Late Move Reductions

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Late Move Reductions (LMR), or its version known as History Pruning and History Reductions, save search by reducing moves that are ordered closer to the end of likely fail-low nodes. Typically, most schemes search the first few moves (say 3-4) at full depth, then if no move fails high, many of the remaining moves are reduced in search depth. The technique has been used for many years in various forms, but it became very popular in 2005 after Fruit and Glaurung used open source implementations based on the History Heuristic. LMR can often reduce the effective branching factor to less than 2, depending on the reduction conditions.

=Common Conditions= Most programs do not reduce these types of moves:
 * Tactical Moves (captures and promotions)
 * Moves while in check
 * Moves which give check
 * Moves that cause a search extension
 * Anytime in a PV-Node in a PVS search
 * Depth < 3 (sometimes depth < 2)

=Less Common Conditions= Less common conditions on moves not to reduce:
 * Passed Pawn Moves
 * Killer Moves
 * Moves threatening the King area
 * Tactically threatening moves
 * Moves with good past relative history
 * Any Pawn Moves
 * Allowing reductions of "bad" captures (SEE < 0)
 * Moves of a threatened piece to safety (often detected via a Null Move search)

=Reduction Depth= Classical implementation reduces by one ply only. Yet modern programs, most notably Stockfish, allow reductions of more than one ply and increase them for later moves. Reduction depth changes according to expected node type (being typically smaller in pv nodes), depth and move number. Here some sample formulas can be viewed:
 * Senpai reduces by one ply for the first 6 moves and by depth / 3 for remaining moves.
 * Fruit Reloaded uses formula: uint8( sqrt(double(depth-1)) + sqrt(double(moves-1))); for non-PV nodes. In PV-nodes it reduces by 2/3 of this value.

=Re-searches= Classical implementation assumes a re-search at full depth if the reduced depth search returns a score above alpha.

=Test Results= Some test results related to LMR can be found on
 * Late Move Reduction Test Results

=See also=
 * Parallelism and Selectivity in Game Tree Search | Video, Talk by Tord Romstad
 * Bobby's Strategic Quiescence Search
 * History Heuristic
 * History Leaf Pruning
 * Move Count Based Pruning (Late Move Pruning)
 * Null Move Pruning
 * Relative History Heuristic
 * SEX Algorithm

=Publications=
 * David Levy, David Broughton, Mark Taylor (1989). The SEX Algorithm in Computer Chess. ICCA Journal, Vol. 12, No. 1
 * Daniel S. Abdi (2013). Analysis of pruned minimax trees. pdf » Alpha-Beta, Null Move Pruning

=Forum Posts=

2004

 * Forward pruning and some related techniques by Sergei Markoff, CCC, March 02, 2004

2005 ...
2006
 * Reductions and null move refutations by Tord Romstad, Winboard Forum, April 18, 2005 » Null Move Pruning
 * What is History Pruning? by David Dahlem, CCC, July 03, 2005
 * History based pruning question by Alvaro Jose Povoa Cardoso, CCC, August 26, 2005
 * About history pruning... by Svein Bjørnar Myrvang, CCC, October 26, 2005
 * What is "history pruning"? by Vladimir Medvedev, Winboard Forum, November 07, 2005
 * History pruning by Frank Phillips, CCC, February 27, 2006
 * late move reductions by Robert Hyatt, CCC, March 01, 2006
 * Re: late move reductions by Alessandro Scotti, CCC, March 01, 2006 » Kiwi
 * PHR (Peak History Reduction) idea by Daniel Mehrmann, CCC, March 01, 2006 » Home, Relative History Heuristic

2007 2008 2009
 * history pruning/ late move pruning by Tom King, Winboard Programming Forum, March 02, 2006
 * LMR question by Uri Blass, Winboard Programming Forum, July 13, 2006
 * LMR in micro-Max by Harm Geert Muller, CCC, April 07, 2007 » Micro-Max
 * Fruit and History Reductions by Ed Schröder, CCC, July 19, 2007 » Fruit
 * LMR other conditions by Mark Lefler, CCC, July 23, 2007
 * Improving history tables by Michael Sherwin, CCC, July 25, 2007
 * LMR: history or not? by Alessandro Scotti, CCC, December 13, 2007 » Hamsters
 * Extreme late move reductions? by Gary, CCC, March 05, 2008
 * LMR? by Martin Giepmans, CCC, April 12, 2008 » Anatoli
 * Adaptative LMR and TT by Fermin Serrano, CCC, December 23, 2008
 * About LMR & History reductions by Joona Kiiski, CCC, February 24, 2009
 * LMR by Robert Hyatt, CCC, March 05, 2009
 * LMR and 'tactically connected' moves by Harm Geert Muller, CCC, March 10, 2009
 * LMR revisited by Robert Hyatt, CCC, April 01, 2009
 * LMR and null move selectivity by Don Dailey, CCC, April 20, 2009
 * LMR by Bas Hamstra, CCC, September 30, 2009
 * May I Ask What You Think Of This? by Christopher Conkie, CCC, October 20, 2009
 * Did people try replacing LMR by pruning by Uri Blass, CCC, December 31, 2009

