一般模型：
给出一个序列，求一个子序列使其符合某个条件
一般解法；
定义子序列的端点l和r，先定l，如果不符合条件r就一直向右拓展，当符合条件时，记录当前的区间，l向右拓展，看是否符合条件，不符合的话，r在继续向右拓展，直至找到最优的区间
例题：
1、 Poj3061
题意：给定一个序列，使得其和大于或等于S，求最短的子序列长度。
分析：首先，序列都是正数，如果一个区间其和大于等于S了，那么不需要在向后推进右端点了，因为其和也肯定大于等于S但长度更长，所以，当区间和小于S时右端点向右移动，和大于等于S时，左端点向右移动以进一步找到最短的区间，如果右端点移动到区间末尾其和还不大于等于S，结束区间的枚举。
这个题目区间和明显是有趋势的：单调变化，所以根据题目要求很容易求解，但是在使用之间需要对区间前缀和进行预处理计算。
#include <iostream> #include <cstdio> #include <sstream> #include <cstring> #include <map> #include <cctype> #include <set> #include <vector> #include <stack> #include <queue> #include <algorithm> #include <list> #include <cmath> #include <bitset> #define rap(i, a, n) for(int i=a; i<=n; i++) #define rep(i, a, n) for(int i=a; i<n; i++) #define lap(i, a, n) for(int i=n; i>=a; i--) #define lep(i, a, n) for(int i=n; i>a; i--) #define rd(a) scanf("%d", &a) #define rlld(a) scanf("%lld", &a) #define rc(a) scanf("%c", &a) #define rs(a) scanf("%s", a) #define rb(a) scanf("%lf", &a) #define rf(a) scanf("%f", &a) #define pd(a) printf("%d ", a) #define plld(a) printf("%lld ", a) #define pc(a) printf("%c ", a) #define ps(a) printf("%s ", a) #define MOD 2018 #define LL long long #define ULL unsigned long long #define Pair pair<int, int> #define mem(a, b) memset(a, b, sizeof(a)) #define _ ios_base::sync_with_stdio(0),cin.tie(0) //freopen("1.txt", "r", stdin); using namespace std; const int maxn = 110000, INF = 0x7fffffff; LL a[maxn]; int main() { int T; rd(T); while(T--) { int n, l = 1, r = 0; LL k; LL tmp, sum = 0; int mi_len = INF; rd(n), rlld(k); for(int i = 1; i <= n; i++) { rlld(a[i]); sum += a[i]; r++; while(sum >= k) sum -= a[l], mi_len = min(mi_len, r - l + 1), l++; } if(mi_len == INF) mi_len = 0; cout << mi_len << endl; } return 0; }
2、 poj3320
题意：一本书有P页，每一页都一个知识点，求去最少的连续页数覆盖所有的知识点。
分析：和上面的题一样的思路，如果一个区间的子区间满足条件，那么在区间推进到该处时，右端点会固定，左端点会向右移动到其子区间，且其子区间会是更短的，只是需要存储所选取的区间的知识点的数量，那么使用map进行映射以快速判断是否所选取的页数是否覆盖了所有的知识点。
#include <cstdio> #include <algorithm> #include <cstring> #include <set> #include <map> #define MAX 1000010 #define LL long long #define INF 0x3f3f3f3f using namespace std; int a[MAX]; map <int, int> cnt; set <int> t; int p, ans = INF, st, en, sum; int main() { scanf("%d", &p); for (int i = 0; i < p; i++) scanf("%d", a+i), t.insert(a[i]); int num = t.size(); while (1){ while (en<p && sum<num) if (cnt[a[en++]]++ == 0) sum++; if (sum < num) break; ans = min(ans, en-st); if (--cnt[a[st++]] == 0) sum--; } printf("%d ", ans); return 0; }
3、 poj2566
题意：给定一个数组和一个值t，求一个子区间使得其和的绝对值与t的差值最小，如果存在多个，任意解都可行。
分析：明显，借用第一题的思路，既然要找到一个子区间使得和最接近t的话，那么不断地找比当前区间的和更大的区间，如果区间和已经大于等于t了，那么不需要在去找更大的区间了，因为其和与t的差值更大，然后区间左端点向右移动推进即可。所以，首先根据计算出所有的区间和，
排序之后按照上面的思路求解即可。
#include <cstdio> #include <algorithm> #include <cstring> #define INF 0x3f3f3f3f #define LL long long #define MAX 100010 using namespace std; typedef pair<LL, int> p; LL a[MAX], t, ans, tmp, b; int n, k, l, u, st, en; p sum[MAX]; LL myabs(LL x) { return x>=0? x:-x; } int main() { while (scanf("%d %d", &n, &k), n+k){ sum[0] = p(0, 0); for (int i = 1; i <= n; i++){ scanf("%I64d", a+i); sum[i] = p(sum[i-1].first+a[i], i); } sort(sum, sum+1+n); while (k--){ scanf("%I64d", &t); tmp = INF; st = 0, en = 1; while(en <= n){ b = sum[en].first-sum[st].first; if(myabs(t-b) < tmp){ tmp = myabs(t-b); ans = b; l = sum[st].second; u = sum[en].second; } if(b > t) st++; else if(b < t) en++; else break; if(st == en) en++; } if (u < l) swap(u, l); printf("%I64d %d %d ", ans, l+1, u); } } return 0; }
4、 poj2739&poj2100
题意：找到某一个区间使得区间内的数的和/平方和等于某一给定值k。
分析：很明显了，几乎之与上面的poj2566又是一样的，当区间右端点不能再向右推进且区间和仍小于k的话就可以结束区间的枚举了。
代码：
poj2739：
