Monthly Archives: November 2013

Ancient Android root attempts on a chip RockChip tablet

Hello there! I’ve probably mentioned earlier that I’ve got two cheap tablets, and I got to root one of them. At least I was able to update it to 2.3 (I guess, don’t remember) with the Uberdroid distro image. Well, the other one has a RockChip chipset, and that’s quite harder to do, since z4root couldn’t trick it. I’m trying some other root methods, but for now, no success.

If after these new sideload attempts I’m still not successful… well, I’ll try some ADB methods with Windows applications. But that would be the last resort. Anyway, I’m playing a lot of music and taking care of pets/home when not at work, so, not much time to write and create new code. Let’s see how it goes. 🙂

JenPop and Genetic Algorithms on the loose

Hey there! I’ve committed a new class on JenPop, and it’s called “Individual”. It was made to represent the rules of typical genetic algorithms:

I’ve read a bit more on the subject and figured that genetic algorithms, although being able to achieve good solutions with little computing, are only suitable for problems where brute-forcing isn’t feasible. For instance, they do a nice job on combinatorial problems such as the classic knapsack problem, where you have a list of items with value v and weight w, and you must figure what’s the maximum value you can carry in a knapsack of capacity wc by combining different amounts of those items.

And that problem was chosen to serve as the first example. At this kind of problem, a naive approach can easily become unfeasible as the input list grows. With an evolutionary approach there’s a bigger probability that it will result in a better solution, given the same time limits. And even if the solution isn’t better, it will be a decent solution. The more time you let it roll, bigger is the chance of getting a great result.

For the classic scenario with a knapsack with wc = 15, and a (v,w) item list of (4, 12), (2, 2), (2, 1), (1, 1), (10, 4), I was able to achieve the following results:

30 iterations: Best result was { 0, 0, 4, 0, 2 } (v 28, w 12)
100 iterations: Best result was { 0, 3, 0, 9, 0 } (v 15, w 15)
500 iterations: Best result was { 0, 7, 1, 0, 0 } (v 16, w 15)
1000 iterations: Best result was { 0, 3, 4, 4, 0 } (v 18, w 14)
3000 iterations: Best result was { 0, 1, 1, 5, 1 } (v 19, w 12)
5000 iterations: Best result was { 0, 0, 5, 2, 2 } (v 32, w 15)

You can see that while none of those runs delivered the optimal solution, which would be { 0, 0, 3, 0, 3 } (v 36, w 15), they got close enough to provide good solutions. Also, it becomes clear that randomism takes a nice part in this, giving us the chance of achieving a great solution with only 30 iterations. These results also show that the more you let it run, bigger is the chance of getting better results.

Anyway, see you! 😀

New project at work = more free time = more hobby development

So, at last, I’ve joined the new project team, reading documentation and “first steps” to develop the new application. They seemed pretty organized so far, and I was able to solve all the hickups I’ve had when starting to install the development environment. Since I started by the end of tthe day, I didn’t have much time to do it and might finish tomorrow.

Since forever, my HabitRPG (they started running ads now?) is telling me that I must do four tasks (and I plan to kill them tomorrow):
– Put Carlin’s Explosion on GitHub;
– Establish a JenPop goal;
– Commit new code into it;
– Put Clonix (my attempt at a 2d physics game engine) on GitHub.

I’ll try to create some new tasks (and honor them, FFS) to better document and maintain those projects. But let’s see how things go. Anyway, tomorrow’s a big day so, see you then!

Projects finished and delivered, new challenges ahead!

Today, at Datacom, we’re celebrating the conclusion of the development of two projects, our little and versatile “CPEs” (both are different models of G.SHDSL bridge modems with Ethernet and PWE3 interfaces). It has been a brief (for me, at least) but intense experience, I’ve learned a lot.

Next week I’ll most likely be joining a new team in a totally different project. This time I’ll focus on developing a desktop application using Java. I really like embedded system development, but it’s time to try something new, and without the need to move to a different company.

