【HDOJ】4080 Stammering Aliens

1. 题目描述
给定一个长为$n \in [1, 4000]$的字符串，求其中长度最长的子串，并且该子串在原串中出现至少$m$次，并求最右起始位置。

2. 基本思路
两种方法：二分+后缀数组，或者二分+哈希。
（1） 二分+后缀数组
对子串长度进行二分，若不同后缀的公共前缀超过这个值，则对计数值累加。若计数值超过m，则证明这个公共前缀是有效的，计数过程中同时维护pos（最右边界），从而更新rpos。

（2）二分+哈希
仍然是对长度进行二分，然后枚举其实位置计算该长度的子串的哈希值，排序后，计算，超过m表示是有效的子串，从而更新最右位置。哈希采用LCP哈希。

3. 代码
（1）后缀数组

 /* 4080 */
 #include <iostream>
 #include <sstream>
 #include <string>
 #include <map>
 #include <queue>
 #include <set>
 #include <stack>
 #include <vector>
 #include <deque>
 #include <algorithm>
 #include <cstdio>
 #include <cmath>
 #include <ctime>
 #include <cstring>
 #include <climits>
 #include <cctype>
 #include <cassert>
 #include <functional>
 #include <iterator>
 #include <iomanip>
 using namespace std;
 //#pragma comment(linker,"/STACK:102400000,1024000")

 #define sti                set<int>
 #define stpii            set<pair<int, int> >
 #define mpii            map<int,int>
 #define vi                vector<int>
 #define pii                pair<int,int>
 #define vpii            vector<pair<int,int> >
 #define rep(i, a, n)     for (int i=a;i<n;++i)
 #define per(i, a, n)     for (int i=n-1;i>=a;--i)
 #define clr                clear
 #define pb                 push_back
 #define mp                 make_pair
 #define fir                first
 #define sec                second
 #define all(x)             (x).begin(),(x).end()
 #define SZ(x)             ((int)(x).size())
 #define lson            l, mid, rt<<1
 #define rson            mid+1, r, rt<<1|1

 const int maxn = ;
 char s[maxn];
 int a[maxn];
 int height[maxn], sa[maxn], rrank[maxn];
 int wa[maxn], wb[maxn], wc[maxn], wv[maxn];
 int rpos, m;

 bool cmp(int *r, int a, int b, int l) {
     return r[a]==r[b] && r[a+l]==r[b+l];
 }

 void da(int *r, int *sa, int n, int m) {
     int i, j, *x=wa, *y=wb, *t, p;

     for (i=; i<m; ++i) wc[i] = ;
     for (i=; i<n; ++i) wc[x[i]=r[i]]++;
     for (i=; i<m; ++i) wc[i] += wc[i-];
     for (i=n-; i>=; --i) sa[--wc[x[i]]] = i;
     for (j=,p=; p<n; j*=, m=p) {
         for (p=, i=n-j; i<n; ++i) y[p++] = i;
         for (i=; i<n; ++i) if (sa[i] >= j) y[p++] = sa[i] - j;
         for (i=; i<n; ++i) wv[i] = x[y[i]];
         for (i=; i<m; ++i) wc[i] = ;
         for (i=; i<n; ++i) wc[wv[i]]++;
         for (i=; i<m; ++i) wc[i] += wc[i-];
         for (i=n-; i>=; --i) sa[--wc[wv[i]]] = y[i];
         for (t=x,x=y,y=t,x[sa[]]=,p=,i=; i<n; ++i)
             x[sa[i]] = cmp(y, sa[i-], sa[i], j) ? p- : p++;
     }
 }

 void calheight(int *r, int *sa, int n) {
     int i, j, k = ;

     for (i=; i<=n; ++i) rrank[sa[i]] = i;
     for (i=; i<n; height[rrank[i++]]=k)
     for (k?k--:, j=sa[rrank[i]-]; r[j+k]==r[i+k]; ++k) ;
 }

 void printSa(int n) {
     for (int i=; i<=n; ++i)
         printf("%d ", sa[i]);
     putchar('\n');
 }

 void printHeight(int n) {
     for (int i=; i<=n; ++i)
         printf("%d ", height[i]);
     putchar('\n');
 }

 bool judge(int bound, int n) {
     int cnt = , pos = -;

     rpos = -;
     rep(i, , n+) {
         if (height[i] >= bound) {
             pos = max(pos, sa[i]);
             ++cnt;
         } else {
             if (cnt >= m)
                 rpos = max(pos, rpos);
             pos = sa[i];
             cnt = ;
         }
     }
     if (cnt >= m)
         rpos = max(pos, rpos);

     return rpos!=-;
 }

 void solve() {
     int n;

