A massively parallel pentago solver

This project implements a strong solution of the board game pentago, which means that perfect play can be efficiently computed starting from any position. The results can be explored at http://perfect-pentago.net. Some high level details are described at http://perfect-pentago.net/details.html.

In the quest towards a strong solution, a total of five separate engines were implemented, three forward tree search engines and two backwards endgame engines. The final strong solution uses only the backwards engines, but the strongest forward engine was essential in developing and testing the backwards engines. Descriptions of the various engines are given below.

The easiest interface to the strong solution is the website at http://perfect-pentago.net, implemented in the web directory. To compute perfect play for a given position, the backend server looks up the answer in a 4 TB database for positions with 17 or fewer stones or computes the answer from scratch using the small scale backwards engine.

The pentago code is released under a BSD license, and the 4 TB data set has been released into the public domain for anyone who wants to tinker around. The data set is generously hosted by Rackspace. If you want links to download the data, see below.

For background on the complexity of pentago and the applicability of a variety of solution algorithms, see Niklas Buscher's 2011 thesis "On Solving Pentago", which analyzed the game but did not solve it.

I will be writing up additional details of the application in paper form soon.

Dependencies

The code is C++, with node.js for the backend server. The main dependencies are

• bazel: Build system
• mpi: OpenMPI, MPICH2, etc.
• node.js >= 8.9: Asynchronous javascript framework

Bazel handles a few extra dependencies automatically (see WORKSPACE for details).

Installation

On Mac:

# Install dependencies
brew install bazel openmpi node

# Build and test C++
bazel test -c opt --copt=-march=native ...

# Build and test node.js server
cd web/server
npm install
node unit.js all

# Build frontend webpage
cd web/client
make public

On Ubuntu: to be written once I have clean Bazel handling of MPI.

Website

The website https://perfect-pentago.net is a static Firebase frontend that talks to a node.js Google Cloud Function. To test and deploy the server:

cd web/server
node unit.js all
./deploy

To deploy the client:

cd pentago/web/client
brew install llvm
npm install
npm run deploy

Algorithm summary

Forward engines

The first forward engine was a naive alpha-beta tree search code, plus a few pentago specific optimizations. Performance was dismal: the code reached only to 4 or 5 ply (here one ply means the combination of stone placement and quadrant rotation). The problem was branching factor, which starts out at a frightening 288 possible first moves (36 empty spots times 8 rotations).

Based on a belief that pentago was likely a tie (wrong!), I modified the solver so that the second player always reversed the rotation of the first player. This can't be any better for the second player, so if the result was a tie the game would be solved. Unfortunately, this "simple" solver declared the game to be a first player win after 15 moves, providing no information about the real value of the game.

The key to the speed of the simple solver was eliminating the branching factor due to rotations. The final forward solver accomplished the same thing without weakening the second player. Instead of computing the value of one board at a time, this "super" solver operated on 256 different rotated version of a given board in parallel using SSE instructions. This also allowed the transposition tables to take advantage of the full 2048 element symmetry group of all global+local transpositions. With rotations eliminated, the supersolver managed to compute out to 17 ply, declaring the game up to that point a tie. See pentago/search/superengine.{h,cpp} for the core algorithm.

What next? Parallelizing the engine would get further, but I still thought the game was a tie, and no forward engine would reach all the way to move 36. The similar games gomoku and renju were solved using threat space search, but it was unclear how to combine this technique with a rotation abstracted solver. So much for going forward.

A massively parallel backward engine

What about backward? Typically, backward (or retrograde) analyses are used to solve convergent games, where the number of positions decreases towards the end of the game. The canonical examples are chess and checkers, where fewer pieces near the end mean fewer positions. Pentago is divergent: the number of positions increases exponentially all the way to ply 24, but the total number of positions is only 3e15.

Unfortunately, the computation required 80 TB of memory at peak. Since the arithmetic intensity of computing pentago positions is fairly low, shuffling all of this data in and out of disks would have destroyed performance. Therefore, the endgame solver was implemented in-core, parallelized using MPI for use on a supercomputer. Like the final forward solver, the endgame solver takes advantage of symmetry by operating on 256 positions at a time and eliminating symmetric positions. The code for the massively parallel endgame solver is contained in pentago/end and pentago/mpi.

