~ubuntu-branches/ubuntu/trusty/dds/trusty-proposed : contents of DDS_alg

~ubuntu-branches/ubuntu/trusty/dds/trusty-proposed : (revision 7)
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\listoverridecount0\ls17}}{\info{\title Bo Haglund}{\author Bo Haglund}{\operator Bo Haglund}{\creatim\yr2008\mo3\dy19\hr10\min23}{\revtim\yr2008\mo3\dy21\hr13\min53}{\version14}{\edmins54}{\nofpages22}{\nofwords5962}{\nofchars-32766}{\*\company  }
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\par }\pard \s16\ri360\nowidctlpar\tqc\tx4536\tqr\tx9072\adjustright {
\par }}{\*\pnseclvl1\pnucrm\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl2\pnucltr\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl3\pndec\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl4\pnlcltr\pnstart1\pnindent720\pnhang{\pntxta )}}
{\*\pnseclvl5\pndec\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl6\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl7\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl8
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\par }\pard\plain \nowidctlpar\adjustright \lang2057 {Rev. E1}{\lang1053 ,  2008-03-21}{
\par }\pard\plain \s1\keepn\nowidctlpar\outlinelevel0\adjustright \lang2057 {\b\fs28\lang1053 
\par 
\par Search Algorithms for a Bridge Double Dummy Solver 
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 
\par This description is intended for anyone interested in the inner workings of a bridge double dummy solver (DDS). It describes my solver implemented in the Win32 environment as a DLL.  
\par 
\par }{DDS algorithm descriptions already exist, see reference list at the end. However, to my knowledge, no document exists that gives an in depth description of all algorithms covered in this document.}{\lang1053 
\par 
\par 
\par 
\par 
\par {\listtext\pard\plain\b \hich\af0\dbch\af0\loch\f0 1.\tab}}\pard \nowidctlpar\jclisttab\tx0\ls16\adjustright {\b\lang1053 The basic search algorithm
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par  The search is based on the zero window search [Pearl 1980]. 
\par  Pseudo code for its application on DD solver search is given.
\par  Cards searched are described as \'94moves\'94 in contrast to cards that are really played.\page 
\par  int  Search(posPoint, target, depth) \{
\par     if (depth==0) \{
\par         tricks=Evaluate;
\par         if  (tricks >= target) 
\par             value=TRUE;
\par         else
\par             value=FALSE;
\par         return value;
\par     \}
\par     else \{
\par         GenerateMoves;
\par         if  (player_side_to_move) \{
\par             value=FALSE;   moveExists=TRUE;
\par             while (moveExists) \{
\par                 Make;
\par                 value=Search(posPoint, target, depth-1);
\par                 Undo;
\par                 if  (value==TRUE) 
\par  \tab goto searchExit;\tab /* Cutoff, current move recorded as \'94best move\'94 */
\par                 moveExists=NextMove;
\par             \}
\par         \}\tab /* Opponents to move */
\par         else \{\tab     
\par              value=TRUE;   moveExists=TRUE;
\par              while (moveExists) \{
\par                  Make;
\par                  value=Search(posPoint, target, depth-1);
\par                  Undo;
\par                  if  (value==FALSE) 
\par  \tab goto searchExit;\tab /* Cutoff, current move recorded as \'94best move\'94 */
\par                  moveExists=NextMove;
\par              \}
\par         \}
\par     \}
\par 
\par     searchExit:
\par     return  value;
\par \}
\par 
\par 
\par }{The Search  parameters are:
\par {\pntext\pard\plain\f3\lang2057 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls5\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls5\adjustright {\b posPoint - }{
 a pointer to a structure containing state information for the position (deal) to be searched, e.g. leading hand,  hand-to-play, cards yet to play etc.
\par {\pntext\pard\plain\f3\lang2057 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls5\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls5\adjustright {\b target}{
 -  the number of tricks the player must take. 
\par {\pntext\pard\plain\f3\lang2057 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls5\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls5\adjustright {\b depth}{ -  the current search depth.

\par }\pard \nowidctlpar\adjustright {
\par Search returns TRUE if the target is reached, otherwise FALSE.
\par }{\lang1053 
\par When Search is called, depth is set to the number of cards left to play minus 4. 
\par GenerateMoves generates a list of alternative moves (=cards) that can be played in the initial position whose state da
ta is pointed to by posPoint. For cards that are equivalent (e.g. AK) only the card with highest rank is generated. Card equivalence is reanalyzed after each trick. 
\par }{E.g. if the hand-to-play has AQ in a suit where K was played in a previous trick, then A and Q become equivalents.}{\lang1053 
\par 
\par If the side of the player has the move, Search tries to find a move that meets the target, i.e that evaluates to TRUE. If such a move is found, search returns TRUE, and saves the move as \'94best\'94.
\par If the other side has the move, Search tries to find a move that defies meeting the target, i.e. that evaluates to FALSE. If such a move is found, search returns FALSE, and saves the move as \'94best\'94.
\par 
\par Each move in the generated move list is handled by first calling Make, which removes the ca
rd from the position state information. Search is then recursively called with a position state that now has excluded the played card, depth has been decremented by one. For each new recursive call to Search, a card is removed from the position state info
r
mation and depth is decremented. This goes on until depth equals 0 in which case only one trick remains. The outcome of this trick is calculated by Evaluate. If the total number of tricks won by the side of the player then reaches target, Search returns T
RUE, otherwise FALSE. This result propagates upwards as Search returns for each level, Undo is called which reinstalls the searched card on this level.\line Finally, Search returns for the top level.
\par 
\par This basic search algorithm is not powerful enough to terminate the search of a typical 52 cards deal in a reasonable time. To accomplish this, a number of search algorithm enhancements are required, which will be described in the following chapters.  

\par \page 
\par The described search algorithm only tells if a predefined target can be reached. It does not tell how many tricks that the side of the player can get. This is accomplished by repeated calls to Search:
\par 
\par g = guessed number of tricks for side of the player
\par iniDepth = number of cards to play minus 4
\par upperbound = 13;
\par lowerbound = 0;
\par do  \{
\par     if  (g==lowerbound)
\par         tricks=g+1;
\par     else
\par         tricks=g;
\par     if  ((Search(posPoint, tricks, iniDepth)==FALSE)  \{
\par         upperbound=tricks-1;
\par         g=upperbound;
\par     \}
\par     else  \{
\par         lowerbound=tricks;
\par         g=lowerbound;
\par     \}
\par \}
\par while (lowerbound < upperbound);
\par g=maximum tricks to be won by side of player.}{\b\lang1053 
\par 
\par 
\par 
\par {\listtext\pard\plain\s3 \b \hich\af0\dbch\af0\loch\f0 2.\tab}}\pard\plain \s3\keepn\nowidctlpar\jclisttab\tx0\ls16\outlinelevel2\adjustright \b\lang1053 {Overview of the search algorithms used in the DD solver 
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\b\lang1053 
\par }{\lang1053 The additional functions in the pseudo code for supporting the search speed enhancements are given in }{\i\lang1053 italics}{\lang1053 .  
