Parallel Programming in Chess Simulations Part 2 Tyler Patton

Discussion: Chess Engine Basics Everything Parallel What Next?

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

Basics : StockFish implementation Minimax Alpha-Beta pruning Bitboards Late Move Reductions Large transposition table

Basics : Minimax search Concept Maximize the evaluation of your move while minimizing opponents move evaluation Reviews each possible move sequence In chess, Minimax has a high time cost since every move sequence is evaluated regardless of the move

Basics : Alpha-Beta Pruning Allows for eliminations of branches Alpha: Our best move so far in the current path Beta: Our Opponent’s best move so far in the current path If the current node has a better beta value we prune the branch (the best beta value is minimum)

Basics : Time complexity Minimax Search: O(b m ) Where b is branch factor and m is move depth For chess, b ≈ 35 and m ≈ 100 Alpha Beta Pruning: O(b m/2 ) Doubles search depth from minimax

Basics : Late Move Reduction (LMR) Alpha-Beta produces an ordered list of effective moves to search Moves toward the end of the list are unlikely to produce values that increase alpha LMR does a reduced depth search on the late move and checks for an increase versus alpha If the score is greater than or equal to alpha we know nothing and complete a full depth search If the score is less than alpha we prune this node

Basics : 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 : Parallelizing Alpha-Beta pruning Goal: Use multiple processors to simultaneously search different branches of the game tree Drawback: Dependency on the alpha value Parallel algorithms tend to be less efficient since the alpha value is not as strong If the best alpha value is the first branch searched then the parallel algorithm has equal iterations to the sequential algorithm Processors are dependent on each other for updated alpha values which cause communication locks

Parallelization : Principal Variation Splitting (PVS) 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 Processors work under the same node

Parallelization : Enhanced Principal Variation Splitting (EPVS) Simple improvement of PVS When a processor runs out of work: Stop all processors at ply P Evaluate the branches 2 ply Split the processors among the tree like PVS Interacts with transposition table further calculations are not redundant Increased communication overhead

Parallelization : Dynamic Tree Splitting (DTS) Assumptions: Shared memory Communication cost = 0 (Cray C916/1024 computer) Steps: One processor searches from ply = 1 to N Each other processer begins processing nodes as in PVS If a processor has no work to do it broadcasts to help and joins another processor with work to share A split position is chosen and the processors divide the node Evaluations and splitting are looped until the node is complete

Parallelization : Speedup Comparisons PVS: EPVS: DTS:

What Next? Some things we didn’t have time for Bitboards Data structure for storing chess positions Younger Brother Wait Concept Master-slave approach; similar to DTS GPU implementations i.e. CPU generates the tree then GPU evaluates Neural Networks Simulation technique which may allow more processors

What Next? Looking to the future The best algorithm for large numbers of processors and indefinite tree size is unknown 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 Explore new algorithms that don’t rely on communication pitfalls or tree structure

Questions?

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