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2902. Count of Sub-Multisets With Bounded Sum

HardArray Preparing judge… LeetCode

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
ArrayHash TableDynamic ProgrammingSliding WindowKnapsack ProblemMultiple Knapsack

Hints

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.