#master method

I learned the Master Method once and then forgot it, which means it needs a blog post.

The Master Method is a way of determining the time complexity of a recursive divide-and-conquer algorithm based on its recurrence T(n).

Formulas for Reference

The formula for the recurrence is:

T(n) = aT(n/b) + O(nd)

And the upper bound on the level of work at a level j is:

cnd * (a/bd)j

Needed Values

a = the branching factor of the recurrence tree. When you divide the problem, how many subproblems do you divide it into? How many recursive calls do you make at each step?

b = the number you divide the problem size by. Each subproblem is 1/b the size of the parent problem.

d = the exponent on the big-oh runtime complexity of the work done outside of the recursive calls. Once you've done your recursive calls, how much work do you do?

bd = the rate at which the amount of work per subproblem decreases

Comparison

Since a is the rate at which the number of subproblems increases, we need to compare a to bd to figure out what dominates the runtime complexity: the proliferation of recursive calls (a), or the work done outside of the recursive calls (bd).

Answers

if a = bd: T(n) = O(nd log(n))

In this case, the number of leaves in the recurrence tree grows at the same rate as the amount of work done outside the recursive calls, so you multiply the amount of work done at one level in the tree, nd, by the number of levels, (log(n)).

if a < bd: T(n) = O(nd)

As you go down levels in the recurrence tree, the rate at which the number of subproblems increases is less than the rate at which the amount of work per subproblem decreases. So you end up with less work the further down you go, and the work in the root dominates the running time. The work done in the root is O(nd).

if a > bd: T(n) = O(nlogb(a))

As you go down levels in the recurrence tree, the rate at which the number of subproblems increases is greater than the rate at which the amount of work per subproblem decreases. So you end up with more work the further down you go, and the work in the leaves dominates the running time. There are logb(n) levels in the tree. We multiply the number of leaves by a at each level. So the number of leaves at the end is alogb(n), and the running time in this case is O(alogb(n)). Weirdly, this is the same as O(nlogb(a)).

Another Way

Another way to do the comparisons is to take the logb of both sides, comparing logb(a) with d. This way, you deal with that more complicated value in the beginning of the calculation, and you can just plug it in if you end up using the third case.

Let's define x as logb(a).

if x = d: T(n) = O(ndlog(n))

if x < d: T(n) = O(nd)

if x > d: T(n) = O(nx)

Sources:

Roughgarden, Tim. Algorithms: Design and Analysis, Part I, Week 2. Coursera www.coursera.com.

http://www.cs.ucdavis.edu/~gusfield/cs222f07/mastermethod.pdf

1. jack of clubs signifies a sincere and lasting friendship founded upon a basis that will endure -- Project in hand (right now, the only project of note going on with my closest friends is Sherri's wedding, which she has asked me to be a bridesmaid in =) I AM SO EXCITED. and this makes absolute perfect sense because Sherri and I have this weird sisterlike bond that doesn't waver with distance or time. And we also have this weird emotional connection... and we both have a thing for the mystical hidden meanings and rely on karma and hope and faith in things, and it makes me happy to see that this spread suggest that her friends, most of whom are involved in the wedding, but some are not directly (that i know of, but are still very dear friends to her--such as Casey, who is in basic right now--i do hope his letter made it there and it made him smile)  - kind and trusty friends will help foreward the project. which is interested because Casey's ex that pretty much emotionally tortured Sherri while they were roommates has decided to pop back up in their circle of friends after all this time--and with the news of her own engagement, and saying how it was before Sherri's and the poor girl is just trying to get the attention put back on her. It's really pathetic. No one likes her anymore after the horrible things she did to everyone. The girl makes ME LOOK LIKE A SAINT. And we all know I am quite the kiniving bitch when it suits me. But she is just legit HEARTLESS. atleast I am only mischeivious when there is a task to be done by it.

2. 7 of diamonds when on square two is strickly taken as an OMEN, and in regards to SATISFACTIONNNN... and THE WORKPLACE... hahahahahaha ....there will be jealousy at work to mar the inquirer's satisfaction. oh god, lets just say that while I liked the personality of the lady my boss hired to help out, I was insanely jealous of her. Like ridiculously. Because... SHE got the actual decent desk/computer/everything. *I* got shoved into the backroom whith no AC, no working computer, and half the time i couldn't even hear what was going on. It was like I was the unwanted foster child. SO THIS IS FUCKING CREEPY ACCURATE, KTHX.

