Optimization Results for a Generalized Coupon Collector Problem

Transcription

1 Optimizatio Results for a Geeralized Coupo Collector Problem Emmauelle Aceaume, Ya Busel, E Schulte-Geers, B Sericola To cite this versio: Emmauelle Aceaume, Ya Busel, E Schulte-Geers, B Sericola. Optimizatio Results for a Geeralized Coupo Collector Problem. Joural of Applied Probability, Applied Probability Trust, 2015, pp.9. <hal > HAL Id: hal Submitted o 1 Sep 2015 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 Applied Probability Trust (15 Jue 2015) OPTIMIZATION RESULTS FOR A GENERALIZED COUPON COLLECTOR PROBLEM EMMANUELLE ANCEAUME, CNRS YANN BUSNEL, Esai ERNST SCHULTE-GEERS, BSI BRUNO SERICOLA, Iria Abstract We study i this paper a geeralized coupo collector problem, which cosists i aalyzig the time eeded to collect a give umber of distict coupos that are draw from a set of coupos with a arbitrary probability distributio. We suppose that a special coupo called the ull coupo ca be draw but ever belogs to ay collectio. I this cotext, we prove that the almost uiform distributio, for which all the o-ull coupos have the same drawig probability, is the distributio which stochastically miimizes the time eeded to collect a fixed umber of distict coupos. Moreover, we show that i a give closed subset of probability distributios, the distributio with all its etries, but oe, equal to the smallest possible value is the oe, which stochastically maximizes the time eeded to collect a fixed umber of distict coupos. Keywords: Coupo collector problem; Optimizatio; Schur-covex fuctios 2010 Mathematics Subect Classificatio: Primary 60C05 Secodary 60J05 Postal address: CNRS, Campus de Beaulieu, Rees Cedex, Frace Postal address: Esai, Campus de Ker-La, BP 37203, Bruz, Cedex, Frace Postal address: BSI, Godesberger Allee , Bo, Germay Postal address: Iria, Campus de Beaulieu, Rees Cedex, Frace 1

3 2 E. Aceaume, Y. Busel, E. Schulte-Geers ad B. Sericola 1. Itroductio The coupo collector problem is a old problem, which cosists i evaluatig the time eeded to get a collectio of differet obects draw radomly usig a give probability distributio. This problem has give rise to a lot of attetio from researchers i various fields sice it has applicatios i may scietific domais icludig computer sciece ad optimizatio, see [3] for several egieerig examples. More formally, cosider a set of coupos, which are draw radomly oe by oe, with replacemet, coupo i beig draw with probability p i. The classical coupo collector problem is to determie the momets or the distributio of the umber of coupos that eed to be draw from the set of coupos to obtai the full collectio of the coupos. A large umber of papers have bee devoted to such aalysis whe teds to ifiity, see [4] ad the refereces therei. We suppose i this paper that p = (p 1,..., p ) is ot ecessarily a probability distributio, i.e. we suppose that i=1 p i 1 ad we defie p 0 = 1 i=1 p i. This meas that there is a ull coupo, deoted by 0, which is draw with probability p 0, but that does ot belog to the collectio. We are iterested, i this settig, i the time eeded to collect c differet coupos amog coupos 1,...,, whe a coupo is draw, with replacemet, at each discrete time 1, 2,... amog coupos 0, 1,...,. This time is deoted by T c, (p) for c = 1,...,. Clearly, T, (p) is the time eeded to get the full collectio. The radom variable T c, (p) has bee cosidered i [7] i the case where the drawig probability distributio is uiform. The expected value E{T c, (p)} has bee obtaied i [5] whe p 0 = 0. Its distributio ad its momets have bee obtaied i [1] usig Markov chais. I this paper, we prove that the almost uiform distributio, defied by v = (v 1,..., v ) with v i = (1 p 0 )/, where p 0 is fixed, is the distributio which stochastically miimizes the time T c, (p) whe p 0 = 1 i=1 p i. This result was expressed as a coecture i [1] where it is proved that the result is true for c = 2 ad for c = extedig the sketch of the proof proposed i [3] to the case p 0 > 0. It has bee proved i [1] that the result is true for the expectatios, that is that E{T c, (u)} E{T c, (v)} E{T c, (p)}, where u = (1/,..., 1/) is the uiform distributio. We first cosider i Sectio 2, the case where p 0 = 0 ad the we exted it to

