>>99
I look forward to seeing it!
[1/14]
Here is the upgraded version of >>76 using stacked self-generation >>35 of the S sequence and layered >>81 partial >>72 fourth-order S-groups >>35 to achieve O(log(log(n))) space consumption and O(n^(1/16)) runtime >>86. Fourth-order S-group formulas were obtained from the general procedure for incrementing the S-group order. To avoid shifting in the main loop the Svec of >>83 is reversed so the 's' value is at index 0. The new Slist is [s, S1, S2 ...] and holds the starting values of the S-groups from the one with the highest order, which is furthest from the base S sequence, to the one with the lowest order, which is closest to the base S sequence, and ending with the S sequence itself. With layers numbered from 1, the first equation that defines the general procedure is:
layer[N+1].count(c,Slist) = layer[N].inc(c,Slist')
where Slist' is Slist with the value at index N-1 decremented. The [N+1].inc equation can be obtained by expressing the [N+2].count both in terms of [N+1].inc and as a sum of the component [N+1].counts:
layer[N+1].inc(c,Slist') = sum(for j in [1...c] of layer[N+1].count(s-2+j,h(j,Slist)))
where h advances the values of Slist[k in 0...N-1] for subgroup j:
h[k](j,Slist) = Slist[k+1]+layer[k+1].inc(j-1,Slist)
One thing that saves on computations is to note that Slist[N] need not be included in the final h because it is a foregone conclusion that in the final result it will appear as a linear term, and its factor will be the full count in the same layer. Implementing these as well as the previous observations yields:
class Layer:
@ classmethod
def count (cls, c, Slist, counts, incs):
pass
@ classmethod
def increment (cls, c, Slist, counts, incs):
pass
class ChainedLayers:
class Layer1 (Layer):
@ classmethod
def count (cls, c, Slist, counts, incs):
counts [0] = c
@ classmethod
def increment (cls, c, Slist, counts, incs):
s = Slist [0]
incs [0] = (c - 1 + s + s) * c // 2
def _access (setasup):
def fun (cls):
cls.up = setasup.increment
return cls
return fun
@ _access (Layer1)
class Layer2 (Layer):
@ classmethod
def count (cls, c, Slist, counts, incs):
s = Slist [0]
# Slist [0] = s - 1
# cls.up (c, Slist, counts, incs)
# Slist [0] = s
# counts [1] = incs [0]
counts [1] = (c - 3 + s + s) * c // 2
@ classmethod
def increment (cls, c, Slist, counts, incs):
countS = counts [1]
incS = countS + c
s = Slist [0]
overR = (incS - 1) * incS // 2
cm1c = (c - 1) * c
backR = (s - 1) * c + (c + 1 + 3 * s) * cm1c // 6
incR = overR - backR + Slist [1] * countS
incs [0] = incS
incs [1] = incR
@ _access (Layer2)
class Layer3 (Layer):
@ classmethod
def count (cls, c, Slist, counts, incs):
S = Slist [1]
Slist [1] = S - 1
cls.up (c, Slist, counts, incs)
Slist [1] = S
counts [2] = incs [1]
@ classmethod
def increment (cls, c, Slist, counts, incs):
# names from seq4_inc
countS1 = counts [1]
countS0 = counts [2]
incS0 = countS0 + countS1
overR = (incS0 - 1) * incS0 // 2
s = Slist [0]
s20 = 20 * s
s30 = 30 * s
s60 = s30 + s30
backR = c*(((((5*c + (s30 - 29))*c + ((s60 - 130)*s + 45))*c + (((s20 + s20 - 140)*s + 90)*s + 85))*c + ((250 - s60)*s - 290))*c + (s20 - 160)*s + 184) // 240
backR = countS1 * (countS1 + 1) // 2 * Slist [1] - countS1 + backR
incR = overR - backR + Slist [2] * countS0
incs [1] = incS0
incs [2] = incR[2/14]
@ _access (Layer3)
