A Simple Proof of the Shallow Packing Lemma

Transcription

1 A Simple Proof of the Shallow Packig Lemma Nabil Mustafa To cite this versio: Nabil Mustafa. A Simple Proof of the Shallow Packig Lemma. Discrete ad Computatioal Geometry, Spriger Verlag, 06, 55 (3), pp <0.007/s >. <hal > HAL Id: hal Submitted o 6 Jul 06 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 A Simple Proof of the Shallow Packig Lemma Nabil H. Mustafa Abstract We show that the shallow packig lemma follows from a simple modificatio of the stadard proof, due to Haussler ad simplified by Chazelle, of the packig lemma. Itroductio I 995 Haussler [5] proved the followig iterestig theorem, improvig upo a earlier result of Dudley []. Theorem A (Packig Lemma [6, Lemma 5.4]). Let (X, P) be a set-system o elemets, ad with VCdimesio at most d. Let be a iteger,, such that (R, S) for every R, S P, where (R, S) = (R \ S) (S \ R). The P = O ( (/) d). Haussler s proof is a beautiful applicatio of the probabilistic method (i particular, coditioal variace), ad was simplified by Chazelle []. Recetly much effort has bee devoted to fidig size-sesitive geeralizatios of this result. Give a set system (X, P) ad a iteger > 0, we say that (X, P) is -separated if (R, S) for every R, S P. For ay set Y X, defie the projectio of P oto Y as the set system P Y = { S Y S P }. After a series of partial bouds [8, 4, 7], the followig statemet has bee recetly established i [3], via two differet proofs (oe buildig o Haussler s origial proof while the other exteds Chazelle s proof): Theorem B (Shallow Packig Lemma). Let (X, P) be a set-system o elemets, ad let d, d, k, > 0 be itegers. Assume that P has VC-dimesio at most d. Further, assume that for ay set Y X the umber of sets i P Y of size at most l is at most f( Y, l) = O ( Y d l d d ). If P is -separated ad S k for all S P, the P = O ( d k d d / d). The objective of this paper is to prove that Theorem B (i fact, a geeralizatio of it), with a simple trick, is a cosequece of Haussler ad Chazelle s Packig Lemma, which we first state below i a slightly more geeral form. Theorem. Let (X, P) be a set-system o elemets. Let d, be two itegers such that the VC-dimesio of P is at most d, ad P is -separated. The P E [ P A ], where A is a uiformly chose radom sample of size 4d. Uiversité Paris-Est, Laboratoire d Iformatique Gaspard-Moge, Equipe A3SI, ESIEE Paris. E- mustafa@esiee.fr. The work of Nabil H. Mustafa i this paper has bee supported by the grat ANR SAGA (JCJC-4-CE ).

3 Usig it, Theorem B ca be prove i a more geeral settig i terms of the so-called shallow-cell complexity of set systems. Give a fuctio ϕ : N N N, a set system (X, P) has shallow-cell complexity ϕ(, ) if for ay Y X, the umber of subsets i P Y of size at most l is at most Y ϕ ( Y, l ). Our mai result is the followig. Theorem. Let (X, P) be a set-system, ad d, k, > 0 be itegers. Assume that X =, the VC-dimesio of P is at most d, ad P has shallow-cell complexity ϕ(, ). If P is -separated ad S k for all S P, the ( P = O ϕ( 4d, dk ) ). Note that if for ay set Y X the umber of sets i P Y of size at most l is O( Y d l d d ), the P has shallow-cell complexity ϕ(, l) = O ( d l d d ), ad so Theorem implies Theorem B. Orgaizatio. We prove the mai theorem i Sectio. As the proof follows from a slight geeralizatio of Haussler ad Chazelle s proof of the packig lemma (as stated i Theorem ), we preset its proof i Sectio 3. Proof of Theorem. The proof i [6, Lemma 5.4] proves Theorem A as follows. By the primal shatter lemma [6], we have P Y = O((/) d ), ad from Theorem we ca coclude that m = P = O((/) d ). Now we show that the proof of Theorem is also a similar step away, by usig istead the shallow-cell complexity of the set system. Proof of Theorem. Let A X be a uiform radom sample of size 4d P = {S P s.t. S A > 3 4dk/}.. Defie Note that E[ S A ] 4dk/ as S k for all S P. By Markov s iequality, for ay S P, Thus Pr[S P ] = Pr[ S A > 3 4dk/] /3. E[ P A ] E[ P ] + E[ (P \ P ) A ] ( Pr[S P ] + A ϕ A, dk ) S P P 3 + 4d ( 4d ϕ, dk ) where the projectio size of P \ P to A is bouded by ϕ(, ). Now the boud follows from Theorem. 3 Proof of Theorem. For the sake of completeess, we reproduce the proof of Haussler ad Chazelle, otig a slight geeralizatio of it as stated i Theorem. Alteratively, it ca be foud i the textbook [6, Lemma 5.4]. Give P = {S,..., S m } o a set X of elemets, we first defie the uit distace graph G U (P) = (P, E P ). The vertex set of G U (P) is P, ad for ay S i, S j P, {S i, S j } E P if ad oly if (S i, S j ) =. We will eed the followig result o uit distace graphs of set systems.

