Parallel Programming in Chess Simulations Tyler Patton

Discussion: Background Sequential Optimizations Parallelization of chess

Background: What is Chess? Strategic 2 player game 64 tiles 16 pieces per player Objective to capture the opponents king

Background: Chess ELO Created by Arpad Elo to improve chess ratings Players of equal ELO have an equal chance to win A difference of 400 ELO gives a win 97% of the time to the higher rated player Basic formula: Performance Rating = Opponents’ Total Ratings + 400*(wins-losses) Number of games Average Player rating: 1200 Highest player rating: 2850 (Magnus Carlson)

Background: Scope First estimate of the number of positions: 64! / 32!*(8!) 2 *(2!) 6 =10 43 (Shannon) Tight upper bound: =10 50 (Dr. Allis) Number of possible game variations: (Shannon number) Given ~10 3 starting moves and 40 move pair average

Background: History of chess engines 1950: Alan Turing develops Turbochamp. 1962: Adam Kotok of MIT develops first “credible” chess program. 1997: Deep blue defeats Gary Kasparov in 6 games Present: Stockfish 6 holds a rating of 3309

Deep blue vs Kasparov

Sequential Optimizations: StockFish StockFish implementation: Alpha-Beta pruning Bitboards Transposition table Late move reductions

Sequential Optimizations: Minimax search Minimax: Evaluate a given move, the opponents responses and your responses to your opponent’s moves… etc. Proceed to the next move and repeat. Choose the tree which yields the best ending evaluation Assume the opponent always chooses the best move

Sequential Optimizations: Alpha-Beta Pruning Alpha is the maximum score the player is guaranteed for a branch Beta is the minimum score the opponent is guaranteed for a branch Allows for eliminations of branches in the search tree Eliminates branches if the opponent would never allow the position Ordered node complexity: O(b d/2 ) Random order complexity: O(b 3d/4 )

Sequential Optimizations: Alpha-Beta Pruning

Sequential Optimizations: Transposition Table Stores the history of search evaluations Positions that been searched are likely to be reached again Before a branch is searched the transpositions table is checked and gives the result if able Implemented as a hash table

Parallelization of Chess: Parallelizing Alpha-Beta pruning Goal: Use multiple processors to simultaneously search different branches of the game tree Drawback: Dependency on the alpha value Processors are dependent on each other for updated alpha values which cause communication locks Parallel algorithms tend to be less efficient since the alpha value is not as strong Parallel implementation only has equivalent efficiency to the sequential algorithm if the first move if the best one examined

Parallelization of Chess: Principal Variation Splitting Early technique for parallelizing alpha-beta Assumptions: The game tree is well ordered The leftmost path is the best Updates alpha after a branch is searched

Parallelization of Chess: Typical Speedup Principal Variation Splitting Implemented using 4 or fewer Processors with a speedup of ~3.5 The Younger Brothers Wait Concept achieved a speedup of 140 on 256 nodes CilkChess parallel engine is shown to be scalable up to 1000 nodes with a similar speedup to The Younger Brothers Wait Concept

Parallelization of Chess: Looking to the future The best algorithm for large numbers of processors and indefinite tree size is unknown Explore new algorithms that don’t rely on communication pitfalls Optimizations to existing algorithms and techniques are still possible i.e. making the new alpha available to each processor when its found as opposed to when a processor finishes a search

Questions?