class Layer4 (Layer):
@ classmethod
def count (cls, c, Slist, counts, incs):
S = Slist [2]
Slist [2] = S - 1
cls.up (c, Slist, counts, incs)
Slist [2] = S
counts [3] = incs [2]
@ classmethod
def increment (cls, c, Slist, counts, incs):
# s A B C
countB = counts [2]
countC = counts [3]
incC = countC + countB
s = Slist [0]
A = Slist [1]
B = Slist [2]
part1 = A*(A*(A*(A*c**3*(c*(c*(c*(3628800*c + 29030400*s - 43545600) + s*(87091200*s - 261273600) + 195955200) + s*(s*(116121600*s - 522547200) + 783820800) - 391910400) + s*(s*(s*(58060800*s - 348364800) + 783820800) - 783820800) + 293932800) + c**2*(c*(c*(c*(c*(c*(c*(3628800*c + 36288000*s - 44755200) + s*(145152000*s - 362880000) + 195955200) + s*(s*(290304000*s - 1103155200) + 1175731200) - 263692800) + s*(s*(s*(290304000*s - 1490227200) + 2264371200) - 667699200) - 572140800) + s*(s*(s*(s*(116121600*s - 754790400) + 1219276800) + 754790400) - 3476390400) + 2340576000) + s*(s*(s*(-348364800*s + 2070835200) - 5080320000) + 5965747200) - 2743372800) + s*(s*(-309657600*s + 1393459200) - 2090188800) + 1045094400)) + c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(1360800*c + 16329600*s - 17236800) + s*(81648000*s - 175996800) + 74239200) + s*(s*(217728000*s - 718502400) + 596332800) - 57093120) + s*(s*(s*(326592000*s - 1466035200) + 1735776000) - 52980480) - 518222880) + s*(s*(s*(s*(261273600*s - 1495065600) + 2036966400) + 1269112320) - 3868421760) + 1589353920) + s*(s*(s*(s*(s*(87091200*s - 609638400) + 462067200) + 3914265600) - 9503222400) + 6943708800) - 754034400) + s*(s*(s*(s*(-503193600*s + 3241728000) - 7576934400) + 6643123200) + 1337817600) - 3952126080) + s*(s*(s*(290304000*s - 2157926400) + 8087385600) - 13536875520) + 7980094080) + s*(s*(s*(-19353600*s + 1645056000) - 7063096320) + 10578677760) - 5430136320) + s*(232243200*s - 696729600) + 503193600) - 38707200*s + 58060800) + c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(226800*c + 3175200*s - 2948400) + s*(19051200*s - 36288000) + 12625200) + s*(s*(63504000*s - 185976000) + 127461600) - 3707760) + s*(s*(s*(127008000*s - 508032000) + 500774400) + 41845440) - 133524720) + s*(s*(s*(s*(152409600*s - 780192000) + 925344000) + 542263680) - 1263376800) + 366997680) + s*(s*(s*(s*(s*(101606400*s - 638668800) + 700358400) + 1967454720) - 4436147520) + 2285216640) + 69970320) + s*(s*(s*(s*(s*(s*(29030400*s - 217728000) - 50803200) + 2999404800) - 6769929600) + 4090443840) + 2021513760) - 1954478160) + s*(s*(s*(s*(s*(-241920000*s + 1678924800) - 3536870400) + 392716800) + 8533123200) - 10621235520) + 3069944640) + s*(s*(s*(s*(464486400*s - 3215923200) + 10204992000) - 15705043200) + 8335474560) + 1164072000) + s*(s*(s*(s*(-19353600*s + 899942400) - 3424619520) + 710277120) + 9379426560) - 8806936320) + s*(s*(s*(58060800*s - 2532096000) + 10668994560) - 16847631360) + 9472619520) + s*(s*(58060800*s - 1403136000) + 3741050880) - 2127928320) + s*(3870720*s + 1033482240) - 1552711680)[3/14]