4 Optimizatio results for a geeralized coupo collector problem 3 oe of p 0 > 0. We show moreover i Sectio 3, that i a give closed subset of probability distributios, the distributio with all its etries, but oe, equal to the smallest possible value is the oe which stochastically maximizes the time T c, (p). This work is motivated by the worst case aalysis of the behavior of streamig algorithms i etwork moitorig applicatios as show i [2]. 2. Distributio miimizig the distributio of T c, (p) The distributio of T c, (p) obtaied i [1] usig Markov chais, is give by c 1 ( ) i 1 P{T c, (p) > k} = ( 1) c 1 i (p 0 + P J ) k, (1) c i=0 J S i, where S i, = {J {1,..., } J = i} ad, for every J {1,..., }, P J is defied by P J = J p. Note that we have S 0, =, P = 0 ad S i, = ( i). This result also shows that the fuctio P{T c, (p) > k}, as a fuctio of p, is symmetric, meaig that it has the same value for ay permutatio of the etries of p. We recall that if X ad Y are two real radom variable the we say that X is stochastically smaller (resp. larger) tha Y, ad we write X st Y (resp. Y st X), if P{X > t} P{Y > t} (resp. P{X > t} P{Y > t}), for all real umbers t. This stochastic order is also referred to as the strog stochastic order The case p 0 = 0 This case correspods the fact that there is o ull coupo, which meas that all the coupos ca belog to the collectio. We thus have i=1 p i = 1. For all 1, i = 1,..., ad k 0, we deote by N (k) i the umber of coupos of type i collected at istats 1,..., k. It is well-kow that the oit distributio of the N (k) i is a multiomial distributio, i.e., for all k 1,..., k 0 such that i=1 k i = k, we have We also deote by U (k) P{N (k) 1 = k 1,..., N (k) = k } = k 1! k! pk1 1 pk. (2) the umber of distict coupo s types, We clearly have, with probability 1, U (0) = 0, U (1) = 1 ad, for i = 0,...,, P{U (k) = i} = P{N u (k) > 0, u J ad N u (k) = 0, u / J}. J S i,

5 4 E. Aceaume, Y. Busel, E. Schulte-Geers ad B. Sericola It is easily checked that T c, (p) > k U (k) < c. We the have usig Relatio (2), P{T c, (p) > k} = P{U (k) < c} = = = c 1 i=0 c 1 P{N u (k) J S i, c 1 i=0 i=0 J S i, k E k,j J P{U (k) = i} > 0, u J ad N (k) u = 0, u / J} p k k!, (3) where E k,j = {k = (k ) J k > 0, for all J ad K J = k}, with K J = J k. Theorem 1. For all 2 ad p = (p 1,..., p ) (0, 1) with i=1 p i = 1, ad for all c = 1,...,, we have T c, (p ) st T c, (p), where p = (p 1,..., p 2, p 1, p ) with p 1 = λp 1 + (1 λ)p ad p = (1 λ)p 1 + λp, for all λ [0, 1]. Proof. The result is trivial for c = 1, sice T 1, (p) = 1 for all p. Moreover, we have P{T c, (p) > 0} = 1 for all p. We thus suppose ow that c 2 ad k 1. The fact that k 1 implies that the term i = 0 i Relatio (3) is equal to 0. We the have P{T c, (p) > k} = c 1 i=1 J S i, k E k,j J p k k!. (4) To simplify the otatio, we deote by T i (p) the ith term of this sum, that is T i (p) = J S i, k E k,j J p k k!. (5) For i = 1, we have S 1, = {{1},..., {}} ad E k,{} = {k}, thus, T 1 (p) = =1 pk. For i 2 we itroduce the followig partitio of the set S i,. S (1) i, = {J {1,..., } J = i with 1 J ad / J}, S (2) i, = {J {1,..., } J = i with 1 / J ad J}, S (3) i, = {J {1,..., } J = i with 1 J ad J}, S (4) i, = {J {1,..., } J = i with 1 / J ad / J}. These subsets ca also be writte as S (1) i, = S i 1, 2 { 1}, S (2) i, = S i 1, 2 {},

6 Optimizatio results for a geeralized coupo collector problem 5 S (3) i, = S i 2, 2 { 1, }, ad S (4) i, = S i, 2. The term T i (p) of Relatio (5) becomes T i (p) = J S i 1, 2 k E k,j { 1} + + J S i 1, 2 k E k,j {} J S i 2, 2 k E k,j { 1,} + J S i, 2 k E k,j p k k! J p k k! J p k. k! J pk 1 k 1! pk p k k! J k! pk 1 k 1! We itroduce the sets L k,j = {k = (k ) J k > 0, for all J ad K J k}. To clarify the otatio, settig k 1 = l ad k = h whe eeded, we obtai T i (p) = p k pk K J 1 k! (k K J )! J S i 1, 2 k L k 1,J J + p k pk K J k! (k K J )! J S i 1, 2 k L k 1,J J + p k pl 1 p h k! l! h! J S i 2, 2 k L k 2,J J l>0,h>0,l+h=k K J + p k, k! J J S i, 2 k E k,j which ca also be writte as T i (p) = p k ( ) p k K J 1 + p k K J (k K J )! k! J S i 1, 2 k L k 1,J J + p k (p 1 + p ) k K J (k K J )! k! J S i 2, 2 k L k 2,J J p k ( ) p k K J 1 + p k K J (k K J )! k! J S i 2, 2 k L k 2,J J + p k. k! J J S i, 2 k E k,j p k k!

