sg函數：sg函數是博弈中的確定一個position性質的一個函數，全稱是sprague-grundy。

性質1：對于所有的p-position，都有sg = 0；對于所有的n-position都有sg ！= 0；

性質2：某點a的sg函數的值由它的后繼的sg函數的值來決定，設后繼為b, c, d, e……則sg(a) = mex(sg(a), sg(b), sg(c), sg(d), sg(e),……）

mex是不屬于這個集合的最小非負整數。

應用范圍：在此無環圖中誰無法再次移動，便是輸。（如果誰無法移動，便是贏，暫時不知如何解決。）

應用：通過判斷該點，sg = 0是p點，sg ！= 0是N點。

構造sg函數的方法：

方法一：打表

例題：hdu-1536-S-nim?點擊打開鏈接

[cpp] view plaincopyprint?

/*?

收獲：?

*/??

#include<iostream> ??

#include<cstdlib> ??

#include<vector> ??

#include<map> ??

#include<cstring> ??

#include<set> ??

#include<string> ??

#include<algorithm> ??

#include<sstream> ??

#include<ctype.h> ??

#include<fstream> ??

#include<string.h> ??

#include<stdio.h> ??

#include<math.h> ??

#include<stack> ??

#include<queue> ??

#include<ctime> ??

//#include<conio.h> ??

using?namespace?std;??

??

const?int?INF_MAX=0x7FFFFFFF;??

const?int?INF_MIN=-(1<<31);??

??

const?double?eps=1e-10;??

const?double?pi=acos(-1.0);??

??

#define?pb?push_back???//a.pb(?) ??

#define?chmin(a,b)?((a)<(b)?(a):(b)) ??

#define?chmax(a,b)?((a)>(b)?(a):(b)) ??

??

??

template<class?T>?inline?T?gcd(T?a,T?b)//NOTES:gcd( ??

??{if(a<0)return?gcd(-a,b);if(b<0)return?gcd(a,-b);return?(b==0)?a:gcd(b,a%b);}??

template<class?T>?inline?T?lcm(T?a,T?b)//NOTES:lcm( ??

??{if(a<0)return?lcm(-a,b);if(b<0)return?lcm(a,-b);return?a*(b/gcd(a,b));}??

??

??

typedef?pair<int,?int>?PII;??

typedef?vector<PII>?VPII;??

typedef?vector<int>?VI;??

typedef?vector<VI>?VVI;??

typedef?long?long?LL;??

int?dir_4[4][2]={{0,1},{-1,0},{0,-1},{1,0}};??

int?dir_8[8][2]={{0,1},{-1,1},{-1,0},{-1,-1},{0,-1},{1,-1},{1,0},{1,1}};??

//下，左下，左，左上，上，右上，右，右下。 ??

??

//*******?WATER?**************************************************************** ??

??

??

const?int?MAXN?=?10500;??

bool?judge[150];??

int?sg[MAXN];??

int?M[150];??

const?int?Init?=?1e7;??

int?Num;??

??

void?input_m()??

{??

????for(int?i?=?0;?i?<?Num;?i++)??

????{??

????????cin>>M[i];??

????}??

????return?;??

}??

??

void?debug()??

{??

????cout<<"sg?function"<<endl;??

????for(int?i?=?0;?i?<?100;?i++)??

????{??

????????cout<<i<<"?"<<sg[i]<<endl;??

????}??

????return?;??

}??

??

void?getsg()??

{??

????for(int?i?=?0;?i?<?MAXN;?i++)??

????{??

????????memset(judge,?false,?sizeof(judge));??

????????//int?tsg?=?Init; ??

????????for(int?j?=?0;?j?<?Num;?j++)??

????????{??

????????????int?ps?=?i?-?M[j];??

????????????if(ps?>=?0)?judge[sg[ps]]?=?true;??

????????}??

????????//if(tsg?==?Init)?tsg?=?0; ??

????????for(int?j?=?0;?j?<?Num?+?1;?j++)??

????????{??

????????????if(judge[j]?==?false)??

????????????{??

????????????????sg[i]?=?j;??

????????????????break;??

????????????}??

????????}??

????}??

????//debug(); ??

????return?;??

}??

int?main()??

{??

????//freopen("input.txt","r",stdin); ??

????//freopen("output.txt","w",stdout); ??

????while(cin>>Num,?Num)??

????{??

????????input_m();??

????????getsg();??

????????int?num;??

????????cin>>num;??

????????while(num--)??

????????{??

????????????int?nn,?tp;??

????????????cin>>nn;??

????????????int?ret?=?0;??

????????????for(int?i?=?0;?i?<?nn;?i++)??

????????????{??

????????????????cin>>tp;??

????????????????ret?^=?sg[tp];??

????????????}??

????????????if(ret?==?0)?cout<<"L";??

????????????else?cout<<"W";??

????????}??

????????cout<<endl;??

????}??

????return?0;??

????//printf("%.6f\n",(double)clock()/CLOCKS_PER_SEC); ??

}??