     for (int i=; ; ++i) {
         if (s[i] == '\0') {
             n = i;
             break;
         }
         a[i] = s[i] - 'a' + ;
     }
     a[n] = ;
     if (m == ) {
         printf("%d 0\n", n);
         return ;
     }

     da(a, sa, n+, );
     calheight(a, sa, n);

     int l = , r = n, mid;
     int ans = , ansp;

     while (l <= r) {
         mid = (l + r) >> ;
         if (judge(mid, n)) {
             ans = mid;
             ansp = rpos;
             l = mid + ;
         } else {
             r = mid - ;
         }
     }

     if (ans)
         printf("%d %d\n", ans, ansp);
     else
         puts("none");
 }

 int main() {
     ios::sync_with_stdio(false);
     #ifndef ONLINE_JUDGE
         freopen("data.in", "r", stdin);
         freopen("data.out", "w", stdout);
     #endif

     while (scanf("%d", &m)!=EOF && m) {
         scanf("%s", s);
         solve();
     }

     #ifndef ONLINE_JUDGE
         printf("time = %d.\n", (int)clock());
     #endif

     return ;
 }

（2）哈希

 /* 4080 */
 #include <iostream>
 #include <sstream>
 #include <string>
 #include <map>
 #include <queue>
 #include <set>
 #include <stack>
 #include <vector>
 #include <deque>
 #include <bitset>
 #include <algorithm>
 #include <cstdio>
 #include <cmath>
 #include <ctime>
 #include <cstring>
 #include <climits>
 #include <cctype>
 #include <cassert>
 #include <functional>
 #include <iterator>
 #include <iomanip>
 using namespace std;
 //#pragma comment(linker,"/STACK:102400000,1024000")

 #define sti                set<int>
 #define stpii            set<pair<int, int> >
 #define mpii            map<int,int>
 #define vi                vector<int>
 #define pii                pair<int,int>
 #define vpii            vector<pair<int,int> >
 #define rep(i, a, n)     for (int i=a;i<n;++i)
 #define per(i, a, n)     for (int i=n-1;i>=a;--i)
 #define clr                clear
 #define pb                 push_back
 #define mp                 make_pair
 #define fir                first
 #define sec                second
 #define all(x)             (x).begin(),(x).end()
 #define SZ(x)             ((int)(x).size())
 #define lson            l, mid, rt<<1
 #define rson            mid+1, r, rt<<1|1

 #define ULL unsigned __int64

 const int MOD = ;
 const int maxn = ;
 ULL H[maxn], base[maxn];
 ULL Hash[maxn];
 int pos[maxn];
 char s[maxn];
 int c[];
 int m, len;
 int rp;

 void init() {
     base[] = ;
     rep(i, , maxn)
         base[i] = base[i-] * MOD;
 }

 bool comp(const int& a, const int& b) {
     if (Hash[a] == Hash[b])
         return a<b;
     return Hash[a] < Hash[b];
 }

 bool judge(int l) {
     int n = len - l + ;
     for (int i=; i<n; ++i) {
         pos[i] = i;
         Hash[i] = H[i] - H[i+l] * base[l];
     }

     sort(pos, pos+n, comp);

     int i = ;
     rp = -;
     while (i < n) {
         int j = i++;
         while (i<n && Hash[pos[i]]==Hash[pos[j]])
             ++i;
         if (i-j >= m)
             rp = max(rp, pos[i-]);
     }

     return rp >= ;
 }

 void solve() {
     len = strlen(s) ;

     memset(c, , sizeof(c));
     rep(i, , len)
         ++c[s[i]-'a'];
     H[len] = ;
     per(i, , len)
         H[i] = H[i+] * MOD + s[i]-'a';

     bool flag = false;

     rep(i, , ) {
         if (c[i] >= m) {
             flag = true;
             break;
         }
     }

     if (!flag) {
         puts("none");
         return ;
     }

     int ans = , ansp = -;
     int l = , r = len, mid;

     while (l <= r) {
         mid = (l + r) >> ;
         if (judge(mid)) {
             ans = mid;
             ansp = rp;
             l = mid + ;
         } else {
             r = mid - ;
         }
     }

     printf("%d %d\n", ans, ansp);
 }

 int main() {
     ios::sync_with_stdio(false);
     #ifndef ONLINE_JUDGE
         freopen("data.in", "r", stdin);
         freopen("data.out", "w", stdout);
     #endif

     init();
     while (scanf("%d", &m)!=EOF && m) {
         scanf("%s", s);
         solve();
     }

     #ifndef ONLINE_JUDGE
         printf("time = %d.\n", (int)clock());
     #endif

     return ;
 }