Massively Parallel Proof-Number Search for Impartial Games and Beyond

2025-11-13

AI Summary

Existing parallel Proof-Number Search (PNS) algorithms exhibit poor scalability on multicore systems, failing to harness the potential of large-scale CPU clusters. To address this, we propose the first scalable PNS algorithm capable of efficient execution on thousand-core systems. Our approach introduces a two-level parallel architecture that integrates lightweight inter-worker information sharing with Grundy-number-based pruning optimization. This marks the first realization of highly scalable, efficient PNS in distributed shared-memory environments. Evaluated on 1,024 cores, our implementation achieves a 332.9× speedup over sequential PNS; its solving efficiency surpasses that of GLOP by four orders of magnitude. Moreover, it increases proof complexity by a factor of 1,000 and successfully verifies 42 previously unsolved Sprouts game positions—significantly advancing the practical frontier of automated fair-game solving.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Multiagent Systems: Distributed Problem Solving

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Proof-Number Search is a best-first search algorithm with many successful applications, especially in game solving. As large-scale computing clusters become increasingly accessible, parallelization is a natural way to accelerate computation. However, existing parallel versions of Proof-Number Search are known to scale poorly on many CPU cores. Using two parallelized levels and shared information among workers, we present the first massively parallel version of Proof-Number Search that scales efficiently even on a large number of CPUs. We apply our solver, enhanced with Grundy numbers for reducing game trees, to the Sprouts game, a case study motivated by the long-standing Sprouts Conjecture. Our solver achieves a significantly improved 332.9$ imes$ speedup when run on 1024 cores, enabling it to outperform the state-of-the-art Sprouts solver GLOP by four orders of magnitude in runtime and to generate proofs 1,000$ imes$ more complex. Despite exponential growth in game tree size, our solver verified the Sprouts Conjecture for 42 new positions, nearly doubling the number of known outcomes.

Problem

• Scaling Proof-Number Search efficiently across many CPU cores
• Reducing game tree complexity using Grundy numbers in Sprouts
• Verifying the Sprouts Conjecture for unsolved game positions

Innovation

• Two-level parallelization with shared worker information
• Grundy number enhancement for game tree reduction
• Massively scalable Proof-Number Search on 1024 cores