POJ 2488 A Knight's Journey

题目链接：https://vjudge.net/problem/POJ-2488

题目大意

　　多组数据，每组给定一个 p * q 的国际象棋棋盘，问国际象棋中的骑士能否选取一个起点，然后从这个起点能够不重复地走到所有格子，如果有，输出字典序最小的路径。

分析

　　先说结论，如果这条路存在，那么字典序最小的路径的起点一定是 A1。
　　这个结论是猜出来的，因为我看很多帖子说的都不合逻辑，我自己写个程序验证了一下，不过 p 和 q 只能枚举到 7，再往后电脑跑不出来了，结果就是，只要以 A1 为起点走不通，那么以其他点为起点也走不通，最后代码提交 AC 也验证了这一点。
　　PS：有大佬如果会证明，恳请赐教！！！
　　验证程序如下：
  1 #include <cmath>
  2 #include <ctime>
  3 #include <iostream>
  4 #include <string>
  5 #include <vector>
  6 #include <cstdio>
  7 #include <cstdlib>
  8 #include <cstring>
  9 #include <queue>
 10 #include <map>
 11 #include <set>
 12 #include <algorithm>
 13 #include <cctype>
 14 #include <stack>
 15 #include <deque>
 16 #include <list>
 17 #include <sstream>
 18 using namespace std;
 19  
 20 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
 21 #define Rep(i,n) for (int i = 0; i < (n); ++i)
 22 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
 23 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
 24 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
 25 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 26 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 27 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 28  
 29 #define pr(x) cout << #x << " = " << x << "  "
 30 #define prln(x) cout << #x << " = " << x << endl
 31  
 32 #define LOWBIT(x) ((x)&(-x))
 33  
 34 #define ALL(x) x.begin(),x.end()
 35 #define INS(x) inserter(x,x.begin())
 36 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 37 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
 38 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 39 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 40  
 41 #define ms0(a) memset(a,0,sizeof(a))
 42 #define msI(a) memset(a,inf,sizeof(a))
 43 #define msM(a) memset(a,-1,sizeof(a))
 44 
 45 #define MP make_pair
 46 #define PB push_back
 47 #define ft first
 48 #define sd second
 49  
 50 template<typename T1, typename T2>
 51 istream &operator>>(istream &in, pair<T1, T2> &p) {
 52     in >> p.first >> p.second;
 53     return in;
 54 }
 55  
 56 template<typename T>
 57 istream &operator>>(istream &in, vector<T> &v) {
 58     for (auto &x: v)
 59         in >> x;
 60     return in;
 61 }
 62  
 63 template<typename T1, typename T2>
 64 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
 65     out << "[" << p.first << ", " << p.second << "]" << "
";
 66     return out;
 67 }
 68 
 69 inline int gc(){
 70     static const int BUF = 1e7;
 71     static char buf[BUF], *bg = buf + BUF, *ed = bg;
 72     
 73     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
 74     return *bg++;
 75 } 
 76 
 77 inline int ri(){
 78     int x = 0, f = 1, c = gc();
 79     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
 80     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
 81     return x*f;
 82 }
 83 
 84 template<class T>
 85 inline string toString(T x) {
 86     ostringstream sout;
 87     sout << x;
 88     return sout.str();
 89 }
 90 
 91 inline int toInt(string s) {
 92     int v;
 93     istringstream sin(s);
 94     sin >> v;
 95     return v;
 96 }
 97 
 98 //min <= aim <= max
 99 template<typename T>
