hdu 6661 Acesrc and String Theory (后缀数组)

大意: 求重复$k$次的子串个数

枚举重复长度$i$, 把整个串分为$n/i$块, 如果每块可以$O(1)$计算, 那么最终复杂度就为$O(nlogn)$

有个结论是: 以$j$开头的子串重复次数最大为$1+\lfloor\frac{lcp(j,j+i)}{i}\rfloor$

先特判掉$k=1$的情况, 然后枚举每个块开头的位置, 计算出$lcp$的值$p$, 由于$k>1$, 合法位置的$lcp$值至少要跨越一个块, 可以得到

• 若$p\ge ki-1$, 那么这个块内所有点都合法
• 若$k(i-1)\le p< ki-1$, 那么这个块内只有前面一部分合法, 后面一部分一定不合法
• 若$p<k(i-1)$, 那么前面一部分一定不合法, 可能合法的点只有该块末尾与下一块的最长公共后缀的长度

#include <iostream>
#include <sstream>
#include <algorithm>
#include <cstdio>
#include <cmath>
#include <set>
#include <map>
#include <queue>
#include <string>
#include <cstring>
#include <bitset>
#include <functional>
#include <random>
#define REP(i,a,n) for(int i=a;i<=n;++i)
#define PER(i,a,n) for(int i=n;i>=a;--i)
#define hr putchar(10)
#define pb push_back
#define lc (o<<1)
#define rc (lc|1)
#define mid ((l+r)>>1)
#define ls lc,l,mid
#define rs rc,mid+1,r
#define x first
#define y second
#define io std::ios::sync_with_stdio(false)
#define endl '\n'
#define DB(a) ({REP(__i,1,n) cout<<a[__i]<<' ';hr;})
using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
const int P = 1e9+7, INF = 0x3f3f3f3f;
ll gcd(ll a,ll b) {return b?gcd(b,a%b):a;}
ll qpow(ll a,ll n) {ll r=1%P;for (a%=P;n;a=a*a%P,n>>=1)if(n&1)r=r*a%P;return r;}
ll inv(ll x){return x<=1?1:inv(P%x)*(P-P/x)%P;}
inline int rd() {int x=0;char p=getchar();while(p<'0'||p>'9')p=getchar();while(p>='0'&&p<='9')x=x*10+p-'0',p=getchar();return x;}
//head

const int N = 1e6+10;
int c[N],rk[N],rk2[N],h[N],sa[N];

const int Max_N = N;
namespace SA_IS {
int *sa;

template <typename _Char>
void push_S(const _Char s[],const int x,int cur[]) { sa[--cur[static_cast<int>(s[x])]] = x; };
template <typename _Char>
void push_L(const _Char s[],const int x,int cur[]) { sa[cur[static_cast<int>(s[x])]++] = x; };
template <typename _Char>
void induced_sort(const int v[], const int n, const int m, const _Char s[], char type[], int cnt[], int n1) {
	std::fill(sa, sa + n, 0);
	int *cur = cnt + m;
	std::copy(cnt, cnt + m, cur);
	for (int i = n1 - 1; i >= 0; --i) push_S(s,v[i],cur);
	std::copy(cnt, cnt + m - 1, cur + 1);
	for (int i = 0; i < n; ++i)
		if (sa[i] > 0 && type[sa[i] - 1] == 0)
			push_L(s,sa[i] - 1,cur);
	std::copy(cnt, cnt + m, cur);
	for (int i = n - 1; i >= 0; --i)
		if (sa[i] > 0 && type[sa[i] - 1])
			push_S(s,sa[i] - 1,cur);
}
template <typename _Char>
bool lms_equal(const _Char s[],char type[],int x, int y) {
	if (s[x] == s[y])
		while (s[++x] == s[++y] && type[x] == type[y])
			if (type[x] == 2 || type[y] == 2)
				return true;
	return false;
}

char *tp;
bool cmp(const int x) {
	return tp[x]!=2;
}
template <typename _Char>
void sais_core(const int n, const int m, const _Char s[], char type[], int lms[], int cnt[]) {
    int n1 = 0, ch = -1;
	tp = type;
    type[n - 1] = 1;
    for (int i = n - 2; i >= 0; --i) type[i] = s[i] == s[i + 1] ? type[i + 1] : s[i] < s[i + 1];

