Component-by-Component Construction of Low-Discrepancy Point Sets of Small Size

Transcription

1 Component-by-Component Contruction of Low-Dicrepancy Point Set of Small Size Benjamin Doerr, Michael Gnewuch, Peter Kritzer, and Friedrich Pillichhammer Abtract We invetigate the problem of contructing mall point et with low tar dicrepancy in the -dimenional unit cube. The ize of the point et hall alway be polynomial in the dimenion. Our particular focu i on extending the dimenion of a given low-dicrepancy point et. Thi reult in a determinitic algorithm that contruct -point et with mall dicrepancy in a component-by-component fahion. The algorithm alo provide the exact tar dicrepancy of the output et. It run-time coniderably improve on the run-time of the currently known determinitic algorithm that generate low-dicrepancy point et of comparable quality. We alo tudy infinite equence of point with infinitely many component uch that all initial ubegment projected down to all finite dimenion have low dicrepancy. To thi end, we introduce the invere of the tar dicrepancy of uch a equence, and derive upper bound for it a well a for the tar dicrepancy of the projection of finite ubequence with explicitly given contant. In particular, we etablih the exitence of equence whoe invere of the tar dicrepancy depend linearly on the dimenion. 1 Introduction In numerical integration, point et with good ditribution propertie are of great interet. One way of meauring the quality of ditribution of a multiet P = {p 0, p 1,..., p } of point in the -dimenional unit cube [0, 1), i to conider it tar dicrepancy, defined by D P ) = up x [0,1] x, P ). Here the dicrepancy function of the point et P i given, for x = x 1,...,x ), by x, P ) = λ [0, x)) 1 1 [0,x) p j ), P.K. and F.P. are upported by the Autrian Science Foundation FWF), Project S9609, that i part of the Autrian ational Reearch etwork Analytic Combinatoric and Probabilitic umber Theory. 1

2 where λ i the -dimenional Lebegue meaure and 1 [0,x) i the characteritic function of the -dimenional half-open box [0, x) = [0, x 1 ) [0, x ). Another ueful quantity in the context of numerical integration i the o-called invere of the tar dicrepancy, ε, ) = min{ : P [0, 1) uch that P = D P ) ε}, where P denote the cardinality of the multiet P. For certain clae of function, point et with mall tar dicrepancy yield cubature formula with a mall error of multivariate integration. Thi i for example illutrated by the well-known Kokma-Hlawka inequality ee [1], but alo [15, 17]). If we approximate the integral fx) dx by a quai- [0,1] Monte Carlo algorithm 1 i=0 fp i), where p 0,...,p P, then the Kokma-Hlawka inequality tate that fx) dx 1 [0,1] i=0 fp i ) V f)d P ), where V f) i the variation of f in the ene of Hardy and Kraue ee, e.g., [17]). Thi implie that, for any function with bounded variation V f), a point et with low tar dicrepancy yield a low integration error for a quai-monte Carlo rule uing thi point et a integration node. However, it i a challenging problem to find point et with good upper bound on the tar dicrepancy. There are many contruction of point et with low tar dicrepancy, for example thoe propoed by iederreiter o called t, m, )-net and t, )-equence, ee [16]). Mot of the known bound on the tar dicrepancy of a t, m, )-net P are of the form D P log ) ) C or D P log ) 1 ) C, where C i a contant depending on the dimenion, ee [14, 16, 17]. Thee bound are excellent with repect to the order of magnitude in. The drawback of uch bound, however, i that the term log ) become large when i very high and that in general the contant C are not mall enough to compenate thi effect. Therefore, the ize of the node et for numerical integration need to be extremely large if one want to obtain good bound on the integration error. In fact it ha to be at leat exponentially in, ince the function log) / increae for e. Since, during the lat year, it ha become a major iue to conider numerical integration in particularly high dimenion e.g., in financial application, where might be in the hundred or thouand), it i of growing interet to find bound on the tar dicrepancy diplaying a better dependence on the dimenion. Thi problem wa uccefully attacked by Heinrich et al. in [9], who howed the exitence of a point et P in [0, 1) uch that D P ) C /, 1) where C i a poitive contant not depending on and. However, the proof of thi reult i non-contructive, and, furthermore, good bound for the contant have not been publihed yet. In another paper [5] the exitence of point et P atifying D P ) C / log, )

