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2902. Count of Sub-Multisets With Bounded Sum

Description

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

Solutions

Solution 1

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class Solution:
    def countSubMultisets(self, nums: List[int], l: int, r: int) -> int:
        kMod = 1_000_000_007
        # dp[i] := # of submultisets of nums with sum i
        dp = [1] + [0] * r
        count = collections.Counter(nums)
        zeros = count.pop(0, 0)

        for num, freq in count.items():
            # stride[i] := dp[i] + dp[i - num] + dp[i - 2 * num] + ...
            stride = dp.copy()
            for i in range(num, r + 1):
                stride[i] += stride[i - num]
            for i in range(r, 0, -1):
                if i >= num * (freq + 1):
                    # dp[i] + dp[i - num] + dp[i - freq * num]
                    dp[i] = stride[i] - stride[i - num * (freq + 1)]
                else:
                    dp[i] = stride[i]

        return (zeros + 1) * sum(dp[l : r + 1]) % kMod

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class Solution {
    static final int MOD = 1_000_000_007;
    public int countSubMultisets(List<Integer> nums, int l, int r) {
        Map<Integer, Integer> count = new HashMap<>();
        int total = 0;
        for (int num : nums) {
            total += num;
            if (num <= r) {
                count.merge(num, 1, Integer::sum);
            }
        }
        if (total < l) {
            return 0;
        }
        r = Math.min(r, total);
        int[] dp = new int[r + 1];
        dp[0] = count.getOrDefault(0, 0) + 1;
        count.remove(Integer.valueOf(0));
        int sum = 0;
        for (Map.Entry<Integer, Integer> e : count.entrySet()) {
            int num = e.getKey();
            int c = e.getValue();
            sum = Math.min(sum + c * num, r);
            // prefix part
            // dp[i] = dp[i] + dp[i - num] + ... + dp[i - c*num] + dp[i-(c+1)*num] + ... + dp[i %
            // num]
            for (int i = num; i <= sum; i++) {
                dp[i] = (dp[i] + dp[i - num]) % MOD;
            }
            int temp = (c + 1) * num;
            // correction part
            // subtract dp[i - (freq + 1) * num] to the end part.
            // leves dp[i] = dp[i] + dp[i-num] +...+ dp[i - c*num];
            for (int i = sum; i >= temp; i--) {
                dp[i] = (dp[i] - dp[i - temp] + MOD) % MOD;
            }
        }
        int ans = 0;
        for (int i = l; i <= r; i++) {
            ans += dp[i];
            ans %= MOD;
        }
        return ans;
    }
}

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class Solution {
public:
    int countSubMultisets(const vector<int> &nums, int l, int r) {
        int cnt[20001] = {};
        int memo[20001] = {};
        const int mod = 1000000007;
        for (int n : nums) {
            ++cnt[n];
        }
        fill_n(memo, cnt[1] + 1, 1);
        for (int n = 2, total = cnt[1]; n <= r; ++n) {
            if (!cnt[n]) {
                continue;
            }
            int top = (cnt[n] + 1) * n;
            total += n * cnt[n];
            for (int i = n, ii = min(total, r); i <= ii; ++i) {
                memo[i] = (memo[i] + memo[i - n]) % mod;
            }
            for (int i = min(total, r); i >= top; --i) {
                memo[i] = (mod + memo[i] - memo[i - top]) % mod;
            }
        }
        return accumulate(memo + l, memo + r + 1, 0LL) * (cnt[0] + 1) % mod;
    }
};

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func countSubMultisets(nums []int, l int, r int) int {
    multiset := make(map[int]int)
    for _, num := range nums {
        multiset[num]++
    }
    mem := make([]int, r+1)
    mem[0] = 1
    prefix := make([]int, len(mem))
    for num, occ := range multiset {
        copy(prefix, mem)
        for sum := num; sum <= r; sum++ {
            prefix[sum] = (prefix[sum] + prefix[sum-num]) % mod
        }
        for sum := r; sum >= 0; sum-- {
            if num > 0 {
                mem[sum] = prefix[sum]
                if sum >= num*(occ+1) {
                    mem[sum] = (mem[sum] - prefix[sum-num*(occ+1)] + mod) % mod
                }
            } else {
                mem[sum] = (mem[sum] * (occ + 1)) % mod
            }
        }
    }
    var result int
    for sum := l; sum <= r; sum++ {
        result = (result + mem[sum]) % mod
    }
    return result
}
var mod int = 1e9 + 7

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function countSubMultisets(nums: number[], l: number, r: number): number {
    const cnt: number[] = Array(20001).fill(0);
    const memo: number[] = Array(20001).fill(0);
    const mod: number = 1000000007;
    for (const n of nums) {
        cnt[n]++;
    }
    memo.fill(1, 0, cnt[1] + 1);
    let total: number = cnt[1];
    for (let n = 2; n <= r; ++n) {
        if (!cnt[n]) {
            continue;
        }
        const top: number = (cnt[n] + 1) * n;
        total += n * cnt[n];
        for (let i = n, ii = Math.min(total, r); i <= ii; ++i) {
            memo[i] = (memo[i] + memo[i - n]) % mod;
        }
        for (let i = Math.min(total, r); i >= top; --i) {
            memo[i] = (mod + memo[i] - memo[i - top]) % mod;
        }
    }
    let result: number = 0;
    for (let i = l; i <= r; i++) {
        result = (result + memo[i]) % mod;
    }
    return (result * (cnt[0] + 1)) % mod;
}

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