That’s it! See you around, and hopefully I’ll have enough spare time to work on hobby projects once again.

Code Jam 2008 Round 1A, Problem C: Numbers

Hello again! This one is one hell of a tricky problem. No wonder it’s called “Numbers”:

In this problem, you have to find the last three digits before the decimal point for the number (3 + √5)n.

For example, when n = 5, (3 + √5)5 = 3935.73982… The answer is 935.

For n = 2, (3 + √5)2 = 27.4164079… The answer is 027.

Wow, that can’t be hard, eh? Indeed! It’s really just an expression, isn’t it? Of course it is! But once you try, you get to see something beautiful happening:

$ python
>>> import math
>>> 3+math.sqrt(5)
5.23606797749979
>>> (3+math.sqrt(5)) ** 18
8751751364607.995
>>> (3+math.sqrt(5)) ** 19
45824765067264.016
>>> (3+math.sqrt(5)) ** 20
239941584945152.1

If I was clear enough, you can see where I’m going. If not, the answers for n = 18, 19 and 20 should be 607, 263 and 151. But, it gets worse as the exponent grows:

>>> (3+math.sqrt(5)) ** 30
3.7167066517539914e+21
>>> (3+math.sqrt(5)) ** 40
5.757196418599164e+28
>>> (3+math.sqrt(5)) ** 50
8.917924847980422e+35

And those results should be 647, 551 and 247, respectively. As you can now clearly see, python can’t naturally handle those big floats, what a bummer! Well, the first thing to handle is the √5 value, since 2.23606797749979 does not give enough sampling to work with the needed precision. Looking a bit, I’ve found that python has a nice Decimal module that can deal with big numbers. So, let’s use it:

>>> from decimal import *
>>> Decimal(5).sqrt()
Decimal('2.236067977499789696409173669')

Hmm… not as much as I was expecting. Digging a little deeper, there’s a way to increase the precision, so things might get better:

>>> from decimal import *
>>> getcontext().prec = 256
>>> Decimal(5).sqrt()
Decimal('2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782274969508176150773783504253267724447073863586360121533452708866778173191879165811276645322639856580535761350417533785003423392414064442086432539097252592627229')

Much better indeed! Let’s test the previous cases that way!

>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(18)
Decimal('8751751364607.992147917157027451941369199107402191688475662016738729585793189576921000417819488022521008035769458236805623355117964654519012995149203145651482621970904733675616603829159073699152262190237686134096053140643178581342535160175294280596644277408')
>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(19)
Decimal('45824765067263.99400154247292878012837438073861701581979206241047287979476762412992360313819778117994498296523948378426713929520188369997784294269662668000818303924272345525577601783001726960434941938734208353272624615875651599971618368131896320524727780674')
>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(20)
Decimal('239941584945151.9954175862094628730047694880020933281648497263958823604254329864718576171579087349895858656483590697583803423507394435817910056755829474974431677475727217968321896916634673228294874675631017566599732643899663816729269614472126021090970897308')
>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(30)
Decimal('3716706651753989275647.999689800241818180639346264792151927705378585007996556972979408800792479361060761238505878803342888805078901553987993893381697765526226152915257781651542284398399012963211473811148141385735168384084704178517042635758936907558116236798')
>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(40)
Decimal('57571964185991590695825047551.99997900148385229515959986036977505867324065830938072692611318337057286921280055552228787014811837140299777965403896929514221392361243371590607709492852574169582705605198217499438837728596606637811005617061688138760057438182845')
>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(50)
Decimal('891792484798041273405092327480885247.9999985785363502863489704179083057190576105613126358868072802757567654991706904766380400266097425885364586125135675324533868405035416842283484903727056986268545269819514444772213000354546446009165968589958054059389753136')

Whoa, now that’s improvement! Since the small input ranges from 2 to 30, we’re covered, yay! To the large input! The first case is 910062006… much bigger! But we have faith in our solution!