3 where C > 0 i a known mall contant and independent of and, wa proved. Thi reult i obtained by making ue of the concept of o-called δ-cover ee below for a precie definition) and by Hoeffding inequality. A lightly better reult, namely D P ) C / log1 + /), wa proved in a imilar manner in [7]. Like the reult of Heinrich et al., the reult in [5, 7] are at firt baed on a probabilitic argument and therefore eem to be non-contructive. On the other hand, in [5] alo a determinitic contruction of point et atifying ) by the mean of a derandomized verion of Hoeffding inequality wa given. However, the run-time of thi algorithm i high, to be more precie, it i exponential in the dimenion. An algorithm uing a different derandomization technique wa preented in the recent paper [4]. It alo generate an -point et in dimenion atifying the bound ), but it run-time improve coniderably over the algorithm from [5]. everthele, it i till exponential in. The above-mentioned probabilitic bound were developed further in [1] to allow infinite equence of point in infinite dimenion. For an infinite equence P of point in [0, 1), let u denote by P the equence of the -dimenional projection of the point from P, and by P, the firt point of P. Then in [1] the following reult were hown: There exit an unknown contant C uch that for every trictly increaing equence m ) m in there i an infinite equence P atifying, for all m,, D m P m,) C / m logm + 1). Furthermore, there exit an explicitly given contant C uch that for every trictly increaing equence m ) m in there i an infinite equence P atifying, for all m,, D m P m,) C m + + log 1 + )) m / m. m + Thu, the reult in [1] are both extenible in the dimenion and in the number of point, which i particularly ueful. Our Reult In thi paper, we firt preent another reult for infinite equence P in [0, 1). At the firt glance it look like a modet improvement of [1, Corollary 3], but it etablihe the exitence of infinite equence P in [0, 1) having the following property: To guarantee D P,) ε for a given ε, we only have to take c ε, where c ε i a contant depending only on ε. ote that thi reult cannot be deduced directly from the reult in [1]. It i known from [9, 11] that we have to take at leat c ε if ε i ufficiently mall. Here c ε depend again only on ε.) In thi ene our reult how that the tatement the invere of the tar dicrepancy depend linearly on the dimenion which i the title of the paper [9]) extend to the projection of infinite equence in [0, 1). To be more precie, let u introduce the invere of the tar dicrepancy of an infinite equence P, Then there exit equence P uch that P ε, ) := min{ : M : D M P M,) ε}. P ε, ) Oε log1 + ε 1 )). 3) 3

4 In fact, if we endow the et [0, 1) with the canonical probability meaure λ = i=1λ 1 and allow the implicit contant in the big-o-notation to depend on the particular equence P, then inequality 3) hold almot urely for a random equence P, ee Corollary 1. In Theorem 1 we provide bound of the form 3) with explicitly given contant and etimate for the meaure of the et of equence atifying uch bound. It would be of great interet to contruct infinite equence whoe tar dicrepancy atifie bound a tated above. In thi paper we contruct, in a component-by-component algorithm i.e., one component i contructed at a time), finite equence P in [0, 1) that atify imilar bound. For a given -point et P 1 = {y 0,...,y } [0, 1) 1 we deduce from concentration of meaure reult that we can find X 0,...,X [0, 1) uch that the extended et P = {y 0, X 0 ),...,y, X )} [0, 1) atifie D P ) O log 1 + ) ) + D P 1 ). We are able to derandomize the probabilitic argument to generate recurively component by component) -point et P atifying DP ) O 3 log 1 + ) ). Thi bound i obviouly lightly weaker than the bound in 1) and ), but the run-time of our derandomized algorithm improve coniderably on the run-time of the algorithm in [4, 5] generating point et atifying ). In Section 5 we compare the run-time of the algorithm to each other and relate the problem of contructing low-dicrepancy ample of mall ize to the problem of approximating the tar dicrepancy of a given point et. Infinite Dimenional Infinite Sequence At the beginning of thi ection we introduce two main tool that will alo be ueful in the following ection. The firt i the concept of o-called δ-cover, a ued in [4, 5, 7]. A δ-cover i defined a follow. Definition 1 A finite et Γ [0, 1] i a δ-cover of [0, 1] if for every x = x 1,...,x ) [0, 1] there exit γ 1 = γ 1) 1,...,γ) 1 ), γ = γ 1),...,γ) ) Γ {0} with λ [0, γ )) λ [0, γ 1 )) δ and γ i) 1 x i γ i) for all 1 i. One can ue δ-cover to approximate the tar dicrepancy of a given et up to ome admiible error δ by a quantity whoe calculation involve only a finite number of tet boxe. Lemma 1 Let Γ be a δ-cover of [0, 1]. Then for any -point et P [0, 1) we have D P ) D Γ P ) + δ, where D Γ P ) = max x Γ x, P ). 4