\par \page 
\par int  Search(posPoint, target, depth) \{
\par       if (no_move_yet_in_trick)  \{}{\cf1\lang1053 
\par }{\lang1053           }{\i\lang1053 TargetTooLowOrHigh;
\par }{\lang1053           if (target_already_obtained)
\par }\pard\plain \s16\nowidctlpar\adjustright \lang2057 {\lang1053               return TRUE;
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053           else if (target_can_no_longer_be_obtained)
\par               return FALSE;}{\i\lang1053  
\par           QuickTricks;
\par           LaterTricks;
\par }{\lang1053           if  (cutoff_for_player_side) 
\par              return TRUE;
\par           else if  (cutoff_for_opponent_side)
\par              return FALSE;}{\i\lang1053 \line           RetrieveTTresult;
\par }\pard\plain \s16\nowidctlpar\adjustright \lang2057 {\lang1053           if (transposition_table_entry_match) \{
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\i\lang1053              }{\lang1053 if  (target_reached)
\par                 return TRUE;
\par             else
\par                 return FALSE;
\par          \}}{\i\lang1053 
\par }\pard\plain \s16\nowidctlpar\adjustright \lang2057 {\lang1053      \}
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 
\par       if (depth==0) \{
\par           }{\i\lang1053 evalRes=Evaluate;
\par }{\lang1053           if  (evalRes.tricks >= target) 
\par              value=TRUE;
\par          else
\par              value=FALSE;
\par          return value;
\par       \}
\par       else \{
\par           GenerateMoves; 
\par }{\i\lang1053           MoveOrdering;
\par           CheckMovesForCutoff;   }{\lang1053 /* For pseudo-code, see chapter 6 */}{\i\lang1053   
\par }{\lang1053           if  (player_side_to_move) \{
\par               value=FALSE;    moveExists=TRUE;
\par               while (moveExists) \{
\par                  Make;
\par                  value=Search(posPoint, target, depth-1);\tab 
\par                  Undo;
\par                  if  (value==TRUE)  \{
\par \tab }{\i\lang1053 MergeMoveData;}{\lang1053  }{\i\lang1053 
\par }{\lang1053  \tab goto searchExit;\tab /* Cutoff, current move recorded as \'94best move\'94 */
\par                  \}
\par                  }{\i\lang1053 MergeAllMovesData;
\par }{\lang1053                  moveExists=}{\i\lang1053 NextMove}{\lang1053 ;
\par              \}
\par          \}\tab /* Opponents to move */
\par          else \{\tab     
\par              value=TRUE;   moveExists=TRUE;
\par              while (moveExists) \{
\par                  Make;
\par                  value=Search(posPoint, target, depth-1);\tab 
\par                  Undo;
\par                  if  (value==FALSE)  \{ 
\par \tab }{\i\lang1053 MergeMoveData;}{\lang1053 
\par  \tab goto searchExit;\tab /* Cutoff, current move recorded as \'94best move\'94 */
\par                  \}
\par }{\i\lang1053                  MergeAllMovesData;}{\lang1053 
\par                  moveExists=}{\i\lang1053 NextMove}{\lang1053 ;
\par              \}
\par          \}
\par      \}
\par      searchExit:
\par      }{\i\lang1053 AddNewTTentry;
\par }{\lang1053      return  value;
\par  \}
\par 
\par 
\par }{\i\lang1053 TargetTooLowOrHigh }{\lang1053  checks the target value against the number of tricks currently won by      side }{\cf1\lang1053 of the player}{\lang1053  and against number of tricks left to play.
\par It is executed at the beginning of each trick, before any card has been played.
\par }{If number of currently won tricks }{\cf1 by player\rquote s side}{ equals or exceeds target, Search returns TRUE.\line If number of currently won tricks }{\cf1 by player\rquote s side}{ plus tricks left to play is less than target Search returns FALSE.

\par Since possible winning cards for the remaining tricks are irrelevant, no winning cards are backed up at cutoff termination.}{\lang1053 
\par }\pard\plain \s16\nowidctlpar\adjustright \lang2057 {\lang1053 
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\i\lang1053 TargetTooLowOrHigh }{\lang1053  search enhancement is described e.g. in [Chang].
\par 
\par }\pard\plain \s1\keepn\nowidctlpar\outlinelevel0\adjustright \lang2057 {\i QuickTricks}{ determines if the side to move can take one or more sure tricks. }{\lang1053 
E.g. if the hand to move has an Ace in an NT contract, at least one sure trick can be taken.
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 It is executed at the beginning of each trick, before any card has been played. A simple quick trick is also executed after the leading card of the trick is played.}{
\par }\pard\plain \s1\keepn\nowidctlpar\outlinelevel0\adjustright \lang2057 {\lang1053 Assuming that the sure tricks are won by the side to move, then the conditions for search cutoff in }{\i\lang1053 TargetTooLowOrHigh}{\lang1053 
 are again tested to produce further search cutoffs.
\par The detailed conditions for determination of sure tricks are described in Chapter 3.
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {When quicktricks win by rank, they are backed up at cutoff termination. 
\par }\pard\plain \s1\keepn\nowidctlpar\outlinelevel0\adjustright \lang2057 {\lang1053 
\par The idea of }{\i\lang1053 QuickTricks }{\lang1053 is described e.g. in [Chang].
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {
\par }{\i LaterTricks}{ determines if the opponents of the side to move can take one or more tricks at their turn or later in the play. It is also executed at the beginning of each trick and uses similar criteria for search cutoff as }{\i Quicktricks}{. 

\par When quicktricks win by rank, they are backed up at cutoff termination.\line For a detailed description, see Chapter 4.
\par }\pard\plain \s1\keepn\nowidctlpar\outlinelevel0\adjustright \lang2057 {\lang1053 
\par }{\i\lang1053 RetrieveTTresult }{\lang1053 scans the set of positions in the transposition table to see if there is a match against the current position. 
\par It is executed at the beginning of each trick, before any card has been played. After detection of a  transposition table entry match, the winning ranks necessary in the remaining cards are backed up. \line For details, see Chapter 7.
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\fs20\lang1053 
\par }{\i Evaluate  }{returns evalResult which updates the position state information. evalResult contains:
\par {\listtext\pard\plain\f3\lang2057 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li780\nowidctlpar\jclisttab\tx780\ls6\adjustright {\b evalResult.tricks}{, the number of tricks won by the side of the player, and
\par {\listtext\pard\plain\f3\lang2057 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}{\b evalResult.winRank}{ which includes the card in the last trick that won by rank.  
\par }\pard \nowidctlpar\adjustright {E.g. if the last trick includes the spades A, Q, 9 and 3, evalResult.winRank returns spade A. But
\par if the last trick was won without a win by rank as for spade 5 (leading and winning card), heart A, heart Q, heart 5, no winning rank is returned. }{\lang1053 
\par 
\par }{Keeping record of cards that win by ranks and subsequently using this information to ignore ranks for other cards is discussed in the Partition Search concept invented by Matthew Ginsberg and described in his paper [Ginsberg]. }{\lang1053 
\par 
\par }{\i MoveOrdering. }{The alternative cards created by MoveGenerate are sorted, with the cards most likely to terminate the search fastest being sorted first in the move list.The allocation of card weights are described in detail in Chapter 5.}{\lang1053 

\par 
\par }{\i\lang1053 CheckMovesForCutoff }{\lang1053 checks if any of the moves just generated will lead to a position that can be found in the transposition table. If so, an immediate S
earch return can be done, saving unnecessary search effort. This is further described in Chapter 6.
\par 
\par To my knowledge this is not described anywhere for usage in a DDS. It is described in [Plaat et al.] and named Enhanced Transposition Cutoffs.
