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if( depth == 0 ) transpositions = MapType( ) ; / / makeEmpty else if( depth >= 3 && depth <= TABLE-DEPTH )
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MapItr itr = transpositionsfind( thisposition if( itr ! = transpositionsend( ) ) return (*itr)second; 1 if ( s == COMPUTER
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opp = H W ; value = alpha; 1 else
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opp = COMPUTER; value = beta;
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Figure 1 1 I 6 The tic-tac-toe algorithm with alpha-beta pruning and transposition table (part 1)
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from lines 54 onward We consolidate the two return statements originally in the code into one by using a goto(If you are one of those who cannot stand any use of a goto,it is easy enough to unconsolidate) Immediately before the return, we store the value of the position in the transposition table The use of the transposition table in this tic-tac-toe algorithm removes about half the positions from consideration, with only a slight cost for the transposition table operations The program's speed is almost doubled
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35 for( int row = 0; row < 3 ; row++ ) 36 for( int column = 0; column < 3 ; column++ ) 37 if( squareIsEmpty( row, column ) ) 38 I 39 place ( row, column, s ) ; 40 reply = chooseMove( opp, dc, dc, 41 alpha, beta, depth + 1 42 place ( row, column, EMPTY ) ; 43 44 if( s == COMPUTER && reply > value I I 45 s == HUMAN && reply < value ) 46 { 47 if ( s == COMPUTER ) 48 alpha = value = reply; 49 else 50 beta = value = reply; 51 bestRow = row; bestColumn = column; 52 53 if( alpha >= beta ) 54 goto Done; / / Refutation 55 1 56 1 57 Done : 58 if( depth <= TABLE-DEPTH ) 59 transpositions[ thisposition I = value; 60 return value; 61 1
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Figure 1117 The tic-tac-toe algorithm with alpha-beta pruning and transposition table (part 2)
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1123 Computer Chess
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Terminal positions besearched in computer chess In the best programs, considerable knowledge is built into the evaluation function
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In a complex game such as Chess or Go, it is infeasible to search all the way to the terminal nodes: Some estimates claim that there are r o u" h l v10'00 ~ legal chess positions, and all the tricks in the world will not bring it down to a manageable level In this case, we have to stop the search after a certain depth of recursion is reached The nodes at which the recursion is stopped become terminal nodes These terminal nodes are evaluated with a function that estimates the value of the position For instance, in a chess program, the evaluation function measures such variables as the relative amount and strength of pieces and other positional factors Computers are especially adept at playing moves involving deep combinations that result in exchanges of material The reason is that the strength of pieces is easily evaluated However, extending the search depth merely one level requires an increase in processing speed by a factor of about 6 (because
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the number of positions increases by about a factor of 36) Each extra level of search greatly enhances the ability of the program, up to a certain limit (which appears to have been reached by the best programs) On the other hand, computers generally are not as good at playing quiet positional games in which more subtle evaluations and knowledge of the game is required However, this shortcoming is apparent only when the computer is playing very strong opposition The mass-marketed computer chess programs are better than all but a small fraction of today's players In 1997, the computer program Deep Blue, using an enormous amount of computational power (evaluating as many as 200 million moves per second) was able to defeat the reigning world chess champion in a six-game match Its evaluation function, although top secret, is known to contain a large number of factors, was aided by several chess grandmasters, and was the result of years of experimentation Writing the top computer chess program is certainly not a trivial task
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In this chapter we introduced an application of binary search and some algorithmic techniques that are commonly used in solving word search puzzles and in game-playing programs such as Chess, Checkers, and Othello The top programs for these games are all world class The game of Go, however, appears too complex for computer searching
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