>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(910062006)
Decimal('8.919448541446674973277872746171245933275138630248130907008894180215290388336023868239234075453030387703632363584380715738200630546564329205017138690712399728915139680036686573669248915790961561461272596637583821147468503082573913305207639264660422490421272E+654339383')

Aaaaaand it fails miserably. You say increase the precision? Go!

>>> getcontext().prec = 1024
>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(910062006)
Decimal('8.919448541446674973277872746171245933275138630248130907008894180215290388336023868239234075453030387703632363584380715738200630546564329205017138690712399728915139680036686573669248915790961561461272596637583821147468503082573913305207639264660422298653727910480913903028526252731111441440780088518055499910461897770689934875196042744680956819167861668586410285893670912335938531514972687567521690574174307478919726903450127130115796805326307689705722233029462717164687160362204672599178459426195515468743692514680427999432359495315725269307474956180961814633487158821252498531057196075527072557513060268997873150760403767129373225601570326857102899047033394123502152602095967110573983281744795256688834356089540911527223279951661861220253786982201657061127666645472375405207936633238083128248926879097462312848916323276895759197636621186062828933585137465868233553517455227951169710940268113754834340838971865855889844812381822071748466341030321954673834742153256457611174551211610695251648386140804719763671673855871318598E+654339383')
>>> getcontext().prec = 1000000000
>>> (Decimal(3) + Decimal(5).sqrt()) ** Decimal(910062006)

It is still computing that last command… maybe it’ll finish before Christmas! 😀 So… nothing we can do about it. Not in a natural way at least. It turns out that Code Jam organizers knew about this (that’s the whole point, indeed) and my approach, even if it’s able to solve the small input, is not the correct one. From the contest analysis:

The key in solving the problem is a mathematical concept called conjugation. In our problem, we simply note that (3 – √5) is a nice conjugate for (3 + √5). Let us define

(1)     α := 3 + √5,   β := 3 – √5,   and Xn := αn + βn.

We first note that Xn is an integer. This can be proved by using the binomial expansion. If you write everything down you’ll notice that the irrational terms of the sums cancel each other out.

(2)     

Another observation is that βn < 1, so Xn is actually the first integer greater than αn. Thus we may just focus on computing the last three digits of X.

Whoa, there’s a ton of math symbols there. What it is trying to explain is that there’s another way around, and we can write a simple algorithm to calculate only what we need. Taking that math apart, this suggested approach is quite interesting:

Solution C. [the periodicity of 3 digits]

For this problem, we have another approach based on the recurrence (7). Notice that we only need to focus on the last 3 digits of Xn, which only depends on the last 3 digits of the previous two terms. The numbers eventually become periodic as soon as we have (Xi, Xi+1) and (Xj, Xj+1) with the same last 3 digits, where i < j. It is clear that we will enter a cycle no later than 106 steps. In fact, for this problem, you can write some code and find out that the cycle has the size 100 and starts at the 3rd element in the sequence. So to solve the problem we can just brute force the results for the first 103 numbers and if n is bigger than 103 return the result computed for the number (n – 3) % 100 + 3.

Hmm… so that means that 103 restarts the cycle?