\par 
\par At move search cutoff,}{\i\lang1053  MergeMoveData }{\lang1053 
collects the union of the backed up accumulated winning ranks and the rank of the made move, assuming it did win by rank. The state data of the position is updated with the collected information.
\par 
\par }{\i\lang1053 MergeAllMovesData }{\lang1053 collects the un
ion of the backed up accumulated winning ranks, the previous accumulated winning ranks of the alternative moves generated on this depth, and the rank of the made move, assuming it did win by rank. When all alternative moves have been searched without a cu
toff, the state data of the position is updated with the collected information.
\par 
\par The information from }{\i\lang1053 MergeMoveData}{\lang1053  and }{\i\lang1053 MergeAllMovesData}{\lang1053  is later stored in the transposition table and determines which ranks that are essential when }{\i\lang1053 RetrieveTTresult}{\lang1053  scans th
e set of positions in the transposition table. A match of ranks with the current position is only needed for winning ranks. See Chapter 7.}{\strike\cf6\lang1053 
\par }{\lang1053 
\par }{\i\lang1053 AddNewTTentry }{\lang1053 adds the evaluated position as a new entry in the transposition table. See Chapter 7.
\par 
\par }{\i\lang1053 NextMove }{\lang1053 filters out all \'94small\'94 cards except one per hand/suit combination. A \'94small\'94 card is a backed up card that is shown to never win by rank. The rest of the \'94small\'94
 card moves for the hand/suit combination are never searched, leading to a smaller search tree.\line This search enhancement was suggested by Hans Kuijf [Kuijf]. 
\par 
\par 
\par 
\par {\listtext\pard\plain\b \hich\af0\dbch\af0\loch\f0 3.\tab}}\pard \nowidctlpar\jclisttab\tx0\ls16\adjustright {\b\lang1053 The Quick Tricks cutoff algorithm
\par }\pard \nowidctlpar\adjustright {\b\lang1053 
\par }{\lang1053 The number of tricks that can immediately be taken by the side to play the leading card of the trick consists of:
\par {\pntext\pard\plain\hich\af0\dbch\af0\loch\f0 a)\tab}}\pard \fi-360\li360\nowidctlpar\tx360{\*\pn \pnlvlbody\ilvl0\ls3\pnrnot0\pnlcltr\pnstart1\pnindent360 {\pntxta )}}\ls3\adjustright {\lang1053 The number of tricks that can be taken 
by the hand-to-play, and
\par {\pntext\pard\plain\hich\af0\dbch\af0\loch\f0 b)\tab}}\pard \fi-360\li360\nowidctlpar\tx360{\*\pn \pnlvlbody\ilvl0\ls3\pnrnot0\pnlcltr\pnstart1\pnindent360 {\pntxta )}}\ls3\adjustright {\lang1053 
The number of tricks that can be taken by the partner of the hand-to-play
\par }\pard \nowidctlpar\adjustright {\lang1053 At return by }{\i\lang1053 QuickTricks}{\lang1053 , the position state information is updated with the winning ranks found. 
\par 
\par }{Of course, in order to add b), there must be an entry from the hand-to-play to the partner\rquote s hand.
\par 
\par For each \'94s\'94 (suit) the following is calculated:}{\lang1053 
\par 
\par If the hand-to-play is the only hand having cards in s, and the opponents have no trumps (when s is not trumps), the number of quick tricks for s is the suit length of the hand-to-play.
\par 
\par If the opponents have no trumps, a check is made to see if quick tricks equal to the maximum of the trumps length for leading hand and the partner causes a search cutoff.
\par 
\par If the hand-to-play has a card in a suit where the partner has a winning rank, and partner is the only hand having cards in s:
\par The number of quick tricks for s is the suit length of partner.
\par 
\par Else:
\par If the winning rank is in hand-to-play, and the opponents can
not ruff, the number of quick tricks is incremented by one. Further, if the second best rank is also in hand-to-play, and the opponents cannot still ruff, the quick tricks is again incremented by one.
\par 
\par Else:
\par If the winning rank is in partner and partner has winning rank as entry, the same applies for the partner as for the hand-to-play described above.
\par 
\par }{If it is a trump contract, the first suit to be investigated is the trump suit. Then if there are trump suit quick tricks for the side to play, those are cashed and quick tricks incremented accordingly.
\par \line When the other suits are investigated for quick tricks, only the remaining opponent trump cards need to be considered.\line }{\lang1053 
\par }{The quick tricks are then summarized from each suit, and the total calculated}{\lang1053 .
\par 
\par A simple Quick Tricks algorithm is also executed after the leading card of the trick has been played:
\par 
\par A quick trick is gained either if the hand-to-play or the partner can win the current trick with the card having the highest rank of the suit played, or if hand-to-play or the partner can win the trick by ruffing.
\par 
\par The idea to also execute Quick Tricks after the leading card has been played was given by Hans Kuijf [Kuijf].
\par 
\par 
\par 
\par {\listtext\pard\plain\b \hich\af0\dbch\af0\loch\f0 4.\tab}}\pard \nowidctlpar\jclisttab\tx0\ls16\adjustright {\b\lang1053 The Later Tricks cutoff algorithm}{\lang1053  
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par Check for search cutoff if the opponents to the trick leading hand have at least a sure trick later. 
\par 
\par }{\b\lang1053 If not trump contract:\line 
\par {\pntext\pard\plain\hich\af0\dbch\af0\loch\f0 1)\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlbody\ilvl0\ls17\pnrnot0\pndec\pnstart1\pnindent360\pnhang{\pntxta )}}\ls17\adjustright {\lang1053 
The opponents have at least a sure trick if for all suits where the trick leading hand has a card, the side of the leading hand does not have the highest rank.\line 
More than one sure trick can be taken by the opponents if they possess the winning rank for more than one suit, or\line 
\par {\pntext\pard\plain\hich\af0\dbch\af0\loch\f0 2)\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlbody\ilvl0\ls17\pnrnot0\pndec\pnstart1\pnindent360\pnhang{\pntxta )}}\ls17\adjustright {\lang1053 
Assume that all suits where the side of the trick leading hand has the winning rank give maximum possible number of tricks, i.e. that the sure trick number is the sum of the\line maximum lengths of these suits.\line 
If this still cannot cause a cutoff for the trick leading side, allocate one sure trick for the opponents side.   
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par }{\b\lang1053 If trump contract:
\par }{\lang1053 
\par Quick tricks for the opponents of the leading hand are added when the opponents have one or more winning trumps. This idea was given by Pedja Stanojevic [Stanojevic].\line \line 
1)   If the opponent side have all the trumps, the number of sure tricks is the maximum suit length\line       length, or\line 
\par 2)   If the opponent side has the highest trump, they have 1 sure trick. If they also have the second\line       highest trump, they have 2 sure tricks, or\line \line 3)  If the opponent side has the second highest trump plus at least one tr
ump more behind the \line      hand with the highest trump, the opponent side has 1 sure trick.
\par \page 
\par {\listtext\pard\plain\s3 \b \hich\af0\dbch\af0\loch\f0 5.\tab}}\pard\plain \s3\keepn\nowidctlpar\jclisttab\tx0\ls16\outlinelevel2\adjustright \b\lang1053 {The Move Ordering algorithm
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\b\lang1053 
\par }{\lang1053 The weight of a card in the move list is affected by the suit and the rank of the card and by the other cards in the same trick.
\par The weights of the cards in the move list are used to sort them, }{with the cards having the highest weight being sorted first in the list}{\lang1053 . 
