the almost unique criteria.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code can be analyzed as follows:

• The initialization of cnt with the first k elements takes O(k) time.
• The calculation of the sum of the first k elements is also O(k).
• The for loop runs for len(nums) - k iterations. Within each iteration, the operations of updating the counter and sum are constant time, i.e., O(1).
• However, the popping of elements from the counter can occur at most once per iteration and is also an O(1) operation in average case for a dictionary in Python.
• Checking the length of cnt and updating ans is O(1).

Considering the for loop is the dominant part, the time complexity is O(n-k), where n is the length of nums. Since k is subtracted from n, and in the worst case k could be much smaller than n, the more generalized form to express the time complexity is O(n).

Space Complexity

The space complexity of the code is mainly due to the counter cnt, which will store at most k unique integers if all elements in the first k elements of nums are unique. Therefore, the space complexity is O(k) in the worst case.

To summarize:

• Time Complexity: O(n)
• Space Complexity: O(k)

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