Randomized ParallelProof-Number SearchACG 12, Pamplona, May 2009

Randomized ParallelProof-Number SearchJahn-T. Saito, Mark Winands, H. Jaap van den Herik

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Motivation

• 10 years ago Kishimoto and Kotani parallelized PDS for distributed memory machines.

• RQHow can standard PNS and PN2 be parallelized for today's common computers?

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Overview

• Proof-Number Search (PNS)• Parallel Search• Randomized Parallel PNS • Experiment• Conclusion and future work

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Proof-Number Search (PNS)• Binary goal: (dis)prove the root node

– E.g., to be a win for the player to move

• Solving positions or whole games: Shogi, LOA, Othello, Fanorona

• Best-First Search (BFS)• Heuristic selection: most-proving node

i. Proof-Number Search (PNS)

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i. Proof-Number Search (PNS)

Proof number and Disproof Number

• Proof number (pn): the minimum number of leaf nodes which have to be proved in order to prove the node

• Disproof number (dn): the minimum number of leaf nodes which have to be disproved in order to disprove the node

• Proof number and disproof number for each node• Assume one expansion for each unexpanded node

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PNS Example

a

b c

ih lk

ed gf

12

12

11

11

12

11

draw ??

loss ?

j

?

11

win

0∞

0∞ 0

∞

0∞

0∞

i. Proof-Number Search (PNS)

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i. Proof-Number Search – PN2

PN2

2-level PN search

Bounded

The sub trees of the children are deleted when the second-level search stops.

PN1

PN2

Second level stops when a certain limit of nodes in memory is reached or its root is (dis) proved.

Leaf node in the first level. Root node in the second level.

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Overview

• Proof-Number Search (PNS)• Parallel Search• Randomized Parallel PNS • Experiment• Conclusion and future work

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Search Overhead

Brockington and Schaeffer, 1997

• Search overhead• Synchronization overhead• Communication overhead

ii. Parallel Search

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ii. Parallel Search

Randomized Best-First Search (RBFS)

• Shoham & Toledo, 2002• Parallelize any Best-First Search (BFS)• Multiple threads operate on same tree• Best-first evaluation to set probability for

selecting a suboptimal child

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Overview

• Proof-Number Search (PNS)• Parallel Search• Randomized Parallel PNS • Experiment• Conclusion and future work

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Multiple threads, one tree

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How do we implement RP for PNS?

• RP-PNS / RP-PN2

• Two kinds of threads:– One PV thread: most-proving heuristic– Alternatives threads: slightly deviate from

most-proving heuristic

iii. Randomized Parallel PNS

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iii. Randomized Parallel PNS

How do the alternative threads deviate from the PV?

• Randomly deviate from PV once to N2 or N3.

• Randomly choose from best n sibbling nodes N1,…,Nn where bsn(Ni) < bsn(N1) + some constant D.

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Randomization

iii. Randomized Parallel PNS

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RP-PNS vs RBFS

iii. Randomized Parallel PNS

Equal chances for n best siblings

Deviate proportional to the BF heuristic

PV and alternative threads

All threads are equal

Proof Number SearchAny Best-First Search

RP-PNSRBFS

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Overview

• Proof-Number Search (PNS)• Parallel Search• Randomized Parallel PNS • Experiment• Conclusion and future work

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Experimental setup

• LOA test set – 143 positions– Average of 4.28 million nodes with PNS

• Two series of experiments– RP-PNS with 1, 2, 4, and 8 threads– RP-PN2 with 1, 2, 4, and 8 threads

iv. Experiment

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LOA

iv. Experiment

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Implementation

• lock per node• locking policy• PN2: limit PN2 tree to S3/4/T nodes

– T is #threads– S is size of PN1 tree– Compromise between speed and space

• C/C++ (pthreads) on 2.66 GHz Xeon

iv. Experiment

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Implementation

iv. Experiment

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Results

iv. Experiment

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Observations

iv. Experiment

• Speed-up for RP-PNS and RP-PN2

• Best RP-PN2 is faster than PNS and uses less memory• RP-PN2 does hardly increase the maximal memory consumption • RP-PN2 scales better than RP-PNS

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Explanations for (3) and (4)

iv. Experiment

• Best RP-PN2 designed to use less memory than serial PNS by size of PN2 trees • RP-PN2 scales better: partially because the size of PN2 depends on T

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Overhead: 8 vs 1 threads

iv. Experiment

RP-PNS ca. 33% more expansions (i.e., search overhead), rest overhead is synchronization overhead.

RP-PN2

ca. 26% more expansions (i.e., search overhead) rest overhead is search overhead

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Overview

• Proof-Number Search (PNS)• Parallel Search• Randomized Parallel PNS • Experiment• Conclusion and future work

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Conclusion

v. Conclusion

RP-PNS viable for parallelizing PNS and PN2 on shared memory architectures

Figures seem comparable to ParaPDS (Kishimoto and Kotani 1999) but performance is not:– Different test sets– Different algorithms – Hash tables in ParaPDS

• Offer a new way to balance time and memory consumption of PNS

RP-PNS and RP-PN2 scale reasonably, but can still be improved

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Future Work

• Smart distribution of parallelization between PN1 and PN2 for PN2 on more processors

• Pooling locks for the implementation• Better distribution function• K-best moves• Apply in small Go boards

v. Future Work

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Thank you.