Transposition Table

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A Transposition Table, first used in Greenblatt's program Mac Hack VI , is a database that stores results of previously performed searches. It is a way to greatly reduce the search space of a chess tree with little negative impact. Chess programs, during their brute-force search, encounter the same positions again and again, but from different sequences of moves, which is called a transposition. Transposition (and refutation) tables are techniques derived from dynamic programming, a term coined by Richard E. Bellman in the 1950s, when programming meant planning, and dynamic programming was conceived to optimally plan multistage processes.

=How it works= When the search encounters a transposition, it is beneficial to 'remember' what was determined last time the position was examined, rather than redoing the entire search again. For this reason, chess programs have a transposition table, which is a large hash table storing information about positions previously searched, how deeply they were searched, and what we concluded about them. Even if the depth (draft) of the related transposition table entry is not big enough, or does not contain the right bound for a cutoff, a best (or good enough) move from a previous search can improve move ordering, and save search time. This is especially true inside an iterative deepening framework, where one gains valuable table hits from previous iterations.

Hash functions
Hash functions convert chess positions into an almost unique, scalar signature, allowing fast index calculation as well as space saving verification of stored positions.
 * Zobrist Hashing
 * BCH Hashing

Both, the more common Zobrist hashing as well BCH hashing use fast hash functions, to provide hash keys or signatures as a kind of Gödel number of chess positions, today typically 64-bit wide. They are updated incrementally during make and unmake move by either own-inverse exclusive or or by addition versus subtraction.

Address Calculation
The index is not based on the entire hash key because this is usually a 64-bit number, and with current hardware limitations, no hash table can be large enough to accommodate it. Therefor to calculate the address or index requires signature modulo number of entries, for power of two sized tables, the lower part of the hash key, masked by an 'and'-instruction accordantly. =Collisions= The surjective mapping from chess positions to a signature and an even more denser index range implies collisions, different positions share same entries, for two different reasons, hopefully rare ambiguous keys (type-1 errors), or regularly ambiguous indices (type-2 errors).

Cardinalities
The typical cardinalities of positions and signatures inside the search, reflects the likelihood of collisions:

Index Collisions
Index collisions or type-2 errors, where different hash keys index same entries, happen regularly. They require detection, realized by storing the signature as part of the hash entry, to check whether a stored entry matches the position while probing. Specially with power of two entry tables, many programmers choose to trade-off space for accuracy and only store that part of the hash key not already considered as index, or even less.

Key Collisions
Key collisions or type-1 errors are inherent in using signatures with far less bits than required to encode all reachable chess positions. A key collision occurs when two different positions map the same hash key or signature. When storing only a partial key, the chance of a collision greatly increases. To accept only hits where stored moves are pseudo-legal decreases the chance of type-1 errors. On his computer chess page, Ken Regan broached on chess engine bugs, anomalies and hash collisions, and mentions a Shredder 9.1 game where a key collision might have caused a strange move.

Bits Required
During the WCCC 1989 Workshop New Directions in Game-Tree Search, James Gillogly, author of Tech, discussed transposition table collisions. He produced the following table using the Birthday Paradox, where the columns are the number of positions stored and the rows are the probability of collision. The entries are the number of bits of combined address and check hash required to reduce the probability of collision to the desired amount.

During the discussion, David Slate and Ken Thompson pointed out that the Birthday Paradox is not applicable to most programs, since the hash table will fill up and not all previous positions will be in the table; thus these figures must be regarded as an upper bound on the number of bits required for safety. The dangers of transposition table collisions were further studied by Robert Hyatt and Anthony Cozzie as published in their 2005 paper Hash Collisions Effect. They gave an surprising answer to the question “Is it really worth all the effort to absolutely minimize signature collisions?”, and concluded that 64 bit signatures are more than sufficient. =What Information is Stored= Typically, the following information is stored as determined by the search :
 * Zobrist- or BCH-key, to look whether the position is the right one while probing
 * Best- or Refutation move
 * Depth (draft)
 * Score, either with Integrated Bound and Value or otherwise with
 * Type of Node
 * PV-Node, Score is Exact
 * All-Node, Score is Upper Bound
 * Cut-Node, Score is Lower Bound


 * Age is used to determine when to overwrite entries from searching previous positions during the game of chess

