The Occupancy and Coupon Collector problems

Transcription

1 Chapter 4 The Occupacy ad Coupo Collector problems By Sarel Har-Peled, Jauary 9, 08 4 Prelmares [ Defto 4 Varace ad Stadard Devato For a radom varable X, let V E [ X [ µ X deote the varace of X, where µ X E X the dstrbuto of X The stadard devato of X, deoted by σ X s the quatty V [ X Observato 4 For ay costat c 0, we have V cx c V X [ For X ad Y depedet varables, we have V X + Y V X + V Y X E [ X µ X Itutvely, ths tells us how cocetrated s Defto 43 Beroull dstrbuto Assume, that oe flps a co ad get heads wth probablty p, ad 0 e, tal wth probablty q p Let X be ths radom varable The varable X s has Beroull dstrbuto wth parameter p We have that E X p + 0 p p, ad V X E [ X µ X E [ X p p p p p pq Defto 44 Bomal dstrbuto Assume that we repeat a Beroull expermet tmes depedetly! Let X,, X be the resultg radom varables, ad let X X + + X The varable X has the bomal dstrbuto wth parameters ad p We deote ths fact by X B, p We have Also, E X p, ad V X b;, p Pr [ V X [ X V [X pq p q Observato 45 Let C,, C be radom evets ot ecessarly depedet Tha [ Pr C Pr[C Ths s usually referred to as the uo boud If C,, C are dsjot evets the [ Pr C Pr[C Ths wor s lcesed uder the Creatve Commos Attrbuto-Nocommercal 30 Lcese To vew a copy of ths lcese, vst or sed a letter to Creatve Commos, 7 Secod Street, Sute 300, Sa Fracsco, Calfora, 9405, USA

2 4 Geometrc dstrbuto Defto 46 Cosder a sequece X, X, of depedet Beroull trals wth probablty p for success Let X be the umber of trals oe has to perform tll ecouterg the frst success The dstrbuto of X s geometrc dstrbuto wth parameter p We deote ths by X Geomp Lemma 47 For a varable X Geomp, we have, for all, that Pr X p p Furthermore, E X /p ad V X p/p Proof: The proof of the expectato ad varace s cluded for the sae of completeess, ad the reader s of course ecouraged to sp readg ths proof So, let f x 0 x x, ad observe that f x x x As such, we have V X E X p p p f p p p p E [ X p p p p p + p p p p We eed to do a smlar trc to what we dd before, to ths ed, we observe that f x x x x 3 As such, we have that x x x + x f x + x f x + x x f x x 3 + x x x 3 + x x x x 3 + x x x x x 3 + x x As such, we have that V X p + p p p p p + p p p p p p3 + p + p p 3 p p p + p p + p p p p p 4 Some eeded math Lemma 48 For ay postve teger, we have: + / e / e

3 ! /e v For ay, we have: e Proof: Ideed, + / exp/, sce + x e x, for x 0 As such + / exp/ e Rewrtg the equalty, we have that we eed to prove e Ths s equvalece to provg e +, whch s our fred from Ideed,! 0! e, by the Taylor expaso of e x x 0! Ths mples that /e!, as requred v Ideed, for ay, we have sce As such,, for As such, +!!! As for the other drecto, we have! e e, by 4 Occupacy Problems Problem 4 We are throwg m balls to bs radomly e, for every ball we radomly ad uformly pc a b from the avalable bs, ad place the ball the b pced There are may atural questos oe ca as here: A What s the maxmum umber of balls ay b? B What s the umber of bs whch are empty? C How may balls do we have to throw, such that all the bs are o-empty, wth reasoable probablty? Let X be the umber of balls the th bs, whe we throw balls to bs e, m Clearly, E [X j Pr[The jth ball fall th b, by learty of expectato The probablty that the frst b has exactly balls s 3 e e

