LeetCode 2539 LintCode 3855 Count the Number of Good Subsequences

Table of contents

Description

Question Links: LeetCode 2539, LintCode 3855

Description

A subsequence of a string is good if it is not empty and the frequency of each one of its characters is the same.

Given a string s, return the number of good subsequences of s. Since the answer may be too large, return it modulo 10^9 + 7.

A subsequence is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters.

Example 1:

Input: s = "aabb"
Output: 11
Explanation: The total number of subsequences is 24. There are five subsequences which are not good: "aabb", "aabb", "aabb", "aabb", and the empty subsequence. Hence, the number of good subsequences is 24-5 = 11.

Example 2:

Input: s = "leet"
Output: 12
Explanation: There are four subsequences which are not good: "leet", "leet", "leet", and the empty subsequence. Hence, the number of good subsequences is 24-4 = 12.
Example 3:

Input: s = "abcd"
Output: 15
Explanation: All of the non-empty subsequences are good subsequences. Hence, the number of good subsequences is 24-1 = 15.

Constraints:

• 1 <= s.length <= 10^4
• s consists of only lowercase English letters.

Idea

let n = s.length;

Let’s consider example 1, the string “aabb”.

1. We count the letters and get {a:2, b:2}.
2. We loop over all possible frequencies. For this example, the frequency could be 1 and 2.
3. For each frequency f, we could choose from the letters with frequency greater than f. We could choose f letters from there or none. We then remove one since empty string does not qualify.
4. For frequency 1, we calculate the number of ways we can form subsequences where each character (‘a’ and ‘b’) appears once. The combination for each character is Comb(2, 1)+1 == 3 (choose one or none). The total possibilities is 3*3 - 1 == 8.
5. For frequency 2, we can choose two or none, so the possibility is Comb(2,2)+1 == 2. The total is 2*2-1==3.

We could precompute the results for calculating the combination function (n choose k).

$\dbinom{n}{k} = \frac{n!}{k!\times(n-k)!}$

Complexity: Time $O(n)$, Space $O(n)$.

Explanation for calculation of the modular multiplicative inverse for $k!$ ($k$‘s factorial).

let $x = \frac{1}{k!}, M = 10^9+7$

$kx \equiv 1 \quad mod \quad M$

$kx \equiv k^{M-1} \quad mod \quad M$ with Fermat’s little theorem.

Multiple both sides by $k^{-1}$,

$x \equiv k^{M-2} \quad mod \quad M$

In python, that is pow(k, M-2, M).

Python

from collections import Counter

N = 10001
MOD = 10**9 + 7
f = [1] * N
g = [1] * N  # g[i] represents (i! % MOD)^(-1)
for i in range(1, N):
    f[i] = f[i - 1] * i % MOD
    g[i] = pow(f[i], MOD - 2, MOD)


def comb(n, k):
    return f[n] * g[k] * g[n - k] % MOD


class Solution:
    """
    467 ms, 5.05 mb with Python math.comb()
    121 ms, 5.66 mb
    """
    def countGoodSubsequences(self, s: str) -> int:
        cnt = Counter(s)
        ans = 0
        for i in range(1, max(cnt.values()) + 1):
            x = 1
            for v in cnt.values():
                if v >= i:
                    x = x * (comb(v, i) + 1) % MOD
            ans = (ans + x - 1) % MOD
        return ans

References

1. modular multiplicative inverse wiki
2. modular multiplicative inverse chinese OI wiki
3. algo monster
4. MMI CP algo