3. 10 of diamonds signifies a journey, and being on this square (and the happenings of my life as of late!!!) I am taking this to mean that this is not so much a physical journey, but my emotional/health/recovery journey from my eating disorder and mental illness. I am pleased to see it show up in the square denoting sucess, but saddened that apparently while I will go very far on the road to recovery, i will not make it there completely (though most people who have suffered with chronic eating disorders never do COMPLETELY AND FULLY RECOVER. so, i am not surprised....)  -- Success - (the success will be, but incomplete)

4. ace of spades is a card of GOOD OMEN! meaning perseverance followed by possession, a happy marriage, success, and advancement in business.... and this particular square concerns Hope; this being covered with a spade means the hopes are wild in nature -- and NO DOUBT, I am pretty out of my gourd and oh wait, this fucking book actually SAYS IT FOR ME?! "TENDING TO MANIA" how fucking appopriate! wowwww. =3

5. ace of hearts represents the HOUSE of the inquirer-- Chance, Luck - good luck will attend the hopes and plans of the inquirer-- and well lets just say that... it is my HOPE AND PLAN that i will be MOVING out of this house and into a new place with Jacob at some point in the next year. (and by next year, I actually mean probably loser to... 9-12 months from now... and that's IF my job seeking goes well and I am able to get myself situated... however, having seen so many GOOD signs regarding things of this NATURE in this spread I am almost willing to go ahead and say it will be probably be so. =3 but I DONT WANT TO PUT ALL MY EGGS IN ONE BASKET... I've already been through a shit fest with a man failing to be a good husband, and I don't want to go through anything near that again. Ever. EVER. so i am very hesitant to put my trust in someone with regards of living with them and have stability... but, for some reason I think it will probably be okay. But I refuse to state it at this point due to the circumstances...

6. king of spades -- Wishes, Desire - bad luck, robbery, financial ruin, disaster-- do i even have to say this? my wishes and desires for my life with you know who fell to pieces... and I am just glad it's in the PAST NOW! but yeah.. D and I.. ugh... we had extremely bad luck with cars and finances... it was, infact, a fucking disaster. nothing else needs to be fucking said. haha. this is legit to a "t" truth and right, right here.

7. 10 of clubs -- Injustice - the wrong already done will require long and courage efforts to efface its effects, and the inquirer will need the support of best friends. oh good lord... what wrongs have i not done? and i am still working my way through the mess. i think i will be at this forever. the support of my best friends? WHAT WOULD I DO WITHOUT YOU GUYS!!! you know who you are!!!! <3333

8. jack of spades -- Ingratitude - ingratitude will bring about sure trouble and serious misfortunes.... hmmmmm.... but WHOS INGRATITUDE?... with the way things have been lately i would think this is about my brother. he sits on his all day everyday. he won't so much as take his clothes from the laundry room to his BEDROOM. he expects my mother to cook him a full 4 course breakfast every day. and he NEVER THANKS HER AT ALL. and here lately i have decided to STOP OVER LOOKING IT! at least i do shit and ask if there's anything else I CAN DO TO HELP OUT AROUND THE HOUSE. He doesn't do SHITTTT and he and I have been fighting.. liteterally AT EACH OTHERS THROATS. I am fucking shocked as hell that I have been able to hold my bipolar anger rage down and not rip his limbs from his body and toss them out inside the yard, douse them in gasoline and set them on fire. THAT IS THE AMOUNT OF RAGE HE INCITES IN ME. YOU HAVE NO IDEA WHAT ITS LIKE TO LIVE WITH THIS LITTLE MONSTER! UGHHH HE JUST WALKS ALL OVER THE WOMEN IN THIS HOUSE DAY AFTER DAY AFTER DAY LIKE ITS HIS GOD GIVEN RIGHT TO HAVE EVERYTHING HANDED TO HIM ON A SILVER FUCKING PLATTER. Now, dont get ME wrong, I am SPOILED AS HELL by my dad. But *I* atleast ALSO know how to work hard to get what I want (which is something that softball/marching band taught me) my BROTHER didn't keep active in team activities that lead to productive workers in the workforce and as such has no work ethic.. literally... just.... lays on his ass. literally. all day.. everyday. and doesn't think anything of it. RAGE RAGE RAGE RAGE RAGE RAGE RAGE

9. king of diamonds -- Association - inquirer will need to use all possible caution and diplomacy, AND EVEN THEN the results will be unsatisfying 

10. 9 of hearts -- Loss - loss of a benefactor, which will be a great grief -- this makes me sad because it goes along with the shorter readings i've done that have had spades come up in places signifying that one of my grandparents will pass away soon. =( and a benefactor would be my nana, which is who most of my readings points to being the one... it's very saddening. and yet again, i need to go see her. very soon. still have a bad feeling regarding her health.....

