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
1 /* 4080 */ 2 #include <iostream> 3 #include <sstream> 4 #include <string> 5 #include <map> 6 #include <queue> 7 #include <set> 8 #include <stack> 9 #include <vector> 10 #include <deque> 11 #include <algorithm> 12 #include <cstdio> 13 #include <cmath> 14 #include <ctime> 15 #include <cstring> 16 #include <climits> 17 #include <cctype> 18 #include <cassert> 19 #include <functional> 20 #include <iterator> 21 #include <iomanip> 22 using namespace std; 23 //#pragma comment(linker,"/STACK:102400000,1024000") 24 25 #define sti set<int> 26 #define stpii set<pair<int, int> > 27 #define mpii map<int,int> 28 #define vi vector<int> 29 #define pii pair<int,int> 30 #define vpii vector<pair<int,int> > 31 #define rep(i, a, n) for (int i=a;i<n;++i) 32 #define per(i, a, n) for (int i=n-1;i>=a;--i) 33 #define clr clear 34 #define pb push_back 35 #define mp make_pair 36 #define fir first 37 #define sec second 38 #define all(x) (x).begin(),(x).end() 39 #define SZ(x) ((int)(x).size()) 40 #define lson l, mid, rt<<1 41 #define rson mid+1, r, rt<<1|1 42 43 const int maxn = 40005; 44 char s[maxn]; 45 int a[maxn]; 46 int height[maxn], sa[maxn], rrank[maxn]; 47 int wa[maxn], wb[maxn], wc[maxn], wv[maxn]; 48 int rpos, m; 49 50 bool cmp(int *r, int a, int b, int l) { 51 return r[a]==r[b] && r[a+l]==r[b+l]; 52 } 53 54 void da(int *r, int *sa, int n, int m) { 55 int i, j, *x=wa, *y=wb, *t, p; 56 57 for (i=0; i<m; ++i) wc[i] = 0; 58 for (i=0; i<n; ++i) wc[x[i]=r[i]]++; 59 for (i=1; i<m; ++i) wc[i] += wc[i-1]; 60 for (i=n-1; i>=0; --i) sa[--wc[x[i]]] = i; 61 for (j=1,p=1; p<n; j*=2, m=p) { 62 for (p=0, i=n-j; i<n; ++i) y[p++] = i; 63 for (i=0; i<n; ++i) if (sa[i] >= j) y[p++] = sa[i] - j; 64 for (i=0; i<n; ++i) wv[i] = x[y[i]]; 65 for (i=0; i<m; ++i) wc[i] = 0; 66 for (i=0; i<n; ++i) wc[wv[i]]++; 67 for (i=1; i<m; ++i) wc[i] += wc[i-1]; 68 for (i=n-1; i>=0; --i) sa[--wc[wv[i]]] = y[i]; 69 for (t=x,x=y,y=t,x[sa[0]]=0,p=1,i=1; i<n; ++i) 70 x[sa[i]] = cmp(y, sa[i-1], sa[i], j) ? p-1 : p++; 71 } 72 } 73 74 void calheight(int *r, int *sa, int n) { 75 int i, j, k = 0; 76 77 for (i=1; i<=n; ++i) rrank[sa[i]] = i; 78 for (i=0; i<n; height[rrank[i++]]=k) 79 for (k?k--:0, j=sa[rrank[i]-1]; r[j+k]==r[i+k]; ++k) ; 80 } 81 82 void printSa(int n) { 83 for (int i=1; i<=n; ++i) 84 printf("%d ", sa[i]); 85 putchar(' '); 86 } 87 88 void printHeight(int n) { 89 for (int i=1; i<=n; ++i) 90 printf("%d ", height[i]); 91 putchar(' '); 92 } 93 94 bool judge(int bound, int n) { 95 int cnt = 1, pos = -1; 96 97 rpos = -1; 98 rep(i, 1, n+1) { 99 if (height[i] >= bound) { 100 pos = max(pos, sa[i]); 101 ++cnt; 102 } else { 103 if (cnt >= m) 104 rpos = max(pos, rpos); 105 pos = sa[i]; 106 cnt = 1; 107 } 108 } 109 if (cnt >= m) 110 rpos = max(pos, rpos); 111 112 return rpos!=-1; 113 } 114 115 void solve() { 116 int n; 117 118 for (int i=0; ; ++i) { 119 if (s[i] == ' ') { 120 n = i; 121 break; 122 } 123 a[i] = s[i] - 'a' + 1; 124 } 125 a[n] = 0; 126 if (m == 1) { 127 printf("%d 0 ", n); 128 return ; 129 } 130 131 da(a, sa, n+1, 30); 132 calheight(a, sa, n); 133 134 int l = 1, r = n, mid; 135 int ans = 0, ansp; 136 137 while (l <= r) { 138 mid = (l + r) >> 1; 139 if (judge(mid, n)) { 140 ans = mid; 141 ansp = rpos; 142 l = mid + 1; 143 } else { 144 r = mid - 1; 145 } 146 } 147 148 if (ans) 149 printf("%d %d ", ans, ansp); 150 else 151 puts("none"); 152 } 153 154 int main() { 155 ios::sync_with_stdio(false); 156 #ifndef ONLINE_JUDGE 157 freopen("data.in", "r", stdin); 158 freopen("data.out", "w", stdout); 159 #endif 160 161 while (scanf("%d", &m)!=EOF && m) { 162 scanf("%s", s); 163 solve(); 164 } 165 166 #ifndef ONLINE_JUDGE 167 printf("time = %d. ", (int)clock()); 168 #endif 169 170 return 0; 171 }
