Find the Sum of the Power of All Subsequences

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Dynamic Programming

You are given an integer array nums of length n and a positive integer k.

The power of an array of integers is defined as the number of subsequences with their sum equal to k.

Return the sum of power of all subsequences of nums.

Since the answer may be very large, return it modulo 10⁹ + 7.

Example 1:

Input: nums = [1,2,3], k = 3

Output: 6

Explanation:

There are 5 subsequences of nums with non-zero power:

The subsequence [1,2,3] has 2 subsequences with sum == 3: [1,2,3] and [1,2,3].
The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].
The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].
The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].
The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].

Hence the answer is 2 + 1 + 1 + 1 + 1 = 6.

Example 2:

Input: nums = [2,3,3], k = 5

Output: 4

Explanation:

There are 3 subsequences of nums with non-zero power:

The subsequence [2,3,3] has 2 subsequences with sum == 5: [2,3,3] and [2,3,3].
The subsequence [2,3,3] has 1 subsequence with sum == 5: [2,3,3].
The subsequence [2,3,3] has 1 subsequence with sum == 5: [2,3,3].

Hence the answer is 2 + 1 + 1 = 4.

Example 3:

Input: nums = [1,2,3], k = 7

Output: 0

Explanation: There exists no subsequence with sum 7. Hence all subsequences of nums have power = 0.

Constraints:

1 <= n <= 100
1 <= nums[i] <= 10⁴
1 <= k <= 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?
What is the maximum size of the input array? What is the expected range of values within the array?
Are duplicate numbers allowed in the input array, and if so, how should they be handled in the subsequences?
By 'power of a subsequence,' do you mean the sum of the elements in the subsequence raised to some power (e.g., squared), or is there another definition?
Should the final sum of the power of all subsequences be returned modulo some value to prevent potential integer overflow?

Brute Force Solution

Approach

The brute force method finds the sum of subsequence powers by looking at every single subsequence we can create. It calculates the power (square) of the sum of elements within each subsequence. Then, it adds all those powers together.

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

First, identify all possible subsequences of the given set of numbers. A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.
For each subsequence, calculate the sum of its elements.
Next, calculate the power of the sum (sum squared).
Finally, add up the powers of all the subsequences. This total sum is your answer.

Code Implementation

def find_sum_of_power_of_all_subsequences_brute_force(numbers):
    number_of_elements = len(numbers)
    total_power_sum = 0

    # Iterate through all possible subsequences
    for i in range(1 << number_of_elements):
        current_subsequence = []

        # Construct subsequence based on the bitmask
        for j in range(number_of_elements):
            if (i >> j) & 1:
                current_subsequence.append(numbers[j])

        # Calculate sum of current subsequence
        subsequence_sum = sum(current_subsequence)

        # Calculate power of the subsequence sum
        power_of_subsequence = subsequence_sum ** 2

        # Accumulate the power to the total
        total_power_sum += power_of_subsequence

    return total_power_sum

Big(O) Analysis

Time Complexity

O(2^n) – The brute force approach iterates through all possible subsequences of the input array of size n. Generating all subsequences takes O(2^n) time because each element has two choices: either be included or excluded in a subsequence. For each subsequence, calculating the sum takes O(n) time in the worst case (when the subsequence includes all n elements). Squaring the sum is O(1). Therefore, the overall time complexity is dominated by the subsequence generation, resulting in O(2^n * n). However, since we can only use the Big O complexity, we can simply say that the total time is O(2^n).

Space Complexity

O(1) – The brute force approach, as described, calculates the sum of each subsequence directly without storing all subsequences explicitly. It computes the sum and then its square, accumulating the squared sums into a final result. No auxiliary data structures that scale with the input size N (the number of elements in the input) are used. Therefore, the space complexity is constant.

Optimal Solution

Approach

Calculating the power sum of all subsequences can be tricky, but there's a simple trick to it. Instead of generating all subsequences, we can realize that each number contributes to the final sum in a predictable way based on its value and the size of the original set.

