Is Find the Sum of the Power of All Subsequences easy or hard?

Find the Sum of the Power of All Subsequences is classified as a Hard problem. It requires understanding subsequence enumeration, combinatorial counting, and dynamic programming optimization. Many candidates initially attempt brute force before recognizing the subset-sum DP pattern.

Find the Sum of the Power of All Subsequences Python/Java solution

Python and Java implementations typically use dynamic programming with a 1D or 2D DP array for subset sums. Each element updates the DP states in reverse order to avoid reuse within the same iteration. The final answer aggregates contributions for subsequences whose sum equals k.

How to solve Find the Sum of the Power of All Subsequences in O(n · k)?

Use dynamic programming similar to subset sum. Maintain a DP array where dp[s] counts subsequences with sum s while iterating through the array. For each number update sums in reverse order, and accumulate the contribution of subsequences that reach k using powers of two based on remaining elements.

What is the best approach for Find the Sum of the Power of All Subsequences?

The most efficient approach uses dynamic programming over subsequence sums. Track how many subsequences produce each sum while iterating through the array, then compute the power contribution for subsequences that reach sum k. This reduces the exponential subsequence search to roughly O(n · k) time.

Is Find the Sum of the Power of All Subsequences asked at Google/Amazon/Meta?

Problems involving subsequence counting, subset sums, and dynamic programming frequently appear in interviews at companies like Google, Amazon, and Meta. This problem combines subset enumeration with combinatorial counting, which is a common advanced DP interview pattern.

What data structure is used in Find the Sum of the Power of All Subsequences?

The core data structure is a dynamic programming array or table indexed by possible sums. It stores counts of subsequences reaching each sum while processing the array. Precomputed power-of-two arrays are often used to quickly calculate contribution weights.

What is the time complexity of Find the Sum of the Power of All Subsequences?

The brute force backtracking solution runs in O(2^n) time because every subsequence must be generated. The optimized dynamic programming approach runs in O(n · k) time, where n is the array length and k is the target sum. Space complexity ranges from O(k) to O(n · k) depending on the DP design.

Description Hints 3 Solution Video 36:13

3082. Find the Sum of the Power of All Subsequences

Hard38.7% acceptance2 companies

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

Asked at 2 companies

D. E. Shaw

Google

Topics

Array Dynamic Programming

Hints open one at a time

Hint 1

If there is a subsequence of length j with the sum of elements k, it contributes 2^{n - j} to the answer.

Hint 2

Let dp[i][j] represent the number of subsequences in the subarray nums[0..i] which have a sum of j.

Hint 3

We can find the dp[i][k] for all 0 <= i <= n-1 and multiply them with 2^{n - j} to get final answer.

36:13

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