Minimum Subsequence in Non-Increasing Order

Easy

Asked by:

16 views

Topics:

ArraysGreedy Algorithms

Given the array nums, obtain a subsequence of the array whose sum of elements is strictly greater than the sum of the non included elements in such subsequence.

If there are multiple solutions, return the subsequence with minimum size and if there still exist multiple solutions, return the subsequence with the maximum total sum of all its elements. A subsequence of an array can be obtained by erasing some (possibly zero) elements from the array.

Note that the solution with the given constraints is guaranteed to be unique. Also return the answer sorted in non-increasing order.

Example 1:

Input: nums = [4,3,10,9,8]
Output: [10,9] 
Explanation: The subsequences [10,9] and [10,8] are minimal such that the sum of their elements is strictly greater than the sum of elements not included. However, the subsequence [10,9] has the maximum total sum of its elements.

Example 2:

Input: nums = [4,4,7,6,7]
Output: [7,7,6] 
Explanation: The subsequence [7,7] has the sum of its elements equal to 14 which is not strictly greater than the sum of elements not included (14 = 4 + 4 + 6). Therefore, the subsequence [7,6,7] is the minimal satisfying the conditions. Note the subsequence has to be returned in non-increasing order.

Constraints:

1 <= nums.length <= 500
1 <= nums[i] <= 100

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:

Can the input array contain negative numbers, zero, or only positive integers?
What should I return if the input array is empty or null?
If multiple subsequences satisfy the conditions, which one should I return? Is there a preference based on length or the value of the elements?
Are duplicate values allowed in the input array, and if so, how should they be handled when constructing the subsequence?
What is the maximum possible size of the input array?

Brute Force Solution

Approach

The brute force method for this task means we'll try out every single possible group of numbers from the original set. We'll look at each group and check if it meets the required condition of being in non-increasing order (meaning each number is less than or equal to the one before it).

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

First, consider a group containing only the first number from the original set.
Then, consider groups containing the first two numbers in every combination.
Continue this pattern, exploring all possible combinations of numbers from the original set, starting with single numbers and growing the group size.
For each group you make, check if the numbers are arranged in non-increasing order (from largest to smallest or staying the same).
If a group is in non-increasing order, remember it.
After checking all possible groups, find the smallest group among those that are in non-increasing order. This smallest group is the answer.

Code Implementation

def minimum_subsequence_brute_force(numbers):
    all_subsequences = []
    number_of_numbers = len(numbers)

    # Generate all possible subsequences
    for i in range(1 << number_of_numbers):
        subsequence = []
        for j in range(number_of_numbers):
            if (i >> j) & 1:
                subsequence.append(numbers[j])
        all_subsequences.append(subsequence)

    non_increasing_subsequences = []

    # Filter for non-increasing subsequences
    for subsequence in all_subsequences:

        is_non_increasing = True
        for k in range(len(subsequence) - 1):
            if subsequence[k] < subsequence[k + 1]:
                is_non_increasing = False
                break

        if is_non_increasing:

            # Only append subsequences that fit the criteria
            non_increasing_subsequences.append(subsequence)

    # Find the minimum length subsequence
    minimum_length = float('inf')
    minimum_subsequence = []

    # If no subsequence exists we return an empty array.
    if not non_increasing_subsequences:
        return []

    for subsequence in non_increasing_subsequences:

        # Keep track of the smallest valid subsequence.
        if len(subsequence) < minimum_length:
            minimum_length = len(subsequence)
            minimum_subsequence = subsequence

    return minimum_subsequence

Big(O) Analysis

Time Complexity

O(2^n) – The provided brute force approach involves generating all possible subsequences of the input array. An array of size n has 2^n possible subsequences (each element can either be included or excluded). For each subsequence, we need to check if it's in non-increasing order, which takes O(n) time in the worst case. Therefore, the overall time complexity is O(n * 2^n). While the sorting check contributes O(n) to each subsequence, the dominant factor remains the generation of all 2^n subsequences, making the overall complexity O(2^n) after simplification by removing constant or lower order factors.