part2 = B*(A*(A*(A*c**2*(c*(c*(29030400*c + 174182400*s - 261273600) + s*(348364800*s - 1045094400) + 783820800) + s*(s*(232243200*s - 1045094400) + 1567641600) - 783820800) + c*(c*(c*(c*(c*(c*(21772800*c + 174182400*s - 203212800) + s*(522547200*s - 1248307200) + 551577600) + s*(s*(696729600*s - 2554675200) + 2090188800) + 246758400) + s*(s*(s*(348364800*s - 1741824000) + 1393459200) + 2815948800) - 3418329600) + s*(s*(-1161216000*s + 5167411200) - 8360755200) + 4833561600) + s*(-696729600*s + 2090188800) - 1567641600)) + c*(c*(c*(c*(c*(c*(c*(c*(5443200*c + 54432000*s - 52617600) + s*(217728000*s - 435456000) + 131846400) + s*(s*(435456000*s - 1349913600) + 742694400) + 256677120) + s*(s*(s*(435456000*s - 1857945600) + 1117670400) + 2228567040) - 1665740160) + s*(s*(s*(s*(174182400*s - 958003200) - 241920000) + 6178636800) - 7575724800) + 1616630400) + s*(s*(s*(-1122508800*s + 5554483200) - 8157542400) + 1485388800) + 3752179200) + s*(s*(1161216000*s - 5786726400) + 12275020800) - 9252472320) + s*(s*(-38707200*s + 2922393600) - 8387850240) + 5904783360) - 464486400*s + 696729600) + B*(A*(A*c*(c*(58060800*c + 232243200*s - 348364800) + s*(232243200*s - 696729600) + 522547200) + c*(c*(c*(c*(29030400*c + 174182400*s - 183859200) + s*(348364800*s - 774144000) + 145152000) + s*(s*(232243200*s - 812851200) - 58060800) + 1112832000) + s*(-928972800*s + 2748211200) - 2032128000)) + c*(c*(c*(c*(c*(c*(3628800*c + 29030400*s - 24192000) + s*(87091200*s - 154828800) + 4032000) + s*(s*(116121600*s - 329011200) - 67737600) + 290304000) + s*(s*(s*(58060800*s - 232243200) - 377395200) + 1354752000) - 473760000) + s*(s*(-464486400*s + 1606348800) - 774144000) - 846720000) + s*(928972800*s - 2709504000) + 1975680000)) + c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(453600*c + 5443200*s - 4536000) + s*(27216000*s - 47174400) + 10735200) + s*(s*(72576000*s - 195955200) + 81043200) + 36281280) + s*(s*(s*(108864000*s - 406425600) + 197164800) + 371589120) - 204876000) + s*(s*(s*(s*(87091200*s - 420940800) + 91929600) + 1411845120) - 1371565440) + 133156800) + s*(s*(s*(s*(s*(29030400*s - 174182400) - 263692800) + 2364364800) - 2937916800) + 19595520) + 935373600) + s*(s*(s*(s*(-270950400*s + 1475712000) - 1703116800) - 2165184000) + 5123865600) - 2034002880) + s*(s*(s*(677376000*s - 3506227200) + 7190668800) - 5481745920) - 251395200) + s*(s*(s*(-19353600*s + 541900800) - 782853120) - 3935554560) + 5662406400) + s*(s*(77414400*s - 3286886400) + 8872980480) - 5715763200) + 928972800*s - 1354752000)
part3 = c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(c*(14175*c + 226800*s - 189000) + s*(1587600*s - 2721600) + 812700) + s*(s*(6350400*s - 16783200) + 9903600) + 52080) + s*(s*(s*(15876000*s - 57456000) + 49215600) + 6347040) - 11050550) + s*(s*(s*(s*(25401600*s - 117936000) + 125193600) + 65677920) - 128034480) + 28664160) + s*(s*(s*(s*(s*(25401600*s - 145152000) + 162388800) + 281756160) - 588183120) + 233335200) + 23121812) + s*(s*(s*(s*(s*(s*(14515200*s - 99187200) + 76204800) + 609799680) - 1329558720) + 628068000) + 363318480) - 231117040) + s*(s*(s*(s*(s*(s*(s*(3628800*s - 29030400) - 37497600) + 662054400) - 1443304800) + 413965440) + 1801674000) - 1744031520) + 303711135) + s*(s*(s*(s*(s*(s*(-38707200*s + 287884800) - 517708800) - 719913600) + 3666835200) - 4386013920) + 1191840000) + 491453160) + s*(s*(s*(s*(s*(122572800*s - 891072000) + 2710512000) - 3678716160) + 179323200) + 3719604000) - 1855381864) + s*(s*(s*(s*(s*(-4838400*s + 75479040) + 11289600) - 2663646720) + 7910091840) - 7185104640) + 1227340800) + s*(s*(s*(s*(25804800*s - 972518400) + 4195215360) - 5737536000) + 766308480) + 2760323440) + s*(s*(s*(-6451200*s + 669312000) - 3513619200) + 7885409280) - 5741832320) + s*(s*(s*(1612800*s - 119347200) + 2311787520) - 5494809600) + 2494340352) + s*(s*(s*(645120*s - 5160960) + 48015360) - 2009825280) + 2832168960