/* 收獲： */ #include<iostream> #include<cstdlib> #include<vector> #include<map> #include<cstring> #include<set> #include<string> #include<algorithm> #include<sstream> #include<ctype.h> #include<fstream> #include<string.h> #include<stdio.h> #include<math.h> #include<stack> #include<queue> #include<ctime> //#include<conio.h> using namespace std;const int INF_MAX=0x7FFFFFFF; const int INF_MIN=-(1<<31);const double eps=1e-10; const double pi=acos(-1.0);#define pb push_back //a.pb( ) #define chmin(a,b) ((a)<(b)?(a):(b)) #define chmax(a,b) ((a)>(b)?(a):(b))template<class T> inline T gcd(T a,T b)//NOTES:gcd({if(a<0)return gcd(-a,b);if(b<0)return gcd(a,-b);return (b==0)?a:gcd(b,a%b);} template<class T> inline T lcm(T a,T b)//NOTES:lcm({if(a<0)return lcm(-a,b);if(b<0)return lcm(a,-b);return a*(b/gcd(a,b));}typedef pair<int, int> PII; typedef vector<PII> VPII; typedef vector<int> VI; typedef vector<VI> VVI; typedef long long LL; int dir_4[4][2]={{0,1},{-1,0},{0,-1},{1,0}}; int dir_8[8][2]={{0,1},{-1,1},{-1,0},{-1,-1},{0,-1},{1,-1},{1,0},{1,1}}; //下，左下，左，左上，上，右上，右，右下。//******* WATER ****************************************************************const int MAXN = 10500; bool judge[150]; int sg[MAXN]; int M[150]; const int Init = 1e7; int Num;void input_m() {for(int i = 0; i < Num; i++){cin>>M[i];}return ; }void debug() {cout<<"sg function"<<endl;for(int i = 0; i < 100; i++){cout<<i<<" "<<sg[i]<<endl;}return ; }void getsg() {for(int i = 0; i < MAXN; i++){memset(judge, false, sizeof(judge));//int tsg = Init;for(int j = 0; j < Num; j++){int ps = i - M[j];if(ps >= 0) judge[sg[ps]] = true;}//if(tsg == Init) tsg = 0;for(int j = 0; j < Num + 1; j++){if(judge[j] == false){sg[i] = j;break;}}}//debug();return ; } int main() {//freopen("input.txt","r",stdin);//freopen("output.txt","w",stdout);while(cin>>Num, Num){input_m();getsg();int num;cin>>num;while(num--){int nn, tp;cin>>nn;int ret = 0;for(int i = 0; i < nn; i++){cin>>tp;ret ^= sg[tp];}if(ret == 0) cout<<"L";else cout<<"W";}cout<<endl;}return 0;//printf("%.6f\n",(double)clock()/CLOCKS_PER_SEC); }

方法二：遞歸迭代

以下

[cpp] view plaincopyprint?

#include"iostream" ??

#include"algorithm" ??

#include"string.h" ??

using?namespace?std;??

int?s[101],sg[10001],k;??

int?getsg(int?m)??

{??

????int?hash[101]={0};??

????int?i;??

????for(i=0;i<k;i++){??

????????if(m-s[i]<0)??

????????????break;??

????????if(sg[m-s[i]]==-1)??

????????????sg[m-s[i]]=getsg(m-s[i]);??

????????hash[sg[m-s[i]]]=1;??

????}??

????for(i=0;;i++)??

????????if(hash[i]==0)??

????????????return?i;??

???

???

}??

int?main()??

{??

????//int?k; ??

???//?freopen("game.in","r",stdin); ??

????//freopen("game.out","w",stdout); ??

????while(cin>>k,k)??

????{??

????????int?i;??

????????for(i=0;i<k;i++)??

????????????cin>>s[i];??

????????sort(s,s+k);??

????????memset(sg,-1,sizeof(sg));??

????????sg[0]=0;??

????????int?t;??

????????cin>>t;??????

????????while(t--)??

????????{??

???????????????

????????????int?n,m;??

????????????cin>>n;??

????????????int?ans=0;??

????????????while(n--)??

????????????{??

????????????????cin>>m;??

????????????????if(sg[m]==-1)??

????????????????????sg[m]=getsg(m);??

????????????????ans^=sg[m];??

????????????}??

????????????if(ans)??

????????????????cout<<'W';??

????????????else?cout<<'L';??

????????}??

????????cout<<endl;??

????}??

????return?0;??

}??

#include"iostream" #include"algorithm" #include"string.h" using namespace std; int s[101],sg[10001],k; int getsg(int m) {int hash[101]={0};int i;for(i=0;i<k;i++){if(m-s[i]<0)break;if(sg[m-s[i]]==-1)sg[m-s[i]]=getsg(m-s[i]);hash[sg[m-s[i]]]=1;}for(i=0;;i++)if(hash[i]==0)return i;} int main() {//int k;// freopen("game.in","r",stdin);//freopen("game.out","w",stdout);while(cin>>k,k){int i;for(i=0;i<k;i++)cin>>s[i];sort(s,s+k);memset(sg,-1,sizeof(sg));sg[0]=0;int t;cin>>t; while(t--){int n,m;cin>>n;int ans=0;while(n--){cin>>m;if(sg[m]==-1)sg[m]=getsg(m);ans^=sg[m];}if(ans)cout<<'W';else cout<<'L';}cout<<endl;}return 0; }
是別人的代碼：

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