Space Complexity

O(2^N) – The brute force approach, as described, generates all possible subsequences. In the worst-case scenario, each element in the input array of size N can either be included or excluded in a subsequence, leading to 2^N possible subsequences. To store these subsequences temporarily for validation and comparison, the algorithm would require memory proportional to the number of subsequences generated. Therefore, the auxiliary space complexity is O(2^N), because we would need memory to store all possible subsequences, each of which could potentially be of length N, although space complexity is dominated by the number of subsequences, not length.

Optimal Solution

Approach

We want to find the smallest group of numbers from the input that, when arranged in decreasing order, are bigger than the rest of the numbers that are left out. We can do this by focusing on the biggest numbers first.

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

First, find the sum of all the numbers in the list.
Next, sort the list in descending order, from largest to smallest.
Then, start adding the largest numbers one by one to a new group.
At each step, check if the sum of the numbers in the new group is greater than the sum of the numbers remaining from the original list. If it is, stop.
The numbers in the new group represent the minimum subsequence that satisfies the problem's condition.

Code Implementation

def minimum_subsequence(nums):
    total_sum = sum(nums)

    # Sort the input list in non-increasing order.
    sorted_nums = sorted(nums, reverse=True)

    subsequence = []
    subsequence_sum = 0

    for number in sorted_nums:
        subsequence.append(number)
        subsequence_sum += number

        # Check if the subsequence sum is greater than the remaining sum.
        remaining_sum = total_sum - subsequence_sum

        if subsequence_sum > remaining_sum:
            return subsequence

    return subsequence

Big(O) Analysis

Time Complexity

O(n log n) – First, the sum of all elements is calculated, which takes O(n) time, where n is the number of elements in the input list. The list is then sorted in descending order, which takes O(n log n) time using an efficient sorting algorithm like merge sort or heap sort. Finally, we iterate through the sorted list, adding elements to the subsequence and updating sums, which takes at most O(n) time. Since O(n log n) dominates O(n), the overall time complexity is O(n log n).

Space Complexity

O(N) – The algorithm sorts the input list of size N in descending order. This sorting operation typically requires auxiliary space, often creating a new sorted list, or using extra space during in-place sorting algorithms in certain cases. The sorted list (or the space used during in-place sorting) contributes O(N) space. Although there are temporary variables, their space usage is constant and negligible compared to the space required for sorting, thus the space complexity is dominated by the sorting step.

Edge Cases

Case	How to Handle
Null or empty input array	Return an empty list as there are no elements to form a subsequence.
Input array with only one element	Return the array itself as it trivially satisfies the non-increasing requirement.
Input array with all elements being identical	The algorithm should still produce a valid non-increasing subsequence containing the necessary elements.
Input array already in non-increasing order	The algorithm should return the entire input array as it is already a valid solution.
Input array with negative numbers and zeros	The algorithm should handle negative numbers and zeros correctly as they are valid numeric values.
Large input array that may cause memory issues with certain data structures	Ensure the solution uses memory-efficient data structures and algorithms to avoid memory overflow.
Input array contains very large integer values that could lead to integer overflow	Utilize appropriate data types or modular arithmetic to prevent integer overflow errors.
Multiple valid minimum subsequences exist	The algorithm should consistently return one of the valid minimum subsequences according to its specific logic, not necessarily all possible subsequences.

Null or empty input array

How to Handle:

Return an empty list as there are no elements to form a subsequence.

Input array with only one element

How to Handle:

Return the array itself as it trivially satisfies the non-increasing requirement.

Input array with all elements being identical

How to Handle:

The algorithm should still produce a valid non-increasing subsequence containing the necessary elements.

Input array already in non-increasing order

How to Handle:

The algorithm should return the entire input array as it is already a valid solution.

Input array with negative numbers and zeros

How to Handle:

The algorithm should handle negative numbers and zeros correctly as they are valid numeric values.

Large input array that may cause memory issues with certain data structures

How to Handle:

Ensure the solution uses memory-efficient data structures and algorithms to avoid memory overflow.

Input array contains very large integer values that could lead to integer overflow

How to Handle:

Utilize appropriate data types or modular arithmetic to prevent integer overflow errors.

Multiple valid minimum subsequences exist

How to Handle:

The algorithm should consistently return one of the valid minimum subsequences according to its specific logic, not necessarily all possible subsequences.