5 The proof i traightforward and can, e.g., be found in [5, Lemma 3.1]. Furthermore, we are going to make ue of a large deviation bound from probability theory called Hoeffding inequality ee [13] or [18, p. 58], [19, p. 191]). Let A 1,..., A r be independent random variable with EA i ) = 0 and u i A i v i for all i {1,..., r}. Then Hoeffding inequality tate that P [ A A r rη] e r η / P r i=1 v i u i ) 4) for any η > 0. In thi ection we improve certain apect of the reult in [1]. Let ζ denote the Riemann zeta function, i.e., ζγ) = m=1 m γ. A mentioned in the introduction, we denote, for an infinite equence P of point in [0, 1), by P the equence of -dimenional projection of the point from P, and by P, the firt point of P. Theorem 1 Let P = p ) be a equence of independent random variable, each of them uniformly ditributed with repect to the infinite) product meaure λ := =1 λ 1 on [0, 1). Let m ) m be a trictly increaing equence in, and let A 1, ). Let γ > ζ 1 ). Then each of the following two event hold with probability at leat 1 ζγ) 1). i) For all m,, with ρ = ρ m, ) := 6emax{1, m / log6e))}) 1/, D m P m,) ) ) 1/ logρ) + γlog1 + m) + log1 + )) + log). 5) m ii) For all ε 0, 1], P ε, ) A ε log1 + Aε 1 ) + loge) ) + + γ log + logε 1 ) loga) ) ) + log1 + ) ) + log1 + A ε ) + log). Proof. Let θ [0, 1). Due to Lemma 1 we have P [D P,) δ] > θ if the inequality P [ D Γ P,) δ ] > θ hold for ome δ-cover Γ of [0, 1]. For each let p ) denote the projection of p onto it firt component. If we define for x [0, 1] and i the random variable ξ x p ) i ) = λ [0, x)) 1 [0,x) p ) i ), then the range of ξ x p ) i ) i contained in an interval of length one and the expectation E[ξ x p ) i )] i zero. Thu Hoeffding inequality implie [ ] 1 P[ x, P, ) > δ] = P ξ x p ) i ) > δ e δ. i=1 6) Thi reult in P [ DP Γ, ) δ ] 1 [ 1 P x Γ ] ξ x p ) i ) > δ > 1 Γ e δ. i=1 5

