相关链接
题目传送门:http://www.lydsy.com/JudgeOnline/problem.php?id=3103
神犇题解:http://foreseeable97.logdown.com/posts/194507-herbicidalontak2010palindromic-equivalence
解题报告
这些字符串相关的计数题已经做过很多了吧?
这题显然就是给定$O(n)$个相等与不等的关系,让你求染色方案数
这样直接做好像是一个$NP$的问题?就是四色定理
那个一般化后的问题
但这题有一个非常好的性质,如果我们把一个等价类看成点,不等关系看成边
那么每一个联通块都是一个完全图!也就是一个弦图!
弦图的染色方案数是可以使用完美消除序列在$O(n)$的时间复杂度内解决的
有因为这题是完全图,所以任意一个$1 \sim n$的排列都是完美消除序列
于是直接从$1 \to n$进行计算即可
至于这图是个弦图图的证明,想一想还是很简单的(虽然不怎么容易观察出来)
我们可以参考:http://foreseeable97.logdown.com/posts/194507-herbicidalontak2010palindromic-equivalence
Code
#include<bits/stdc++.h> #define LL long long using namespace std; const int N = 2000009; const int M = N << 1; const int MOD = 1000000007; char pat[N],s[N]; int n,ans=1,fa[N],rid[N],col[N]; int vis[N],head[N],nxt[M],to[M]; vector<pair<int,int> > e; inline int read() { char c=getchar(); int f=1,ret=0; while (c<'0'||c>'9') {if(c=='-')f=-1;c=getchar();} while (c<='9'&&c>='0') {ret=ret*10+c-'0';c=getchar();} return ret * f; } int find(int x) { return x == fa[x]? x: fa[x] = find(fa[x]); } inline void manacher() { for (int i=1,r=1,p=1,t;i<=n*2;i++) { t = min(r - i, rid[p - (i - p)]); while (s[i+t+1] == s[i-t-1]) t++, fa[find(i+t)] = find(fa[i-t]); if ((rid[i] = t) + i > r) r = t + i, p = i; e.push_back(make_pair(i - t - 1, i + t + 1)); } } inline void AddEdge(int u, int v) { static int E = 1; if (u == v || !(u & 1) || !(v & 1)) return; to[++E] = v; nxt[E] = head[u]; head[u] = E; to[++E] = u; nxt[E] = head[v]; head[v] = E; } int main() { scanf("%s",pat+1); n = strlen(pat+1); s[0] = '#'; s[n*2] = '@'; for (int i=1;i<=n;i++) s[i*2-1] = pat[i], s[i*2] = '*'; for (int i=1;i<=n*2;i++) fa[i] = i; manacher(); for (int i=0;i<e.size();i++) AddEdge(find(e[i].first), find(e[i].second)); for (int i=1,cnt;cnt=26,i<=n*2;i+=2) { if (find(i) != i) continue; col[i] = 1; for (int j=head[i];j;j=nxt[j]) { if (col[to[j]] && vis[to[j]] < i) --cnt, vis[to[j]] = i; } if (cnt <= 0) ans = 0; else ans = (LL)ans * cnt % MOD; } cout<<ans; return 0; }
—————————— UPD 2017.4.26 ——————————
今天我们机房讨论了一下,这货是个弦图,不一定是一个完全图
哇塞,居然是沙发?留个名
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