=Table Entry Types= In an alpha-beta search, we usually do not find the exact value of a position. But we are happy to know that the value is either too low or too high for us to be concerned with searching any further. If we have the exact value, of course we store that in the transposition table. But if the value of our position is either high enough to set the lower bound, or low enough to set the upper bound, it is good to store that information also. So each entry in the transposition table is identified with the type of node, often referred to as exact, lower- or upper bound. =Replacement Strategies= Because there are a limited number of entries in a transposition table, and because in modern chess programs they can fill up very quickly, it is necessary to have a scheme by which the program can decide which entries would be most valuable to keep, i.e. a replacement scheme. Replacement schemes are used to solve an index collision, when a program attempts to store a position in a table slot that already has a different entry in it. There are two opposing considerations to replacement schemes:
 * Entries that were searched to a high depth save more work per table hit than those searched to a low depth.
 * Entries that are closer to the leaves of the tree are more likely to be searched multiple times, making the table hits of them higher. Also, entries that were searched recently are more likely to be searched again.
 * Most well-performing replacement strategies use a mix of these considerations.

Always Replace
This replacement strategy is very simple, placing all emphasis on the second consideration. Any old entries are replaced immediately when a new entry is stored.

Priority by Searched Nodes Count
This replacement strategy uses number of nodes searched spent to obtain an entry, as replacement priority.

Priority by Move Ordering Position
This replacement strategy uses position of entry move in move ordering list as replacement priority. The main idea is that if the best move was not considered as good cut-off candidate by move-ordering algorithm, storing it in TT should provide better help for the search.

Depth-Preferred
This replacement strategy puts all emphasis on the first consideration. The only criteria in deciding whether to overwrite an entry is whether the new entry has a higher depth than the old entry.

Two-tier System
This strategy, devised by Ken Thompson and Joe Condon, uses two tables, side by side. For each table slot, there is a depth-preferred and an always-replace entry.

Bucket Systems
This family of strategies is similar to the two-tier system, but any number of tiers (known as "buckets") can be used (typically the number is based on the size of a cacheline). The difference is that the buckets are not specific to one consideration, but rather the new entry overwrites the entry in the bucket with the lowest depth.

Aging
Aging considers searches of different chess positions during game play. While early implementations and programs relying on root pre-processing to guide search and evaluation were obligated to clear the hash table between root positions, most todays programs do not, to profit from entries of previous searches. Nevertheless, to avoid persistence of old entries which may no longer occur from the current root, aging is used to likely replace those entries by new ones, even if their draft and flags would otherwise protect them. To implement aging, one often stores and compares the current halfmove clock as age, likely modulo some power two constant, depending on how many bits are used to store it inside an entry.

=TT and Parallel Search= A global transposition table, shared by multiple threads or processes is essential for effective parallel search algorithms on modern multi core cpus, and might be accessed lock-less, as proposed by Robert Hyatt and Tim Mann.

=Further Hash Tables= Besides storing the best move and scores of the search trees, further hash tables are often used to cache other features.
 * Evaluation Hash Table
 * Material Hash Table
 * Pawn Hash Table
 * Separate TT for the PV

=Maximizing Transpositions=
 * Enhanced Transposition Cutoff
 * Repetitions

=See also=
 * CPW-Engine_transposition
 * Fifty-move Rule
 * Graph History Interaction
 * Hamming Distance
 * Hash Move
 * Hash Table
 * Huge Pages
 * Integrated Bounds and Values
 * Interior Node Recognizer
 * Iterative Deepening
 * Lasker-Reichhelm Position (Fine #70)
 * Move Ordering
 * Path-Dependency
 * Persistent Hash Table
 * Refutation Table
 * Repetitions
 * Search Instability
 * Shared Hash Table
 * Transposition
 * TT Statistics

=Publications=

1967 ...

 * Richard Greenblatt, Donald Eastlake, Stephen D. Crocker (1967). The Greenblatt Chess Program. Proceedings of the AfiPs Fall Joint Computer Conference, Vol. 31, pp. 801-810. Reprinted (1988) in Computer Chess Compendium, pdf from The Computer History Museum or as pdf or ps from DSpace at MIT

1970 ...