11. 9 of spades -- Trouble - trouble arising from jealousy -- i'm willing to say this could involve my jealousy against the other secretary and how she got to take my desk and all my space at the office and I got shoved off into the back room where there's no AC or internet or nasdjbfoekldbjfgorkd yeah. I really fucking pissed me off. I just kept my mouth shut =( but thought I was jealous of her, I still thought she was cool as a person. baaah

12. queen of diamonds -- State or Condition - inquirer will only attain a satisfactory position in life after he/she has overcome numerous and powerful enemies, though may never get very far... --- OH NO FUCKING SHIT. my enemies are of the physical human and social/emotion/mentalhealth/health sense. =( I was actually talking to my both my parents earlier today about how I feel like a huge disappopintment and they said that I WAS NOT! and that i, having atleast LEFT THE CITY I WAS BORN IN FOR A WHILE, have done more than many young women in the south do-- I realized there are more options that being barefoot and pregnant in a kitchen. hahaha. I can't believe my dad ACTUALLY SAID THAT. I was like "wow, dad. THANK YOU. FOR THE LOVE OF GOD. FINALLY THANK YOU." and my mom also said something to the effect of-- that I just haven't found my niche in life yet. that i'm still looking for my "thing" that will be what i need to be doing with myself, that will keep me content and on my feet financially at the same time. granted i have to defeat my ENEMIES of.. you know, my EATING DISORDER that is really really out of hand. =( 

13. 10 of hearts on this square signifies nothing more than what it is... which is awesome in and of itself, if i dont say so myself!-- Joy - deep unruffled delight, joy of a pure nature!!!! =)

14. 2 of spades -- Love - love will be slighted and betrayed--- hmmmm this worries me! as i know there are some rude people online who, after finding out the name and somethings about Jacob, would probably find some amusement in trying to get him to leave me or dislike me. slighted and betrayed? haha. but here's the good part-- i have already warned him of the childishness of my ex and his little troll friends. and while he himself is very much into... internet culture lol (lets just say I showed Martin /b/ in spring 2007, and that summer he showed Jacob and the rest is.. history.. i somehow planted a seed that made this happen, i think. ISNT THAT FUCKING WICKED COOL. FUCK YEAH IMMA GYPSY BIATCH WHAT WHAT WHAT WOT WOT) he HIMSELVES does not find their method of heartless trolling amusing. At all. it's one thing to poke fun. It's another to intentionally cause serious emotional distress to an innocent human being. point made. case closed. moving on.

15. 8 of diamonds is supposed to signify a short journey.. hmmm and being on the Prosperity square.... i suppose i could interpret this to mean my JOURNEY to prosperity will be short (it doesss seem as if things might line up for Jacob and I to be more serious and possibly living together sooner rather than later, butttt i don't take this to heart because it really all depends on how we feel and finances and other such random things)--- and "prosperity will be damaged by the jealousy of others" , i'd be willing to say this has to do with my brother again, being a little dick, my brother is jealous of how i am an adult and get to do what i want with who i want when i want. that i get to stay out all night with my boyfriend and such... like, DUH i'm 25 fucking years old, dude. you're 16. THERES A REASON YOU DONT GET TO STAY OUT ALL NIGHT. YOURE A KID FFS. he acts like he should already be getting to do everything i can do as an adult.. completely and totally unrealistic and unacceptable.

16. ace of clubs -- Marriage - fortells a marriage promptued by practical or financial considerations alone -----..........................

17. 7 of hearts -- Sorrow, Affliction - inquirer will pass through a love trouble but it will only be of short duration -- this obviously has to be the whole ordeal with Jacob and I having that little.... spat or sorts that WASNT REALLY A SPAT AT ALL. just me being mopey and stupid and thinking he was going to leave me. but he told me he waaaaasnt and that i had no SAY SO IN THE MATTER WHATSOEVER =3

18. 10 of spades -- Pleasure, Enjoyment - love will be unreal and evanescent, unable to bear the test of time, or to survive the first disagreement-- i have no doubt in my mind that this is in reference to the PAST LOVE, aka Dennis. I don't even want to go into details and explain how this is absolutely spot on and relevant-- so. Anybody who reads this and knows me well will already know what I mean anyways. =)

19. 9 of clubs signifies a PRESENT/GIFT-- and since this square denotes Inheritance and Prosperty - a friend will bequeath a portion of his property or money to the inquirer... however, since this card is AFTER a spade-- the recipient of the gift will derive NO pleasure from it.

hmmmm i wonder.... this is something i am unsure about-- most of the monetary gain related cards i've had come up point to a RELATIVE leaving me such things in a WILL. but this mentions a FRIEND... i am more apt to think this will end up being "property" than money. and that i woudl derive no pleasure from it... i wonder if this is isn't something like... a hand me down object from someone i know.

