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1. 题目描述
给定$n \times m， n、m \in [1, 10]$的方格，求不同形状的$[1 \cdots 10]$联通块的个数？
所谓不同形状，表示不能通过平移、旋转、镜像实现相同的形状。
2. 基本思路
显然数据不大， 那么可以打表。首先考虑，这个表怎么打？不妨使用$cn$表示连通块数。
那么对于$n \times m, m、n \ge cn$的种类数与$cn \times cn$中$cn$连通块数完全相同。
否则搜索$cn \times cn$中$cn$中的连通块，找出那些在边界$n \times m$的数量即可。
那么，关键问题是怎么对连通块抽象，怎么进行状态压缩。显然存在着平移、旋转、镜像等操作。
这也意味着同一个连通块的表达形式可能不同。
这里采用正则化的思想，即找到$x,y$的最小值，然后对所有坐标减去这个值。然后在对坐标值排序。
即可以得到标准化的连通块。那么如何状态压缩呢？
对排好序的连通块的$x,y$进行压缩，因此pair<int,int>即可以表示一个连通块，使用map充当visit。
假定我们已经得到了$cn \times cn$中$n-1$连通的情况，那么对其中的每个块向四个方向拓展既可以得到所有$n$连通的情况。
这里，首先注意边界范围不能超过$cn \times cn$，同时，仅考虑镜像和旋转得到的状态visit中是否包含。
平移不用考虑是因为使用了正则化，大大减少了计算量。
3. 代码
(1) 打表代码，其实不打表也能过
1 /* 2170 */ 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 #define INF 0x3f3f3f3f 44 #define mset(a, val) memset(a, (val), sizeof(a)) 45 46 #define LL unsigned long long 47 48 typedef struct node_t { 49 vpii vp; 50 51 void sorted() { 52 sort(all(vp)); 53 } 54 55 void push_back(pii p) { 56 vp.pb(p); 57 } 58 59 void clear() { 60 vp.clr(); 61 } 62 63 int size() const { 64 return vp.size(); 65 } 66 67 void regular() { 68 int mnx = INT_MAX, mny = INT_MAX; 69 70 int sz = SZ(vp); 71 72 rep(i, 0, sz) { 73 mnx = min(vp[i].fir, mnx); 74 mny = min(vp[i].sec, mny); 75 } 76 77 rep(i, 0, sz) { 78 vp[i].fir -= mnx; 79 vp[i].sec -= mny; 80 } 81 } 82 83 pair<pii,pii> calBound() const { 84 int mnx, mny, mxx, mxy; 85 int sz = SZ(vp); 86 87 if (sz == 0) 88 return mp(mp(0, 0), mp(0,0)); 89 90 mnx = mny = INT_MAX; 91 mxx = mxy = INT_MIN; 92 rep(i, 0, sz) { 93 mnx = min(mnx, vp[i].fir); 94 mxx = max(mxx, vp[i].fir); 95 mny = min(mny, vp[i].sec); 96 mxy = max(mxy, vp[i].sec); 97 } 98 99 return mp(mp(mnx, mxx), mp(mny, mxy)); 100 } 101 102 pii calL() const { 103 pair<pii,pii> ppii = calBound(); 104 return mp(ppii.fir.sec-ppii.fir.fir+1, ppii.sec.sec-ppii.sec.fir+1); 105 } 106 } node_t; 107 108 const int maxn = 11; 109 vector<node_t> E[maxn][maxn]; 110 set<pair<LL,LL> > has; 111 int ans[maxn][maxn][maxn]; 112 int dir[4][2] = { 113 -1, 0, 1,0, 0,-1, 0,1 114 }; 115 int n, m, cn; 116 bool printInfo = false; 117 118 pair<LL,LL> zip(const node_t& d, int sz) { 119 node_t nd = d; 120 LL x = 0, y = 0; 121 122 nd.regular(); 123 nd.sorted(); 124 125 rep(i, 0, sz) { 126 x = 10 * x + nd.vp[i].fir; 127 y = 10 * y + nd.vp[i].sec; 128 } 129 130 return mp(x, y); 131 } 132 133 void unzip(pair<LL,LL> p, node_t& nd, int sz) { 134 LL &x = p.fir, &y = p.sec; 135 136 per(i, 0, sz) { 137 nd.vp[i].fir = x % 10; 138 nd.vp[i].sec = y % 10; 139 x /= 10; 140 y /= 10; 141 } 142 } 143 144 bool check(const node_t& b) { 145 pair<LL,LL> p = zip(b, cn); 146 147 return has.find(p) != has.end(); 148 } 149 150 void rotate(node_t& d) { 151 rep(i, 0, cn) { 152 swap(d.vp[i].fir, d.vp[i].sec); 153 d.vp[i].sec = -d.vp[i].sec; 154 } 155 } 156 157 void mirror(node_t& d) { 158 rep(i, 0, cn) 159 d.vp[i].fir = -d.vp[i].fir; 160 } 161 162 bool judge(node_t& d) { 163 rep(i, 0, cn) { 164 rep(j, i+1, cn) { 165 if (d.vp[i] == d.vp[j]) 166 return false; 167 } 168 } 169 170 node_t dd; 171 172 dd = d; 173 174 rep(i, 0, 2) { 175 rep(j, 0, 4) { 176 if (check(dd)) 177 return false; 178 rotate(dd); 179 } 180 mirror(dd); 