Problem Description

You have an array of integers, nums, and a positive integer, k. Your task is to find out how many subarrays exist within this array where the highest number in the array, which we can call mx, shows up at least k times. A subarray is defined simply as a continuous portion of the array; it does not need to include all elements, but the elements it does include must be in the same order they appear in the full nums array.

Intuition

To solve this problem, we need to think about how we can track occurrences of the maximum element in each subarray. A direct approach to identifying subarrays would involve looking at every possible subarray, which would be inefficient especially for larger arrays. A smarter way to do this is by using the concept of a moving window, often known as the two-pointer technique.

We define a variable mx to hold the maximum value in the array and then attempt to find subarrays that have at least k occurrences of mx. Starting with the first element, we expand our window to the right until we have k instances of mx. When we reach that point, any larger subarray that includes this window will also have at least k instances of mx. Thus, we add the total possible subarrays to our answer by calculating the number of elements from the end of our window to the end of the array, inclusive. We then slide our window to the right by one and reduce our count of mx occurrences if the first element of our previous window was a mx.

The beauty of this approach lies in its linear time complexity, as each element is visited only once by each of the two pointers. We count the occurrences of mx within a current window and continuously adjust the window size, keeping track of how many subarrays meet our requirement. This helps us avoid reexamining portions of the array unnecessarily, turning what could be an overwhelmingly complex problem into a manageable one.

Learn more about Sliding Window patterns.

Solution Approach

The solution implements a two-pointer algorithm to efficiently find the number of subarrays fulfilling the given condition. We use the variables i and j to represent our two pointers, which define the edges of our current window on the array. The variable cnt serves as a counter for how many times mx (the maximum element of the entire array) appears within this window. The solution also initializes an answer variable ans to accumulate the number of valid subarrays.

We start by finding the maximum element in the array, mx = max(nums), since the problem specifically concerns occurrences of this element.

Next, we iterate through each element in the array with a for-loop, thinking of each element as the potential start of a subarray (this is the left pointer, i). As we iterate:

1. We use the right pointer j to extend the window to the right, incrementing cnt whenever we encounter the maximum value mx.
2. We continue extending the window until cnt is at least k, meaning we've found a subarray where mx appears at least k times.
3. At this point, any subarray starting from i and ending at or past j-1 will contain at least k instances of mx. The total such subarrays can be counted as n - (j - 1), and we add this count to our answer ans.
4. Before moving i to the next position, reducing our window from the left, we need to decrease cnt if the element nums[i] that's being removed is equal to mx.

The loop breaks if j reaches the end of the array, or if cnt falls below k, because further iteration of i cannot possibly find a sufficient count of mx to fulfill the condition.

This approach only passes through the array once with each pointer and hence, has a linear time complexity of O(n), where n is the number of elements in the array.

Here's the implementation walkthrough based on the given solution:

• mx = max(nums): Find the maximum element in nums.
• n = len(nums): Get the length of the array to know when we've reached the end.
• ans = cnt = j = 0: Initialize the answer (ans), the counter for the maximum value (cnt), and the right pointer (j) to zero.
• for x in nums: Start the main loop over nums, considering each x to be the potential beginning of a subarray.
• while j < n and cnt < k: Expand the window to the right with the right pointer j until k instances of mx are within it or the end of the array is reached.
• cnt += nums[j] == mx: Increment cnt when the current element is mx.
• j += 1: Move the right pointer j to the next position.
• if cnt < k: If k or more instances of mx cannot be found, no more valid subarrays are possible, break the loop.
• ans += n - j + 1: Add the number of valid subarrays to ans.
• cnt -= x == mx: Before the next iteration of the loop, decrease cnt if the element removed from the current window is mx.

By using these steps, we obtain a seamless, efficient algorithm that finds the total number of valid subarrays without resorting to brute force.

Example Walkthrough

Let's consider a small example to illustrate the solution approach. Suppose the array nums is [1, 2, 2, 3, 2], and k is 2. We want to count the subarrays where the maximum element appears at least twice.

First, we find the maximum element mx of nums: mx = max(nums) = 3.

Now, we initialize ans = 0, cnt = 0, and j = 0, and start iterating over the array with our left pointer, represented by the elements in the for-loop.

Here's the step-by-step process:

1. We start at i = 0, with nums[i] = 1.
2. We look to expand our window to the right; since cnt < k, our inner while-loop starts.
3. We increment j until we encounter mx at least k times. During the process:, we increment cnt whenever nums[j] == mx.
   • j = 0, nums[j] = 1, cnt remains 0.
   • j = 1, nums[j] = 2, cnt remains 0.
   • j = 2, nums[j] = 2, cnt remains 0.
   • j = 3, nums[j] = 3, cnt becomes 1.
   • j = 4, nums[j] = 2, cnt remains 1.
   • As j has reached the end of the array and we have not found k instances of mx, we break from the loop and no subarrays are counted from this position ( ans remains 0 ).
4. We move to the next starting position, i = 1.
   • Repeat steps 2-3, but since j is already at the end, we don't enter the while-loop, and no subarrays are counted from this position either.
5. We continue this process moving i from 0 to n-1 (end of the array), but in this case, since mx appears only once and our k condition is at least twice, we don't find any subarray that matches the criteria.

In the end, the ans is 0 because there are no subarrays within [1, 2, 2, 3, 2] where the maximum element 3 appears at least 2 times.

Had mx appeared at least k = 2 times, we would add up all the possible subarrays from i up to j - 1 to our answer ans each time, and if nums[i] is an instance of mx, we would decrease cnt before moving on to the next i.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n), where n is the length of the nums array. This is because the algorithm uses a sliding window that iterates over each element of the array at most twice - once when extending the window and once when moving the starting point of the window forward. Thus, each element is visited a constant number of times, leading to a linear time complexity with respect to the size of the input array.

The space complexity is O(1) as the algorithm only uses a constant amount of additional space for variables like mx, n, ans, cnt, and j, irrespective of the size of the input array.

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