Sliding Subarray Beauty

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class Solution:
    def getSubarrayBeauty(self, nums: List[int], k: int, x: int) -> List[int]:
        counter, ans = [0] * 51, [0] * (len(nums) - k + 1)

        for i in range(k):  # first k elements
            if nums[i] < 0: counter[nums[i] + 50] += 1

        for i in range(k, len(nums) + 1):  # start from k
            count = 0
            for j in range(50):
                count += counter[j]
                if count >= x:
                    ans[i - k] = j - 50
                    break

            if i < len(nums):  # to avoid index out of range error
                if nums[i] < 0: counter[nums[i] + 50] += 1
                if nums[i - k] < 0: counter[nums[i - k] + 50] -= 1

        return ans

Problem Classification

It specifically focuses on the manipulation and querying of arrays and subarrays.

‘What’ Components:

  1. Input: The problem takes three inputs:

    • An integer array ’nums’ of size ’n’.
    • An integer ‘k’ that represents the size of the subarrays to consider.
    • An integer ‘x’ that represents the number of smallest negative numbers to consider in each subarray.

  2. Subarrays: A primary component of the problem is to identify all possible subarrays of size ‘k’ within the given array.

  3. Beauty of Subarray: The problem requires us to calculate the beauty of each subarray, defined as the ‘x’th smallest negative number in the subarray, or 0 if there are fewer than ‘x’ negative integers.

  4. Output: The output should be an array of integers representing the beauty of each subarray, from the first index in the array.

This problem can be classified as a “Sliding Window” type problem, where the window is of a fixed size ‘k’, and we have to calculate a certain value (in this case, the beauty) for each window. These types of problems are common in array-based problems where we need to perform some operation on a subset of the array and then slide the subset window across the array.

Thought Process

In this problem statement, the range of numbers being small and fixed allows us to adopt a counting sort-like strategy where we can keep track of the frequency of negative numbers in a counter array. The fact that the xth smallest negative number is required, which is essentially an order statistic, also hints towards this strategy.

Understanding the range and nature of input data is a significant part of problem-solving in programming. It allows you to make informed decisions about the type of data structures and algorithms that can be used to solve the problem efficiently. In this case, because the numbers fall within a small and fixed range, a simple count array can be used instead of a more complex data structure like a balanced binary search tree or a heap, which might be necessary for larger or variable ranges.

The counting sort technique works here due to the limited range of values. It’s possible to keep track of the counts of negative numbers in a running window, which can be efficiently updated as the window slides. By maintaining a count array representing the sorted negative values, we can quickly find the xth smallest negative value. This indeed leverages the constraints given in the problem and makes the solution more efficient.

Here are the key coding constructs used by this code:

  1. Sliding Window Technique: The sliding window technique is used for efficiently handling subarray problems. A “window” of a fixed size (here it’s k) moves through the list, and calculations are performed for each position of the window. In this case, the calculations are about finding the ‘beauty’ of the subarray.

  2. Frequency Counting: The use of a counter array to keep track of the occurrence of each negative number in the current window is an example of the frequency counting pattern. This pattern is commonly used when we want to know the frequency of certain elements within a collection (like an array or a list).

  3. Accumulation: This code uses an accumulation pattern to find the xth smallest negative number in the counter array. This is done by iterating over the array and accumulating the counts until it reaches x.

  4. Incremental Update: When the window moves, the code doesn’t recompute the counts of all the numbers in the window. Instead, it updates the counter array incrementally, by adjusting the counts of just the number that has moved out of the window and the number that has moved into the window. This strategy significantly improves the efficiency of the code.

Counting Sort

Here’s a simple example of the counting sort algorithm in Python 3:

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def counting_sort(arr):
    max_val = max(arr)
    min_val = min(arr)
    range_of_elements = max_val - min_val + 1  # Find range of elements

    # Create a count array to store the count of individual elements and initialize count array as 0
    count_arr = [0 for _ in range(range_of_elements)]
    output_arr = [0 for _ in range(len(arr))]

    # Store count of each character
    for num in arr:
        count_arr[num-min_val] += 1

    # Change count_arr[i] so that count_arr[i] now contains actual position of this character in output array
    for i in range(1, len(count_arr)):
        count_arr[i] += count_arr[i-1]

    # Build the output array
    for i in range(len(arr)-1, -1, -1):
        output_arr[count_arr[arr[i] - min_val] - 1] = arr[i]
        count_arr[arr[i] - min_val] -= 1

    # Return the sorted output array
    return output_arr

# Testing the function
arr = [4, 2, 2, 8, 3, 3, 1]
print("Original Array: ", arr)
sorted_arr = counting_sort(arr)
print("Sorted Array: ", sorted_arr)

In the code above, we first calculate the range of elements in the array. Then, we create a count array to store the count of each element in the array. We use this count array to place the elements in their correct position in the sorted output array.