 * Albert Zobrist (1970). A New Hashing Method with Application for Game Playing. Technical Report #88, Computer Science Department, The University of Wisconsin, Madison, WI, USA. Reprinted (1990) in ICCA Journal, Vol. 13, No. 2., pdf
 * David Slate, Larry Atkin (1977). CHESS 4.5 - The Northwestern University Chess Program. Chess Skill in Man and Machine (ed. Peter W. Frey), pp. 82-118. Springer-Verlag, New York, N.Y. 2nd ed. 1983. ISBN 0-387-90815-3. Reprinted (1988) in Computer Chess Compendium » Chess

1980 ...

 * Harry Nelson (1985). Hash Tables in Cray Blitz. ICCA Journal, Vol. 8, No. 1
 * Tony Warnock, Burton Wendroff (1988). Search Tables in Computer Chess. ICCA Journal, Vol. 11, No. 1
 * James Gillogly (1989). Transposition Table Collisions. Workshop on New Directions in Game-Tree Search
 * Harry Nelson (1989). Some Observations about Hash Tables in Cray Blitz. Workshop on New Directions in Game-Tree Search

1990 ...

 * Dennis Breuker, Jos Uiterwijk, Jaap van den Herik (1996). Replacement Schemes and Two-Level Tables. ICCA Journal, Vol. 19, No. 3.
 * Don Beal and Martin C. Smith (1996). Multiple Probes of Transposition Tables. ICCA Journal, Vol. 19, No. 4
 * Dennis Breuker, Jos Uiterwijk, Jaap van den Herik (1997). Information in Transposition Tables. Advances in Computer Chess 8
 * Dennis Breuker (1998). Memory versus Search in Games. Ph.D. Thesis, Universiteit Maastricht, The Netherlands. ISBN 90-9012006-8.

2000 ...

 * John Romein, Henri Bal, Jonathan Schaeffer, Aske Plaat (2002). A Performance Analysis of Transposition-Table-Driven Scheduling in Distributed Search. IEEE Transactions on Parallel and Distributed Systems, Vol. 13, No. 5, pp. 447–459. pdf » Parallel Search
 * Robert Hyatt, Tim Mann (2002). A lock-less transposition table implementation for parallel search chess engines. ICGA Journal, Vol. 25, No. 1
 * Robert Hyatt, Anthony Cozzie (2005). The Effect of Hash Signature Collisions in a Chess Program. ICGA Journal, Vol. 28, No. 3
 * Borko Bošković, Sašo Greiner, Janez Brest, Viljem Žumer (2005). The Representation of Chess Game. Proceedings of the 27th International Conference on Information Technology Interfaces
 * Joel Veness, Alan Blair (2007). Effective Use of Transposition Tables in Stochastic Game Tree Search. IEEE Symposium on Computational Intelligence and Games, pdf
 * Rune Rasmussen, Frédéric Maire, Ross Hayward (2007). A Template Matching Table for Speeding-Up Game-Tree Searches for Hex. 20th Australian Joint Conference on Artificial Intelligence » Hex

2010 ...

 * Timothy Furtak, Michael Buro (2011). Using Payoff-Similarity to Speed Up Search. IJCAI 2011, pdf » Skat
 * Robert Hyatt (2014). A Solution to Short PVs Caused by Exact Hash Matches. ICGA Journal, Vol. 37, No. 3 » Separate TT for the PV

=Forum Posts=

1990 ...

 * Hash tables - Clash!!! What happens next? by Valavan Manohararajah, rec.games.chess, March 14, 1994
 * Hash table question by John Stanback, rec.games.chess, March 23, 1994

1995 ...
1996 1997
 * hash mem in win-chess progs by Pc Sol, rgcc, September 28, 1995 » Windows
 * Transposition table by Matthew Bengtson, rgcc, January 25, 1996
 * Collision probability by Dennis Breuker, rgcc, April 15, 1996
 * Alpha-Beta window in transposition tables? by Marty Bochane, rgcc, October 25, 1996
 * Re: Berliner vs. Botvinnik Some interesting points, post 8 by Bradley C. Kuszmaul, rgcc, November 06, 1996 » Transposition Table in Mac Hack
 * hash table fail high/fail low problem by Robert Hyatt, rgcc, November 26, 1996
 * Hash functions for use with a transition table by Tim Foden, rgcc, March 5, 1997 » Transposition Table
 * Re: Hash functions for use with a transition table by Ronald de Man, rgcc, March 7, 1997 » BCH Hashing