20. 8 of spades -- Fraud, Deciet - deceit and underhand dealings will culminate in calumny which will cost the inquirer many friends, and have serious consequences for her. -- ahaha well no fucking duh. this is obviously having to do with James/Dennis/that whole fucking mess. abso-fucking-lutely. like, no way to argue any other meaning.  NO FURTHER EXPLINATION NEEDED.

21. queen of spades -- Rivals - the rival will triumph over the inquirer, robbing her and plunging her into disgrace both with her benefactors and with members of her own immediate circle -- 

22. jack of hearts  ON THIS PARTICULAR SQAURE specifically signifies the FIANCE of the enquirer (i couldn't make this up if I tried, it's in the god damn book lol. CREEEPY SHITTTTT MAN)... GIVEN THAT I DONT HAVE ONE.. YETTT... AAAAAAAAAAAANDDDD this on the present/gift square... omggg hahaha-- A Present, Gift - the inquirer will have a very ***handsome and unexpected*** present. unexpected? YESSSSS, handsome?! YOU HAVE NO IDEA. *clings to her Jacob* miiiiineeee

23. 8 of hearts -- Lover - the lover of the inquirer will be both kind and faithful (and INDEED HE ISSSSSSSSSSSSSSSSSSSSSS =3 ) Jacob is... the kindest most patient person I've ever known. and okay, faithful? he hasnt went anywhere despite my crazyness. AND guess what? just found out YESTERDAY he's CHANGING HIS CELL PHONE PLANS' "FAVE FIVE" with T-Mobile so that I'm in it. so he doesn't get charged to talk to me. Which means I am being talked to enough to need to be there to avoid using up tons of cell phone minutes... AND it also mean, well, you know, while it's not REALY hard to change you fave five... i'm sure he wouldn't do it if he wasnt intent on keeping me around for.. quite a while. so that seems like it would be a sign of faithfulness, does it not? lol =P

24. 9 of diamonds -- Advancement, A Rise in the World - advancement will only be obtained after a hard struggle against difficulties, caused by the jealous ill-will of others.. and SINCE THE 9 of diamonds represents news, so i am to gather that there will be news of my advancement in the world (this must mean my CAREER/JOB! SURELY!!!) haha.... but after a hard struggle? i can ONLY IMAGINE this must signify the fact that it's HELL ON EARTH GETTING HIRED RIGHT NOW!!! AAAAHHH but! since the card BEFORE THIS ONE is of the HEARTS suite-- the advancement will STILL BE GAINED, AND I WILL BE VICTORIOUS,... just might take a while =P

25. 2 of diamonds -- Kindness, A Good Turn - the inquirer will only obtain a modicum of kindness, and that is only after she has surmounted serious obstacles built up by the jealousy of her enemies-- ohh this totally fucking goes along with everything the other cards have said! I don't even think I NEED to be explaining this!!! =3

26. ace of diamonds -- Undertaking, Enterprises - the success of the undertaking will be hindered and decreased by the jealousy and self-seeking of some people concerned in it.************************

27. king of clubs -- Changes - a change very much for the  better will take place in the inquirer's circumstances, owing to the good offices of friends-- hmmm indeeeeeeeeeeeeeeed. i think this pretty much speaks for itself haha.

28. 2 of clubs -- The End (of Life) - a handsome legacy from a friend may be expected by the inquirer -- and the two of clubs is a trusty friend. and yet again, this being here makes me think again of my nana, because while she is very mentally ill-- she is one of the people who i can talk to about my problems, if i really really needed to. and i am actually getting close to the point that i might need to. she opened up to me and told me about her strange childhood that caused her severely warped sense of self and anxiety =/ she told me how insane her sisters were and now that they're older she tries to pretend its not like that but... it totally is. they're just... horrible people. and my grandmother really is just being used and manipulated by them and yes, she does bad things too, but she does it in response to the way they treated her all her life. =(

29. queen of clubs -- Rewards - a due and fitting reward will be meted out to industry and perseverance .... surely.. SURELY this has to do with me seeking a job  oh please please haha. rewards. i mean, surely. =3 and ESPECIALLY THE PERSEVERANCE PART!!! I CANNOT GIVE UP. I MUST KEEP SEARCHING... AND I WILL FIND THE PERFECT JOB FOR ME! =3 

