Transposition tests

Data sets:

1. FEN dataset: All positions from http://data.lczero.org/files/training_pgns/pgns-run1-20190419-1854.tar.bz2 (12'582'379 positions).
2. Tree nodes after go nodes 1000000 from startpos with network id42482.
3. Tree nodes after go nodes 1000000 from (middlegame) r2br3/ppq2pkp/2b2np1/4p3/6N1/2PB1Q1P/PP1B1PP1/R3R1K1 w - - 1 2 with network id42482.
4. Tree nodes after go nodes 1000000 from (late middlegame) br5r/2bpkp2/B6p/2p4q/1PN1np2/P3P3/1BQ3PP/R4RK1 w - - 0 26 with network id42482.
5. Tree nodes after go nodes 1000000 from (endgame) 8/8/1p1n2k1/3P2p1/3P1b2/1P1K1B2/8/4B3 w - - 2 55 with network id42482.
6. Same as dataset 6, but with 10'000'000 nodes (10 times more).
7. Perpetual check position r6k/pp4pp/2p5/5Q2/1q5P/8/2P2PP1/1K1R3R w - - 1 25 from Leko vs Kramnik game. go nodes 1000000 with network id42482.
8. KRN vs KR draw enggame, 6r1/6k1/8/8/8/8/3N4/2KR4 w - - 0 1. go nodes 1000000 with network id42482.

Stats

Note: Ignore MaterialKey for now, that was crem's idea for Lc2

• Percentage of positions requiring MaterialKey to be computed:
  • Dataset 1: 29.8%
  • Dataset 2: 46.7%
  • Dataset 3: 48.1%
  • Dataset 4: 33.1%
  • Dataset 5: 7.1%
  • Dataset 6: 46.6%
• Bucket size distribution (Remaining bits encode pawns at ranks 2 and 3):
  • Dataset 1: (just for completeness; not really useful stats as positions are sampled from different games).
    • 2312346 buckets. 5.44 positions per bucket (4.7 if only unique).
    • Fun fact: 4.8% of all positions in training data is RKvsRK.
  • Dataset 6 (unique only):
    • 313978 buckets, 11.8 posisions per bucket.
    • Top bucket countains 0.672% of nodes (24951).
• Bucket size distribution (Remainig bits encode hash of pawns)
  • Dataset 6 (unique only): 388226 buckets (9.5)
    • 98.5% of larger buckets contain 50% of nodes (remaining 50% nodes are contained in size < 100 buckets).
    • Size < 16 buckets still contain 25% of Nodes.
  • Dataset 5 (unique only): 573 buckets, (240.0)
  • Dataset 1 (unique only): 2491831 buckets, (4.36)

Number of transpositions with exact and non-exact history

Conclusions

• Keeping one copy of unique position leads to 50% tree reduction in most cases (and closer to endgames even more).
• Including history into the position key destroys large part of benefit.
• On the other hand, including rule50 counter into the position key doesn't spoil things much, and removes need of an NN cache (also adds a nice benefit of non-loop PV, e.g. move graph becomes DAG, although it's not really needed).
• Threefold and twofold repetitions are very rare, and it's worthwhile to have slower logic when handling them. Also it's fine to only support threefold and not twofold.
• Memory footprint for handling repetitions ("pseudonodes") is very reasonable.

Dataset	3 midg.	4 late midg.	5 endg.	6 startp.	8 shuf.
Positions	1000350	1000107	1000000	10111141	1000145
Unique positions	537946 (53.8%)	692459 (69.2%)	103738 (10.4%)	3290287 (32.5%)	243723
Unique pos with history	702473 (70.2%)	875692 (87.6%)	897186 (89.7%)	4535616 (44.9%)	949180
Unique pos rep2 history	538307 (53.8%)	700134 (70.0%)	111754 (11.2%)	3291930 (32.6%)	245762
Unique +rule50	584265 (58.4%)	750441 (75.0%)	129460 (12.9%)	3685536 (36.5%)	278887
Threefold	69	3128	404	323	33
Twofold	413	3438	20606	3794	6853
Norep	999223 (99.9%)	970586 (97.0%)	918339 (91.8%)	10097097 (99.9%)	983750
Pseudonodes	878	8002	21020	2651	10415

Dataset	2 startp.	7 perpet.
Positions	1000232	1001502
Unique positions	409988 (41.0%)	23244
Unique pos with history	535639 (53.6%)	64001
Unique pos rep2 history	410229 (41.0%)	34520
Unique +rule50	449766 (45.0%)	29254
Threefold	60	109884
Twofold	355	99294
Norep	999036 (99.9%)	88295
Pseudonodes	363	436

• Dataset -- title of dataset.
• Positions -- number of positions in the set.
• Unique positions -- number of unique positions in the set (what makes a cache hit in Lc0).
• Unique positions with history -- number of unique positions in the set, where all positions from the last capture or pawn move (in any order) are included into positions.
• Unique positions with rep2 history -- the same, but only history positions which appeared more than once are included into the position.
• Unique +rule50 -- Number of unique positions (without history) if rule50 count is included into position hash.
• Threefold -- Number of three-fold repetition positions.
• Threefold -- Number of exact two-fold repetition positions.
• Norep -- Number of positions which do not include any repetitions in PV.
• Pseudonodes -- Number of additional "pseudo nodes" due to repeptition handles.