[DMS-75] Store black height in type

[Sereja313]

Profiling branch: https://github.com/serokell/canister-profiling/tree/sereja/set-profiling
Baseline: https://github.com/serokell/motoko-base/tree/sereja/specialize-persistentorderedset (9ebd08f)

Collection benchmarks

	binary_size	generate	max mem	batch_get 50	batch_put 50	batch_remove 50	upgrade
trieset+100	211_008	574_022	47_652	131_218	288_429	268_499	352_101
persistentset_baseline+100	196_855	196_106	37_784	49_615	128_635	130_668	210_758
persistentset+100	198_472	202_891	40_856	49_622	133_081	145_795	165_064
trieset+1000	211_008	7_374_045	633_440	162_806	383_594	375_264	731_650
persistentset_baseline+1000	196_855	2_804_787	532_992	66_718	172_961	177_698	302_292
persistentset+1000	198_472	2_901_910	577_624	66_725	179_379	200_287	251_326
trieset+10000	211_008	105_695_670	682_792	192_931	457_923	462_594	3_986_772
persistentset_baseline+10000	196_855	44_279_117	360_508	82_339	214_958	227_419	374_743
persistentset+10000	198_472	45_938_541	400_508	82_346	221_497	262_370	331_581
trieset+100000	211_008	1_234_038_235	6_826_516	222_247	560_440	549_813	178_864_274
persistentset_baseline+100000	196_855	534_152_351	3_600_508	96_540	260_077	273_659	486_063
persistentset+100000	198_472	553_354_610	4_000_508	96_547	267_902	316_477	419_377
trieset+1000000	211_008	13_990_048_548	68_228_312	252_211	650_405	642_099	1_782_978_890
persistentset_baseline+1000000	196_855	6_241_486_671	36_000_508	114_567	305_638	327_353	551_652
persistentset+1000000	198_472	6_458_837_421	40_000_544	114_574	314_842	382_569	505_973

set API

	size	intersect	union	diff	equals	isSubset
trieset+100	100	352_496	411_306	350_935	201_896	201_456
persistentset_baseline+100	100	157_677	170_099	219_483	156_755	157_147
persistentset+100	100	137_470	137_822	163_025	157_191	156_752
trieset+1000	1000	731_650	1_079_906	912_629	2_589_090	4_023_673
persistentset_baseline+1000	1000	339_879	516_679	946_277	1_851_541	1_851_538
persistentset+1000	1000	257_226	255_676	359_488	1_851_541	1_851_538
trieset+10000	10000	3_986_854	21_412_306	5_984_106	46_174_710	31_885_381
persistentset_baseline+10000	10000	450_437	1_056_800	2_154_884	28_585_151	28_585_107
persistentset+10000	10000	366_799	388_139	565_939	29_005_151	29_005_107
trieset+100000	100000	178_863_894	209_889_623	199_028_396	521_399_350	521_399_346
persistentset_baseline+100000	100000	589_221	1_922_651	3_769_853	317_013_739	317_013_654
persistentset+100000	100000	464_937	530_657	757_591	321_213_698	321_213_531
trieset+1000000	1000000	1_782_977_198	2_092_850_787	1_984_818_266	5_813_335_155	5_813_335_151
persistentset_baseline+1000000	1000000	675_690	3_032_485	6_064_414	3_473_642_146	3_473_641_487
persistentset+1000000	1000000	571_282	683_965	950_978	3_515_640_465	3_515_640_175

new set API

	size	foldLeft	foldRight	mapfilter	map
persistentset_baseline	100	85_458	86_734	158_197	315_166
persistentset	100	85_458	86_734	161_478	324_783
persistentset_baseline	1000	821_920	833_624	3_050_847	4_468_816
persistentset	1000	821_961	834_019	3_153_082	4_607_544
persistentset_baseline	10000	15_241_830	15_354_246	38_010_778	65_203_581
persistentset	10000	15_661_830	15_774_205	39_261_257	67_405_966
persistentset_baseline	100000	152_314_883	153_422_513	450_944_413	789_670_737
persistentset	100000	156_514_924	157_622_882	465_500_521	815_720_516
persistentset_baseline	1000000	1_523_084_378	1_534_107_566	5_197_944_386	9_265_810_970
persistentset	1000000	1_565_084_255	1_576_108_427	5_363_388_523	9_566_516_387

[GoPavel]

I see that all operations except intersect, union, diff become slower on ~3-4%. However, the speed up of the intersect, union, and diff is significant.

I am wondering if can we precalculate black depth and only there.
Or maybe we can make functions like join take and return a pair (blackDepth, tree) instead of tree
WDYT?

[Sereja313]

I tried to calculate bh in these functions in #31 but it's not so easy to track what change we need to make to the height because it depends on the color of the parent/child node and therefore requires a lot of case analysis in internal functions which leads to significant slowdown. As for precalculating bh in just these functions, this will require going through all the paths of both trees, even those paths that we don't actually need. I tried to find examples of something like this, but there doesn't seem to be a better way than just storing the bh in the type, this is also suggested in Nipkow's book.

[GoPavel]

I think this table makes it more clear

	size	intersect	union	diff	equals	isSubset
persistentset_baseline+100	100	157_677	170_099	219_483	156_755	157_147
persistentset_26+100	100	137_470	137_822	163_025	157_191	156_752
persistentset_31+100	100	186_143	193_149	234_366	156_755	156_752
persistentset_baseline+1000	1000	339_879	516_679	946_277	1_851_541	1_851_538
persistentset_26+1000	1000	257_226	255_676	359_488	1_851_541	1_851_538
persistentset_31+1000	1000	432_457	429_026	720_916	1_851_541	2_565_361
persistentset_baseline+10000	10000	450_437	1_056_800	2_154_884	28_585_151	28_585_107
persistentset_26+10000	10000	366_799	388_139	565_939	29_005_151	29_005_107
persistentset_31+10000	10000	676_919	691_488	1_452_936	28_584_115	28_584_235
persistentset_baseline+100000	100000	589_221	1_922_651	3_769_853	317_013_739	317_013_654
persistentset_26+100000	100000	464_937	530_657	757_591	321_213_698	321_213_531
persistentset_31+100000	100000	889_597	961_645	2_178_331	317_012_457	317_012_495
persistentset_baseline+1000000	1000000	675_690	3_032_485	6_064_414	3_473_642_146	3_473_641_487
persistentset_26+1000000	1000000	571_282	683_965	950_978	3_515_640_465	3_515_640_175
persistentset_31+1000000	1000000	1_137_064	1_267_131	3_149_146	3_473_639_306	3_473_638_647