    std::fill(cnt, cnt + m, 0);
    for (int i = 0; i < n; ++i) ++cnt[static_cast<int>(s[i])];
    std::partial_sum(cnt, cnt + m, cnt);

    for (int i = 1; i < n; ++i)
        if (type[i - 1] == 0 && type[i] == 1)
            type[i] = 2, lms[n1++] = i;
    induced_sort(lms,n,m,s,type,cnt,n1);

    int *s1 = std::remove_if(sa, sa + n, cmp);
    for (int i = 0; i < n1; ++i) s1[sa[i] >> 1] = ch += ch <= 0 || !lms_equal(s,type,sa[i], sa[i - 1]);
    for (int i = 0; i < n1; ++i) s1[i] = s1[lms[i] >> 1];

    if (ch + 1 < n1)
        sais_core(n1, ch + 1, s1, type + n, lms + n1, cnt + m);
    else
        for (int i = 0; i < n1; ++i) sa[s1[i]] = i;

    for (int i = 0; i < n1; ++i) lms[n1 + i] = lms[sa[i]];
    induced_sort(lms + n1,n,m,s,type,cnt,n1);
}
template <typename _Char>
void main(const _Char s[], const int n, const int m) {
    static int _lms[Max_N], _cnt[Max_N << 1];
    static char _type[Max_N << 1];
    sais_core(n + 1, m, s, _type, _lms, _cnt);
}
}

void calc(int *s,int *rk,int n) {
	REP(i,1,n) ++sa[i];
    for(int i=1;i<=n;i++) rk[sa[i]]=i;
    for(int i=1,j,k=0;i<=n;i++) {
        if(k) k--;
        j=sa[rk[i]-1];
        while (s[i+k]==s[j+k]) k++;
        h[rk[i]] = k;
    }
}

void build(int *a,int *rk,int n) {
	SA_IS::sa = sa;
	a[n+1] = 0;
	SA_IS::main(a+1,n,128);
	calc(a,rk,n);
}

int Log[N],f[20][N],g[20][N];
void init(int a[N],int f[20][N],int n) {
    Log[0] = -1;
    REP(i,1,n) f[0][i] = a[i], Log[i]=Log[i>>1]+1;
    REP(j,1,19) for (int i=0;i+(1<<j-1)-1<=n; ++i) {
        f[j][i] = min(f[j-1][i],f[j-1][i+(1<<j-1)]);
    }
}
int RMQ(int f[20][N], int l, int r) {
    int t = Log[r-l+1];
    return min(f[t][l],f[t][r-(1<<t)+1]);
}
int n, k, a[N];
char s[N];
int lcp(int x, int y) {
	x = rk[x], y = rk[y];
	if (x>y) swap(x,y);
	return RMQ(f,x+1,y);
}
int lcs(int x, int y) {
	x = rk2[n-x+1], y = rk2[n-y+1];
	if (x>y) swap(x,y);
	return RMQ(g,x+1,y);
}

int main() {
	int t;
	scanf("%d", &t);
	while (t--) {
		scanf("%d%s", &k, s+1);
		n = strlen(s+1);
		if (k==1) {
			printf("%lld\n",(ll)n*(n+1)/2);
			continue;
		}
		REP(i,1,n) a[i]=s[i]-'a'+1;
		build(a,rk,n);
		init(h,f,n);
		reverse(a+1,a+1+n);
		build(a,rk2,n);
		init(h,g,n);
		ll ans = 0;
		REP(i,1,n) for (int j=1; j+i<=n; j+=i) {
			int p = lcp(j,j+i);
			if (p>=k*i) ans += i;
			else if (p>=k*i-i) ans += p-k*i+i+1;
			else {
				if (j+2*i-1>n) break;
				int s = lcs(j+i-1,j+2*i-1);
				if (!s) continue;
				s = min(s, i-1);
				int p2 = lcp(j+i-s,j+2*i-s);
				if (p2>=k*i-i) ans += min(p2-k*i+i+1,s);
			}
		}
		printf("%lld\n", ans);
	}
}