181 } 182 183 return true; 184 } 185 186 int calc() { 187 if (n>=cn && m>=cn) 188 return SZ(E[cn][cn]); 189 190 const vector<node_t>& vc = E[cn][cn]; 191 int sz = SZ(vc); 192 int ret = 0; 193 194 rep(i, 0, sz) { 195 pair<pii,pii> ppii = vc[i].calBound(); 196 int lx = ppii.fir.sec - ppii.fir.fir + 1; 197 int ly = ppii.sec.sec - ppii.sec.fir + 1; 198 if ((lx<=n && ly<=m) || (lx<=m && ly<=n)) 199 ++ret; 200 } 201 202 return ret; 203 } 204 205 void printAns() { 206 puts("int ans[10][100] = {"); 207 rep(i, 1, 11) { 208 putchar(‘\t‘); 209 putchar(‘{‘); 210 rep(j, 1, 11) { 211 rep(k, 1, 11) { 212 if (j==1 && k==1) 213 printf("%d", ans[i][j][k]); 214 else 215 printf(",%d", ans[i][j][k]); 216 } 217 } 218 putchar(‘}‘); 219 if (i != 10) 220 putchar(‘,‘); 221 putchar(‘\n‘); 222 } 223 puts("};"); 224 } 225 226 void solve() { 227 vector<node_t>& vc = E[n][cn]; 228 const vector<node_t>& ovc = E[n][cn-1]; 229 230 has.clr(); 231 int osz = SZ(ovc); 232 233 rep(i, 0, osz) { 234 const node_t& nd = ovc[i]; 235 pair<pii,pii> ppii = nd.calBound(); 236 int mxx, mxy, mnx, mny; 237 238 node_t d; 239 240 rep(j, 0, cn-1) d.pb(nd.vp[j]); 241 d.pb(mp(0, 0)); 242 243 rep(j, 0, cn-1) { 244 const int& x = nd.vp[j].fir; 245 const int& y = nd.vp[j].sec; 246 int xx, yy; 247 248 rep(k, 0, 4) { 249 xx = x + dir[k][0]; 250 yy = y + dir[k][1]; 251 d.vp[cn-1].fir = xx; 252 d.vp[cn-1].sec = yy; 253 254 mnx = min(ppii.fir.fir, xx); 255 mxx = max(ppii.fir.sec, xx); 256 mny = min(ppii.sec.fir, yy); 257 mxy = max(ppii.sec.sec, yy); 258 259 if (mxx-mnx+1>n || mxy-mny+1>n) 260 continue; 261 262 if (judge(d)) { 263 vc.pb(d); 264 pair<LL, LL> p = zip(d, cn); 265 has.insert(p); 266 } 267 } 268 } 269 } 270 } 271 272 void init() { 273 for (n=1; n<maxn; ++n) { 274 for (cn=1; cn<maxn; ++cn) { 275 if (cn == 1) { 276 node_t nd; 277 nd.pb(mp(0, 0)); 278 E[n][cn].pb(nd); 279 } else if (cn < n) { 280 int sz = SZ(E[cn][cn]); 281 rep(i, 0, sz) 282 E[n][cn].pb(E[cn][cn][i]); 283 } else { 284 solve(); 285 } 286 287 printf("E[%d][%d] = %d\n", n, cn, SZ(E[n][cn])); 288 fflush(stdout); 289 } 290 } 291 292 for (cn=1; cn<=10; ++cn) { 293 for (n=1; n<=cn; ++n) { 294 for (m=n; m<=cn; ++m) { 295 ans[cn][n][m] = ans[cn][m][n] = calc(); 296 } 297 } 298 } 299 300 printAns(); 301 } 302 303 int main() { 304 ios::sync_with_stdio(false); 305 #ifndef ONLINE_JUDGE 306 freopen("data.in", "r", stdin); 307 freopen("data.out", "w", stdout); 308 #endif 309 310 init(); 311 312 #ifndef ONLINE_JUDGE 313 printf("time = %d.\n", (int)clock()); 314 #endif 315 316 return 0; 317 }
(2) AC程序
1 /* 2170 */ 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 #define INF 0x3f3f3f3f 44 #define mset(a, val) memset(a, (val), sizeof(a)) 45 46 int ans[10][100] = { 47 {1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 48 {0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 49 {0,0,1,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,1,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 50 {0,0,0,1,0,0,0,0,0,0,0,1,4,5,0,0,0,0,0,0,0,4,4,5,0,0,0,0,0,0,1,5,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 51 {0,0,0,0,1,0,0,0,0,0,0,0,2,5,6,0,0,0,0,0,0,2,8,11,12,0,0,0,0,0,0,5,11,11,12,0,0,0,0,0,1,6,12,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 