4 Ths follows by Lemma 48 v Let C j be the evet that the jth b has or more balls t The, Pr C Let 3 l /l l The, Pr C e e e e/ + e + e + e 3 l /l l e exp l l + l l l l l l exp 3 l + 6 l exp 5 l l l, e/ exp l 3 l l + l l l for large eough We coclude, that sce there are bs ad they have detcal dstrbutos that Pr[ay b cotas more tha balls C Theorem 4 Wth probablty at least /, o b has more tha 3 l l l balls t Exercse 43 Show that for m l, wth probablty o, every b has Olog balls It s terestg to ote, that f at each terato we radomly pc d bs, ad throw the ball to the b wth the smallest umber of balls, the oe ca do much better We curretly do ot have the machery to prove the followg theorem, but hopefully we would prove t later the course Theorem 44 Suppose that balls are sequetally places to bs the followg maer For each ball, d bs are chose depedetly ad uformly at radom wth replacemet Each ball s placed the least full of the d bs at the tme of placemet, wth tes broe radomly After all the balls are places, the maxmum load of ay b s at most l l /l d + O, wth probablty at least o/ Note, eve by settg d, we get cosderable mprovemet A proof of ths theorem ca be foud the wor by Azar et al [ABKU00 4 The Probablty of all bs to have exactly oe ball Next, we are terested the probablty that all m balls fall dstct bs Let X be the evet that the th ball fell a dstct b from the frst balls We have: [ m m m Pr X Pr[X Pr X + m X j 3 j m e / mm exp, 4

5 thus for m +, the probablty that all the m balls fall dfferet bs s smaller tha /e Ths s sometme referred to as the brthday paradox You have m 30 people the room, ad you as them for the date day ad moth of ther brthday e, 365 The above shows that the probablty of all brthdays to be dstct s exp 30 9/730 /e Namely, there s more tha 50% chace for a brthday collso, a smple but coutertutve pheomea 43 The Marov ad Chebyshev s equaltes We remd the reader that for a radom varable X assumg real values, ts expectato s E[Y y y Pr[Y y Smlarly, for a fucto f, we have E[ f Y y f y Pr[Y y Theorem 43 Marov s Iequalty Let Y be a radom varable assumg oly o-egatve values The for all t > 0, we have Pr Y t E[ Y t Proof: Ideed, E Y y t y Pr Y y t Pr Y y y t + y Pr[Y y y<t t Pr y t y Pr Y y Y t Marov equalty s tght, as the followg exercse testfes Exercse 43 For ay teger >, defe a radom postve varable X such that Pr[X E[X Theorem 433 Chebyshev s equalty Pr [ X µ X tσ X V [ X Proof: Note that Set Y X µ X Clearly, E Y Pr [ X µ X tσ X Pr [ X µ X t σ X t, where µ X E X ad σ X σx Now, apply Marov s equalty to Y 44 The Coupo Collector s Problem There are types of coupos, ad at each tral oe coupo s pced radom How may trals oe has to perform before pcg all coupos? Let m be the umber of trals performed We would le to boud the probablty that m exceeds a certa umber, ad we stll dd ot pc all coupos 5

6 { } Let C,, be the coupo pced the th tral The jth tral s a success, f C j was ot pced before the frst j trals Let X deote the umber of trals from the th success, tll after the + th success Clearly, the umber of trals performed s X 0 Now, the probablty of X to succeed a tral s p /, ad X has the geometrc dstrbuto wth probablty p As such E [X /p, ad V [X q/p p /p Thus, E X 0 E [X 0 X H l + Θ l + O, where H / s the th Harmoc umber As for varace, usg the depedece of X 0,, X, we have V X Sce, lm 0 0 V [X 0 p p 0 π /6, we have lm V[X / π 6 0 / 0 H Corollary 44 Let X be the umber of rouds tll we collecto all coupos The, V ad ts stadard devato s σ X π 6 X π 6 Ths mples a wea boud o the cocetrato of X, usg Chebyshev equalty, but ths s gog to be qute weaer tha what we mpled we ca do Ideed, we have Pr [X log + + t 6 π X Pr[ E[X tσ X t, Note, that ths s somewhat approxmate, ad hold for suffcetly large 45 Notes The materal ths ote covers parts of [MR95, sectos 3,3,36 Bblography [ABKU00 Y Azar, A Z Broder, A R Karl, ad E Upfal Balaced allocatos SIAM J Comput, 9:80 00, 000 [MR95 R Motwa ad P Raghava Radomzed Algorthms Cambrdge Uversty Press, Cambrdge, UK, 995 6