\par There are 2 exceptions though:
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls12\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls12\adjustright {\lang1053 
If the hand-to-play is trick leading hand, cards having the suit decided as best by the move ordering algorithm will occupy the first two positions in the move list, provided that there are at least 2 cards with that suit.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls13\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls13\adjustright {\lang1053 
If the hand-to-play is void in the suit played by leading hand, a card having the suit decided as best by the move ordering algorithm will occupy the first position in the move list.
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par A "best move" is maintained for each searched depth. At a search (alpha-beta) cutoff, the move causing the cutoff overwrites the present "best move" for the current depth. When a Transpos
ition Table entry is created, the current best move is stored in that entry if:
\par The target is met and the leading hand belongs to the player\rquote s side, or target is not met and the leading hand belongs to the other side. Otherwise the best move is not stored in the Transposition Table entry. 
\par At a Transposition Table entry match, }{its stored best move will be best move for the current search depth.}{\lang1053 
\par 
\par By \'94card move\'94 in the following pseudo code is meant the card by the hand-to-play that is getting a weight in the move list. }{\cf1\lang1053 The \'94card rank\'94 is a value in the range 2-14, corresponding to the card ranks 2 to the Ace. 
\par 
\par }{\lang1053 For the determination of the weight, it is calculated whether or not the current card move is a card that wins the current trick for the side of the hand-to-play, assuming that both sides play their optimum cards. }{\cf1\lang1053 
\par }{\fs20\lang1053 
\par 
\par }\pard\plain \s2\keepn\nowidctlpar\outlinelevel1\adjustright \lang2057 {\ul\lang1053 Hand-to-play is trick leading hand
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 
\par The contribution of the suit to the weight:
\par  
\par suitWeightDelta = suitBonus - (countLH+countRH) * 2
\par 
\par If trump contract, and the suit is not trump, then there is a (negative) suitBonus of \endash 12  if 
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\tx360{\*\pn \pnlvlblt\ilvl0\ls2\pnrnot0\pnf3\pnindent360 {\pntxtb \'b7}}\ls2\adjustright {\lang1053 LHO is void and LHO has trump card(s), or
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\tx360{\*\pn \pnlvlblt\ilvl0\ls2\pnrnot0\pnf3\pnindent360 {\pntxtb \'b7}}\ls2\adjustright {\lang1053 RHO is void and RHO has trump card(s)
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par Otherwise, suitBonus = 0.
\par 
\par countLH = (suit length of LHO) * 4, if LHO is not void in the suit,
\par countLH = (depth + 4), if LHO is void in the suit
\par 
\par countRH = (suit length of RHO) * 4, if RHO is not void in the suit,
\par countRH = (depth + 4), if RHO is void in the suit
\par 
\par Suits are thus favoured where the opponents have as few move alternatives as possible. 
\par 
\par 
\par if (trick winning card move) \{
\par     if (one of the opponents has a singleton highest rank in the suit)
\par         weight = suitWeightDelta + 40 \endash  (rank of card move)
\par     else if (hand-to-play has highest rank in suit)  \{
\par         if (partner has second highest rank in suit)
\par             weight = suitWeightDelta + 50 \endash  (rank of card move)
\par         else if  (the card move is the card with highest rank in the suit)
\par             weight = suitWeightDelta + 31
\par         else
\par             weight = suitWeightDelta + 19 \endash  (rank of card move)
\par     \}
\par     else if  (partner  has highest rank in suit)  \{
\par         if (hand-to-play has second highest rank in suit)
\par             weight = suitWeightDelta + 50 \endash  (rank of card move)
\par         else
\par             weight = suitWeightDelta + 35 \endash  (rank of card move)  
\par     \}
\par     else if (card move include equivalent card(s) in the suit)
\par         weight = suitWeightDelta + 40 \endash  (rank of card move)
\par     else
\par         weight = suitWeightDelta + 30 \endash  (rank of card move)
\par     if  (the card move  is \'94best move\'94 as obtained at search cutoff)
\par         weight = weight + 50;
\par \}
\par else \{\tab /* Not a trick winning move */
\par     if  (either LHO or RHO has singleton in suit which has highest rank)
\par         weight = suitWeightDelta + 20 \endash  (rank of card move)
\par     else if (hand-to-play has highest rank in suit)  \{
\par         if (partner has second highest rank in suit)
\par             weight = suitWeightDelta + 35 \endash  (rank of card move)
\par         else if  (the card move is the card with highest rank in the suit)
\par             weight = suitWeightDelta + 16
\par         else
\par             weight = suitWeightDelta + 4 \endash  (rank of card move)
\par     \}
\par     else if  (partner  has highest rank in suit)  \{
\par         if (hand-to-play has second highest rank in suit)
\par             weight = suitWeightDelta + 35 \endash  (rank of card move)
\par         else
\par             weight = suitWeightDelta + 20 \endash  (rank of card move)  
\par     \}
\par     else if  (hand-to-play has second highest rank together with equivalent card(s) in suit)
\par         weight = suitWeightDelta + 20 \endash  (rank of card move)
\par     else
\par         weight = suitWeightDelta + 4 \endash  (rank of card move)
\par     if  (the card move  is \'94best move\'94 as obtained at search cutoff)
\par         weight = weight + 30;
\par \}
\par 
\par 
\par  }{\ul\lang1053 Hand-to-play is left hand opponent (LHO) to leading hand
\par }{\lang1053 
\par if (trick winning card move) \{
\par     if  (hand-to-play void in the suit played by the leading hand)  \{
\par         if  (trump contract and trump is equal to card move suit)
\par             }{\cf1\lang1053 weight = 30 - (rank of card move) + 2 * (suit length for card move suit)
\par         else
\par             weight = 60 - (rank of card move) + 2 * (suit length for card move suit)
\par     \}
\par     else if (lowest card for partner to leading hand is higher than LHO played card)
\par         weight = 45 - (rank of card move)
\par     else if (RHO has a card in the leading suit that is higher than the trick leading card\line                but lower than the highest rank of the leading hand)
\par         weight = 60 - (rank of card move)
\par     else if (LHO played card is higher than card played by the leading hand) \{
\par         if (played card by LHO is lower than any card for RHO in the same suit)
\par             weight = 75 - (rank of card move)
\par         else if (played card by LHO is higher than any card in the same suit for the leading hand)
\par             weight = 70 - (rank of card move)
\par         else  \{
\par             }{\lang1053 if  (LHO move card has at least one equivalent card) \{
\par                 }{\cf1\lang1053 weight = 60 - (rank of card move)}{\lang1053  
\par             else
\par                 }{\cf1\lang1053 weight = 45 - (rank of card move)
\par         \}
\par     \}
\par     else if (RHO is not void in the suit played by the leading hand) \{
\par         }{\lang1053 if  (LHO move card has at least one equivalent card) }{\cf1\lang1053  }{\lang1053   }{\cf1\lang1053 
\par             weight = 50 - (rank of card move)
\par         else
\par             weight = 45 - (rank of card move)
\par     \}
\par     else
\par         weight = 45 - (rank of card move)
\par \}
\par else \{\tab /* card move is not trick winning */
\par     }{\lang1053 if  (hand-to-play void in the suit played by the leading hand)  \{