4 Lemma 3 (Haussler [5]). Give a set system (X, P) with VC-dimesio d, let G U (P) = (P, E P ) be its uit distace graph. The E P d P. Proof of Theorem. Let P = {S,..., S m }, where m = P. Choose A to be a radom sample of X of size s = 4d/, picked uiformly from all s-sized subsets of X. Let G U (P A ) = (P A, E P A ) be the uit distace graph o P A. For ay set S P A, defie w(s ) = {S P s.t. S A = S }. Defie the weight of a edge {S i, S j } E P A to be w({s i, S j }) = mi{w(s i ), w(s j )}. Let W = e E P A w(e). Claim 3.. W d m. Proof. By Lemma 3, E P A d P A. Hece there exists a set S P A with degree at most d i G U (P A ). Furthermore the weight of each edge icidet to S is at most w(s ), thus the total weight of edges icidet to S is at most d w(s ). Remove S from G U (P A ), ad recursively boud the weight of edges i the remaiig graph. Thus the total weight of edges is at most d S w(s ) = d m. A alterate way of pickig A is by first choosig radomly a set A of s elemets of X, ad the choosig the last elemet uiformly from X \ A. Let W be the radom variable deotig the weight of the edges i G P A for which the elemet i A \ A is the symmetric differece. By symmetry, we have E[W ] = s E[W ] () To compute E[W ], fix the set of first s vertices. Now coditioed o this fixed choice of A, we show the followig statemet. Claim 3.. E [ W A = Y ] ( ) m P Y. Proof. Cosider a set S P Y, ad let P S be the sets of P whose projectio to Y is S. Oce the choice of the last elemet a has bee made, S will be split ito two sets S ad S i P A, where S = S ad S = S {a}. Similarly P S will be partitioed ito P S, cosistig of the sets of P whose projectio to A is S ad P S, cosistig of the sets of P whose projectio to A is P S. Let b = P S ad b = P S. The weight of the edge i G U (P A ) betwee the two sets S ad S is mi{b, b }, ad it is show i [6, Lemma 5.4] that E [ mi{b, b } ] ( ) P S Summig up over all sets of P Y, E [ W A = Y ] S P Y ( ) P S = ( ) m P Y. From Equatio () ad Claim 3., we get a lower-boud o E[W ]: E[W ] = s E[W ] = s E[W A = Y ] Pr[A = Y ] s s Y =s ( m Y =s ( m P Y Y =s = 4dm 4d E[ P A ] Pr[A = Y ] 3 ) Pr[A = Y ] Y =s P Y Pr[A = Y ] )

5 where the secod iequality follows from Claim 3.. Together with the upper-boud o W from Claim 3., we get dm E[W ] 4dm 4d E[ P A ], implyig that m E[ P A ] as desired. Discussio. Besides a shorter proof, we have show that a geeralizatio of the shallow packig lemma follows without ay modificatio or additio to the proof of Haussler ad Chazelle. The somewhat subtle key idea that was missed by earlier work [3, 4] is that it is fie if E[ P A ] is bouded i terms of c P for a small-eough costat c, as i ay case it would be absorbed by the LHS of the equatio i Theorem. This allows us to replace the complicated techical machiery developed i earlier work (iterative processes, Cheroff bouds for hypergeometric series, complicated probabilistic computatios) by a mere Markov s iequality. Refereces [] Berard Chazelle. A ote o Haussler s packig lemma. 99. [] Richard M. Dudley. Cetral limit theorems for empirical measures. A. Probab., 6(6): , 978. [3] Kual Dutta, Esther Ezra, ad Arijit Ghosh. Two proofs for shallow packigs. I 3st Iteratioal Symposium o Computatioal Geometry (SoCG), pages 96 0, 05. [4] Esther Ezra. A size-sesitive discrepacy boud for set systems of bouded primal shatter dimesio. I Proc. of the Twety-Fifth Aual ACM-SIAM Symposium o Discrete Algorithms (SODA), pages , 04. [5] David Haussler. Sphere packig umbers for subsets of the boolea -cube with bouded Vapik- Chervoekis dimesio. 69():7 3, 995. [6] Jiří Matoušek. Geometric Discrepacy : A Illustrated Guide. Algorithms ad combiatorics. Spriger, Berli, New York, 999. [7] Nabil H. Mustafa ad Saurabh Ray. Near-optimal geeralisatios of a theorem of Macbeath. I 3st Iteratioal Symposium o Theoretical Aspects of Computer Sciece (STACS), pages , 04. [8] Evagelia Pyrga ad Saurabh Ray. New existece proofs for epsilo-ets. I Symposium o Computatioal Geometry, pages 99 07,