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Number of Subsequences That Satisfy the Given Sum Condition

Medium

Number of Subsequences That Satisfy the Given Sum Condition

Medium

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 is the range of values within the input array, and can the array contain negative numbers or zero?
What is the maximum length of the input array?
Are duplicate numbers allowed in the input array, and if so, how do they affect the counting of subsequences?
If no such subsequence exists, what value should I return?
Is the order of elements in the subsequence important? (e.g., is [1, 2] different from [2, 1])?

Brute Force Solution

Approach

The brute force method involves checking every possible combination of numbers in the list to see if they satisfy the sum condition. We essentially try every possible subsequence to find the ones that work. If the sum condition is met, we count it and repeat this process until all possible subsequence combinations have been checked.

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

Consider all possible groups of numbers from the input list.
For each group, find the smallest and largest numbers within that group.
Check if the sum of the smallest and largest number in the group is less than or equal to the given target.
If the sum is within the limit, then count this group as a valid subsequence.
Repeat the above steps for every possible group of numbers.
The total count of valid groups is the final answer.

Code Implementation

def count_subsequences_brute_force(numbers, target):
    list_length = len(numbers)
    count = 0

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

        # Skip empty subsequence
        if not subsequence:
            continue

        # Find the minimum and maximum element of subsequence
        minimum_element = min(subsequence)
        maximum_element = max(subsequence)

        # Check if the sum condition is satisfied
        if minimum_element + maximum_element <= target:
            # Increment count if sum condition is met
            count += 1

    return count

Big(O) Analysis

Time Complexity

O(2^n) – The described brute force approach generates all possible subsequences from the input list. A list of size n has 2^n possible subsequences (each element can either be present or absent in a subsequence). For each subsequence, we find the minimum and maximum elements, which takes O(n) in the worst case. Therefore, the time complexity is dominated by the generation of all subsequences, resulting in O(2^n) since the check for each subsequence takes O(n) but the subsequences are the cost driver. Since 2^n grows much faster than any polynomial function of n, we focus only on 2^n for the overall complexity.

Space Complexity

O(1) – The provided brute force approach iterates through all possible subsequences without using any auxiliary data structures that scale with the input size N (the number of elements in the input list). It only requires a few constant space variables for tracking the smallest and largest numbers within each subsequence and a counter for the valid subsequences. Therefore, the space complexity remains constant, irrespective of the input size. No significant extra memory is used beyond a few integer variables.

Optimal Solution

Approach

The trick here is realizing we don't need to check every possible group of numbers. By sorting the numbers first, we can quickly find pairs that definitely meet the requirement or definitely don't, greatly reducing our work.

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

First, sort the numbers from smallest to largest. This makes it easier to find pairs that add up correctly.
Start with the smallest number. Find the largest number that, when added to the smallest number, is still less than or equal to the target sum.
Count all the subsequences that can be formed using the numbers between the smallest and largest numbers you just found. Each number in between can either be included or excluded in a subsequence, so there are two possibilities for each number.
Since each number has two choices, the total number of subsequences will be 2 raised to the power of the number of elements between the smallest and largest numbers.
Move to the next smallest number, and repeat the process. Keep a running total of the number of subsequences found so far.
Continue until you've checked all the possible smallest numbers. The final total will be the answer.

Code Implementation

def number_of_subsequences_that_satisfy_the_given_sum_condition(numbers, target):
    numbers.sort()
    total_subsequences_count = 0
    left_index = 0
    right_index = len(numbers) - 1
    modulo = 10**9 + 7

    while left_index <= right_index:
        # Find the largest element that satisfies the sum condition with the smallest.
        while left_index <= right_index and numbers[left_index] + numbers[right_index] > target:
            right_index -= 1

        if left_index > right_index:
            break

        # Calculate the number of subsequences based on elements between left and right.
        elements_between = right_index - left_index

        # Each element between can be included or excluded, thus 2^elements_between.
        subsequences_count = pow(2, elements_between, modulo)
        total_subsequences_count = (total_subsequences_count + subsequences_count) % modulo

        left_index += 1

    return total_subsequences_count

Big(O) Analysis

Time Complexity

O(n log n) – The dominant operation is sorting the input array of size n, which takes O(n log n) time. The subsequent steps involve iterating through the sorted array. For each element, we perform a binary search (implicitly or explicitly) to find the largest element that satisfies the sum condition. Binary search takes O(log n) time, and we perform this operation for each of the n elements in the array. Thus the time complexity due to iteration and binary search is O(n log n). Therefore the overall time complexity is O(n log n) + O(n log n), which simplifies to O(n log n).