\par         if  (trump contract and trump is equal to card move suit)}{\cf1\lang1053 
\par }{\lang1053             }{\cf1\lang1053 weight = 15 - (rank of card move) + 2 * (suit length for card move suit)
\par         else
\par             weight = - (rank of card move) + 2 * (suit length for card move suit)
\par     \}
\par     else if (lowest card for partner to leading hand or for RHO in the suit played is higher \line                than played card for LHO) 
\par         weight = - (rank of card move) 
\par     else if (LHO played card is higher than card played by the leading hand) \{ 
\par         }{\lang1053 if  (LHO move card has at least one equivalent card)
\par             }{\cf1\lang1053 weight = 20 - (rank of card move) 
\par          else}{\lang1053  
\par              }{\cf1\lang1053 weight = 10 - (rank of card move)
\par      \} }{\lang1053  
\par      else 
\par          }{\cf1\lang1053 weight = - (rank of card move)
\par \} }{\lang1053  }{\cf1\lang1053  }{\lang1053   }{\cf1\lang1053  
\par }{\lang1053 
\par 
\par 
\par }\pard\plain \s2\keepn\nowidctlpar\outlinelevel1\adjustright \lang2057 {\ul\lang1053 Hand-to-play is partner to trick leading hand
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 
\par if (trick winning card move) \{
\par     if  (hand-to-play void in the suit played by the leading hand)  \{
\par         if (card played by the leading hand is highest so far) \{
\par             if (card by hand-to-play is trump and the suit played by the leading hand is not trump) 
\par                 }{\cf1\lang1053 weight = 30 - (rank of card move) + 2 * (suit length for card move suit)
\par             else
\par                 weight = 60 - (rank of card move) + 2 * (suit length for card move suit)
\par         \}
\par         else if (hand-to-play is on top by ruffing)
\par             weight = 70 - (rank of card move) + 2 * (suit length for card move suit)
\par         else if (hand-to-play discards a trump but still loses)
\par             weight = 15 - (rank of card move) + 2 * (suit length for card move suit) 
\par         else       }{\lang1053 
\par             }{\cf1\lang1053 weight = 30 - (rank of card move) + 2 * (suit length for card move suit)
\par     \}
\par     else }{\lang1053 
\par         }{\cf1\lang1053 weight = 60 - (rank of card move) 
\par \}
\par else \{               /* card move is not trick winning */
\par     }{\lang1053 if  (hand-to-play void in the suit played by the leading hand)  \{
\par }{\cf1\lang1053         if (hand-to-play is on top by ruffing)
\par             weight = 40 - (rank of card move) + 2 * (suit length for card move suit)
\par         else if (hand-to-play underruffs */
\par             weight = -15 - (rank of card move) + 2 * (suit length for card move suit)
\par         else
\par             weight = - (rank of card move) + 2 * (suit length for card move suit)
\par     \}
\par     else \{
\par          if (the card by hand-to-play is highest so far) \{
\par              if (rank of played card is second highest in the suit)
\par                 weight = 25  
\par              else }{\lang1053 if (hand-to-play card has at least one equivalent card)
\par                  }{\cf1\lang1053 weight = 20 - (rank of card move)
\par              else
\par                  weight = 10 - (rank of card move)
\par          \}
\par          else
\par              weight = -10 - (rank of card move)
\par     \}
\par \}}{\lang1053 
\par 
\par }{\ul\lang1053 Hand-to-play is right hand opponent (RHO) to leading hand
\par 
\par }{\lang1053 if  (hand-to-play is void in leading suit)  \{
\par     if  (LHO has current highest rank of the trick)  \{
\par        if  (card move ruffs)
\par           weight = 14- (rank of card move) + 2 * (suit length for card move)
\par        else
\par           weight = 30- (rank of card move) + 2 * (suit length for card move) 
\par }{\fs20\lang1053      \}
\par }{\lang1053     else if  (hand-to-play ruffs and wins) 
\par }{\fs20\lang1053          }{\lang1053 weight = 30- (rank of card move) + 2 * (suit length for card move)
\par     else if  (card move suit is trump, but not winning)
\par         weight = - (rank of card move)
\par     else
\par         weight = 14- (rank of card move) + 2 * (suit length for card move)
\par \}
\par else if  (LHO has current winning move)  \{
\par     if  (RHO ruffs LHO\rquote s winner)
\par         weight = 24 - (rank of card move) 
\par     else
\par         weight = 30- (rank of card move) 
\par \}
\par else if  (card move superior to present winning move not by LHO)  \{        }{\fs20\lang1053 
\par      }{\lang1053 weight = 30- (rank of card move)
\par else  \{
\par     if  (card move ruffs but still losing)
\par         weight = - (rank of card move)
\par     else
\par         weight = 14- (rank of card move)
\par \}}{\fs20\lang1053 
\par }{\lang1053 
\par }\pard \fi1304\nowidctlpar\adjustright {\lang1053 
\par \page 
\par {\listtext\pard\plain\b \hich\af0\dbch\af0\loch\f0 6.\tab}}\pard \nowidctlpar\jclisttab\tx0\ls16\adjustright {\b\lang1053 Algorithm to try early cutoff for generated moves
\par }\pard \nowidctlpar\adjustright {\b\lang1053 
\par }\pard\plain \s1\keepn\nowidctlpar\outlinelevel0\adjustright \lang2057 {\lang1053 After generating moves at the end of a trick, they are each checked to see if one of them will lead to a position that already is stored in the Transposition Table.
\par Due to the processing overhead, this check is only made if the depth is 29 or more (i.e there are at least 33 cards in the position). 
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 Pseudo code:
\par }\pard\plain \s1\keepn\nowidctlpar\outlinelevel0\adjustright \lang2057 {\lang1053 
\par moveExists = TRUE;
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 while  (moveExists)  \{
\par      Make;
\par      depth = depth \endash 1;
\par      RetrieveTTresult;
\par      if (hit in the transposition table)  \{
\par           Search returnsTRUE if value of the position is TRUE and player side to move, or 
\par           FALSE if value of the position is FALSE and opponents side to move.  
\par           Else: Increment weight of move with 100.  
\par       \}
\par        depth = depth +1;
\par        Undo;
\par        moveExists = NextMove;
\par \}
\par     
\par The performance improvement for this enhancement is less than for the other enhancements. The number of generated nodes is roughly decreased by 10% and the search time is slighly decreased.
\par 
\par 
\par 
\par {\listtext\pard\plain\b \hich\af0\dbch\af0\loch\f0 7.\tab}}\pard \nowidctlpar\jclisttab\tx0\ls16\adjustright {\b\lang1053 Storage and retrieval of position state data in the Transposition Table
\par }\pard\plain \s16\nowidctlpar\tx360\adjustright \lang2057 {\lang1053  }{\b\lang1053 \line }{\cf1\lang1053 
Positions stored in the Transposition Table always consist of completed tricks. Positions stored start at depth=4, then 8,12, and so on. The information stored is information on won cards, the suit lengths of the hands, the hand to play the l
eading card in the position and upper and lower bounds for the number of future tricks to be taken by the side of the player.
\par 
\par Starting from issue 1.1.8, each \'94winning cards node\'94 contain all winning cards for one suit after an idea by Jo\'ebl Bradmetz. This new solution is faster.}{\b\cf1 \line }{\b\cf1\lang1053 
\par }\pard\plain \nowidctlpar\adjustright \lang2057 {\b\lang1053 7.1   Transposition Table storing winning card ranks
\par }\pard \nowidctlpar\tx360\adjustright {\b\lang1053 
\par }{\lang1053 For the outcome of played tricks, only card ranks that are winning due to their ranks matter:
\par Assume that the last two tricks of a deal without trumps looks like the following:
\par Trick 12: Leading hand North plays heart A, East, South and West follow by hearts Q, 9 and 7 respectively. 