Please note that Counting Sort can only be used for integers and it’s not a comparison sort. Also, it is not suited for sorting arrays with large ranges of numbers, as it requires auxiliary space proportional to the range of numbers.

Sliding Window + Counting

In this problem, the range of the numbers (-50 <= nums[i] <= 50) allows for a particular optimization - using a counting technique. We leverage this property to reduce the time complexity of our solution.

Here, the counting technique is implemented using a HashMap or a list (if using Python) that can efficiently track the frequency of each negative number in the current sliding window.

In essence, this algorithm is a combination of the “Sliding Window” and “Counting” techniques. The Sliding Window technique helps efficiently iterate over the subarrays, and the Counting technique enables quick access to the xth smallest negative integer within each subarray.

This combination is especially powerful as it brings the time complexity down to O(n*50), where n is the size of the array and 50 is the maximum possible count for any number. The space complexity is also minimized to O(50), which represents the maximum possible distinct negative numbers in the array.

Remember, while this is an optimal solution for this particular problem, the applicability depends heavily on the constraints mentioned in the problem statement (here -50 <= nums[i] <= 50). Always ensure to understand the problem constraints fully and consider how they can be leveraged to optimize the solution.

Sliding Window

Let’s consider a simple problem to illustrate the Sliding Window approach. The problem is to find the maximum sum of any contiguous subarray of size ‘k’ from a given list of numbers.

Here’s a Python3 program for it:

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def max_sum_subarray_of_size_k(arr, k):
    max_sum = float('-inf')
    window_sum = 0

    for i in range(len(arr)):
        window_sum += arr[i]  # add the next element into the window

        # slide the window when we've hit the size 'k'
        if i >= k - 1:
            max_sum = max(max_sum, window_sum)  # update the result
            window_sum -= arr[i - (k - 1)]  # subtract the element going out of the window

    return max_sum

# Usage:
nums = [2, 3, 4, 1, 5]
k = 3
print(max_sum_subarray_of_size_k(nums, k))  # Outputs: 10

In this code, the window size is initially less than ‘k’. It’s expanded by adding elements until its size becomes ‘k’. After that point, for each new element added into the window (from the right), an element is removed (from the left). Thus, the window slides through the array, keeping its size constant at ‘k’. The maximum sum found during this process is returned as the result.

Optimization

Instead of linearly searching through the counter array to find the Xth smallest, you can update where the Xth smallest is in constant time each time an element is added or removed.

Keeping track of the Xth smallest number in constant time is an excellent optimization strategy! It leverages the principle of “maintaining the state of the window”, which is a key part of solving many sliding window problems.

For instance, when an element is added to the window, you can update the Xth smallest number only if the added element is smaller than the current Xth smallest number. Similarly, when an element is removed from the window, you update the Xth smallest number only if the removed element is equal to the current Xth smallest number.

This way, you eliminate the need to iterate over the counter array to find the Xth smallest number for each window, thus reducing the time complexity from O(n*50) to O(n).

Here’s a rough illustration of the idea in Python-like pseudocode. Please note that the actual implementation may vary:

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def getSubarrayBeauty(nums, k, x):
    counter = [0] * 51
    ans = [0] * (len(nums) - k + 1)
    xth_smallest = None

    for i in range(k):
        if nums[i] < 0:
            counter[nums[i] + 50] += 1
            # Update the xth smallest if necessary
            if xth_smallest is None or nums[i] < xth_smallest:
                xth_smallest = nums[i]

    for i in range(k, len(nums) + 1):
        ans[i - k] = xth_smallest

        if i < len(nums):  # to avoid index out of range error
            if nums[i] < 0:
                counter[nums[i] + 50] += 1
                if nums[i] < xth_smallest:
                    xth_smallest = nums[i]
            if nums[i - k] < 0:
                counter[nums[i - k] + 50] -= 1
                # If the removed number is the xth smallest, need to update xth smallest
                if nums[i - k] == xth_smallest:
                    # Find the next smallest number
                    for j in range(50):
                        if counter[j] > 0:
                            xth_smallest = j - 50
                            break

    return ans

In this code, xth_smallest always stores the Xth smallest number of the current window. It gets updated only when an element that affects it is added or removed from the window, resulting in a more efficient solution.