7 6 E. Aceaume, Y. Busel, E. Schulte-Geers ad B. Sericola We deote these four terms respectively by A i (p), B i (p), C i (p) ad D i (p). We thus have, for i 2, T i (p) = A i (p) + B i (p) C i (p) + D i (p). We have already show that T 1 (p) = A 1 (p) + D 1 (p), so we set B 1 (p) = C 1 (p) = 0. We the have c 2 c 1 c 1 P{T c, (p) > k} = A c 1 (p) + (A i (p) C i+1 (p)) + B i (p) + D i (p). (6) For i 1, we obtai A i (p) C i+1 (p) = J S i 1, 2 i=1 i=2 p k ( p k K J 1 (k K J )! k! k E k 1,J J i=1 ) + p k K J. By defiitio of the set E k 1,J, we have K J = k 1 i the previous equality. Thus, A i (p) C i+1 (p) = J S i 1, 2 k E k 1,J p k (p 1 + p ). k! J The fuctio x x s beig covex o iterval [0, 1] for every s N, we have p k K J 1 + p k K J = (λp 1 + (1 λ)p ) k K J + ((1 λ)p 1 + λp ) k K J λp k K J 1 = p k K J 1 + (1 λ)p k K J + p k K J, + (1 λ)p k K J 1 + λp k K J ad i particular p 1 + p = p 1 + p. It follows that A c 1 (p ) A c 1 (p), A i (p ) C i+1 (p ) = A i (p) C i+1 (p), B i (p ) = B i (p), D i (p ) = D i (p), ad from (6) that P{T c, (p ) > k} P{T c, (p) > k}, which cocludes the proof. The fuctio P{T c, (p) > k} beig symmetric, this theorem easily exteds to the case where the two etries p 1 ad p of p are ay p i, p {p 1,..., p }, with i. I fact we have show i this theorem that for fixed ad k, the fuctio of p, P{T c, (p) k} is a Schur-covex fuctio, that is, a fuctio that preserves the order of maorizatio; see [6] for more details o this subect. Theorem 2. For all 1, p (0, 1) ad c = 1,..., we have T c, (u) st T c, (p). Proof. We apply successively ad at most 1 times Theorem 1 as follows. We first choose two differet etries of p, say p i ad p such that p i < 1/ < p ad the we defie p i ad p by p i = 1/ ad p = p i + p 1/. This leads us to write

8 Optimizatio results for a geeralized coupo collector problem 7 p i = λp i + (1 λ)p ad p = (1 λ)p i + λp, with λ = p 1/ p p i. From Theorem 1 vector p, which is obtaied by takig the other etries equal to those of p, i.e. by takig p l = p l, for l i,, is such that P{T c, (p ) > k} P{T c, (p) > k}. Note that at this poit vector p has at least oe etry equal to 1/, so repeatig at most 1 this procedure, we get vector u, which cocludes the proof The case p 0 > 0 We cosider ow the case where p 0 > 0. We have p = (p 1,..., p ) with i=1 p i < 1 ad p 0 = 1 i=1 p i. Recall that i this case T c, (p) is the time or the umber of steps eeded to collect a subset of c differet coupos amog coupos 1,...,. Coupo 0 is ot allowed to belog to the collectio. The umber N (k) i of coupos of type i collected at istats 1,..., k follows the biomial distributio with parameters k ad p i. Moreover, for all k 0, k 1,..., k 0 such that i=0 k i = k, we have P{N (k) 0 = k 0, N (k) 1 = k 1,..., N (k) = k } = k 0!k 1! k! pk0 0 pk1 1 pk. It follows that for all k 1,..., k 0 such that i=1 k i = k k 0, P{N (k) 1 = k 1,..., N (k) = k N (k) 0 = k 0 } = (k k ( ) k1 ( ) k 0)! p1 p. k 1! k! 1 p 0 1 p 0 (7) As i Subsectio 2.1, we have T c, (p) > k U (k) P{T c, (p) > k N (k) 0 = k 0 } = = c 1 i=0 c 1 P{N u (k) J S i, i=0 J S i, k E k k0,j > 0, u J ad N (k) u (k k 0 )! J < c ad so usig (7), we obtai ( ) k p 1 p 0 k! = 0, u / J N (k) 0 = k 0 }. (8) Theorem 3. For all 1, p (0, 1) with i=1 p i < 1, ad for all c = 1,...,, we have T c, (u) st T c, (v) st T c, (p), Proof. From Relatio (3) ad Relatio (8), we obtai, for all k 0 = 0,..., k, P{T c, (p) > k N (k) 0 = k 0 } = P{T c, (p/(1 p 0 )) > k k 0 },