\par Trick 13: North then leads spade A, the other hands plays diamonds  J, 8,3 in that order.
\par 
\par In trick 12, heart A wins by rank. In trick 13, spade A wins but not by rank.
\par The sequence of cards could have been the following without  changing the outcome:
\par Trick 12:  heart A, heart x, heart x, heart x
\par Trick 13:  spade x, diamond x, diamond x, diamond x
\par where x is any rank below lowest winning rank.
\par 
\par The ca
rds that win by rank are recorded during the search and backed up similarly to the search value. If a card wins by rank and there are equivalent cards, e.g. only spade A is searched from a sequence of AKQ, then also the other cards K and Q must be recorde
d as having won by rank.
\par 
\par The cards winning by rank are stored in the Transposition Table as relative ranks, however any rank larger than the lowest winning rank in the suit are also stored as \'94winning ranks\'94. Using relative ranks rather than absolute ranks
 considerably increases the number of positions that match this Transposition Table entry:\line 
As an example, assume that there are only 4 cards left in a suit, A, Q, 9, 7 where each hand has one card in the suit. Then any combination of ranks, e.g. 8, 6, 3, 2 that preserves the relative order of ranks between hands will cause a match.
\par 
\par In the state position information absolute ranks are used, it is only in the Transposition Table where the ranks are stored as relatives. 
\par  
\par }\pard \fi-420\li420\nowidctlpar\tx420\adjustright {\b 
\par }\pard \nowidctlpar\adjustright {\b 7.2  Backing up the winning ranks
\par }\pard \nowidctlpar\tx420\adjustright {\b 
\par }{At the search termination, either at the last trick or at a cutoff, the cards that have won by rank are backed up in the search tree together with the search value.
\par As this information propagates upwards, it is aggregated  with backed up information from other tree branches.
\par At a search cutoff, }{\i\lang1053 MergeMoveData }{\lang1053 merges the information (V is a union):
\par  }{  
\par }\pard \fi-420\li420\nowidctlpar\tx420\adjustright {(winning ranks of all suits for current depth) = (winning ranks of all suits for depth - 1)  V  (possible winning rank for the current move causing the cutoff)
\par 
\par For each new move not causing cutoff, }{\i\lang1053 MergeAllMovesData }{\lang1053 merges:
\par 
\par }{(winning ranks of all suits for current depth) = (winning ranks of all suits for current depth)  V (winning ranks of all suits for depth - 1)  V  (possible winning rank for the current move) 
\par }\pard \nowidctlpar\tx420\adjustright {
\par }\pard \fi-420\li420\nowidctlpar\tx420\adjustright {\b 
\par 7.3\tab }{\b\lang1053 Checking the current position for a Transposition Table entry match 
\par 
\par }\pard \nowidctlpar\tx420\adjustright {\cf1\lang1053 The "Transposition Table" has a tree structure rather than a table, consisting of 2 interconnected trees. 
\par For deciding if there is a match, input is the position state data, including the cards left to play and the current leading hand. 
\par There are \'94root pointers\'94 per number of tricks left and per leading hand which  each points to the root of a tree of  \'94suit lengths combination\'94 nodes. Each such node includes a 64-bit code that
 uniquely defines one combination of suit lengths for the hands. The nodes are ordered such that the value of the 64-bit code in a parent node is higher than the 64-bit code of its left child but lower than the 64-bit code of its right child. So to find t
he node with the suit lengths combination for the actual position, a binary search is made. The basic binary search algorithm is described in [Knuth]. 
\par Each \'94suit length combination node\'94 points to the root of a tree consisting of \'94winning cards nodes\'94, ie. cards that win by rank. (So the Transposition Table is really a number of trees, a forest.)
\par When a position is checked for a possible Transposition Table match, a tree branch is selected consisting of 4 subsequent \'94winning cards nodes\'94, each \'94winning cards node\'94
 includes an aggregate of all winning cards for one suit. This branch is followed as long as the \'94winning cards\'94 also can be found in the current position. (Note that the ranks of the \'94winning card nodes\'94
 are relative, so the ranks of the current position must first be translated from absolute to relative ranks.)  When the \'94winning cards node\'94 no longer matches with the current position and there is no other alternative \'94winning cards node\'94
 that fits, then the search backs up and tries an alternative \'94winning cards node\'94 on a higher level. 
\par \line When the last of the 4 subsequent \'94winning cards nodes\'94 containing clubs is reached, it points to a \'94set of positions node\'94. Its stored upper and lower value bounds are checked against the number of tricks won so far by the player
\rquote s side and the target value. The following conditions are then checked, assuming that it is the North/South side that is the player\rquote s side: 
\par 
\par If the sum of the stored lower value bound and the number of tricks won so far for the player\rquote s side is equal or larger than target, then target can be reached for the player\rquote 
s side in the current position. Search on this depth is terminated and TRUE is returned.
\par 
\par If the sum of the stored upper value bound and the number of tricks won so far for the player\rquote s side is less than target, then reaching target can be prevented by the opponents to the player\rquote 
s side in the current position. Search on this depth is terminated and FALSE is returned.
\par 
\par If instead it is East/West that is the player\rquote s side, the following conditions apply:
\par 
\par If the sum of number of tricks remaining and the number of tricks won so far for the player\rquote s side minus the upper value bound is equal or larger than target, then target can be reached for the player\rquote 
s side in the current position. Search on this depth is terminated and TRUE is returned.
\par 
\par If the sum of number of tricks remaining and the number of tricks won so far for the player\rquote s side minus the lower value bound is less than target, then reaching target can be prevented by the opponents to the player\rquote 
s side in the current position. Search on this depth is terminated and FALSE is returned.
\par 
\par For all other cases, the search continues for the current depth.
\par 
\par For example, take the previous example with 2 tricks remaining with spade rank order 1 at North. (Rank order 1 is highest rank.) The hearts have rank orders 1 at North, 2 at East, 3 at South and 4 at West.  The diamond rank orders are orders 1 at
 East, 2 at South and 3 at West.  North is leading hand.
\par The \'94root pointer\'94 is now defined by the number of tricks remaining (=2) and North as leading hand.
\par The \'94root pointer\'94 points to the root node of its \'94suit lengths combination\'94 tree. The 64-bit integ
er coded from the suit lengths for all suits and hands is now searched within the tree. When the node is found with matching 64-bit suit lengths code, this node will point to the root of its \'94winning card\'94 tree.
\par This pointer points to a "winning cards node" containing spade rank order 1 at North which fits with the current position. This \'94winning cards node\'94
 points to another "winning cards node" containing hearts rank orders 1 at North and 2 at East  which also fits the current position. Next \'94winning cards node\'94
 pointed to contains diamonds order 1 at South, which does not match the current position. However, there is an alternative \'94winning cards node\'94 that has diamonds order 1 at East, which fits. (If there had been no alternative \'94winning cards node
\'94 which fitted, the search had backed up to the previous \'94winning cards node\'94 to see if there was an alternative \'94winning cards node\'94 on this level which also fitted.) The next \'94winning cards node\'94
 pointed to is for clubs. This node is empty, which fits the current position which have no clubs. 
\par This \'94winning cards node\'94 points to a "set of positions node\'94 which have upper and lower value bounds defined. The conditions for these bounds are assumed to be fulfilled causing search termination on this depth, as described earlier. 