30. jack of diamonds -- Misfortune, Disgrace - misfortune will be brought about by jealousy, and will indirectly affect the inquirer-- jealous is coming up a LOT IN THIS SPREAD. this could mean MY jealousy of others =( omggg..., or also.. the opposite. so i am left wondering who is so jealous of me. i get the feeling that my mother is jealous of how thin i am-- because she has gained weight and she keeps.. being weird and underhandedly slighting me about my eating disorder. like, she's jealous and wishes she could do atleast SOME of the same things i do but lacks the self-control. i mean, yes okay. it's weird. i have tons of self control over this aspect of my life-- over food and regulating that. but i have ABSOLUTELY NOT CONTROL over so many other things--like my MOODS and my hypersexuality. and i think that's WHY i am so stuck in my eating disorder. i need to control something so i control the only thing i can... and its.. more or less slowly killing me.

31. 7 of clubs -- Happiness - the inquirer will experience unexpected happiness which will be both deep and lasting to her. expected happiness? can we say RANDOMLY BUMPING INTO JACOB ON TWITTER?!?!?!?!?!? CAN I GET A HELL YES!? and i sure hope it's lasting -3 I'm not sure what i would do if it WASNT

32. king of hearts, being on square 32 is representative of a lover. and since this square also involves MONEY&FORTUNE - the inquirer will rapidly acquire a large fortune (OF LOOOOOOOOVEEEEE) by a lucky speculation (indeed, something told me after following J on twitter that there was something very, very on target about how our personalities would mesh-- which is WHY i jumped on the chance to try and hang out with him and offered to talk to him on AIM... which QUICKLY developed into talking to him.. all the time... =3 >.> and... the decision to go out on a date almost immediately.... AND THEN after talking to Martin and hearing J talk ABOUT him and Martin.. everything really did CREEPILY line up in regards to our interests and pasts and other random thingsss ahaha) =3

33. queen of hearts signifies a married woman who desires the happiness of the inquirer and does her best to promote it... being in THIS square which is Indifference, and being covered by a heart says that thanks to her indifference and the want/caring of heart, the inquirer will lead an unruffled --though somewhat joyless life. (my mom is obviously the married woman here. she does want me to be happy but she stays out of my love life and always has tried to be indifferent about who i am dating (which is how i let myself be stupid enough to leave with D, pretty much... she knew i shouldn't have but her indifference and refusal to intervene made things go this way) and now you know, i really do wish she would be more active in telling me what she had thought of the men i date. because i often overlook the things that i SHOULD NOTICED THAT ARE EVENTUALLY THE DOWN FALL OF THE RELATIONSHIPS... AND if she were more into my love life she would MOST DEFINITELY BE CELEBRATING WITH ME EVERYTIME JACOB AND I HAVE AN ADORABLY CUTE DATE/MOMENT/JOKE/WHATEVER. But she has always remained very STOIC in that realm of her parenting and that is not going to change. 

34. 7 of spades -- signifies all troubles and worries concerning tender passions of the inquirer... and this being in this square is very interesting as this square reguards Favor and that no effort of any kind will be required in the tender passionate favor i am seeking >.> teeeheee i wonderrrrr <3 =3 (i don't talk about mine and J's sex life much but *ahem* lets just say that he is the best match for me in the bedroom I've come across as of yet -- we just work well, everything is perfect aaaaand he is very much into pleasing ME which is nice for a chance considering how little my exs have done that, haha -- though I will give D credit, he was the best out of all my past partners at the time we were together, so I was pleased with him. But that's irrelevant NOW obviously... and now I am with J and he is  so sos o so much more.. patient? and ATTENTIVE!!!!!... >.>)

35. 8 of clubs -- Ambition - the moderate ambition of the inquirer will be realized

36. 2 of Hearts, represents me, Tabitha (the inquirer); on this square I am affected ENTIRELY and thoroughly by the fact that is is on the square for Ill-Health, Sickness (this does not surprise me at all, given my eating disorder and illhealth) - however, I am thankful to know that it will pass-- (RECOVERY IS POSSIBLE! .... okay, now I really wanna try and get back my therapy appointment after all lol, damn ittt)  inquirer will suffer from passing ailments that will leave no bad results. So hopefully this is accurate and my worries (which concern my ability to bear children after so much malnutrition) are actually nothing to worry about.. yay!

_______________________________________

It took me FOREVER TO FINISH THIS!!!! It's a huuuuge spread. VERY THOROUGHHH.

I dunno.. I hope somebody finds this as interesting as I do. Hahaha.