Language Agnostic Coding Drills

  1. Dissecting the code:

Here are the distinct conceptual building blocks or “coding drills” that this code uses:

a. Variable Initialization: The code starts by setting up two arrays: counter and ans.

b. Looping: There are several loops in this code that iterate over the elements in nums.

c. Condition Checking: The code frequently checks whether a number is less than 0 using if statements.

d. Index Manipulation: The code uses the indices of the nums list to access specific elements and update the counter array.

e. Array Manipulation: The code makes use of arrays to store intermediate results (counter) and the final output (ans).

  1. Coding Concepts/Drills Listed in Order of Increasing Difficulty:

    a. Variable Initialization: This is an essential concept in any programming language and is typically one of the first things learners encounter. Difficulty: Low.

    b. Looping: Again, a fundamental concept. It’s slightly more complex than variable initialization because loops require understanding flow control. Difficulty: Low.

    c. Condition Checking: Along with loops, if statements are a core part of controlling program flow. Difficulty: Low.

    d. Array Manipulation: While not overly complex, this concept does require understanding how arrays work and how to perform operations on them. Difficulty: Medium.

    e. Index Manipulation: This is one of the more complex aspects of this code. It requires a good understanding of how indices work, and how they can be manipulated to access or modify elements in arrays or lists. Difficulty: High.

  2. Problem-Solving Approach:

The problem-solving approach to this problem involves several steps:

a. Understand the Problem: Before we can start coding, we need to have a solid understanding of what the problem is asking us to do. In this case, we’re being asked to calculate the “beauty” of each subarray of a given size in an array of numbers.

b. Plan the Solution: We need to devise a strategy to solve the problem. We observe that the beauty of a subarray is the xth smallest negative number in the subarray. This leads us to consider a “sliding window” approach, which is effective for problems involving subarrays.

c. Set Up the Data Structures: We decide to use two arrays: one to keep track of the counts of negative numbers (counter), and another to store the results (ans).

d. Implement the Solution: We then use our “coding drills” to implement the solution. We use loops to iterate over the nums list, if statements to check whether a number is negative, and array and index manipulation to update our counter and ans arrays as necessary.

e. Review and Refine the Solution: After coding the initial solution, we review it for correctness and efficiency. If needed, we refine the solution to make it more efficient or easier to understand. For instance, we could start the second loop from k instead of 0 to avoid checking i - k + 1 < 0 each time.

Each of these “drills” contributes to the final solution by performing a specific task that’s part of the overall problem-solving process. By mastering these smaller tasks, we can more easily tackle complex problems like this one.

High Level Coding Construct

This code is like a specialized calculator. It takes in a list of numbers, a window size, and a rank number. Then, it moves a “window” (or a sub-list of a specific size) across the original list of numbers from beginning to end. For each position of this window, the code tries to find the rank-th smallest negative number within that window, let’s call it ‘beauty’. If there are less than rank negative numbers, it assigns the beauty as zero.

So essentially, this code is finding the ‘beauty’ of all possible windows in the list and returning these ‘beauty’ values in order. For example, if we had a list like [1, -2, 3, -4, 5], a window size of 3, and a rank of 2, the program would tell us the ‘beauty’ of each window of size 3 in this list. A window size of 3 means it’s looking at 3 numbers at a time, so the first window would be [1, -2, 3], the next window would be [-2, 3, -4], and so on. In each of these windows, it’s looking for the 2nd smallest negative number, and if there isn’t one, it’s just zero. So, for our list, the ‘beauty’ values would be [0, -2, -4], and that’s what the code would give us back.

Below are the logical constructs used in the given code:

  1. Iteration: The code uses loops to repeatedly perform actions on elements in a list. This is seen in the multiple for-loops present in the code.

  2. Conditional Statements: The code checks certain conditions using if-statements. Depending on whether a condition is true or not, different actions are performed.

  3. Indexing: The code accesses specific elements of an array by their position (index).

  4. Counting: The code keeps track of counts of elements using an array as a counter.

  5. Window Sliding: The code maintains a ‘window’ of a certain size over the list of numbers, sliding it one position at a time and performing calculations on the elements within the window.

  6. Accumulation: The code sums the counts to find the xth smallest negative number in a window.

Remember, each of these logical constructs are basic building blocks in programming and can be implemented in any modern programming language.

Here’s a high-level description of the algorithm in plain English:

  1. We start with a list of numbers and two given numbers, k and x. We want to calculate the ‘beauty’ of each subarray (or chunk) of k numbers in the list.