extra = c * (part1 + part2 + part3) // 464486400
incR = extra + Slist [3] * countC
incs [2] = incC
incs [3] = incR[4/14]
class Group:
def counts (self, c, Slist, counts, incs):
pass
def increments (self, c, Slist, counts, incs):
pass
class LayeredGroups:
class Group1 (Group):
def __init__ (self, layer1):
self.layer1 = layer1
def counts (self, c, Slist, counts, incs):
self.layer1.count (c, Slist, counts, incs)
def increments (self, c, Slist, counts, incs):
self.layer1.increment (c, Slist, counts, incs)
class GroupN (Group):
def __init__ (self, order, layers):
self.order = order
self.layers = layers
self.cache_coufun = [l.count for l in layers [:order]]
self.cache_incfun = layers [order - 1].increment
def counts (self, c, Slist, counts, incs):
for fun in self.cache_coufun:
fun (c, Slist, counts, incs)
def increments (self, c, Slist, counts, incs):
self.cache_incfun (c, Slist, counts, incs)
def work_layers_partial (c, Slist, counts, incs, nleft, countfun, order):
order -= 1
countfun (1, Slist, counts, incs)
want = counts [order]
if want > nleft:
return 0
low, high = 1, c
diff = high - low
savec = counts [:]
savei = incs [:]
while diff > 1:
now = (low + high) // 2
countfun (now, Slist, counts, incs)
want = counts [order]
if want <= nleft:
low = now
savec [:] = counts
savei [:] = incs
else:
high = now
diff = high - low
counts [:] = savec
incs [:] = savei
return low
def work_layers_loop (n, groups):
glen = len (groups)
gm1 = glen - 1
R = 3
Slist = [ 4] * glen
counts = [None] * glen
incs = [None] * glen
levels = []
s, c = 4, 3
done = 2
next = seq2_next
g = groups [-1]
gcounts = g.counts
increments = g.increments
slistrange = range (gm1)
gcounts (c, Slist, counts, incs)
advance = counts [gm1]
done += advance
while done <= n:
increments (c, Slist, counts, incs)
R += incs [gm1]
for k in slistrange:
Slist [k + 1] += incs [k]
s, c = next (levels, 0)
Slist [0] = s
gcounts (c, Slist, counts, incs)
advance = counts [gm1]
done += advance
done -= advance
print ("levels:", len (levels))
for gorder in range (glen, 1, -1):
g = groups [gorder - 1]
used = work_layers_partial (
c, Slist, counts, incs,
n - done, g.counts, gorder)
if used > 0:
advance = counts [gorder - 1]
g.increments (used, Slist, counts, incs)
R += incs [gorder - 1]
for k in range (gorder - 1):
Slist [k + 1] += incs [k]
done += advance
levels = [[s, c, used, Slist [1]]]
s, c = next (levels, 0)
Slist [0] = s
del Slist [1]
if done < n:
g = groups [0]
c = n - done
g.counts (c, Slist, counts, incs)
g.increments (c, Slist, counts, incs)
R += incs [0]
return R
def work_layers (n):
layers = [
ChainedLayers.Layer1,
ChainedLayers.Layer2,
ChainedLayers.Layer3,
ChainedLayers.Layer4,
]
groups = [
LayeredGroups.Group1 (layers [0]),
LayeredGroups.GroupN (2, layers),
LayeredGroups.GroupN (3, layers),
LayeredGroups.GroupN (4, layers),
]
return work_layers_loop (n, groups)[5/14]
I do not recommend looking at the fourth-order formula for too long, it might start looking back at you. Here are the updated timings, all in vanilla python.
- The u64 overflow R(10^10) is too small for fourth-order S-groups to pay off and the third-order timing >>76 of under 100 microseconds remains as reference.