100 inline bool BETWEEN(const T aim, const T min, const T max) {
101     return min <= aim && aim <= max;
102 }
103  
104 typedef long long LL;
105 typedef unsigned long long uLL;
106 typedef pair< double, double > PDD;
107 typedef pair< int, int > PII;
108 typedef pair< int, PII > PIPII;
109 typedef pair< string, int > PSI;
110 typedef pair< int, PSI > PIPSI;
111 typedef set< int > SI;
112 typedef set< PII > SPII;
113 typedef vector< int > VI;
114 typedef vector< double > VD;
115 typedef vector< VI > VVI;
116 typedef vector< SI > VSI;
117 typedef vector< PII > VPII;
118 typedef map< int, int > MII;
119 typedef map< int, string > MIS;
120 typedef map< int, PII > MIPII;
121 typedef map< PII, int > MPIII;
122 typedef map< string, int > MSI;
123 typedef map< string, string > MSS;
124 typedef map< PII, string > MPIIS;
125 typedef map< PII, PII > MPIIPII;
126 typedef multimap< int, int > MMII;
127 typedef multimap< string, int > MMSI;
128 //typedef unordered_map< int, int > uMII;
129 typedef pair< LL, LL > PLL;
130 typedef vector< LL > VL;
131 typedef vector< VL > VVL;
132 typedef priority_queue< int > PQIMax;
133 typedef priority_queue< int, VI, greater< int > > PQIMin;
134 const double EPS = 1e-8;
135 const LL inf = 0x7fffffff;
136 const LL infLL = 0x7fffffffffffffffLL;
137 const LL mod = 1e9 + 7;
138 const int maxN = 4e4 + 7;
139 const LL ONE = 1;
140 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
141 const LL oddBits = 0x5555555555555555;
142 
143 int T;
144 int p, q; 
145 string ans;
146 int board[27][27];
147 int dx[] = {-2, -2, -1, -1, 1, 1, 2, 2};
148 int dy[] = {-1, 1, -2, 2, -2, 2, -1, 1};
149 
150 inline void dfs(int x, int y) {
151     ans.PB('A' + x - 1);
152     ans.PB(y + '0');
153     board[x][y] = 1;
154     
155     Rep(i, 8) {
156         int newx = x + dx[i], newy = y + dy[i];
157         if(BETWEEN(newx, 1, p) && BETWEEN(newy, 1, q) && !board[newx][newy]) {
158             dfs(newx, newy);
159             if(ans.size() == q * p * 2) break;
160             board[newx][newy] = 0;
161             ans.resize(ans.size() - 2);
162         }
163     }
164 }
165 
166 int main(){
167     //freopen("MyOutput.txt","w",stdout);
168     //freopen("input.txt","r",stdin);
169     //INIT();
170     For(i, 1, 7) {
171         For(j, 1, 7) {
172             p = i;
173             q = j;
174             ans = "";
175             ms0(board);
176             dfs(1, 1);
177             
178             if(ans.size() == q * p * 2) continue;
179             
180             // 暴力枚举其他起点 
181             For(a, 1, (p + 1) >> 1) {
182                 For(b, 1, (q + 1) >> 1) {
183                     if(a == 1 && b == 1) continue;
184                     ans = "";
185                     ms0(board);
186                     dfs(a, b);
187                     if(ans.size() == q * p * 2) {
188                         printf("Find other road!
");
189                         printf("Scenario #%d, p = %d, q = %d:
", (i - 1) * 100 + j, p, q);
190                         cout << ans << endl << endl;
191                         return 0;
192                     }
193                 }
194             }
195         }
196     }
197     return 0;
198 }
View Code

