[BZOJ3238][Ahoi2013]差异

3238: [Ahoi2013]差异

Time Limit: 20 Sec  Memory Limit: 512 MB
Submit: 3394  Solved: 1542
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Description

Input

一行，一个字符串S

Output

一行，一个整数，表示所求值

Sample Input

cacao

Sample Output

54

HINT

2<=N<=500000,S由小写英文字母组成

先把答案加的那部分弄出来，可以$O(1)$计算。。

然后剩下的部分可以通过建反串的后缀自动机，那么$Parent$树就是原串的后缀树。。

然后树形DP即可

#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
const int maxn = 500000 + 10;
struct State {
    int len, link, tot;
    int son[26];
}st[maxn * 2];
int cnt, last;
void sam_init() {
    cnt = last = 0;
    st[0].len = 0;
    st[0].link = -1;
    st[0].tot = 0;
    memset(st[0].son, 0, sizeof(st[0].son));
}
void sam_extend(char c) {
    int cur = ++cnt, idx = c - 'a';
    st[cur].len = st[last].len + 1;
    memset(st[cur].son, 0, sizeof(st[cur].son));
    st[cur].tot = 1;
    int p;
    for (p = last; p != -1 && !st[p].son[idx]; p = st[p].link)
        st[p].son[idx] = cur;
    if (p == -1) st[cur].link = 0;
    else {
        int q = st[p].son[idx];
        if (st[p].len + 1 == st[q].len) st[cur].link = q;
        else {
            int clone = ++cnt;
            st[clone].len = st[p].len + 1;
            memcpy(st[clone].son, st[q].son, sizeof(st[q].son));
            st[clone].link = st[q].link;
            st[clone].tot = 0;
            for (; p != -1 && st[p].son[idx] == q; p = st[p].link)
                st[p].son[idx] = clone;
            st[q].link = st[cur].link = clone;
        }
    }
    last = cur;
}
char S[maxn];
int len;
struct Edge {
    int to, next;
    Edge() {}
    Edge(int _t, int _n) : to(_t), next(_n) {}
}e[maxn * 2];
int fir[maxn * 2] = { 0 }, ecnt = 0;
inline void add(int u, int v) {
    e[++ecnt] = Edge(v, fir[u]); fir[u] = ecnt;
}
void build() {
    for (int i = 1; i <= cnt; i++)
        add(st[i].link, i);
}
long long ans;
void dfs(int u) {
    for (int v, i = fir[u]; i; i = e[i].next) {
        v = e[i].to;
        dfs(v);
        ans -= 2LL * st[u].len * st[u].tot * st[v].tot;
        st[u].tot += st[v].tot;
    }
}
int main() {
    scanf("%s", S);
    sam_init();
    len = strlen(S);
    for (int i = 0; i < len; i++)
        sam_extend(S[len - i - 1]);
    build();
    ans = (long long)(len + 1) * len * (len - 1) / 2;
    dfs(0);
    printf("%lld
", ans);
    return 0;
}

 时隔两年来更新

前面两项可以$O(1)$计算

主要是计算后面一项

注意到$sa$数组是一个$1$到$n$的全排列且$lcp(T_i,T_j)=lcp(T_j,T_i)$

因此$sum_{1le i<jle n}lcp(T_i,T_j)=sum_{1le i<jle n}lcp(T_{sa[i]},T_{sa[j]})$

而$lcp(T_{sa[i]},T_{sa[j]})=min_{i+1le kle j}(height[k])$

因此原式可变形为$sum_{2le ile jle n}min_{ile kle j}(height[k])$

可以考虑计算每个$height[i]$作为最小值时被多少个区间包含

显然可以用单调栈维护

注意一下相等数值之间的影响

时间复杂度$O(nlogn)$

#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
typedef long long ll;
const int maxn = 500000 + 10;
char s[maxn];
int n, m;
int sa[maxn], rank[maxn], height[maxn];
int tax[maxn], tp[maxn];
void qsort(){
    for(int i = 1; i <= m; i++) tax[i] = 0;
    for(int i = 1; i <= n; i++) tax[rank[i]]++;
    for(int i = 2; i <= m; i++) tax[i] += tax[i - 1];
    for(int i = n; i; i--) sa[tax[rank[tp[i]]]--] = tp[i];
}
bool cmp(int *a, int l, int r, int w){
    return a[l] == a[r] && a[l + w] == a[r + w];
}
void suffix_sort(){
    m = 128;
    for(int i = 1; i <= n; i++) rank[i] = s[i];
    for(int i = 1; i <= n; i++) tp[i] = i;
    qsort();
    for(int k = 1, p = 0; p < n; m = p, k <<= 1){
        p = 0;
        for(int i = 1; i <= k; i++) tp[++p] = n - k + i;
        for(int i = 1; i <= n; i++) if(sa[i] > k) tp[++p] = sa[i] - k;
        qsort();
        swap(tp, rank);
        p = rank[sa[1]] = 1;
        for(int i = 2; i <= n; i++)
            rank[sa[i]] = cmp(tp, sa[i - 1], sa[i], k) ? p : ++p;
    }
    for(int i = 1, j, k = 0; i <= n; height[rank[i++]] = k)
        for(k ? k-- : 0, j = sa[rank[i] - 1]; s[i + k] == s[j + k]; k++);
}
int f[maxn], g[maxn], sta[maxn], top;
void calc(){
    ll ans = (ll)n * (n - 1) * (n + 1) / 2;
    f[2] = 1;
    top = 0;
    for(int i = 2; i <= n; i++){
        while(top && height[sta[top]] > height[i]) top--;
        if(!top) f[i] = i - 1;
        else f[i] = i - sta[top];
        sta[++top] = i;
    }
    g[n] = 1;
    top = 0;
    for(int i = n; i >= 2; i--){
        while(top && height[sta[top]] >= height[i]) top--;
        if(!top) g[i] = n - i + 1;
        else g[i] = sta[top] - i;
        sta[++top] = i;
    }
    for(int i = 2; i <= n; i++)
        ans -= (ll)2 * f[i] * g[i] * height[i];    
    printf("%lld
", ans);
}
int main(){
    scanf("%s", s + 1);
    n = strlen(s + 1);
    suffix_sort();
    calc();
    return 0;
}