- The u128 overflow R(10^20) is computed in under one millisecond.
- The u256 overflow R(10^39) is computed in 16 milliseconds.
- The first integer k for which R(10^k) overflows an u512 is k=78. The value of R(10^78) is computed in under 5 seconds.
$ python3 -m timeit -s 'import flake.hofs as mod' 'mod.work_layers (100000000000000000000)'
1000 loops, best of 3: 945 usec per loop
$ python3 -m timeit -s 'import flake.hofs as mod' 'mod.work_layers (1000000000000000000000000000000000000000)'
100 loops, best of 3: 15.6 msec per loop
$ python3 -m timeit -s 'import flake.hofs as mod' 'mod.work_layers (1000000000000000000000000000000000000000000000000000000000000000000000000000000)'
10 loops, best of 3: 4.79 sec per loopR(10^78) is the appropriate value for comparing implementations with this runtime order of growth. Here is the k<=120 table in >>38 format. This is most likely the last such table I'll post in full so please excuse the length. Consider it raw research data.
[6/14]
$ cat table4.txt
1 69 7 0
2 5764 13 0
3 526334 20 0
4 50874761 26 1
5 5028514748 33 1
6 500918449417 39 1
7 50029364025646 46 1
8 5000934571821489 53 2
9 500029664631299282 59 2
10 50000940106333921296 66 2
11 5000029765607020319048 73 2
12 500000941936492050650505 79 2
13 50000029798618763894670256 86 2
14 5000000942529842698007077786 93 2
15 500000029809255627825531266625 99 3[7/14]
16 50000000942720152238624116795401 106 3
17 5000000029812655507343465281696595 112 3
18 500000000942780823112495107784753816 119 3
19 50000000029813737262126730811322149673 126 3
20 5000000000942800098290022420982686040347 132 3
21 500000000029814080548392288266955229183571 139 3
22 50000000000942806209878293665994446398371544 146 3
23 5000000000029814189323670710814676032031444555 152 3
24 500000000000942808145472258657037814775197247031 159 3
25 50000000000029814223760934912839828327249688721190 166 3
26 5000000000000942808758091081952084125868709915711089 172 3
27 500000000000029814234658067055252766458087914346630413 179 3
28 50000000000000942808951913512756782469229223199209677641 186 3
29 5000000000000029814238105320163714566005337846445873165675 192 3
30 500000000000000942809013222649956573426148951590151576716407 199 4[8/14]
31 50000000000000029814239195666197369782628797813806672178028303 205 4
32 5000000000000000942809032613363227344483520545074111347580106666 212 4
33 500000000000000029814239540504829346639521751526121067855028048469 219 4
34 50000000000000000942809038745792444395013325500575549841178605176222 225 4
35 5000000000000000029814239649559693010895101002840721426848127286038482 232 4
36 500000000000000000942809040685134541874198687080125967207639651015633313 239 4
37 50000000000000000029814239684047173947897406203055196803310606765583116781 245 4
38 5000000000000000000942809041298425788194699108125613517442497904284636150437 252 4
39 500000000000000000029814239694953305682145634502486118475654808671411192582303 259 4
40 50000000000000000000942809041492368615124254362375085779982992292343853681977675 265 4
41 5000000000000000000029814239698402168816728284052931912470389462588984951289354475 272 4
42 500000000000000000000942809041553699275352369083555067298879759994519671549809149522 279 4
43 50000000000000000000029814239699492802485126708959655333290571225887358803916579127627 285 4