1998
 * Handling of repetition (draw) in transposition table by Bjarke Dahl Ebert, rgcc, June 9, 1997 » Repetitions
 * double hash tables by Chris Whittington, CCC, October 01, 1997
 * Hash collisions (was Re: Let's analyze move 36) by Daniel Homan, CCC, October 08, 1997 » Collisions
 * Using values from transition tables by Tim Foden, rgcc, October 13, 1997
 * Fast hash algorithm by John Scalo, CCC, January 08, 1998
 * Fast hash key method - Revisited! by John Scalo, CCC, January 14, 1998
 * Hash tables and data-cache, some programmer stuff... by Ed Schröder, CCC, January 17, 1998
 * Hash Table Size Versus Performance by Steven Juchnowski, CCC, May 30, 1998
 * Selective deepening and Hashtables by Werner Inmann, CCC, June 30, 1998
 * Using too-shallow mate scores from the hash table by David Eppstein, CCC, July 05, 1998 » Mate Scores
 * Re: Using too-shallow mate scores from the hash table by David Eppstein, CCC, July 06, 1998
 * Re: Using too-shallow mate scores from the hash table by Don Dailey, CCC, July 07, 1998

1999
 * draw by repetition and hash tables by Werner Inmann, CCC, July 24, 1998
 * Hash collisions and a little maths by Rémi Coulom, CCC, February 18, 1999
 * Hash Collisions by KarinsDad, CCC, February 21, 1999
 * Problem with draws by rep and hash table by James Robertson, CCC, October 07, 1999

2000 ...
2001 2002 2003 2004
 * Can we use hash table at root? by Tim, CCC, January 31, 2000 » Root
 * Re: Atomic write of 64 bits by Frans Morsch, comp.lang.asm.x86, September 25, 2000 » MMX, Parallel Search
 * hashing function by Vladimir R. Medvedev, rgcc, October 17, 2000
 * Hash table efficiency by Miguel A. Ballicora, CCC, December 05, 2000 » Gaviota, Search Statistics
 * Why not store both a lower and an upper bound in a hashtable? by Leen Ammeraal, CCC, December 06, 2000
 * A fast hash -- assuming you are not planning to do incremental updates by Dann Corbit, CCC, June 15, 2001
 * "Don't trust draw score" <=Is it true? by Teerapong Tovirat, CCC, August 08, 2001 » Repetitions, Path-Dependency
 * About random numbers and hashing by Severi Salminen, CCC, December 04, 2001
 * Re: About random numbers and hashing by Sven Reichard, CCC, December 05, 2001
 * When to RecordHash? by Sune Fischer, CCC, December 25, 2001
 * Non power of two hash table sizes by Alvaro Jose Povoa Cardoso, CCC, February 18, 2002
 * Detecting Draws using a Small Hash Table? by Steve Maughan, CCC, February 20, 2002
 * hash entry replacement schemes by Scott Farrell, CCC, September 14, 2002
 * hash numbers requested: authors please read by James Swafford, CCC, September 17, 2002
 * How important is a big hash table? Measurements... by Tom Kerrigan, CCC, March 29, 2003
 * Another memory latency test by Dieter Bürßner, CCC, July 17, 2003
 * Replacement shemes by Sune Fischer, CCC, July 27, 2003
 * Is it necessary to include empty fields in the hash key of a position? by Frank Hablizel, rgcc, December 25, 2003
 * I need your opinion about this hash entry structure by Federico Corigliano, CCC, January 02, 2004
 * Fruit - Question for Fabien by Dan Honeycutt, CCC, March 11, 2004 » Fruit, Node Types, Principal Variation, Principal Variation Search
 * Hashkey collisions (typical numbers) by Renze Steenhuisen, CCC, April 07, 2004 » Transposition Table - Collisions
 * A move to search 2nd... Keep in the trans table? by Eric Oldre, CCC, July 21, 2004
 * Transposition table replacement by Eric Oldre, CCC, July 29, 2004

2005 ...

