Is Count of Sub-Multisets With Bounded Sum easy or hard?

Count of Sub-Multisets With Bounded Sum is a Hard problem. The difficulty comes from recognizing the bounded knapsack structure and optimizing transitions with prefix sums or sliding window techniques to avoid O(n * r * freq) complexity.

Count of Sub-Multisets With Bounded Sum Python/Java solution

Implement a DP array of size r+1 and process values grouped by frequency. For each value, update reachable sums using prefix-sum sliding window optimization so you respect the maximum count of that number. The same logic works in Python, Java, C++, C#, and JavaScript.

How to solve Count of Sub-Multisets With Bounded Sum in O(n)?

The problem cannot generally be solved in strict O(n) time because every reachable sum up to r must be considered. The closest practical bound is O(U * r) using dynamic programming with prefix-sum sliding window optimization, which avoids the extra factor caused by element frequency.

What is the best approach for Count of Sub-Multisets With Bounded Sum?

The best approach uses dynamic programming with frequency compression and sliding window optimization. Treat the problem as a bounded knapsack where each value can be used up to its frequency. By maintaining prefix sums while updating the DP array, transitions can be computed in O(r) per unique value instead of iterating over all counts.

Is Count of Sub-Multisets With Bounded Sum asked at Google/Amazon/Meta?

Problems based on bounded subset sum and multiset counting frequently appear in interviews at companies like Google, Amazon, and Meta. Variants test dynamic programming, frequency compression, and optimization techniques such as prefix sums or sliding window transitions.

What data structure is used in Count of Sub-Multisets With Bounded Sum?

The main data structures are a hash table for counting frequencies and a dynamic programming array for storing the number of ways to achieve each sum. Prefix sums or sliding window accumulators are used to optimize transitions between DP states.

What is the time complexity of Count of Sub-Multisets With Bounded Sum?

The optimized solution runs in O(U * r) time where U is the number of unique values in the array and r is the upper bound of the sum range. Space complexity is O(r) because only a single DP array of size r+1 is maintained.

Description Hints 4 Solution Video 35:16

2902. Count of Sub-Multisets With Bounded Sum

Hard22.3% acceptance

You are given a 0-indexed array nums of non-negative integers, and two integers l and r.

Return the count of sub-multisets within nums where the sum of elements in each subset falls within the inclusive range of [l, r].

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

A sub-multiset is an unordered collection of elements of the array in which a given value x can occur 0, 1, ..., occ[x] times, where occ[x] is the number of occurrences of x in the array.

Note that:

Two sub-multisets are the same if sorting both sub-multisets results in identical multisets.
The sum of an empty multiset is 0.

Example 1:

Input: nums = [1,2,2,3], l = 6, r = 6
Output: 1
Explanation: The only subset of nums that has a sum of 6 is {1, 2, 3}.

Example 2:

Input: nums = [2,1,4,2,7], l = 1, r = 5
Output: 7
Explanation: The subsets of nums that have a sum within the range [1, 5] are {1}, {2}, {4}, {2, 2}, {1, 2}, {1, 4}, and {1, 2, 2}.

Example 3:

Input: nums = [1,2,1,3,5,2], l = 3, r = 5
Output: 9
Explanation: The subsets of nums that have a sum within the range [3, 5] are {3}, {5}, {1, 2}, {1, 3}, {2, 2}, {2, 3}, {1, 1, 2}, {1, 1, 3}, and {1, 2, 2}.

Constraints:

1 <= nums.length <= 2 * 10⁴
0 <= nums[i] <= 2 * 10⁴
Sum of nums does not exceed 2 * 10⁴.
0 <= l <= r <= 2 * 10⁴

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Hints open one at a time

Hint 1

Since the sum of numsis at most 20000, the number of distinct elements of nums is 200.

Hint 2

Let dp[x] be the number of submultisets of nums with sum x.

Hint 3

The answer to the problem is dp[l] + dp[l+1] + … + dp[r].

Hint 4

Use coin change dp to transition between states.

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2902. Count of Sub-Multisets With Bounded Sum

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