44 5000000000000000000000942809041573093831553392834637054332803551711977694994007009790372 292 4
45 500000000000000000000029814239699837692448235838620313436065493069882669969298165276420932 298 4[9/14]
46 50000000000000000000000942809041579226946272526629650233821592037083538450290978659280817924 305 4
47 5000000000000000000000029814239699946756464770327217558477196037097548855222595700238539054865 312 4
48 500000000000000000000000942809041581166410560146744996087637489747926009831507674531860092771246 318 4
49 50000000000000000000000029814239699981245576707388365411525341190531682837962715909950754430162916 325 4
50 5000000000000000000000000942809041581779723574457735429357187680244102657245323510219796435077511331 332 4
51 500000000000000000000000029814239699992151998934816806362938022380717694881944763055115621651771537182 338 4
52 50000000000000000000000000942809041581973670277645147248950274174680459501036705469450283128240202936345 345 4
53 5000000000000000000000000029814239699995600913788820676216348225539327181484799269054153129487836460510789 352 4
54 500000000000000000000000000942809041582035001627896095619939474929936571762744906288012004273501460096218956 358 4
55 50000000000000000000000000029814239699996691556662690868009166942952903517096871912188005052332391102154504541 365 4
56 5000000000000000000000000000942809041582054396306898966297540047317634200313888303569207665197822737010723880453 372 4
57 500000000000000000000000000029814239699997036448263909956827998702688566561312699712649393246034276487069718675623 378 4
58 50000000000000000000000000000942809041582060529443468690126506942010712319519060683290576216509510379646108458690288 385 5
59 5000000000000000000000000000029814239699997145512571896807332586675028913049270348666820028707699723929284676993408671 391 5
60 500000000000000000000000000000942809041582062468911643702033358845623678711109326514636451791089982268036336668374054905 398 5[10/14]
61 50000000000000000000000000000029814239699997180001735682481622436253734484378611829704069855253539840822385586766235223775 405 5
62 5000000000000000000000000000000942809041582063082225349534414976370874538365419773853994503610373186092493285764626069077930 411 5
63 500000000000000000000000000000029814239699997190908167132758797213601135850071440613709845156968123373754366585399707913755854 418 5
64 50000000000000000000000000000000942809041582063276172175725114596742736120123079153718814281207807930795334286345090895730008850 425 5
65 5000000000000000000000000000000029814239699997194357083627207181322355525685599913243509599846712885018947248295200321875155512194 431 5
66 500000000000000000000000000000000942809041582063337503547853702069293508176406619791980181270066919789232260704875309077658871313179 438 5
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Since I think R(10^100) is about as high as a reasonable person would want to go, I'll only be adding other improvements if they are interesting enough. For example, the numbers involved are well over the limit that the gmpy2 documentation states as the start of its speed advantage over python ints, so it might be time for a gmpy2 or GMP port to compare with vanilla python. So for possibly the last time, please verify the value of R(10^120) with other implementations.