2010 ...
2011 2012 2013 2014
 * The problem with LMR in suprtactical positions by Oliver Brausch, CCC, January 05, 2010
 * Researching if LMR-affected search improves Alpha? by John Merlino, CCC, January 22, 2010
 * Problems with LMR in late endgames by Luca Hemmerich, CCC, March 01, 2010
 * LMR algorithm by kenny stanley, CCC, March 15, 2010
 * LMR Conditions by kenny stanley, CCC, March 23, 2010
 * LMR at root of search tree - worthwhile? by Tom King, CCC, May 21, 2010
 * EPR, even better than LMR! by Michael Sherwin, CCC, May 27, 2010
 * Re: "Automated Discovery of Search Extensions" by Ed Schröder, OpenChess Forum, June 26, 2010
 * lmr at PV nodes? by Tom King, CCC, July 23, 2010
 * move count based pruning by Tom King, CCC, September 02, 2010
 * LMR differences in long vs short games by Jacob Børs Lind, CCC, November 08, 2010
 * fulitiy + lmr; funny results by Volker Böhm, CCC, December 28, 2010
 * lmr in PV by Larry Kaufman, CCC, January 03, 2011
 * Should reduction depend on depth? by Larry Kaufman, CCC, January 14, 2012 » Komodo
 * Possible LMR improvement using AB_FOOL by Ed Schroder, CCC, April 24, 2012
 * LMR Research by Matthew R. Brades, CCC, July 02, 2012
 * Adjustable search pruning depending on time control by Jerry Donald, CCC, December 20, 2012 » Time Management
 * ROC analysis of Late Move Reductions by Gerard van Ewijk, CCC, March 22, 2013
 * Is LMR Sound by Henk van den Belt, CCC, May 29, 2013
 * Is LMR safe within NULL move reduction by Henk van den Belt, CCC, May 30, 2013
 * LMR at CUT nodes can be arbitrarily bad! by Michel Van den Bergh, CCC, June 20, 2013 » Node Types, Python
 * How much to reduce ? by Henk van den Belt, CCC, October 03, 2013 » R
 * Null move and LMR by Laurie Tunnicliffe, CCC, March 12, 2014 » Null Move Pruning
 * LMR & TT interaction by Natale Galioto, CCC, April 29, 2014 » Transposition Table
 * A question about Stockfish and LMR or other pruning... by Forrest Hoch, CCC, November 20, 2014

2015 ...
2016 2017
 * About LMR, again :-) by Daniel Anulliero, CCC, March 01, 2015
 * LMR tuning by Shawn Chidester, CCC, May 11, 2015
 * Interpretation of LMR by Alexandru Mosoi, CCC, May 29, 2015
 * What's Your Engine's Maximum LMR Reduction? by Steve Maughan, CCC, June 08, 2015
 * New "smoothing" issue by Robert Hyatt, CCC, July 20, 2015
 * On LMR, and statically predicting moves by Matthew Lai, CCC, August 25, 2015
 * LMR by another name by Steven Edwards, CCC, September 02, 2015 » Spector
 * Ratio reduction by Steven Edwards, CCC, September 20, 2015 » Symbolic
 * late move reduction by Folkert van Heusden, CCC, January 21, 2016
 * Extensions in the days of LMR? by Martin Fierz, CCC, March 22, 2016 » Extensions
 * LMR problems by Alvaro Cardoso, CCC, May 16, 2016
 * Reductions by Harm Geert Muller, CCC, May 22, 2016 » R
 * Disappointing LMR Results by David Cimbalista, CCC, August 04, 2016
 * Null Move in LMR by Laurie Tunnicliffe, CCC, August 10, 2016 » Null Move Pruning
 * LMR and PVS by thevinenator, OpenChess Forum, February 10, 2017 » Principal Variation Search
 * Late move reductions ? by Mahmoud Uthman, CCC, March 22, 2017
 * Check extension vs LMR by Harm Geert Muller, CCC, April 04, 2017 » Check Extensions
 * LMR - or starters - Advance and Expert by Ed Schroder, CCC, May 01, 2017
 * LMR and killer by Harm Geert Muller, CCC, September 14, 2017 » Killer Heuristic
 * LMR prescription by Evert Glebbeek, CCC, September 24, 2017

=External Links=
 * Late Move Reductions from Wikipedia
 * An Introduction to Late Move Reductions by Tord Romstad (dead link)
 * Mediocre Chess: [Guide Late move reduction (LMR)] by Jonatan Pettersson, March 26, 2007 » Mediocre
 * History Reductions in Pro Deo by Ed Schröder » Pro Deo
 * LMR advanced from Rebel 13 by Ed Schroder » Rebel

=References=

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