6 In the latter etimate we get >, ince necearily 1 := 1,..., 1) Γ and 1, P) = 0 for all finite et P [0, 1).) Hence P [D P,) δ] > θ i atified if ) δ log Γ + log 7) hold. In [7, Theorem 1.15] a δ-cover Γ wa contructed atifying Γ! δ 1 + 1) e) δ 1 + 1). 8) If we chooe thi δ-cover and retrict ourelve to δ 0, 1/], then it i eaily verified that δ := 1 )) 1/ logρ) + log 9) atifie 7). Thu D P,) ))) 1/ logρ) + log with probability at leat θ. ow let m ) m be a trictly increaing equence in and define, for γ > 1, θ = θm,, γ) = m)1 + ) ) γ. 10) The probability that there exit a pair m, ) uch that D m P m,) > m logρ) + log m,, γ) ))) 1/ i bounded from above by ) ) m,, γ)) = 1 + m) γ 1 + ) γ = ζγ) 1). m,=1 m=1 Since the Riemann zeta function i trictly decreaing, the latter expreion i trictly le than one if γ i trictly larger than ζ 1 ). Thi prove 5). One might be tempted to ue inequality 5) directly with the equence m ) m = ) to derive a bound like 6) for P ε, ). But thi i not eaily done, ince it i hard to olve the reulting inequality with repect to. For that reaon we again ue 7) to derive tatement ii) of the theorem. Let u put ε := δ. We know that each with ε log 1 + ε 1) + loge) ) )) + log implie P [D P,) ε] > θ, ee 7) and 8). ow put ε m := A m and θ = θ m,, γ) = 1 6 =1 1 + m)1 + )) γ. 1 + ε m

7 For fixed m and put m, := ε m For all m, chooe ϑ) uch that that i, = ε m log1 + ε 1 m ) + loge)) + log log1 + ε 1 m ) + loge)) + log m,, γ) )), 1 ϑ) ϑ) = 1 exp 1 ) e)1 ε m + ε 1 m )) θ m,, γ). )). Hence the probability that there exit an m, uch that D P,) > ε m i bounded from above by 1 ϑ)) = e)1 + ε 1 m )) exp 1 ) εm = m, = m, e)1 + ε 1 m )) exp 1 ) ε m m, + exp 1 ) ) εm x dx m, = e)1 + ε 1 m ) ) 1 + ε m ) exp 1 ) εm m, 1 + ε m ) m,, γ)) = 1 + m)1 + )) γ. Thu the probability that there exit a pair m, ) uch that P ε m, ) > m, i again bounded by ζγ) 1). If γ > ζ 1 ), there exit an infinite equence P uch that P ε m, ) i majorized by m, for all m,. Furthermore, we ee that for ε [A m, A 1 m ) the imple etimate P ε, ) P ε m, ) implie 6). Thi complete the proof of Theorem 1. Theorem 1 immediately implie the following corollary. Corollary 1 For an infinite equence P in [0, 1) the following hold with probability one: There exit contant C P, C P uch that DP, ) C P log 1 + )) 1/ and Pε, ) C Pε log 1 + ε 1) for all, and all ε 0, 1]. 3 Extenion in the Dimenion ow we will try to relate the tar dicrepancie of an arbitrary 1)-dimenional point et and a uitably choen -dimenional extenion of it. To be more precie, let P 1 = {y 0,...,y } [0, 1) 1 be a point et with tar dicrepancy D P 1). Let m be an integer, and let { } 1 G = Gm) := m, 3 m 1,...,. m m 7

8 For a 0,...,a G we conider the point et given by P = P a 0,..., a ) = {y i, a i ) : 0 i < }. 11) The following theorem how that there i alway a choice of a 0,...,a Gm) uch that we can bound the tar dicrepancy of P a contructed in 11) in term of m,, and the tar dicrepancy of P 1. Theorem Let m be an integer and let 0 θ < 1 be a real. Aume that a 0,...,a are choen independently and uniformly ditributed from the et Gm). Then with probability greater than θ we have for all D P ) logρ, )) + log where ρ = ρ, ) := 6emax{1, / log6e))}) 1/. )) 1/ + 1 m + D P 1 ), Proof. For x = x 1,..., x ) [0, 1), y = y 1,...,y 1 ) P 1 and a G let ξ x y, a) = λ [0, x)) 1 [0,x) y, a). Aume that A i choen according to a uniform ditribution from the et G. Then we have 1 E [ξ x y, A))] = x d 1 [0,xd )y d )E [ 1 [0,x) A) ]. A E [ 1 [0,x) A) ] = 1 m it follow that 1 ) 1 x x d 1 [0,xd )y d ) d=1 d=1 a G d=1 d=1 1 [0,x) a) = mx 1/ m = 1 1 m E [ξ x y, A)] x d=1 { x + 1/m), x 1/m), ) 1 x d 1 [0,xd )y d ) d=1 + 1 m. If A 0,...,A are choen independently and if P = P A 0,...,A ), then we obtain x 1 x 1,...,x 1 ), P 1 ) 1 m E [ x, P )] x 1 x 1,...,x 1 ), P 1 ) + 1 m, where the expectation i taken with repect to A 0,..., A. Therefore we get E [ x, P )] 1 m + x 1 x 1,...,x 1 ), P 1 ) 1 m + D P 1). 1) Let now 0 < δ 1/. Aume that x, P ) D P 1) + 1 m + δ. Thi implie x, P ) E [ x, P )] δ or equivalently ξ x y i, A i ) E [ξ x y i, A i )]) δ. i=0 8