  2. The ‘beauty’ of a subarray is defined as the xth smallest negative number in that subarray. If there are fewer than x negative numbers, then the beauty is zero.

  3. To compute this efficiently, we use an approach called the sliding window technique. Picture a window of k numbers moving along the list, one number at a time. For each position of the window, we calculate the beauty of the numbers in the window.

  4. To keep track of the negative numbers, we use a counter array. Each element of the counter array records the number of times a particular negative number appears in the current window.

  5. At each step, we use the counter array to find the xth smallest negative number (if it exists). We do this by going through the counter array, accumulating the counts until we reach x. The position in the counter array gives us the xth smallest negative number.

  6. Finally, as we slide the window along the list, we adjust our counter array by subtracting one for the number leaving the window and adding one for the new number entering the window.

  7. By the end, we have a list that shows the ‘beauty’ of each subarray of k numbers in the original list.

The key steps or operations that this code is performing on the input data are:

  1. Initialization: Creating a counter array to keep track of the count of each negative number in the current window and an answer array to store the beauty of each subarray.

  2. Preprocessing: Counting the occurrences of each negative number in the first window of size k in the list.

  3. Sliding the window and calculating the beauty: For each subsequent window of size k, the code performs the following operations:

    • It calculates the beauty of the current window. This is done by iterating over the counter array and accumulating the counts until it reaches x. The index in the counter array at this point gives the xth smallest negative number, which is the beauty of the current window.
    • The beauty of the current window is then added to the answer array.
    • If we’re not at the end of the list, the code slides the window one step to the right. It updates the counter array by decrementing the count of the number that has moved out of the window and incrementing the count of the new number that has moved into the window.

The reason why these operations are performed is to efficiently calculate the beauty of each subarray of size k in the list. The use of a counter array to keep track of negative numbers and a sliding window technique to move through the list enables this to be done in a computationally efficient manner.

Expanding Window

The expanding window technique is used to gradually consider more elements in a sequence, typically within a loop. Let’s consider a simple example where we want to find the first subarray from the input list that sums to a given target.

Here is a Python code for it:

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def expanding_window(arr, target):
    window_start = 0
    window_sum = 0
    for window_end in range(len(arr)):
        window_sum += arr[window_end]  # here we are expanding the window
        if window_sum == target:
            return arr[window_start:window_end+1]
    return []

print(expanding_window([1, 2, 3, 7, 5], 12))  # Outputs: [2, 3, 7]

Shrinking Window

The shrinking window technique is used to gradually consider fewer elements in a sequence. This is typically done once a certain condition is met, and we want to optimize the result further.

Let’s consider a simple example where we want to find the smallest subarray from the input list that sums to a given target or more.

Here is a Python code for it:

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def shrinking_window(arr, target):
    window_start = 0
    window_sum = 0
    min_length = float('inf')

    for window_end in range(len(arr)):
        window_sum += arr[window_end]

        while window_sum >= target:  # here we are shrinking the window
            min_length = min(min_length, window_end - window_start + 1)
            window_sum -= arr[window_start]
            window_start += 1

    return min_length if min_length != float('inf') else 0

print(shrinking_window([2, 3, 1, 2, 4, 3], 7))  # Outputs: 2

In this example, we initially expand the window until the window sum is at least as large as the target. Then we start to shrink the window from the start until the window sum is less than the target. During this shrinking process, we are constantly checking and updating the minimum length of the subarray that meets the condition.

Expanding and Shrinking Windows

Let’s use a simple problem to illustrate the concepts of Expanding Window and Shrinking Window.

Problem: Given an array of integers, find the maximum sum of any contiguous subarray of size ‘k’.

For simplicity, let’s assume the array is [2, 1, 5, 1, 3, 2] and k = 3.

Here’s the Python code implementing Expanding Window and Shrinking Window concepts to solve this problem:

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def max_subarray_sum(arr, k):
    max_sum = 0
    window_sum = 0

    for i in range(len(arr)):
        window_sum += arr[i]  # expanding the window

        # once we've reached the size 'k', we should start shrinking the window
        if i >= k-1:
            max_sum = max(max_sum, window_sum)
            window_sum -= arr[i - k + 1]  # shrinking the window

    return max_sum

print(max_subarray_sum([2, 1, 5, 1, 3, 2], 3))  # outputs 9

In this example, the window sum is calculated by adding each element in the array (expanding window). When the window size is equal to ‘k’, we start to subtract the first element of the window while moving to the next element (shrinking window). This way, we are sliding the window of size ‘k’ through the array.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.