Space Complexity

O(1) – The provided algorithm sorts the input array in-place, which doesn't require additional memory proportional to the input size. It only utilizes a few constant space variables for the two-pointer approach and calculating the power of 2. Therefore, the auxiliary space used is constant, irrespective of the input array's size N.

Edge Cases

Case	How to Handle
Empty input array	Return 0, as no subsequences can be formed.
Array with a single element	Return 0, as a subsequence of at least length 2 is needed.
Array with all elements greater than the target	Return 0, since no subsequence will satisfy the condition.
Array with all elements equal to zero and a positive target	Return 0, as the sum will always be zero and can't exceed the target.
Array contains very large positive numbers that could lead to integer overflow when summed.	Use a data type with a larger range, such as long, or perform modulo operation if applicable and stated.
Target value is very small or negative	Ensure algorithm handles negative target correctly and does not cause unexpected behavior (e.g., consider all possible sub-sequences).
Input array is extremely large (close to memory limits).	Ensure that the solution's space complexity is as efficient as possible (e.g., avoid creating unnecessarily large auxiliary data structures) and consider in-place operations.
Presence of duplicate numbers in the input array.	The solution must correctly handle duplicate numbers when counting the number of valid subsequences, potentially requiring combinations of the duplicates.

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 is the range of values within the input array, and can the array contain negative numbers or zero?
What is the maximum length of the input array?
Are duplicate numbers allowed in the input array, and if so, how do they affect the counting of subsequences?
If no such subsequence exists, what value should I return?
Is the order of elements in the subsequence important? (e.g., is [1, 2] different from [2, 1])?

Brute Force Solution

Approach

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

Consider all possible groups of numbers from the input list.
For each group, find the smallest and largest numbers within that group.
Check if the sum of the smallest and largest number in the group is less than or equal to the given target.
If the sum is within the limit, then count this group as a valid subsequence.
Repeat the above steps for every possible group of numbers.
The total count of valid groups is the final answer.

Code Implementation

def count_subsequences_brute_force(numbers, target):
    list_length = len(numbers)
    count = 0

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

        # Skip empty subsequence
        if not subsequence:
            continue

        # Find the minimum and maximum element of subsequence
        minimum_element = min(subsequence)
        maximum_element = max(subsequence)

        # Check if the sum condition is satisfied
        if minimum_element + maximum_element <= target:
            # Increment count if sum condition is met
            count += 1

    return count

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

First, sort the numbers from smallest to largest. This makes it easier to find pairs that add up correctly.
Start with the smallest number. Find the largest number that, when added to the smallest number, is still less than or equal to the target sum.
Count all the subsequences that can be formed using the numbers between the smallest and largest numbers you just found. Each number in between can either be included or excluded in a subsequence, so there are two possibilities for each number.
Since each number has two choices, the total number of subsequences will be 2 raised to the power of the number of elements between the smallest and largest numbers.
Move to the next smallest number, and repeat the process. Keep a running total of the number of subsequences found so far.
Continue until you've checked all the possible smallest numbers. The final total will be the answer.

Code Implementation

def number_of_subsequences_that_satisfy_the_given_sum_condition(numbers, target):
    numbers.sort()
    total_subsequences_count = 0
    left_index = 0
    right_index = len(numbers) - 1
    modulo = 10**9 + 7

    while left_index <= right_index:
        # Find the largest element that satisfies the sum condition with the smallest.
        while left_index <= right_index and numbers[left_index] + numbers[right_index] > target:
            right_index -= 1

        if left_index > right_index:
            break

        # Calculate the number of subsequences based on elements between left and right.
        elements_between = right_index - left_index

        # Each element between can be included or excluded, thus 2^elements_between.
        subsequences_count = pow(2, elements_between, modulo)
        total_subsequences_count = (total_subsequences_count + subsequences_count) % modulo

        left_index += 1

    return total_subsequences_count

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty input array	Return 0, as no subsequences can be formed.
Array with a single element	Return 0, as a subsequence of at least length 2 is needed.
Array with all elements greater than the target	Return 0, since no subsequence will satisfy the condition.
Array with all elements equal to zero and a positive target	Return 0, as the sum will always be zero and can't exceed the target.
Array contains very large positive numbers that could lead to integer overflow when summed.	Use a data type with a larger range, such as long, or perform modulo operation if applicable and stated.
Target value is very small or negative	Ensure algorithm handles negative target correctly and does not cause unexpected behavior (e.g., consider all possible sub-sequences).
Input array is extremely large (close to memory limits).	Ensure that the solution's space complexity is as efficient as possible (e.g., avoid creating unnecessarily large auxiliary data structures) and consider in-place operations.
Presence of duplicate numbers in the input array.	The solution must correctly handle duplicate numbers when counting the number of valid subsequences, potentially requiring combinations of the duplicates.