52 {0,0,0,0,0,1,0,0,0,0,0,0,1,7,12,13,0,0,0,0,0,1,8,29,34,35,0,0,0,0,0,7,29,29,34,35,0,0,0,0,0,12,34,34,34,35,0,0,0,0,1,13,35,35,35,35,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 53 {0,0,0,0,0,0,1,0,0,0,0,0,0,2,13,18,19,0,0,0,0,0,7,48,84,89,90,0,0,0,0,2,48,66,102,107,108,0,0,0,0,13,84,102,102,107,108,0,0,0,0,18,89,107,107,107,108,0,0,0,1,19,90,108,108,108,108,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 54 {0,0,0,0,0,0,0,1,0,0,0,0,0,1,11,30,37,38,0,0,0,0,3,63,169,223,230,231,0,0,0,1,63,140,307,361,368,369,0,0,0,11,169,307,307,361,368,369,0,0,0,30,223,361,361,361,368,369,0,0,0,37,230,368,368,368,368,369,0,0,1,38,231,369,369,369,369,369,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, 55 {0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,25,53,60,61,0,0,0,1,43,256,466,543,550,551,0,0,0,43,224,820,1127,1204,1211,1212,0,0,3,256,820,893,1200,1277,1284,1285,0,0,25,466,1127,1200,1200,1277,1284,1285,0,0,53,543,1204,1277,1277,1277,1284,1285,0,0,60,550,1211,1284,1284,1284,1284,1285,0,1,61,551,1212,1285,1285,1285,1285,1285,0,0,0,0,0,0,0,0,0,0,0}, 56 {0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,16,68,108,117,118,0,0,0,21,277,842,1226,1329,1338,1339,0,0,21,287,1847,3234,3773,3876,3885,3886,0,1,277,1847,2376,4003,4542,4645,4654,4655,0,16,842,3234,4003,4003,4542,4645,4654,4655,0,68,1226,3773,4542,4542,4542,4645,4654,4655,0,108,1329,3876,4645,4645,4645,4645,4654,4655,0,117,1338,3885,4654,4654,4654,4654,4654,4655,1,118,1339,3886,4655,4655,4655,4655,4655,4655} 57 }; 58 59 int main() { 60 ios::sync_with_stdio(false); 61 #ifndef ONLINE_JUDGE 62 freopen("data.in", "r", stdin); 63 freopen("data.out", "w", stdout); 64 #endif 65 66 int n, w, h; 67 68 while (scanf("%d%d%d",&n,&w,&h)!=EOF) { 69 --n; 70 --w; 71 --h; 72 cout << ans[n][w*10+h] << endl; 73 } 74 75 #ifndef ONLINE_JUDGE 76 printf("time = %d.\n", (int)clock()); 77 #endif 78 79 return 0; 80 }
4. 数据生成器
1 import sys 2 import string 3 from random import randint, shuffle 4 5 6 def GenData(fileName): 7 with open(fileName, "w") as fout: 8 t = 1 9 for tt in xrange(t): 10 for j in xrange(1, 11): 11 for i in xrange(1, j+1): 12 for k in xrange(1, j+1): 13 fout.write("%d %d %d\n" % (j, i, k)) 14 15 16 def MovData(srcFileName, desFileName): 17 with open(srcFileName, "r") as fin: 18 lines = fin.readlines() 19 with open(desFileName, "w") as fout: 20 fout.write("".join(lines)) 21 22 23 def CompData(): 24 print "comp" 25 srcFileName = "F:\Qt_prj\hdoj\data.out" 26 desFileName = "F:\workspace\cpp_hdoj\data.out" 27 srcLines = [] 28 desLines = [] 29 with open(srcFileName, "r") as fin: 30 srcLines = fin.readlines() 31 with open(desFileName, "r") as fin: 32 desLines = fin.readlines() 33 n = min(len(srcLines), len(desLines))-1 34 for i in xrange(n): 35 ans2 = int(desLines[i]) 36 ans1 = int(srcLines[i]) 37 if ans1 > ans2: 38 print "%d: wrong" % i 39 40 41 if __name__ == "__main__": 42 srcFileName = "F:\Qt_prj\hdoj\data.in" 43 desFileName = "F:\workspace\cpp_hdoj\data.in" 44 GenData(srcFileName) 45 MovData(srcFileName, desFileName)
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原文地址：http://www.cnblogs.com/bombe1013/p/5322678.html