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

Consider each number in the original set one by one.
Realize that each number will appear in exactly half of all possible subsequences.
Calculate 2 raised to the power of (the number of elements in the set minus 1). This result represents how many times each individual number appears in the total sum across all subsequences.
Multiply each number by this calculated power of 2.
Add up all of the multiplied results together. This final sum is the power sum of all subsequences.

Code Implementation

def find_the_sum_of_power_of_all_subsequences(number_list):
    list_length = len(number_list)
    
    #If the list is empty, the power sum is zero.
    if list_length == 0:
        return 0

    # This calculates 2^(n-1), which is how many times each number appears
    number_of_times_each_appears = 2**(list_length - 1)

    total_power_sum = 0
    for number in number_list:
        #Each number contributes this much to the total power sum.
        total_power_sum += number * number_of_times_each_appears

    return total_power_sum

Big(O) Analysis

Time Complexity

O(n) – The algorithm iterates through each of the n elements in the input array once. Inside the loop, a constant number of operations are performed: calculating 2 to the power of (n-1), multiplying by the current element, and adding to the running sum. The power calculation can be done in O(1) time if we precompute or use bit shifting. Therefore, the overall time complexity is directly proportional to the number of elements n, resulting in O(n).

Space Complexity

O(1) – The provided solution calculates the sum of the power of all subsequences without using any auxiliary data structures that scale with the input size. It involves calculating 2 raised to the power of (N-1), multiplying each number in the input set by this value, and adding up the multiplied results. These calculations are performed using constant space for variables, meaning the extra memory used does not depend on N, where N is the number of elements in the input set. Therefore, the space complexity is constant.

Edge Cases

Case	How to Handle
Null or empty input array	Return 0, as there are no subsequences, thus no sum of powers.
Array with a single element	Return 2 raised to the power of the single element (2^element).
Array with all elements equal to zero	The power of any zero value is 1, so sum of powers of all subsequences becomes equivalent to calculating number of subsequences, which can get very large and overflow.
Array with large positive integers	The power function can result in integer overflow, so use modular arithmetic or consider using larger data types.
Array containing negative integers	The power function applied to negative numbers may result in different behaviors depending on the exponent, so handle integer power carefully or clarify problem constraints.
Array with duplicate numbers	The problem statement defines subsequence and duplicates are allowed, so the duplicates will be counted multiple times in the subsequences.
Maximum sized input array	The number of subsequences will be 2^n, so the sum of powers may become extremely large, which requires using modular arithmetic to avoid overflow.
Language-specific integer overflow in power or sum calculations	Use modular arithmetic with a large prime number or use larger data types (e.g., long long in C++, BigInteger in Java) to prevent overflow.

Null or empty input array

How to Handle:

Return 0, as there are no subsequences, thus no sum of powers.

Array with a single element

How to Handle:

Return 2 raised to the power of the single element (2^element).

Array with all elements equal to zero

How to Handle:

The power of any zero value is 1, so sum of powers of all subsequences becomes equivalent to calculating number of subsequences, which can get very large and overflow.

Array with large positive integers

How to Handle:

The power function can result in integer overflow, so use modular arithmetic or consider using larger data types.

Array containing negative integers

How to Handle:

The power function applied to negative numbers may result in different behaviors depending on the exponent, so handle integer power carefully or clarify problem constraints.

Array with duplicate numbers

How to Handle:

The problem statement defines subsequence and duplicates are allowed, so the duplicates will be counted multiple times in the subsequences.

Maximum sized input array

How to Handle:

The number of subsequences will be 2^n, so the sum of powers may become extremely large, which requires using modular arithmetic to avoid overflow.

Language-specific integer overflow in power or sum calculations

How to Handle:

Use modular arithmetic with a large prime number or use larger data types (e.g., long long in C++, BigInteger in Java) to prevent overflow.