Algoritmen en Datastructuren (ALDAT)

EVMINX4 Week 5

GamesSpel met 2 spelers, die elk afwisselend aan

de beurt zijn.Boter Kaas en Eieren (Tic-Tac-Toe).4 op een rij.ZeeslagDammen, Schaken, Go, …

Hoe vind je de beste zet?Bouw een game tree.Pas het min-max algoritme toe.Gebruik alfa-beta pruning om min-max sneller te

maken.

Game tree

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Zie Handouts

Game tree

Backtracking algoritme met minimax strategie!

Zie Handouts

TicTacToe::chooseMoveint TicTacToe::chooseMove(Side s, int& bestRow, int& bestColumn) { Side opp(s==COMPUTER ? HUMAN : COMPUTER); int value(s==COMPUTER ? HUMAN_WIN : COMPUTER_WIN); int simpleEval(positionValue()); if (simpleEval!=UNCLEAR) return simpleEval; for (int r(0); r<board.numrows(); ++r) for (int c(0); c<board.numcols(); ++c) if (squareIsEmpty(r, c)) { place(r, c, s); int dc; int reply(chooseMove(opp, dc, dc)); place(r, c, EMPTY); if (s==COMPUTER && reply>value || s==HUMAN &&

reply<value) { value=reply; bestRow=r; bestColumn=c; } } return value;}

Tic-Tac-Toeaantal aanroepen chooseMove bij eerste zet

(computer begint)Maximaal: 1+9+9x8+9x8x7+ ... +

9x8x7x6x5x4x3x2x1 = 986410Stoppen als er een winnaar is =

549946Toepassen alpha-beta pruning =

18297Toepassen transposition table =

7954Zoek identieke stellingen (draaien en spiegelen)

= 5204

Alfa-Beta pruning

Zie p. 398 (Weiss). After H2A is evaluated, C2, which is the minimum of the H2’s, is at best a draw. Consequently, it cannot be an improvement over C1. We therefore do not need to evaluate H2B, H2C, and H2D, and can proceed directly to C3.

Minmax algoritme voorbeeld

MAX

MIN

14

14

13

12

2

210

1

9

8

7

62

35

4

4 2 6

8

8 1 11

2

12

14

14

4

4

4

2

Alfa-beta pruning voorbeeld

14

14

13

12

2

210

1

9

8

7

63

35

4

4 2 6

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8 1 11

2

12

14

14

4

4

4

2

<=

>=

<= >=4

<=4

>=4>=4

>=4

MAX

MIN

Alfa-beta pruningpair<int, int> cComp(int p, int a, int b, int d);pair<int, int> cHuman(int p, int a, int b, int d);

pair<int, int> cComp(int p, int a, int b, int d) { int bestPos(p); int value(positionValue(p)); if (value==UNCLEAR) { value=a; for (int i(1); i<3 && a<b; ++i) { pair<int, int> r(cHuman(2*p+i, a, b, d+1)); if (r.first>a) { value=r.first; a=value; bestPos=2*p+i; } } } return make_pair(value, bestPos);}

return pair = best value, best position.p=positiona=alfab=betad=dept

Alfa-beta pruningpair<int, int> cComp(int p, int a, int b, int d);pair<int, int> cHuman(int p, int a, int b, int d);

pair<int, int> cHuman(int p, int a, int b, int d) { int bestPos(p); int value(positionValue(p)); if (value==UNCLEAR) { value=b; for (int i(1); i<3 && b>a; ++i) { pair<int, int> r(cComp(2*p+i, a, b, d+1)); if (r.first<b) { value=r.first; b=value; bestPos=2*p+i; } } } return make_pair(value, bestPos);}

return pair = best value, best position.p=positiona=alfab=betad=dept

Alfa-beta pruningenum S {H, C};

pair<int, int> cMove(S s, int p, int a, int b, int d) { int bestPos(p); int value(positionValue(p)); if (value==UNCLEAR) { value=(s==C)?a:b; for (int i(1); i<3 && a<b; ++i) { pair<int, int> r( cMove((s==C)?H:C, 2*p+i, a, b, d+1) ); if (s==C&&r.first>a || s==H&&r.first<b) { value=r.first; if (s==C) a=value; else b=value; bestPos=2*p+i; } } return make_pair(value, bestPos);}

return pair = best value, best position.p=positiona=alfab=betad=depts=sideH= HumanC = Computer

Alpha-beta pruningint TicTacToe::chooseMove(Side s, int& bestRow, int& bestColumn, int alpha, int beta) { Side opp(s==COMPUTER ? HUMAN : COMPUTER); int value(s==COMPUTER ? alpha : beta); int simpleEval(positionValue()); if (simpleEval!=UNCLEAR) return simpleEval; for (int r(0); r<board.numrows(); ++r) for (int c(0); c<board.numcols(); ++c) if (squareIsEmpty(r, c)) { place(r, c, s); int dc; int reply(chooseMove(opp, dc, dc, alpha, beta)); place(r, c, EMPTY); if (s==COMPUTER && reply>value || s==HUMAN && reply<value) { value=reply; if (s==COMPUTER) alpha=value; else beta=value; bestRow=r; bestColumn=c; if (alpha>=beta) return value; } } return value;}

Transpostions

Zie p. 400 (Weiss). Two searches that arrive at identical positions.

Transpositionsclass Position {public: Position(const matrix<int>& theBoard): board(theBoard) { } bool operator<(const Position& rhs) const;private: matrix<int> board;};

bool Position::operator<(const Position & rhs) const { for (int i(0); i<board.numrows(); ++i) for (int j(0); j<board.numcols(); ++j) if (board[i][j]!=rhs.board[i][j]) return board[i][j]<rhs.board[i][j]; return false;}

class TicTacToe {//...private: map<Position, int> transpositions;};

Position is een wrapper (inpakker) voor

matrix<int> board.

Waarom is dit nodig?

Transpositionsint TicTacToe::chooseMove(Side s, int& bestRow, int& bestColumn, int alpha, int beta, int depth) { Position thisPosition(board); if (depth>=3 && depth<=5) { MapItr itr(transpositions.find(thisPosition)); if (itr!=transpositions.end()) return (*itr).second; }// idem ... int reply(chooseMove(opp, dc, dc, alpha, beta, depth+1));// idem ... if (alpha>=beta) goto Done; } }Done: if (depth>=3 && depth<=5) transpositions[thisPosition]=value; return value;}

Verklaar?