 * Side effects of transposition tables by Alessandro Scotti, Winboard Forum, March 04, 2005
 * Hashing double bounds by Rasjid Chan, CCC, June 20, 2005
 * Mate scores in the hash table by Eduardo Waghabi, Winboard Forum, September 02, 2005 » Mate Scores
 * Re: Mate scores in the hash table by Bruce Moreland, Winboard Forum, September 04, 2005
 * Re: Mate scores in the hash table by Josué Forte, Winboard Forum, September 04, 2005

2007 2008 2009
 * Hash table hit rate by Stef Luijten, Winboard Forum, November 16, 2005
 * Transposition Tables by Colin, CCC, July 15, 2007
 * fail high handling with tranposition tables by Uri Blass, CCC, August 01, 2007
 * Problem with Transposition Table and Repitition-Draw by Oliver Brausch, CCC, January 11, 2008 » Repetitions
 * Transposition Table and nps drop by Mathieu Pagé, CCC, February 27, 2008 » Nodes per Second
 * Semi-Path Dependent Hashing: a semi-useless idea by Zach Wegner, CCC, May 24, 2008 » Path-Dependency
 * Is 64 bits enough as a hash length by Mathieu Pagé, CCC, July 10, 2008
 * 31 bit hash values. How often will it fail? by Carey, CCC, September 07, 2008 » Transposition Table - Collisions
 * Adaptative LMR and TT by Fermin Serrano, CCC, December 23, 2008 » Late Move Reductions
 * transposition table details by Don Dailey, CCC, May 26, 2009
 * Transposition Table by Pablo Vazquez, CCC, June 26, 2009
 * Cache pollution when reading/writing hash table by Marco Costalba, CCC, August 09, 2009
 * Transposition table pruning by Luca Hemmerich, CCC, November 25, 2009

2010 ...
2011 2012 2013 2014
 * Correcting Evaluation with the hash table by Mark Lefler, CCC, February 05, 2010 » Evaluation
 * TT hit/miss rates by Vlad Stamate, CCC, February 25, 2010
 * Zero window TT entry by Vlad Stamate, CCC, March 05, 2010 » Null Window
 * Null-move pruning and the hash table by Robert Purves, CCC, March 28, 2010 » Null Move Pruning
 * Crafty Transpostion Table Question by Eric Stock, CCC, May 30, 2010 » Crafty, Lockless Hashing
 * TT behavior by Karlo Bala Jr., CCC, June 25, 2011
 * working! by Robert Hyatt, CCC, September 17, 2010 » Separate TT for the PV
 * Is a querying the hash tables such a huge bottleneck? by Oliver Uwira, CCC, October 28, 2010
 * Puzzle with mate scores in TT by Robert Purves, CCC, December 10, 2010 » Mate Scores
 * Transposition Table: addressing slots by Michael Hoffmann, CCC, March 18, 2011
 * Transposition Table updates in Stockfish by Onno Garms, CCC, April 12, 2011 » Stockfish
 * Hashing the PV by Stef Luijten, CCC, June 06, 2011
 * Dual Bound TT by João Guerra, CCC, July 28, 2011
 * TT Key Collisions, Workarounds? by Clemens Pruell, CCC, August 16, 2011
 * Repetitions/50 moves and TT by Sergei Markoff, CCC, September 13, 2011 » Fifty-move Rule
 * Determining the age of TT buckets by Julien Marcel, CCC, Sepember 18, 2011
 * Key collision handling by Jonatan Pettersson, CCC, October 21, 2011
 * Mate score in TT by Marco Costalba, CCC, December 28, 2011 » Mate Scores
 * Question on TT by Giuseppe Cannella, Winboard Forum, January 06, 2012
 * Stockfish hash implementation by Jon Dart, CCC, January 10, 2012 » Stockfish
 * cache alignment of tt by Daniel Shawul, CCC, March 11, 2012
 * Hash table division by Ed Schröder, CCC, April 05, 2012
 * TT question? by Fermin Serrano, CCC, May 19, 2012 » Quiescence Search
 * hashtables by Daniel Anulliero, Winboard Forum, May 22, 2012
 * how to measure frequency of hash collisions by Daniel Shawul, CCC, June 16, 2012
 * TT avoid nullmove flag by Matthew R. Brades, CCC, August 02, 2012 » Null Move Pruning
 * Transposition Table - Replacement and PV by Cheney Nattress, CCC, December 04, 2012
 * Speaking of the hash table by Ed Schröder, CCC, December 09, 2012
 * Transposition table usage in quiescent search? by Jerry Donald, CCC, March 01, 2013 » Quiescence Search
 * Transposition driven scheduling by Daniel Shawul, CCC, April 04, 2013
 * The effect of larger hash size by Jacob Børs Lind, CCC, April 13, 2013
 * Effect of the Hash Table size by Kai Laskos, CCC, April 28, 2013
 * Hash cutoffs and analysis by Harm Geert Muller, CCC, June 17, 2013
 * Hashing based on move lists by Matthew R. Brades, CCC, June 30, 2013
 * transposition tables by Folkert van Heusden, CCC, July 23, 2013
 * ep and castle rights hashing by Natale Galioto, CCC, September 15, 2013 » Castling Rights, En passant, Repetitions
 * Hash tables: one bound or two bounds? by Alberto Sanjuan, CCC, December 11, 2013
 * LMR & TT interaction by Natale Galioto, CCC, April 29, 2014 » Late Move Reductions
 * transposition and multithreading question by Marco Belli, CCC, May 04, 2014 » Parallel Search
 * transposition table implementation help by lazyguy123, OpenChess Forum, August 09, 2014
 * Transposition table chaining and replacement strategy by Alex Ferguson, CCC, August 28, 2014
 * Speculative prefetch by Peter Österlund, CCC, September 27, 2014 » Memory
 * Transposition Table Oddity by Thomas Kolarik, CCC, September 28, 2014
 * Can someone explain what advantage there is to... by Forrest Hoch, CCC, October 06, 2014
 * Cache over-writing and PV's by Forrest Hoch, CCC, October 16, 2014 » Principal Variation
 * Mate score from the transposition table by Evert Glebbeek, CCC, October 25, 2014 » Mate Scores
 * Two hash functions for distributed transposition table by Daniel Shawul, CCC, December 16, 2014
 * Transposition table in Q-search by Alex Ferguson, CCC, December 26, 2014 » Quiescence Search

