BZOJ 1396: 识别子串( 后缀数组 + 线段树 )

Posted 2020-06-13

这道题各位大神好像都是用后缀自动机做的?.....蒟蒻就秀秀智商写一写后缀数组解法.....

求出Height数组后, 我们枚举每一位当做子串的开头.

如上图(x, y是height值), Heights数组中相邻的3个后缀, 假如我们枚举s2的第一个字符为开头, 那我们发现, 长度至少为len = max(x, y)+1, 才能满足题意(仅出现一次). 这个很好脑补...因为s2和其他串的LCP是RMQ, 肯定会<=LCP(s1,s2)或<=LCP(s2,s3). 然后就用len去更新s2中前len个字符的答案, 线段树维护. 然后对于长度lth>len的也肯定是合法的, 他们对s2的前lth个字符都有贡献...但是事实上lth对前lth-1个字符c的贡献是没有卵用的....(因为小于同样字符开头的以c结尾的串的贡献或者是len的贡献), 所以lth>=len对第lth个字符有贡献.

容易看出这样的贡献是成等差数列的。。。。线段树维护就OK了.

时间复杂度O(N log N), 空间复杂度O(N) 

-------------------------------------------------------------------------

#include<cstdio>

#include<cstring>

#include<algorithm>

 

using namespace std;

 

const int maxn = 100009;

 

char S[maxn];

int N, L, R, Val;

int Height[maxn], Rank[maxn], Sa[maxn], cnt[maxn];

 

inline void Min(int &x, int t) {

if(t < x) x = t;

}

inline void Max(int &x, int t) {

if(t > x) x = t;

}

 

void BuildSA(int m) {

int *x = Height, *y = Rank;

for(int i = 0; i < m; i++) cnt[i] = 0;

for(int i = 0; i < N; i++) cnt[x[i] = S[i]]++;

for(int i = 1; i < m; i++) cnt[i] += cnt[i - 1];

for(int i = N; i--; ) Sa[--cnt[x[i]]] = i;

for(int k = 1, p = 0; k <= N; k <<= 1, p = 0) {

for(int i = N - k; i < N; i++) y[p++] = i;

for(int i = 0; i < N; i++)

if(Sa[i] >= k) y[p++] = Sa[i] - k;

for(int i = 0; i < m; i++) cnt[i] = 0;

for(int i = 0; i < N; i++) cnt[x[y[i]]]++;

for(int i = 1; i < m; i++) cnt[i] += cnt[i - 1];

for(int i = N; i--; ) Sa[--cnt[x[y[i]]]] = y[i];

swap(x, y);

p = (x[Sa[0]] = 0) + 1;

for(int i = 1; i < N; i++) {

if(y[Sa[i]] != y[Sa[i - 1]] || y[Sa[i] + k] != y[Sa[i - 1] + k]) p++;

x[Sa[i]] = p - 1;

}

if((m = p) >= N) break;

}

for(int i = 0; i < N; i++) Rank[Sa[i]] = i;

Height[0] = 0;

for(int i = 0, h = 0; i < N; i++) if(Rank[i]) {

if(h) h--;

while(S[i + h] == S[Sa[Rank[i] - 1] + h]) h++;

Height[Rank[i]] = h;

}

}

 

struct Node {

Node *lc, *rc;

int n, d;

inline void pd(int len) {

if(n != maxn) {

Min(lc->n, n);

Min(rc->n, n);

}

if(d != maxn) {

Min(lc->d, d);

Min(rc->d, d + ((len + 1) >> 1));

}

}

} pool[maxn << 1], *pt = pool, *Root;

 

void Build(Node* t, int l, int r) {

t->n = t->d = maxn;

if(l != r) {

int m = (l + r) >> 1;

Build(t->lc = pt++, l, m);

Build(t->rc = pt++, m + 1, r);

}

}

 

void Modify(Node* t, int l, int r) {

if(L <= l && r <= R) {

Min(t->n, Val);

} else {

int m = (l + r) >> 1;

if(L <= m) Modify(t->lc, l, m);

if(m < R) Modify(t->rc, m + 1, r);

}

}

 

void Change(Node* t, int l, int r) {

if(L <= l && r <= R) {

Min(t->d, Val + l - L);

} else {

int m = (l + r) >> 1;

if(L <= m) Change(t->lc, l, m);

if(m < R) Change(t->rc, m + 1, r);

}

}

 

void DFS(Node* t, int l, int r) {

if(l != r) {

int m = (l + r) >> 1;

t->pd(r - l + 1);

DFS(t->lc, l, m);

DFS(t->rc, m + 1, r);

} else 

printf("%d\n", min(t->d, t->n));

}

 

int main() {

scanf("%s", S);

N = strlen(S);

S[N++] = ‘$‘;

BuildSA(‘z‘ + 1);

int n = N - 1;

Build(Root = pt++, 1, n);

Height[N] = 0;

for(int i = 1; i < N; i++) {

Val = max(Height[i], Height[i + 1]) + 1;

if(Val > 1) {

if(Sa[i] + Val > n) continue;

L = Sa[i] + 1, R = L + Val - 2;

Modify(Root, 1, n);

}

L = Sa[i] + Val, R = n;

if(L > R) continue;

Change(Root, 1, n);

}

DFS(Root, 1, n);

return 0;

}

-------------------------------------------------------------------------

1396: 识别子串

Time Limit: 10 Sec  Memory Limit: 162 MB
Submit: 201  Solved: 119
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Description

Input

一行,一个由小写字母组成的字符串S,长度不超过10^5

Output

L行,每行一个整数,第i行的数据表示关于S的第i个元素的最短识别子串有多长.

Sample Input

agoodcookcooksgoodfood

Sample Output

1
2
3
3
2
2
3
3
2
2
3
3
2
1
2
3
3
2
1
2
3
4

HINT

Source