9 8 E. Aceaume, Y. Busel, E. Schulte-Geers ad B. Sericola ad ucoditioig, P{T c, (p) > k} = k l=0 ( ) k p l l 0(1 p 0 ) k l P{T c, (p/(1 p 0 )) > k l}. (9) Sice p/(1 p 0 ) is a probability distributio, applyig Theorem 2 to this distributio, observig that u = v/(1 p 0 ) ad applyig (9) to v, we obtai P{T c, (p) > k} = k ( ) k p l l 0(1 p 0 ) k l P{T c, (u) > k l} k ( ) k p l l 0(1 p 0 ) k l P{T c, (v/(1 p 0 )) > k l} l=0 l=0 = P{T c, (v) > k}. This proves the secod iequality. To prove the first oe, observe that P{T c, (p/(1 p 0 )) > l} is decreasig with l. This leads, usig (9), to P{T c, (p) > k} P{T c, (p/(1 p 0 )) > k}. Takig p = v gives P{T c, (v) > k} P{T c, (u) > k}, which completes the proof. 3. Distributio maximizig the distributio of T c, (p) We fix a parameter θ (0, (1 p 0 )/] ad we are lookig for the distributios p which stochastically maximize the time T c, (p) o the set A θ defied by A θ = {p (0, 1) p i = 1 p 0 ad p θ, for every = 1,..., }. i=1 We first itroduce the set B θ defied by the distributios of A θ with all their etries, except oe, are equal to θ. The set B θ has elemets give by B θ = {(γ, θ,..., θ), (θ, γ, θ,..., θ),..., (θ,..., θ, γ)}, with γ = 1 p 0 ( 1)θ. Sice θ (0, (1 p 0 )/], we have 1 p 0 ( 1)θ θ which meas that B θ A θ. Theorem 4. For every 1, p A θ ad c = 1,...,, we have T c, (p) st T c, (q), for every q B θ. Proof. By symmetry, P{T c, (q) > k} has the same value for every q B θ, so we suppose that q l = θ for every l ad q = γ. Let p A θ \ B θ ad let i

10 Optimizatio results for a geeralized coupo collector problem 9 be the first etry of p such that i ad p i > θ. We defie p (1) as p (1) i = θ, p (1) = p i + p θ > p ad p (1) l = p l, for l i,. Thus p i = λp (1) i + (1 λ)p (1) ad p = (1 λ)p (1) i + λp (1), with λ = (p θ)/(p θ + p i θ) [0, 1). From Theorem 1, we get P{T c, (p) > k} P{T c, (p (1) ) > k}. Repeatig the same procedure from distributio p (1) ad so o, we get, after at most 1 steps, distributio q, that is P{T c, (p) > k} P{T c, (q) > k}, which completes the proof. Refereces [1] Aceaume, E., Busel, Y. ad Sericola, B. (2015). New results o a geeralized coupo collector problem usig Markov chais. J. Appl. Prob. 52. [2] Aceaume, E., Busel, Y., Rivetti, N. ad Sericola, B. (2015). Idetifyig global icebergs i distributed streams. Research Report. [3] Boeh, A. ad Hofri, M. (1997). The coupo-collector problem revisited-a survey of egieerig problems ad computatioal methods. Stoch. Models 13, pp [4] Doumas, A. V. ad Papaicolaou, V. G. (2012). Asymptotics of the risig momets for the coupo collector s problem. Electro. J. Prob. 18, pp [5] Flaolet, P., Gardy, D. ad Thimoier, L (1992). Birthday paradox, coupo collectors, cachig algorithms ad self-orgaizig search. Discrete Appl. Math. 39, pp [6] Marshall, A. W., Olki, I. ad Arold, B. C. (2011). Iequalities : Theory of Maorizatio ad Its Applicatios. 2d. ed. Spriger-Verlag, New York. [7] Rubi, H. ad Zidek, J. (1965). A waitig time distributio arisig from the coupo collector s problem, Techical Report No. 107, Departmet of Statistics, Staford Uiversity, Califoria, USA.