2015 ...
2016 2017 2018 2019
 * Legality Check on TT move by Jordan Bray, CCC, January 11, 2015
 * (iteration+depth) TT replacement policy by Aleks Peshkov, CCC, February 27, 2015
 * Idea #9228: Clearing stale transtable entries by Steven Edwards, CCC, April 06, 2015
 * TTable questions by Henk van den Belt, CCC, April 11, 2015
 * Fractional plies and transposition tables by Alexandru Mosoi, CCC, April 18, 2015 » Depth - Fractional Plies
 * Re: Worst advice by Alexandru Mosoi, CCC, August 10, 2015
 * most similar hashes of two positions by Alexandru Mosoi, CCC, August 12, 2015
 * The cost of transposition table instrumentation by Steven Edwards, CCC, August 23, 2015
 * atomic TT by lucasart, CCC, September 13, 2015
 * Transposition table test positons by Robert Pope, CCC, September 11, 2015 » Test-Positions
 * Hash cache by Harm Geert Muller, CCC, October 12, 2015
 * Perft and hash with legal move generator by Peterpan, OpenChess Forum, November 12, 2015 » Perft
 * Transposition table utilizattion by Shawn Chidester, CCC, December 29, 2015
 * On-the fly hash key generation? by Evert Glebbeek, CCC, January 12, 2016
 * Transposition Age Tutorial by Dennis Sceviour, CCC, January 25, 2016
 * TT Mate scores by rgoomes, OpenChess Forum, February 10, 2016 » Mate Scores
 * Hashing question by Kenneth Jones, CCC, February 23, 2016
 * TT and Iterative Deepening by kuket15, OpenChess Forum, February 26, 2016 » Iterative Deepening
 * Transposition table replacement strategies by Shawn Chidester, CCC, March 01, 2016
 * about hashtable by Daniel Anulliero, CCC, March 04, 2016
 * Hashing in Qsearch? by Martin Fierz, CCC, April 03, 2016 » Quiescence Search
 * Shifting alpha/beta on hash hit by Martin Fierz, CCC, April 14, 2016 » Search Instability
 * What happens during a hash collision in say Stockfish ... by Isaac Haïk Dunn, CCC, April 24, 2016
 * Hashtable aging by Martin Fierz, CCC, April 25, 2016
 * Transposition Table replacement schemes by Andrew Grant, CCC, May 05, 2016
 * lockless hashing by Lucas Braesch, CCC, May 10, 2016 » Shared Hash Table
 * Suggestion for hash tables and analysis by J. Wesley Cleveland, May 22, 2016
 * Hashtable and 50 move rule draws by Stan Arts, CCC, May 24, 2016 » Fifty-move Rule
 * Transposition Table - Updating entries by Andrew Grant, CCC, June 24, 2016
 * Transposition Table Usage by theturk1234, OpenChess Forum, July 12, 2016
 * Aspiration window with TT question by sandermvdb, OpenChess Forum, August 01, 2016 » Aspiration Windows
 * Storing both alpha and beta scores in TT by Thomas Dybdahl Ahle, CCC, August 02, 2016
 * Using side to move as a separate bit in hash key by J. Wesley Cleveland, CCC, August 06, 2016 » Side to move
 * Question about TT by stefano.c, FishCooking, August 21, 2016
 * transposition table pseudocode by Erin Dame, CCC, August 22, 2016
 * transposition tables and three-fold repetition by Erin Dame, CCC, September 10, 2016 » Repetitions
