题目链接

题意：给定一个字符串\(|S|\le 3\times 10^5\)
对于每个 \(i\in [1,|S|]\) 求有多少子串\(s_ls_{l+1}\cdots s_r\)满足下面条件

• \(r-l+1 = i\)
• \(s_ls_{l+1}\cdots s_r\)是一个回文串
• \(s_ls_{l+1}\cdots s_{\lfloor(l+r)/2\rfloor}\)也是一个回文串

回文树学习:https://blog.csdn.net/Clove_unique/article/details/53750322

根据条件可知\(s_ls_{l+1}\cdots s_{\lfloor(l+r)/2\rfloor-1} == s_{\lfloor(l+r)/2\rfloor}\cdots s_r\)
先用回文自动机求出本质不同的回文串

• 第一种方法是回文树中创建新节点（意味着新的回文串）时判断该回文串是否满足上面条件，再建好树之后从后向前计算个数时累加即可
• 第二种是利用回文树中fail指针特性，每次循环找fail所指的回文串是否满足条件，直到根节点或找到为止，此次寻找结果可以更新到下一层fail指针所指的节点中（避免多次重复寻找，不然会T）

第一种方法代码：

#include <bits/stdc++.h>
using namespace std;
const int N = 3e5+10;
typedef unsigned long long ull;
typedef long long ll;
ull has[N],pn[N],base = 131;
ull getVal(int l,int r){return has[r] - has[l-1] * pn[r-l+1];
}
ll res[N];
char s[N];
namespace PAT{const int SZ = 6e5+10;int ch[SZ][26],fail[SZ],cnt[SZ],len[SZ],tot,last,pos[SZ],satisfy[SZ];void init(int n){for(int i=0;i<=n+10;i++){fail[i] = cnt[i] = len[i] = 0;for(int j=0;j<26;j++)ch[i][j] = 0;}s[0] = -1;fail[0] = 1;last = 0;len[0] = 0;len[1] = -1,tot = 1;}inline int newnode(int x){len[++tot] = x;return tot;}inline int getfail(int x,int n){while(s[n-len[x]-1] != s[n])x = fail[x];return x;}void create(char *s,int n){s[0] = -1;for(int i=1;i<=n;++i){int t = s[i]- 'a';int p = getfail(last,i);if(!ch[p][t]){int q = newnode(len[p]+2);fail[q] = ch[getfail(fail[p],i)][t];ch[p][t] = q;int need = (len[q] + 1) /2;if(len[q] == 1 || getVal(i-len[q]+1,i-len[q] + need) == getVal(i-need +1,i))satisfy[q] = 1;else satisfy[q] = 0;}++cnt[last = ch[p][t]];}}void solve(){for(int i=tot;i>=2;i--){cnt[fail[i]] += cnt[i];if(satisfy[i])res[len[i]] += cnt[i];}}
}
int main(){pn[0] = 1;for(int i=1;i<N;i++)pn[i] = pn[i-1] * base;while(~scanf("%s",s+1)){int n = strlen(s+1);has[0] = 1;for(int i=1;i<=n;i++){has[i] = has[i-1] * base + s[i];}for(int i=1;i<=n;i++)res[i] = 0;PAT::init(n);PAT::create(s,n);PAT::solve();for(int i=1;i<n;i++)printf("%lld ",res[i]);printf("%lld\n",res[n]);}return 0;
}

第二种方法code

#include <bits/stdc++.h>
using namespace std;
const int N = 3e5+10;
typedef long long ll;
char s[N];
ll res[N];
namespace PAT{const int SZ = 6e5+10;int ch[SZ][26],fail[SZ],cnt[SZ],len[SZ],tot,last;int be[SZ],ok[SZ];void init(int n){for(int i=0;i<=n+10;i++){fail[i] = cnt[i] = len[i] = 0;for(int j=0;j<26;j++)ch[i][j] = 0;be[i] = ok[i] = 0;}s[0] = -1;fail[0] = 1;last = 0;len[0] = 0;len[1] = -1;tot = 1;}inline int newnode(int x){len[++tot] = x;return tot;}inline int getfail(int x,int n){while(s[n-len[x]-1] != s[n])x = fail[x];return x;}void create(char *s){s[0] = -1;for(int i=1;s[i];++i){int t = s[i]- 'a';int p = getfail(last,i);if(!ch[p][t]){int q = newnode(len[p]+2);fail[q] = ch[getfail(fail[p],i)][t];ch[p][t] = q;}++cnt[last = ch[p][t]];}}void solve(){for(int i=tot;i>=2;i--){if(be[i] == 0)be[i] = i;while(be[i] >= 2 && len[be[i]] > (len[i] + 1)/2)be[i] = fail[be[i]];if(len[be[i]] == (len[i]+1)/2)ok[i] = 1;be[fail[i]] = be[i];}for(int i=tot;i>=2;i--){cnt[fail[i]] += cnt[i];if(ok[i]) res[len[i]] += cnt[i];}}
}
int main(){while(~scanf("%s",s+1)){int n = strlen(s+1);for(int i=1;i<=n;i++)res[i] = 0;PAT::init(n);PAT::create(s);PAT::solve();for(int i=1;i<n;i++)printf("%lld ",res[i]);printf("%lld\n",res[n]);}return 0;
}