Skip to content

730. Count Different Palindromic Subsequences

Description

Given a string s, return the number of different non-empty palindromic subsequences in s. Since the answer may be very large, return it modulo 109 + 7.

A subsequence of a string is obtained by deleting zero or more characters from the string.

A sequence is palindromic if it is equal to the sequence reversed.

Two sequences a1, a2, ... and b1, b2, ... are different if there is some i for which ai != bi.

 

Example 1:

Input: s = "bccb"
Output: 6
Explanation: The 6 different non-empty palindromic subsequences are 'b', 'c', 'bb', 'cc', 'bcb', 'bccb'.
Note that 'bcb' is counted only once, even though it occurs twice.

Example 2:

Input: s = "abcdabcdabcdabcdabcdabcdabcdabcddcbadcbadcbadcbadcbadcbadcbadcba"
Output: 104860361
Explanation: There are 3104860382 different non-empty palindromic subsequences, which is 104860361 modulo 109 + 7.

 

Constraints:

  • 1 <= s.length <= 1000
  • s[i] is either 'a', 'b', 'c', or 'd'.

Solutions

Solution 1

Thinking

Count distinct palindromic subsequences; \(n\le 1000\) and four letters. Listing subsequences is impossible, and a plain palindrome DP overcounts the same string from different spans.

Classify by the end letter: palindromes wrapped in \(c\) are one plus the four kinds inside, or just \(c\) itself. Shorter intervals yield a standard interval DP.

\(dp[i][j][k]\) is the count in \(s[i..j]\) that start and end with letter \(k\). Both ends equal \(c\) gives \(2+\sum dp[i+1][j-1]\); otherwise shrink the side that is not \(c\). Sum the four values on \([0,n-1]\).

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
class Solution:
    def countPalindromicSubsequences(self, s: str) -> int:
        mod = 10**9 + 7
        n = len(s)
        dp = [[[0] * 4 for _ in range(n)] for _ in range(n)]
        for i, c in enumerate(s):
            dp[i][i][ord(c) - ord('a')] = 1
        for l in range(2, n + 1):
            for i in range(n - l + 1):
                j = i + l - 1
                for c in 'abcd':
                    k = ord(c) - ord('a')
                    if s[i] == s[j] == c:
                        dp[i][j][k] = 2 + sum(dp[i + 1][j - 1])
                    elif s[i] == c:
                        dp[i][j][k] = dp[i][j - 1][k]
                    elif s[j] == c:
                        dp[i][j][k] = dp[i + 1][j][k]
                    else:
                        dp[i][j][k] = dp[i + 1][j - 1][k]
        return sum(dp[0][-1]) % mod

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
class Solution {
    private final int MOD = (int) 1e9 + 7;

    public int countPalindromicSubsequences(String s) {
        int n = s.length();
        long[][][] dp = new long[n][n][4];
        for (int i = 0; i < n; ++i) {
            dp[i][i][s.charAt(i) - 'a'] = 1;
        }
        for (int l = 2; l <= n; ++l) {
            for (int i = 0; i + l <= n; ++i) {
                int j = i + l - 1;
                for (char c = 'a'; c <= 'd'; ++c) {
                    int k = c - 'a';
                    if (s.charAt(i) == c && s.charAt(j) == c) {
                        dp[i][j][k] = 2 + dp[i + 1][j - 1][0] + dp[i + 1][j - 1][1]
                            + dp[i + 1][j - 1][2] + dp[i + 1][j - 1][3];
                        dp[i][j][k] %= MOD;
                    } else if (s.charAt(i) == c) {
                        dp[i][j][k] = dp[i][j - 1][k];
                    } else if (s.charAt(j) == c) {
                        dp[i][j][k] = dp[i + 1][j][k];
                    } else {
                        dp[i][j][k] = dp[i + 1][j - 1][k];
                    }
                }
            }
        }
        long ans = 0;
        for (int k = 0; k < 4; ++k) {
            ans += dp[0][n - 1][k];
        }
        return (int) (ans % MOD);
    }
}

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
using ll = long long;

class Solution {
public:
    int countPalindromicSubsequences(string s) {
        int mod = 1e9 + 7;
        int n = s.size();
        vector<vector<vector<ll>>> dp(n, vector<vector<ll>>(n, vector<ll>(4)));
        for (int i = 0; i < n; ++i) dp[i][i][s[i] - 'a'] = 1;
        for (int l = 2; l <= n; ++l) {
            for (int i = 0; i + l <= n; ++i) {
                int j = i + l - 1;
                for (char c = 'a'; c <= 'd'; ++c) {
                    int k = c - 'a';
                    if (s[i] == c && s[j] == c)
                        dp[i][j][k] = 2 + accumulate(dp[i + 1][j - 1].begin(), dp[i + 1][j - 1].end(), 0ll) % mod;
                    else if (s[i] == c)
                        dp[i][j][k] = dp[i][j - 1][k];
                    else if (s[j] == c)
                        dp[i][j][k] = dp[i + 1][j][k];
                    else
                        dp[i][j][k] = dp[i + 1][j - 1][k];
                }
            }
        }
        ll ans = accumulate(dp[0][n - 1].begin(), dp[0][n - 1].end(), 0ll);
        return (int) (ans % mod);
    }
};

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
func countPalindromicSubsequences(s string) int {
    mod := int(1e9) + 7
    n := len(s)
    dp := make([][][]int, n)
    for i := range dp {
        dp[i] = make([][]int, n)
        for j := range dp[i] {
            dp[i][j] = make([]int, 4)
        }
    }
    for i, c := range s {
        dp[i][i][c-'a'] = 1
    }
    for l := 2; l <= n; l++ {
        for i := 0; i+l <= n; i++ {
            j := i + l - 1
            for _, c := range [4]byte{'a', 'b', 'c', 'd'} {
                k := int(c - 'a')
                if s[i] == c && s[j] == c {
                    dp[i][j][k] = 2 + (dp[i+1][j-1][0]+dp[i+1][j-1][1]+dp[i+1][j-1][2]+dp[i+1][j-1][3])%mod
                } else if s[i] == c {
                    dp[i][j][k] = dp[i][j-1][k]
                } else if s[j] == c {
                    dp[i][j][k] = dp[i+1][j][k]
                } else {
                    dp[i][j][k] = dp[i+1][j-1][k]
                }
            }
        }
    }
    ans := 0
    for _, v := range dp[0][n-1] {
        ans += v
    }
    return ans % mod
}

Comments