 * Transposition tables in Cray Blitz by Erin Dame, CCC, September 26, 2016 » Cray Blitz
 * Hash table transformation by Dann Corbit, CCC, October 05, 2016
 * Modify hash probing code to pvs by Fernando Tenorio, CCC, December 05, 2016
 * What is wrong with doing Nulls & ttcuts in a pv node ? by Mahmoud Uthman, CCC, January 21, 2017 » Null Move Pruning, PV-Nodes
 * Interesting hash-replacement scheme by Harm Geert Muller, CCC, April 08, 2017
 * Problems with TT, sometimes makes blunder moves by Tony Soares, CCC, April 12, 2017
 * TT Table replacement strategy by kickstone, OpenChess Forum, April 29, 2017
 * Hash Tables Deep Blue by Gustavo Mallada, CCC, May 18, 2017 » Deep Blue
 * Hash table by Daniel José Queraltó, CCC, June 12, 2017
 * Improving hash replacing schema for analysis mode by Daniel José Queraltó, CCC, July 05, 2017 » Replacement Strategy, Persistent Hash Table
 * TT aging by Marco Pampaloni, CCC, July 09, 2017
 * Equidistributed draft by Alvaro Cardoso, CCC, July 14, 2017
 * 2-level TT by Rein Halbersma, CCC, July 14, 2017
 * cutechess-cli: not restarting an engine because of tt by Folkert van Heusden, CCC, July 22, 2017 » Cutechess-cli
 * Size of Transposition Table Entry by Jason Fernandez, CCC, September 29, 2017
 * TT in Qsearch by Laurie Tunnicliffe, CCC, December 05, 2017 » Quiescence Search
 * Transposition table based pruning idea by Jerry Donald, CCC, December 25, 2017 » Pruning
 * hashing in chess4j by James Swafford, CCC, December 30, 2017 » chess4j
 * Marcel Duchamp endgame "splits" engines / hash phenomenon by Kenneth Regan, CCC, February 19, 2018 » Marcel Duchamp
 * Do You Track Hash Table Efficiency? by OneMoreTime, OpenChess Forum, April 07, 2018
 * Mate scores in TT (a new?... approach) by Vince Sempronio, CCC, April 09, 2018
 * Yet another Mate scores in TT thread by Andrew Grant, CCC, April 12, 2018 » Checkmate, Score
 * Draw scores in TT by Srdja Matovic, CCC, April 14, 2018 » Draw, Draw Score
 * Depth extensions and effect on transposition queries by Kenneth Jones, CCC, April 16, 2018 » Extensions, Check Extensions
 * Playing transposition table moves in the Quiescence search by Andrew Grant, CCC, January 17, 2019 » Quiescence Search
 * Prefetch and Threading by Dennis Sceviour, CCC, April 25, 2019 » Memory, Thread
 * Hash collision? by Tom King, CCC, June 05, 2019
 * Looking for TT policy advice by Vivien Clauzon, CCC, October 04, 2019 » Replacement Strategy
 * Probing the transposition table at depth 0 by Gonzalo Arro, CCC, November 29, 2019

=External Links=
 * The Main Transposition Table from Bruce Moreland's Programming Topics
 * Transposition table from Wikipedia
 * Zobrist hashing from Wikipedia
 * Gödel numbering from Wikipedia
 * Computer Chess - Hash Collisions in Chess Engines, and What They May Mean... by Kenneth W. Regan
 * Hashing in chess4j by James Swafford, December 30, 2017» chess4j
 * Tal Wilkenfeld - Table For One, YouTube Video

=References= Up one Level