Find X-Sum of All K-Long Subarrays I

You are given an array nums of n integers and two integers k and x. The x-sum of an array is calculated by the following procedure:

1. Count the occurrences of all elements in the array.
2. Keep only the occurrences of the top x most frequent elements. If two elements have the same number of occurrences, the element with the bigger value is considered more frequent.
3. Calculate the sum of the resulting array.

Note that if an array has less than x distinct elements, its x-sum is the sum of the array.

Return an integer array answer of length n - k + 1 where answer[i] is the x-sum of the subarray nums[i..i + k - 1].

Example 1:

• Input: nums = [1,1,2,2,3,4,2,3], k = 6, x = 2
• Output: [6,10,12]
• Explanation:
  • For subarray [1, 1, 2, 2, 3, 4], only elements 1 and 2 will be kept in the resulting array. Hence, answer[0] = 1 + 1 + 2 + 2.
  • For subarray [1, 2, 2, 3, 4, 2], only elements 2 and 4 will be kept in the resulting array. Hence, answer[1] = 2 + 2 + 2 + 4. Note that 4 is kept in the array since it is bigger than 3 and 1 which occur the same number of times.
  • For subarray [2, 2, 3, 4, 2, 3], only elements 2 and 3 are kept in the resulting array. Hence, answer[2] = 2 + 2 + 2 + 3 + 3.

Example 2:

• Input: nums = [3,8,7,8,7,5], k = 2, x = 2
• Output: [11,15,15,15,12]
• Explanation: Since k == x, answer[i] is equal to the sum of the subarray nums[i..i + k - 1].

from collections import Counter


def calculate_x_sum(arr, x):
    """Calculates the x-sum of an array.

    Args:
        arr: The input array.
        x: The number of most frequent elements to consider.

    Returns:
        The x-sum of the array.
    """
    counts = Counter(arr)
    sorted_counts = sorted(counts.items(), key=lambda item: (item[1], item[0]), reverse=True)
    
    top_x_elements = sorted_counts[:x]
    
    top_x_values = [element for element, count in top_x_elements]
    
    x_sum = 0
    for num in arr:
        if num in top_x_values:
            x_sum += num
            
    return x_sum


def x_sum_subarrays(nums, k, x):
    """Calculates the x-sum of all subarrays of size k in nums.

    Args:
        nums: The input array of integers.
        k: The size of the subarrays.
        x: The number of most frequent elements to consider for the x-sum.

    Returns:
        An array containing the x-sums of all subarrays of size k.
    """
    n = len(nums)
    answer = []
    for i in range(n - k + 1):
        subarray = nums[i:i + k]
        answer.append(calculate_x_sum(subarray, x))
    return answer


# Example usage:
nums1 = [1, 1, 2, 2, 3, 4, 2, 3]
k1 = 6
x1 = 2
print(f"Example 1: {x_sum_subarrays(nums1, k1, x1)}")  # Output: [6, 10, 12]

nums2 = [3, 8, 7, 8, 7, 5]
k2 = 2
x2 = 2
print(f"Example 2: {x_sum_subarrays(nums2, k2, x2)}")  # Output: [11, 15, 15, 15, 12]

Explanation:

1. calculate_x_sum(arr, x) function:

   • Takes an array arr and an integer x as input.
   • Counts the occurrences of each element in the array using Counter.
   • Sorts the elements based on their counts in descending order. If counts are equal, it sorts by the element's value in descending order.
   • Selects the top x most frequent elements.
   • Calculates the sum of the elements in the original array, considering only the top x most frequent elements.
   • Returns the calculated x-sum.

2. x_sum_subarrays(nums, k, x) function:

   • Takes the main array nums, subarray size k, and the top element count x as input.
   • Iterates through the nums array to create subarrays of size k.
   • Calculates the x-sum for each subarray using the calculate_x_sum function.
   • Appends the x-sum to the answer list.
   • Returns the answer list, which contains the x-sums of all subarrays.

Big O Analysis:

Time Complexity:

• calculate_x_sum(arr, x): O(n + m log m), where n is the length of the input array arr and m is the number of distinct elements in arr. The O(n) comes from counting the elements using Counter, and the O(m log m) comes from sorting the distinct elements by frequency.
• x_sum_subarrays(nums, k, x): O((n - k + 1) * (k + m log m)), where n is the length of nums, k is the subarray size, and m is the number of distinct elements in each subarray. The outer loop iterates (n - k + 1) times. Inside the loop, calculate_x_sum is called on a subarray of size k, leading to O(k + m log m) complexity for each subarray.

Space Complexity:

• calculate_x_sum(arr, x): O(m), where m is the number of distinct elements in arr. This is due to the Counter object and the sorted list of counts.
• x_sum_subarrays(nums, k, x): O(n - k + 1 + m), where n is the length of nums, k is the subarray size, and m is the maximum number of distinct elements in any subarray. The O(n - k + 1) comes from storing the answer list, and O(m) comes from the space used within the calculate_x_sum function.

Edge Cases:

1. Empty Input Array:
   • If the input array nums is empty, the function should return an empty array since there are no subarrays to process.
2. k is greater than n:
   • If k is greater than the length of nums, there are no valid subarrays of size k. The function should return an empty array.
3. x is greater than the number of distinct elements in a subarray:
   • If x is greater than the number of distinct elements in a subarray, the x-sum should be the sum of all elements in the subarray.
4. All elements in a subarray are the same:
   • If all elements in a subarray are the same, the x-sum should be the sum of all elements in the subarray, regardless of the value of x.
5. Negative numbers in the input array:
   • The algorithm should work correctly with negative numbers in the input array.
6. Large input array:
   • For very large input arrays, the time complexity of the algorithm might become a concern. In such cases, optimization techniques such as using more efficient data structures or parallel processing might be considered.