>>> getcontext().prec = 1024
>>> str((Decimal(3) + Decimal(5).sqrt()) ** Decimal(3))
'143.5541752799932702850935573994008395340997875075688231766687118531366696204097567812612610681210327990242616368247610721361045671823063636347635238890704866837369015421401333059608526503244754105771443632133610811201095485572500621467658412511120829640713240415845568781018115768290851039517558845033513247514268365293660766111722141944646319577243016028721322545978616614879992775936489738916811506453866403999620857004610560179974673677933515377954018029702474909866234538867118832726784429359876544843489516123361404699859563411214642284786110581426213620665031024800019372553994881602971764760169511058165201173143835473121683945817118282695117541643928648522370457633421861064234721885791403592429823579671023036790457862064776462707851047085181514532031422824259553334558658255398529105918381922193776593783203964365329603283979620022727823878012615898529089431909156945831424556056508947493929464193144955007377329721849300605964711369201940159834308432696751549585124838734830577613026312919174891129200931587878779'
>>> str((Decimal(3) + Decimal(5).sqrt()) ** Decimal(103))
'114167750723954075118506909152243184891636055812752354180398291472449798143.9999999999990991848908257663970808178511998207346190405521007125450976083864218743136022578058573724307353453054670910157005516822203809922099339999229730921853127935894210481666862007195197029531387123303773842565764868785959347563566969625431650825652899007545905552208934231424236695732218626707231897771500030743392334437529104775115629703470534218197873009424685334262220812905849999596760216536020750239094398005209138812883762761775627540601688798157222970582315547073108170909501918275307282209081948400177828835529550843580664303363677674984334542177398232468766365174239306558484619915875621147584977655598495728316487075831729616962367129848658475956289730636910101894486413362652617806007673400475350422005999251984236600004538944250397180445073308202368906957039259759659759171424708223454172257556499138474886346739987815229848744135302653447948550170944394443810710713254841099059181243617896686076589736820366175425243225873342031095'
>>> str((Decimal(3) + Decimal(5).sqrt()) ** Decimal(4))
'751.6594202199646689967411763468544075540238844147363216775107372289675155071512231016216206076354221948773735933299956287145489777071084090825085004176200550896187330962356998562944764142034959055300079068701456758805751299255628262705206665683384355613744512183189236100345107783526967957467183936425944549449908917791719022086541245209393177780525834150786943366387737228119962073666571129313260408882798620998009499274205440944867036809150955734258594655937993276797731329052373871815618254139351860428319959647647374674262707908876871995127080552487621508491412880200101705908473128415601764990889933055367306159005136233888840715539870984149367093630625404742444902575464770587232289900404868860256573793272870943149903775840076429216217997197202951293164969827362655006432955840842277806071505091517327117361820812917980417240893005119321075359566233467277719517523073965614978919296671974343129687014011013788730981039708828181314734688310185839130119271657945635321905403357860532468388142825668178428304890836363588'
>>> str((Decimal(3) + Decimal(5).sqrt()) ** Decimal(104))
'597790103628874365020709731601266597564608080186061135989016966589686218751.9999999999993118384917497799741823573220669410860125985943311312283719331038685561426361821548229163694601416604360587022213664498323168042300721157774333881227779113388111518769679330555901306111571209984513448987672187738925735173356845673100783866660482880393505457702913561140292994651458434392515606071335149084501581993111115430324514143366418268949755088875955878162140123915031739484539675175716477662141640774955323553027076342238845088259277403081453735076960587935422087441774083950451065986208934485511159638170169172967026214058721899896878428807562496126156550849295831738165499903388396019937912975536800221981964584521002823218293972856046263138381804404409768650023404607326059948575617153491987096854714709481244777860640664620489694413768067522712403081007060606347733537888589375255218262079999570621589681646821848243321335219275048192774559691932753194312696801857829138795089010920052048921614192197989676705158844531382420174'

Yay! To the git-pushed solution!

from decimal import *                            # Improved math

getcontext().prec = 1024                         # Adjust the precision
base = (Decimal(3) + Decimal(5).sqrt())          # Calculate the base

for case in range(1,int(raw_input())+1):         # For all test cases
  exp = (int(raw_input()) - 3) % 100 + 3         # Get the recurrent exponent

  num = str(base ** Decimal(exp))                # Calculate
  ans = num[:num.find('.')]                      # Get only the integer part

  print "Case #%d: %03d" % (case, int(ans[-3:])) # Zero-lead the last 3 digits

One thing that’s really interesting about this is that I actually solved this (of course, not from scratch) while posting! I was already OK with the idea of only solving the small input, but reading the analysis brought some new light into this problem. Blogging FTW! See ya!