\par 
\par The usage of upper and lower value bounds in transposition tables is described in [Chang] and [Kupferschmid, Helmert].}{\b\lang1053 
\par }\pard \nowidctlpar\adjustright {\b\lang1053 
\par 
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000024030300a606bb07e7069a07a6067a07040000002d010200040000002d01040004000000f00105000800000026060f000600ffffffff0100100000002606
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0000fc020000000000000000040000002d0105000a000000240303003e0739095f077a097f073909040000002d010200040000002d01040004000000f0010500
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00002d0105000a000000240303002601dd0a47011e0b6701dd0a040000002d010200040000002d01040004000000f00105000800000026060f000600ffffffff
0100040000002d010600040000002d010300070000001b04bb056403ca043702040000002d010400040000002d010200030000001e000700000016049c052803
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14026f089b07050000002e010100000011000000320a6f089b0704000400000000002a10cc0d4e6578743c00250029001700050000002e010000000005000000
14020902640e05000000020101000000040000002701ffff1000000026060f001600ffffffff00002e040000660a000078040000230b000008000000fa020000
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000000000000040000002d0105000a000000240303003204dd0a53041e0b7304dd0a040000002d010200040000002d01040004000000f0010500080000002606
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0500040000002d01060008000000250302005f076a0a5f07df0a040000002d01030004000000f001050007000000fc020000000000000000040000002d010500
0a000000240303003e07dd0a5f071e0b7f07dd0a040000002d010200040000002d01040004000000f00105000800000026060f000600ffffffff010007000000
fc020000ffffff000000040000002d01050008000000fa0200000600000000000002040000002d010700070000001b049f03340672020b01040000002d010400
04000000f0010500040000002d01020004000000f0010700030000001e000700000016047d03f50593024a0105000000020101000000040000002d0101000500
000009020000000005000000140294024a01050000002e010100000017000000320a94024a0108000400000000002a10cc0d54726565206f662034001c002500
240015002a001b001500050000002e01000000000500000014020902640e05000000140294025202050000002e010100000014000000320a9402520206000400
000000002a10cc0d6e6f6465732029002a002a00250020001500050000002e01000000000500000014020000000005000000140294022903050000002e010100
00001f000000320a940229030d000400000000002a10cc0d656163682073746f72696e67200025002500250029001500200017002a001c0017002a0029001500
050000002e0100000000050000001402000000000500000014029402d204050000002e010100000011000000320a9402d20404000400000000002a10cc0d6f6e
65202a00290025001500050000002e01000000000500000014020000000005000000140294025f05050000002e010100000011000000320a94025f0504000400
000000002a10cc0d737569742100290017001700050000002e010000000005000000140200000000050000000201010000000500000002010100000005000000
1402f6024a01050000002e010100000028000000320af6024a0113000400000000002a10cc0d6c656e6774687320636f6d62696e6174696f6e03170025002900
2a0017002a002000150025002b003f002a00180029002500170017002a002900050000002e010000000005000000140200000000050000000201010000000400
00002701ffff1000000026060f001600ffffffff0000430100009a03000078020000cf04000008000000fa0200000600000000000000040000002d0105000400
00002d010600080000002503020073029e0373019e04040000002d01030004000000f001050007000000fc020000000000000000040000002d0105000a000000
240303005e0185044701ca048c01b304040000002d010200040000002d01040004000000f00105000800000026060f000600ffffffff0100040000002d0100000300000000000000000000000000000000000000}}}}}{\lang1053 
\par The \'94suit lengths combination\'94 node includes:
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls14\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls14\adjustright {\lang1053 
      The suit lengths combination as a 64-bit integer.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls14\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls14\adjustright {\lang1053       A pointer to the top \'94
winning cards node\'94.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls14\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls14\adjustright {\lang1053       A pointer to next left \'94
suit lengths combination node\'94 in the binary tree.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls14\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls14\adjustright {\lang1053       A pointer to next right \'94
suit lengths combination node\'94 in the binary tree.
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par 
\par The \'94winning cards node\'94 includes:
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-720\li720\nowidctlpar{\*\pn \pnlvlblt\ilvl0\ls4\pnrnot0\pnf3 {\pntxtb \'b7}}\ls4\adjustright {\lang1053 The hands of the relative ranks for each winning card of the actual suit.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-720\li720\nowidctlpar{\*\pn \pnlvlblt\ilvl0\ls4\pnrnot0\pnf3 {\pntxtb \'b7}}\ls4\adjustright {\lang1053 
A pointer to the next winning cards node required to achieve a Transposition Table match for this branch of the tree.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-720\li720\nowidctlpar{\*\pn \pnlvlblt\ilvl0\ls4\pnrnot0\pnf3 {\pntxtb \'b7}}\ls4\adjustright {\lang1053 A pointer to the previous winning cards node.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-720\li720\nowidctlpar{\*\pn \pnlvlblt\ilvl0\ls4\pnrnot0\pnf3 {\pntxtb \'b7}}\ls4\adjustright {\lang1053 
A pointer to the next alternative winning cards node that leads to a Transposition Table match in an alternative tree branch.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-360\li360\nowidctlpar\jclisttab\tx360{\*\pn \pnlvlblt\ilvl0\ls10\pnrnot0\pnf3\pnstart1\pnindent360\pnhang{\pntxtb \'b7}}\ls10\adjustright {\lang1053 
      A pointer to the "set of positions node". 
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par 
\par The "set of positions node\'94 includes:
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-720\li720\nowidctlpar{\*\pn \pnlvlblt\ilvl0\ls4\pnrnot0\pnf3 {\pntxtb \'b7}}\ls4\adjustright {\lang1053 
An upper and a lower bound for the winning tricks of the North/South side. These values\line are used to determine whether or not a search cutoff can be done.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-720\li720\nowidctlpar{\*\pn \pnlvlblt\ilvl0\ls4\pnrnot0\pnf3 {\pntxtb \'b7}}\ls4\adjustright {\lang1053 The lowest winning rank per suit, expressed as relative rank.
\par {\pntext\pard\plain\f3 \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi-720\li720\nowidctlpar{\*\pn \pnlvlblt\ilvl0\ls4\pnrnot0\pnf3 {\pntxtb \'b7}}\ls4\adjustright {\lang1053 The suit and rank for the currently best move.
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par  
\par }\pard \nowidctlpar\tx420\adjustright {\cf1\lang1053 After a Transposition Table match, the current position may later be part of a position that will be stor
ed in the Transposition Table. Therefore, the stored winning ranks in the Transposition Table must be included in the state information of the current position. However, the winning ranks cannot be taken as is, because they are stored as relative ranks wh
ich now must be converted to absolute ranks in the current position.
\par This is done using the lowest winning relative rank for each suit that is stored in the \'94set of positions\'94 node that gave the Transposition Table match:
\par The aggregated set of (absolute) ra
nks for each suit in the current position is filtered using the stored information on the lowest winning relative rank. The winning ranks for each suit is then the aggregated set with only the number of highest ranks implied by the stored lowest winning r
elative rank in the \'94set of positions\'94 node.
\par E.g. if the aggregated rank set for spades is A J 9 4 2 and lowest winning relative rank is order=2, then winning ranks are A J.