9 We can apply Hoeffding inequality to obtain [ P x, P ) DP 1 ) + 1 ] m + δ [ ] P ξ x y i, A i ) E[ξ x y i, A i )]) δ e δ. i=0 Similar to the proof of Theorem 1 we get, for a δ-cover Γ of [0, 1], [ P DP ) δ + 1 ] [ m + D P 1 ) P max x, P ) δ + 1 ] x Γ m + D P 1 ) The latter term i greater than or equal to θ if δ log Γ + log > 1 Γ e δ. ). 13) From [7, Theorem 1.15] we get Γ e) δ 1 + 1), hence any olution of δ 1 log 3eδ 1) )) + log atifie 13). Put now δ = δ, ) := 1 log ρ) + log It i eay to check that thi δ atifie 14). 14) )) 1/. 15) The reult in Theorem give u the opportunity to contruct point et with low tar dicrepancy in a component-by-component fahion. The following corollary how an upper bound on the dicrepancy of uch a point et. ote that thi bound diplay a lightly wore dependence on the dimenion. The advantage of the contruction i that the point lie on a relativly mall meh with not necearily the ame reolution in each dimenion). Thi allow a cheaper computation of the precie dicrepancy ee Equation 17) in Section 4) and alo a more efficient derandomization ee the remainder of Section 4). Corollary For any 1 and any equence m d ) d 1 of poitive integer there exit a point et P = {x 0,...,x } [0, 1) with i + 1 x i =, a i,1 + 1, a ) i, + 1,... m 1 m where a i,d {0,..., m d 1} for all d 1 and 0 i < uch that for all 1 for the point et P, coniting of the projection of the point from P to the firt component, we have D P ) where ρ, ) i a in Theorem. 1) log ρ, )) + log)) 1/ d=1 9 1 m d + 1, 16)

10 Proof. It i eay to check that D P 1) = 1/) ee [17, Theorem.6]). Due to Theorem, we can recurively chooe et {x 0,d,...,x,d } uch that D P d ) d j logρ, j)) + log)) 1/ + 1 j= d 1 j=1 1 m j + 1 hold for all d =,...,. Elementary calculu how that logρ, )) d logρ, d)) for d. 4 A Contruction Algorithm One may ue Theorem and Corollary to contruct low-dicrepancy point et via a derandomized algorithm imilar to what ha been done in [5]. Here, we would like to ue a lightly different algorithm, ince the et we want to contruct are contained in a grid whoe cardinality i of the ame order a the mallet δ-cover contructed in [7]; intead of only approximating the dicrepancy of uch a et up to ome admiible error δ with the help of a δ-cover, we can calculate it dicrepancy exactly with at mot the ame effort. We make ue of the following imple obervation: Let P = {p 0, p 1,...,p } [0, 1) with p i = p i,1,..., p i, ). Define for d = 1,..., Furthermore, et Γ d P ) := {p 0,d,...,p,d } and Γ d P ) := Γ d P ) {1}. ΓP ) := Γ 1 P ) Γ P ) and ΓP ) := Γ 1 P ) Γ P ). It i eay to ee that DP ) = { max max x ΓP ) 1 i=0 1 [0,x] p i ) λ [0, x)) ), max x ΓP ) λ [0, x)) 1 i=0 1 [0,x) p i ) Oberve that for general more preciely: almot all) -point et P the evaluation of the right hand ide of 17) involve Θ + 1) ) tet boxe, while the approximation of their dicrepancy up to δ choen a in 15) via a δ-cover a contructed in [7, Theorem 1.15] require at mot O e) log 1 + )) ) 1/ ) ) )}. 17) tet boxe. In [8] a maller δ-cover ha been contructed in dimenion =. There it i conjectured that the method can be extended to arbitrary and would give δ- cover of ize δ + O δ +1 ), which would be coniderably maller than the δ-cover contructed in [7].) However, in the derandomized contruction we have in mind we will confine ourelve to et P having the property that the cardinality of ΓP ) and ΓP ) 10