#include <cstdio> #include <cstring> #include <algorithm> #include <vector> #include <utility> #include <queue> #define INF 0x3f3f3f3f #define LL long long using namespace std; int prime[] = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909,2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,3541,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,3659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769,3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,3907,3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793,4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933,4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021,5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153,5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281,5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,5417,5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519,5521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651,5653,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749,5779,5783,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861,5867,5869,5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257,6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353,6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,6481,6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619,6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737,6761,6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863,6869,6871,6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,7307,7309,7321,7331,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,7477,7481,7487,7489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,7573,7577,7583,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,7703,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263,8269,8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,8389,8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539,8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669,8677,8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761,8779,8783,8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887,8893,8923,8929,8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013,9029,9041,9043,9049,9059,9067,9091,9103,9109,9127,9133,9137,9151,9157,9161,9173,9181,9187,9199,9203,9209,9221,9227,9239,9241,9257,9277,9281,9283,9293,9311,9319,9323,9337,9341,9343,9349,9371,9377,9391,9397,9403,9413,9419,9421,9431,9433,9437,9439,9461,9463,9467,9473,9479,9491,9497,9511,9521,9533,9539,9547,9551,9587,9601,9613,9619,9623,9629,9631,9643,9649,9661,9677,9679,9689,9697,9719,9721,9733,9739,9743,9749,9767,9769,9781,9787,9791,9803,9811,9817,9829,9833,9839,9851,9857,9859,9871,9883,9887,9901,9907,9923,9929,9931,9941,9949,9967,9973}; int main() { int n; while (scanf("%d", &n), n){ int ans, st, en, sum; st = en = ans = sum = 0; while (1){ if (sum == n) ans++; if (sum >= n) sum -= prime[st++]; else{ if (prime[en] <= n) sum += prime[en++]; else break; } } printf("%d ", ans); } }
poj2100：
#include <cstdio> #include <cstring> #include <algorithm> #include <vector> #include <utility> #include <queue> #define INF 0x3f3f3f3f #define LL long long using namespace std; typedef pair<LL, pair<LL, LL> > p; p ans[1010]; int main() { LL n, st, en, sum; while (~scanf("%I64d", &n)){ st = 1, en = 1, sum = 0; int k = 0; while (1){ if (sum == n) ans[k++] = p(en-st, pair<LL, LL>(st, en-1)); if (sum >= n) sum -= st*st, st++; else{ if (en*en <= n) sum += en*en, en++; else break; } } printf("%d ", k); for (int i = 0; i < k; i++){ printf("%I64d ", ans[i].first); for (int j = ans[i].second.first; j <= ans[i].second.second; j++) printf("%I64d ", j); puts(""); } } return 0; }
总结：尺取法的模型便是这样：根据区间的特征交替推进左右端点求解问题，其高效的原因在于避免了大量的无效枚举，其区间枚举都是根据区间特征有方向的枚举，如果胡乱使用尺取法的话会使得枚举量减少，因而很大可能会错误，所以关键的一步是进行问题的分析！
例题均转自：https://blog.csdn.net/lxt_lucia/article/details/81091597