Find X-Sum of All K-Long Subarrays I

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ArraysSliding WindowsGreedy Algorithms

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:

Count the occurrences of all elements in the array.
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.
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].

Constraints:

1 <= n == nums.length <= 50
1 <= nums[i] <= 50
1 <= x <= k <= nums.length

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible value ranges for elements in the input array?
Can the input array be empty or null? What should I return in those cases?
Can `k` be larger than the size of the input array? If so, what should the function return?
Should I return a new array containing the X-Sums, or modify the existing array in place?
What data type should I use to represent the X-Sum (e.g., int, long) to avoid potential overflow issues?

Brute Force Solution

Approach

The brute force strategy for this problem involves examining every possible group of numbers of a specific size (K) within the larger set. For each of these groups, we'll calculate a special sum, and then find the total of all those sums. It's like checking every single possible combination to find the answer.

Here's how the algorithm would work step-by-step:

Start at the beginning of the list of numbers.
Take the first group of K numbers in order.
Calculate the special sum for this group of K numbers (the 'X-Sum').
Move one number forward in the list. Now, take the next group of K numbers.
Again, calculate the special sum for this new group.
Keep repeating this process of shifting one number forward and calculating the X-Sum for each new group of K numbers until you reach the end of the list.
Finally, add up all the individual X-Sums you calculated for each group to get the final total X-Sum.

Code Implementation

def find_x_sum_of_all_k_long_subarrays_i_brute_force(numbers, subarray_length):
    total_x_sum = 0

    # Iterate through all possible subarrays of length K
    for i in range(len(numbers) - subarray_length + 1):
        current_x_sum = 0
        
        # Calculate the X-Sum for the current subarray
        for j in range(subarray_length):
            current_x_sum += numbers[i + j]

        total_x_sum += current_x_sum

    return total_x_sum

Big(O) Analysis

Time Complexity

O(n*k) – The brute force approach iterates through the input array of size n, creating subarrays of size k in each iteration. Calculating the X-Sum for each subarray of size k takes O(k) time, since it processes each element within the subarray once. Since we iterate through n-k+1 subarrays each of size k, the total runtime is (n-k+1)*O(k). This can be approximated as n*k, therefore the time complexity is O(n*k).

Space Complexity

O(1) – The algorithm iterates through subarrays of size K, calculating the X-Sum for each. The plain English description doesn't mention storing these X-Sums or any other intermediate results in auxiliary data structures beyond what's required for a single X-Sum calculation. It only specifies adding up the X-Sums to find the final total, implying a single variable is sufficient for this purpose. Therefore, the space used is constant, regardless of the input size N (the size of the list of numbers) or K.

Optimal Solution

Approach

We need to find a special sum for every group of consecutive numbers of a fixed length within a larger list. Instead of recalculating the sum for each new group from scratch, we'll use a clever shortcut to update the sum efficiently.

Here's how the algorithm would work step-by-step:

First, calculate the sum of the initial group of numbers with the specified length.
Store this initial sum as the current sum.
Now, for each subsequent group, instead of calculating the entire sum again, subtract the number that is leaving the group (the leftmost number of the previous group).
Then, add the number that is entering the group (the rightmost number of the new group).
This updated sum is the sum for the current group.
Repeat the subtract-and-add process for each subsequent group, calculating and tracking the special sum each time.
Keep track of the overall sum of all the group sums to determine the final answer.

Code Implementation

def find_x_sum_of_all_k_long_subarrays(number_array, subarray_length):
    array_length = len(number_array)
    if array_length < subarray_length or subarray_length <= 0:
        return 0

    total_x_sum = 0
    current_subarray_sum = 0

    # Calculate the sum of the initial subarray.
    for i in range(subarray_length):
        current_subarray_sum += number_array[i]

    total_x_sum += current_subarray_sum

    # Iterate through the remaining subarrays using
    # the sliding window technique.
    for i in range(subarray_length, array_length):
        current_subarray_sum -= number_array[i - subarray_length]

        # Subtract the leftmost element of the previous subarray.
        current_subarray_sum += number_array[i]

        # Add the rightmost element of the current subarray.
        total_x_sum += current_subarray_sum

    return total_x_sum

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through the input array of size n once to compute the X-Sum for each k-long subarray. The initial sum of the first subarray of length k is calculated in O(k) time but this is done only once. Subsequent subarray sums are calculated by subtracting the leaving element and adding the entering element, which takes constant time, O(1), for each subarray. Since there are approximately n-k+1 such subarrays (which is bounded by n), and each subarray's X-Sum calculation takes O(1) time, the overall time complexity is dominated by the single pass through the array, leading to O(n).

Space Complexity

O(1) – The algorithm uses a fixed number of variables to store the initial sum, the current sum, and potentially a few loop counters or index variables. The number of these variables does not depend on the size of the input array, N. Therefore, the algorithm's auxiliary space complexity is constant.

Edge Cases

Case	How to Handle
Null or empty input array	Return an empty list immediately as there are no subarrays to process.
k is zero or negative	Return an empty list immediately, as a subarray of zero or negative length is invalid.
k is greater than the length of the array	Return an empty list immediately because no subarray of length k exists.
Array contains only one element and k is 1	The single element subarray will contribute that element's value to the X-Sum.
Array contains very large numbers that could cause integer overflow when summed.	Use a data type with a larger range like long or handle potential overflow during the summation process.
Array contains negative numbers.	The algorithm should handle negative numbers correctly during subarray summation.
All elements in the array are zero.	The X-Sum will simply be the sum of zeroes within each k-sized subarray, potentially resulting in zero.
Large array size with a small k. The sliding window approach is most efficient.	Implement sliding window technique to compute x-sum with time complexity O(n).

Null or empty input array

How to Handle:

Return an empty list immediately as there are no subarrays to process.

k is zero or negative

How to Handle:

Return an empty list immediately, as a subarray of zero or negative length is invalid.

k is greater than the length of the array

How to Handle:

Return an empty list immediately because no subarray of length k exists.

Array contains only one element and k is 1

How to Handle:

The single element subarray will contribute that element's value to the X-Sum.

Array contains very large numbers that could cause integer overflow when summed.

How to Handle:

Use a data type with a larger range like long or handle potential overflow during the summation process.

Array contains negative numbers.

How to Handle:

The algorithm should handle negative numbers correctly during subarray summation.

All elements in the array are zero.

How to Handle:

The X-Sum will simply be the sum of zeroes within each k-sized subarray, potentially resulting in zero.

Large array size with a small k. The sliding window approach is most efficient.

How to Handle:

Implement sliding window technique to compute x-sum with time complexity O(n).