This backward engine was sufficient to strongly solve the game, requiring about four hours of time on 98304 threads of Edison, a Cray supercomputer at NERSC.

Like endgame solvers for other games, the backwards engine starts at the end of the game and walks backwards towards the beginning, computing the value of each position. Unlike previous endgame solvers, nearly all of the data is discarded once it is used, and only positions with 18 or fewer stones are kept and written to disk. To my knowledge, this is the first use of an endgame solver to directly compute an opening book for a nontrivial game.

A tiny backward engine

Since the massively parallel computation discarded all positions with more than 18 stones, the game isn't strongly solved unless the missing values can be quickly recomputed. Before the massively parallel computation, I tried the forward solver on a variety of 18 stone positions and got satisfactory performance: the solver was capable of reaching forwards to the end of the game. This turned out to be naive: random positions are much easier than interesting positions, and I quickly found interesting positions once I began exploring the results. The first time I came across one of these the solve took an hour; I managed to get it down to 10 minutes, but this was still much too slow for an interactive website.

To solve the problem, I implemented yet another backward engine, this one specialized to traverse only positions downstream of a given root board with 18 or more stones. In addition to tackling a much smaller problem than the massively parallel engine, this "midgame" engine is also much simpler: since rotations are abstracted away, the set of empty positions is fixed by the root board. As a perk, we can use parity to operate on only 128 different rotated positions instead of 256, as rotating the board flips the parity of the local rotation group. Conveniently, the performance of the midgame engine is about 15 seconds for all 18 stone positions; unlike the forward solver performance does not vary with the board. The code for the midgame engine is in pentago/mid.

Together, the 4 TB data set and the midgame engine comprise a strong solution of pentago. Go to https://perfect-pentago.net to see the results.

Code structure

The code is organized into the following directories:

• utility: Thread pools, memory allocation, and other non-pentago-specific utilities
• base: Pentago boards, moves, scores, symmetries, counts, etc., used by the various solvers.
• search: Forward search engines
• data: File formats and I/O for the backwards engine
• end: Everything in the backwards engine that doesn't directly touch mpi
• mpi: Everything in the backwards engine that does touch mpi
• high: High level interface used in the website backend
• mid: "Midgame" backward solver used to complete the strong solution for 18 or more stone boards.

Most files have summarizing comments at the top, but otherwise documentation is fairly scant. Email me if you have questions at [email protected]!

Data

Update: For some reason CDN bandwidth costs went above my open source limit, so I've temporarily disabled access to the data. This is probably due to a bug in the server, and I'll hopefully open it back up once I find time to diagnose the bug. If you want the data in the mean time, please email me.

The computed data set of perfect pentago results up to slice 18 can be downloaded directly from Rackspace. If you do download it and play around, I'd love to get an email, but you should feel no obligation to do so. Also available are sparse sample files and win/losss/tie counts for all slices, which would be useful for cross-validation. The data is all in custom formats, so the easiest way to access it is via compiling this repository.

Warning: The total size of the data is 3.7 TB. If you want something more manageable, you can download only the first few slices (say up to move 12). To see the sizes of all slices before downloading, look at data/edison/final.txt. If you've gotten past this warning, you can download the data via

# Base url
url="http://582aa28f4f000f497ad5-81c103f827ca6373fd889208ea864720.r52.cf5.rackcdn.com"

# Download a sparse sample or count file.  Replace <n> with the desired slice in [0,35].
wget $url/sparse-<n>.npy
wget $url/counts-<n>.npy

# Download a slice file.  Replace <n> with the desired slice in [0,18]
wget $url/slice-<n>.pentago

Slices 12 through 18 are chunked into 5GB-1B chunks on Rackspace, so if you have difficulty downloading them all at once you can do it one chunk at a time. Chunks look like slice-<n>.pentago.<chunk>, where <chunk> is 1-based and zero padded. Thus, all the chunks look like

12:  slice-12.pentago.1    to  slice-12.pentago.2
13:  slice-13.pentago.1    to  slice-13.pentago.8
14:  slice-14.pentago.01   to  slice-14.pentago.13
15:  slice-15.pentago.01   to  slice-15.pentago.55
16:  slice-16.pentago.01   to  slice-16.pentago.86
17:  slice-17.pentago.001  to  slice-17.pentago.275
18:  slice-18.pentago.001  to  slice-18.pentago.365

Once you have all the chunks, just cat them together.