代码如下

  1 #include <cmath>
  2 #include <ctime>
  3 #include <iostream>
  4 #include <string>
  5 #include <vector>
  6 #include <cstdio>
  7 #include <cstdlib>
  8 #include <cstring>
  9 #include <queue>
 10 #include <map>
 11 #include <set>
 12 #include <algorithm>
 13 #include <cctype>
 14 #include <stack>
 15 #include <deque>
 16 #include <list>
 17 #include <sstream>
 18 using namespace std;
 19  
 20 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
 21 #define Rep(i,n) for (int i = 0; i < (n); ++i)
 22 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
 23 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
 24 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
 25 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 26 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 27 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 28  
 29 #define pr(x) cout << #x << " = " << x << "  "
 30 #define prln(x) cout << #x << " = " << x << endl
 31  
 32 #define LOWBIT(x) ((x)&(-x))
 33  
 34 #define ALL(x) x.begin(),x.end()
 35 #define INS(x) inserter(x,x.begin())
 36 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 37 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
 38 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 39 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 40  
 41 #define ms0(a) memset(a,0,sizeof(a))
 42 #define msI(a) memset(a,inf,sizeof(a))
 43 #define msM(a) memset(a,-1,sizeof(a))
 44 
 45 #define MP make_pair
 46 #define PB push_back
 47 #define ft first
 48 #define sd second
 49  
 50 template<typename T1, typename T2>
 51 istream &operator>>(istream &in, pair<T1, T2> &p) {
 52     in >> p.first >> p.second;
 53     return in;
 54 }
 55  
 56 template<typename T>
 57 istream &operator>>(istream &in, vector<T> &v) {
 58     for (auto &x: v)
 59         in >> x;
 60     return in;
 61 }
 62  
 63 template<typename T1, typename T2>
 64 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
 65     out << "[" << p.first << ", " << p.second << "]" << "
";
 66     return out;
 67 }
 68 
 69 inline int gc(){
 70     static const int BUF = 1e7;
 71     static char buf[BUF], *bg = buf + BUF, *ed = bg;
 72     
 73     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
 74     return *bg++;
 75 } 
 76 
 77 inline int ri(){
 78     int x = 0, f = 1, c = gc();
 79     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
 80     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
 81     return x*f;
 82 }
 83 
 84 template<class T>
 85 inline string toString(T x) {
 86     ostringstream sout;
 87     sout << x;
 88     return sout.str();
 89 }
 90 
 91 inline int toInt(string s) {
 92     int v;
 93     istringstream sin(s);
 94     sin >> v;
 95     return v;
 96 }
 97 
 98 //min <= aim <= max
 99 template<typename T>
100 inline bool BETWEEN(const T aim, const T min, const T max) {
101     return min <= aim && aim <= max;
102 }
103  
104 typedef long long LL;
105 typedef unsigned long long uLL;
106 typedef pair< double, double > PDD;
107 typedef pair< int, int > PII;
108 typedef pair< int, PII > PIPII;
109 typedef pair< string, int > PSI;
110 typedef pair< int, PSI > PIPSI;
111 typedef set< int > SI;
112 typedef set< PII > SPII;
113 typedef vector< int > VI;
114 typedef vector< double > VD;
115 typedef vector< VI > VVI;
116 typedef vector< SI > VSI;
117 typedef vector< PII > VPII;
118 typedef map< int, int > MII;
119 typedef map< int, string > MIS;
120 typedef map< int, PII > MIPII;
121 typedef map< PII, int > MPIII;
122 typedef map< string, int > MSI;
123 typedef map< string, string > MSS;
124 typedef map< PII, string > MPIIS;
125 typedef map< PII, PII > MPIIPII;
126 typedef multimap< int, int > MMII;
127 typedef multimap< string, int > MMSI;
128 //typedef unordered_map< int, int > uMII;
129 typedef pair< LL, LL > PLL;
130 typedef vector< LL > VL;
131 typedef vector< VL > VVL;
132 typedef priority_queue< int > PQIMax;
133 typedef priority_queue< int, VI, greater< int > > PQIMin;
134 const double EPS = 1e-8;
135 const LL inf = 0x7fffffff;
136 const LL infLL = 0x7fffffffffffffffLL;
137 const LL mod = 1e9 + 7;
138 const int maxN = 4e4 + 7;
139 const LL ONE = 1;
140 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
141 const LL oddBits = 0x5555555555555555;
142 
143 int T;
144 int p, q; 
145 string ans;
146 int board[27][27];
147 int dx[] = {-2, -2, -1, -1, 1, 1, 2, 2};
148 int dy[] = {-1, 1, -2, 2, -2, 2, -1, 1};
149 
150 void dfs(int x, int y) {
151     ans.PB('A' + x - 1);
152     ans.PB(y + '0');
153     board[x][y] = 1;
154     
155     Rep(i, 8) {
156         int newx = x + dx[i], newy = y + dy[i];
157         if(BETWEEN(newx, 1, p) && BETWEEN(newy, 1, q) && !board[newx][newy]) {
158             dfs(newx, newy);
159             if(ans.size() == q * p * 2) break;
160             board[newx][newy] = 0;
161             ans.resize(ans.size() - 2);
162         }
163     }
164 }
165 
166 int main(){
167     //freopen("MyOutput.txt","w",stdout);
168     //freopen("input.txt","r",stdin);
169     //INIT();
170     cin >> T;
171     For(cases, 1, T) {
172         cin >> q >> p;
173         ans = "";
174         ms0(board);
175         
176         dfs(1, 1);
177         
178         printf("Scenario #%d:
", cases);
179         if(ans.size() != q * p * 2) ans = "impossible";
180         cout << ans << endl;
181         if(cases != T) cout << endl;
182     }
183     return 0;
184 }
View Code
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原文地址：https://www.cnblogs.com/zaq19970105/p/11133551.html