（2）哈希
1 /* 4080 */ 2 #include <iostream> 3 #include <sstream> 4 #include <string> 5 #include <map> 6 #include <queue> 7 #include <set> 8 #include <stack> 9 #include <vector> 10 #include <deque> 11 #include <bitset> 12 #include <algorithm> 13 #include <cstdio> 14 #include <cmath> 15 #include <ctime> 16 #include <cstring> 17 #include <climits> 18 #include <cctype> 19 #include <cassert> 20 #include <functional> 21 #include <iterator> 22 #include <iomanip> 23 using namespace std; 24 //#pragma comment(linker,"/STACK:102400000,1024000") 25 26 #define sti set<int> 27 #define stpii set<pair<int, int> > 28 #define mpii map<int,int> 29 #define vi vector<int> 30 #define pii pair<int,int> 31 #define vpii vector<pair<int,int> > 32 #define rep(i, a, n) for (int i=a;i<n;++i) 33 #define per(i, a, n) for (int i=n-1;i>=a;--i) 34 #define clr clear 35 #define pb push_back 36 #define mp make_pair 37 #define fir first 38 #define sec second 39 #define all(x) (x).begin(),(x).end() 40 #define SZ(x) ((int)(x).size()) 41 #define lson l, mid, rt<<1 42 #define rson mid+1, r, rt<<1|1 43 44 #define ULL unsigned __int64 45 46 const int MOD = 754283; 47 const int maxn = 40005; 48 ULL H[maxn], base[maxn]; 49 ULL Hash[maxn]; 50 int pos[maxn]; 51 char s[maxn]; 52 int c[26]; 53 int m, len; 54 int rp; 55 56 void init() { 57 base[0] = 1; 58 rep(i, 1, maxn) 59 base[i] = base[i-1] * MOD; 60 } 61 62 bool comp(const int& a, const int& b) { 63 if (Hash[a] == Hash[b]) 64 return a<b; 65 return Hash[a] < Hash[b]; 66 } 67 68 bool judge(int l) { 69 int n = len - l + 1; 70 for (int i=0; i<n; ++i) { 71 pos[i] = i; 72 Hash[i] = H[i] - H[i+l] * base[l]; 73 } 74 75 sort(pos, pos+n, comp); 76 77 int i = 0; 78 rp = -1; 79 while (i < n) { 80 int j = i++; 81 while (i<n && Hash[pos[i]]==Hash[pos[j]]) 82 ++i; 83 if (i-j >= m) 84 rp = max(rp, pos[i-1]); 85 } 86 87 return rp >= 0; 88 } 89 90 void solve() { 91 len = strlen(s) ; 92 93 memset(c, 0, sizeof(c)); 94 rep(i, 0, len) 95 ++c[s[i]-'a']; 96 H[len] = 0; 97 per(i, 0, len) 98 H[i] = H[i+1] * MOD + s[i]-'a'; 99 100 bool flag = false; 101 102 rep(i, 0, 26) { 103 if (c[i] >= m) { 104 flag = true; 105 break; 106 } 107 } 108 109 if (!flag) { 110 puts("none"); 111 return ; 112 } 113 114 int ans = 1, ansp = -1; 115 int l = 1, r = len, mid; 116 117 while (l <= r) { 118 mid = (l + r) >> 1; 119 if (judge(mid)) { 120 ans = mid; 121 ansp = rp; 122 l = mid + 1; 123 } else { 124 r = mid - 1; 125 } 126 } 127 128 printf("%d %d ", ans, ansp); 129 } 130 131 int main() { 132 ios::sync_with_stdio(false); 133 #ifndef ONLINE_JUDGE 134 freopen("data.in", "r", stdin); 135 freopen("data.out", "w", stdout); 136 #endif 137 138 init(); 139 while (scanf("%d", &m)!=EOF && m) { 140 scanf("%s", s); 141 solve(); 142 } 143 144 145 #ifndef ONLINE_JUDGE 146 printf("time = %d. ", (int)clock()); 147 #endif 148 149 return 0; 150 }