\par }\pard \nowidctlpar\adjustright {\lang1053 
\par 
\par }\pard \nowidctlpar\tx420\adjustright {\b\cf1\lang1053 7.4  Building a new entry in the Transposition Table}{\cf1\lang1053 
\par 
\par When the value of the current position is known and it is the end of a trick (except the last),  position state information is collected for storage in the Transposition Table. 
\par The ranks of the backed up winning cards are converted from absolute to relative.
\par For each suit, it is determined which winning rank that is lowest. The relative ranks then stored in the new Transposition Table entry are all ranks above and including the lowest rank, filling out any \'94holes\'94
 in the ranks that might have been present.
\par The trees of the Transposition Table are searched starting from the \'94root pointer\'94 and additional nodes are inserted corresponding to the current position. 
\par First, the suit lengths of the current position are used to find a \'94suit lengths combination node\'94 or to create a new such node if it does not exist already.
\par The next step is to search for a \'94winning card node\'94 that has the \'94suit length combination node\'94 as parent. This \'94winning card node\'94 has then winning cards for spades.
\par If no such node yet exists, \'94winning card nodes\'94, one for each suit, are created using the winning cards of the current position. Each such node includes all winning cards for one of the suits. Then, a \'94set of positions\'94
 node is created. This node is pointed to from the last created \'94winning card node\'94 created for the winning cards of clubs. 
\par Otherwise, if there already exists a matching \'94winning card node\'94 with the \'94suit length combination node\'94 as parent, it is checked whether or not the \'94winning card nodes\'94 in a subsequent tree branch already created 
for hearts, diamonds and clubs also are matched with the current position.
\par If such a sequence of nodes can be found, the upper or lower bound in the connected \'94set of positions node\'94 may be updated to allow for an increased number of cutoffs:
\par 
\par If the current  position upper value bound is less than the stored upper value bound, the stored value is updated with the current position value.   
\par If the current  position lower value bound is larger than the stored lower value bound, the stored value is updated with the current position value.
\par 
\par In case a matching \'94winning card node\'94 cannot be found, a new \'94winning card node\'94 is created and linked to the last matching node. E.g. if existing \'94winning card nodes\'94 for spades and hearts match the current position, but n
o node match for diamonds, then a \'94winning cards node\'94 for diamonds is created and linked to the previous \'94winning cards node\'94 for hearts. Then a clubs \'94winning cards node\'94 and a \'94set of positions node\'94 are created.
\par 
\par  
\par }\pard \nowidctlpar\adjustright {\b\lang1053 
\par 
\par 
\par References
\par }{\lang1053 
\par James Dow Allen:
\par Source code for a simple DDS.
\par }{\field\fldedit{\*\fldinst {\lang1053  HYPERLINK  \\l "_Hlk134153111" }{\fs20\lang1053 {\*\datafield 
08d0c9ea79f9bace118c8200aa004ba90b02000000080000000e0000005f0048006c006b00310033003400310035003300310031003100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00}}}{\fldrslt {\cs15\ul\cf2 http://freepages.genealogy.rootsweb.com/~jamesdow/Tech/dbldum.htm}}}{\lang1053 
\par 
\par Matthias Brill:
\par DDS algorithms description (in German) and DDS source code.
\par }{\field\fldedit{\*\fldinst {\lang1053  HYPERLINK  \\l "_Hlk134153080" }{\fs20\lang1053 {\*\datafield 
08d0c9ea79f9bace118c8200aa004ba90b02000000080000000e0000005f0048006c006b00310033003400310035003300300038003000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00}}}{\fldrslt {\ul\cf2  http://linux.softpedia.com/get/Science-and-Engineering/Artificial-Intelligence/cddsolve-20055.shtml}{\lang1053  }}}{\lang1053 
\par 
\par Ming-Sheng Chang:
\par DDS algorithms description.
\par {\*\bkmkstart _Hlt193293965}}{\field\fldedit{\*\fldinst {\lang1053 HYPERLINK "\\\\l "_Hlk132979785" "}{\fs20\lang1053 {\*\datafield 
00d0c9ea79f9bace118c8200aa004ba90b02000000010000000303000000000000c0000000000000460000040000005c6c2000ffffadde000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000}}}{\fldrslt {cs.nyu.edu/web/Research/TechReports/TR1996-725/TR1996-725.ps.gz
\par }}}\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 {\*\bkmkend _Hlt193293965}
\par 
\par Ed Colley:
\par DDS source code and DDS executable.
\par }{\field\flddirty{\*\fldinst {\lang1053 HYPERLINK "\\\\l "_Hlk133040134" "}{\fs20\lang1053 {\*\datafield 
00d0c9ea79f9bace118c8200aa004ba90b02000000010000000303000000000000c0000000000000460000040000005c6c2000ffffadde000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000}}}{\fldrslt {\ul\cf2\lang1053 http://freefinesse.sourceforge.net/}}}{\lang1053 
\par 
\par Matthew L. Ginsberg:
\par DDS algorithms description.
\par }{\field\fldedit{\*\fldinst {\lang1053  HYPERLINK  \\l "_Hlk134152954" }{\fs20\lang1053 {\*\datafield 
08d0c9ea79f9bace118c8200aa004ba90b02000000080000000e0000005f0048006c006b00310033003400310035003200390035003400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00}}}{\fldrslt {\cs15\ul\cf2 http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume14/ginsberg01a.pdf}}}{\lang1053 
\par 
\par Dan Hirschberg:
\par DDS algorithms description and DDS executable (MS DOS, cannot run in XP?)
\par }{\field\flddirty{\*\fldinst {\lang1053 HYPERLINK "\\\\l "_Hlk132979763" "}{\fs20\lang1053 {\*\datafield 
00d0c9ea79f9bace118c8200aa004ba90b02000000010000000303000000000000c0000000000000460000040000005c6c2000ffffadde000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000}}}{\fldrslt {\ul\cf2\lang1053 http://www.ics.uci.edu/~dan/bridge/index.html
\par }}}\pard\plain \nowidctlpar\adjustright \lang2057 {\lang1053 
\par Alexey Slovesnov:
\par DDS source code and DDS executable.
\par }{\field\flddirty{\*\fldinst {\lang1053 HYPERLINK "\\\\l "_Hlk132986169" "}{\fs20\lang1053 {\*\datafield 
00d0c9ea79f9bace118c8200aa004ba90b02000000010000000303000000000000c0000000000000460000040000005c6c2000ffffadde000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000}}}{\fldrslt {\ul\cf2\lang1053 http://slovesnov.narod.ru/bridge/en/index.html}}}{\lang1053 
\par 
\par Judea Pearl: Asymptotic properties of minimax trees and game search precedures.\line    Artificial Intelligence 14(2):113-138. [Pearl 1980]
\par 
\par Aske Plaat, Jonathan Schaeffer, Wim Pijls and Arie de Bruin:  Exploiting graph properties of game trees. In }{\i\lang1053 Proceedings of the Thirteenth National Conference on Artificial Intelligence}{\lang1053 , pages 234-239, 1996  [Plaat et al.]
\par 
\par Hans Kuijf, personal communication.
\par 
\par Pedja Stanojevic, personal communication.
\par 
\par Knuth: The art of computer programming, Vol. 3, Searching and Sorting, chapter 6.2.2, Algorithm T.
\par 
\par Sebastian Kupferschmid, Malte Helmert: A Skat Player Based on Monte Carlo Simulation.
\par 
\par Jo\'ebl Bradmetz, personal communication.
\par http://jibe-bridge.perso.cegetel.net/
\par }}