DP Interval¶
Table of Contents¶
- 516. Longest Palindromic Subsequence (Medium)
- 647. Palindromic Substrings (Medium)
- 5. Longest Palindromic Substring (Medium)
516. Longest Palindromic Subsequence¶
"""
- Return the length of the longest palindromic subsequence in `s`.
- Bottom-up DP table
| dp | b | b | b | a | b |
| :-: | :-: | :-: | :-: | :--------------: | :----------: |
| b | 1 | 2 | 3 | 3 | 4 |
| b | 0 | 1 | 2 | 2 | 3 `dp[i][j]` |
| b | 0 | 0 | 1 | 1 `dp[i+1][j-1]` | 2 |
| a | 0 | 0 | 0 | 1 | 1 |
| b | 0 | 0 | 0 | 0 | 1 |
"""
def longestPalindromeSubseq(s: str) -> int:
n = len(s)
dp = [[0] * n for _ in range(n)]
for i in range(n):
dp[i][i] = 1
for i in range(n - 1, -1, -1):
for j in range(i + 1, n):
if s[i] == s[j]:
dp[i][j] = dp[i + 1][j - 1] + 2
else:
dp[i][j] = max(dp[i][j - 1], dp[i + 1][j])
return dp[0][-1]
print(longestPalindromeSubseq("bbbab")) # 4
647. Palindromic Substrings¶
"""
- Return the number of palindromic substrings in `s`.
- Bottom-up DP table
| dp | a | b | b | a | e |
| :---: | :-: | :-: | :-: | :-: | :-: |
| **a** | 1 | 0 | 0 | 1 | 0 |
| **b** | 0 | 1 | 1 | 0 | 0 |
| **b** | 0 | 0 | 1 | 0 | 0 |
| **a** | 0 | 0 | 0 | 1 | 0 |
| **e** | 0 | 0 | 0 | 0 | 1 |
"""
def countSubstrings(s: str) -> int:
n = len(s)
dp = [[0] * n for _ in range(n)]
res = 0
for i in range(n - 1, -1, -1):
for j in range(i, n):
if s[i] == s[j]:
if j - i <= 1:
dp[i][j] = 1
res += 1
elif dp[i + 1][j - 1]:
dp[i][j] = 1
res += 1
return res
print(countSubstrings("abbae")) # 7
5. Longest Palindromic Substring¶
class longestPalindrome:
"""Return the longest palindromic substring in s."""
@staticmethod
def dp(s: str) -> str:
n = len(s)
if n <= 1:
return s
start, maxLen = 0, 1
# Init
dp = [[0] * n for _ in range(n)]
for i in range(n):
dp[i][i] = 1
# update
for j in range(1, n):
for i in range(j):
if s[i] == s[j]:
if j - i <= 2:
dp[i][j] = 1
else:
dp[i][j] = dp[i + 1][j - 1]
if dp[i][j] and j - i + 1 > maxLen:
maxLen = j - i + 1
start = i
return s[start : start + maxLen]
@staticmethod
def expand_around_center(s: str) -> str:
def expand(left, right):
while left >= 0 and right < len(s) and s[left] == s[right]:
left -= 1
right += 1
return right - left - 1
if len(s) <= 1:
return s
start, end = 0, 0
for i in range(len(s)):
odd = expand(i, i)
even = expand(i, i + 1)
maxLen = max(odd, even)
if maxLen > end - start:
start = i - (maxLen - 1) // 2
end = i + maxLen // 2
return s[start : end + 1]
if __name__ == "__main__":
s = "babad"
assert longestPalindrome.dp(s) in ["bab", "aba"]
assert longestPalindrome.expand_around_center(s) in ["bab", "aba"]