11 i at mot of the order of 18). To decribe our approach in detail, we have to introduce further notation. Let m 1, m,...,m, and, for d = 1,...,, let { 1 3 G d := Gm d ) =,,..., m } { } d 1 and m d m d m G 1 d := G d {1}) \. d m d Furthermore, put G := G 1 G and G := G 1 G. Let P 1 = {y 0, y 1,...,y } be a ubet of G 1. For X 0,...,X G we conider the point et P = P X 0,...,X ) = {y i, X i ) : 0 i < }. 19) To avoid having to ditinguih between open and cloed tet boxe and to reduce the number of event we have to control in our random experiment, we reformulate 17). It i eay to verify D P ) =max max x eg { max x G 1 i=0 λ [0, x)) 1 1 [0,x] y i, X i )) λ [0, x)) i=0 1 [0,x) y i, X i )) ), ), } 1, D m P 1). 0) For d = 1,...,, put Additionally, put G d := { 1,,..., m } d 1, 1. m d m d m d G := G 1 G. Then for each x G and each x G we find uniquely determined t, t G uch that [0, x] S = [0, t) S and [0, x) S = [0, t) S for all ubet S of G. Thee relation define a mapping Φ from the et B := {[0, x] : x G } {[0, x) : x G } onto G. ow let n := G = d=1 m d, and let t 1,...,t n be an enumeration of G. For i = 1,...,n, y P 1, and a randomly choen X G let Then ξ i y, X) = 1 [0,ti )y, X) λ [0, t i )). Ξ i y, X) := ξ i y, X) E[ξ i y, X)] = 1 [0,ti )y, X) t i, 1 [0,t i )y), 1) where we ued the convention to denote the projection of an -dimenional vector t onto it firt 1 component by t. We ue the correponding convention for ubet of [0, 1]. 11

12 Let B B and t i = ΦB). Denote by B ) the projection of B to the -th component. Then 1 1 B y j, X j )) λ B) = 1 1 [0,ti )y j, X j )) λ 1 B ) )λ 1 B ) ) = 1 1 Ξ i y j, X j ) + t i, 1 [0,t i )y j ) λ 1 B ) + λ 1 B )t i, λ 1 B ) )). Due to 1 [0,t i )y j ) = 1 B y j ) for 0 j < and t i, λ 1 B ) ) 1/m ) we obtain from 0) DP ) 1 max Ξ i y 1 i n j, X j ) + D P 1 ) + 1. ) m From Hoeffding inequality we get [ ] P max Ξ 1 i n i y j, X j ) δ ne δ. 3) Thi reult in [ P D P ) δ + D P 1) + 1 ] 1 ne δ. m ow the latter term i greater than or equal to θ if ) δ logn) + log. 4) The choice δ := 1 ) ) 1/ logm d ) + log d=1 5) give u the following theorem. Theorem 3 Let θ [0, 1). Aume that X 0,...,X are choen independently and uniformly ditributed from G. Then with probability θ we have DP ) 1 ) ) 1/ logm d ) + log D m P 1 ). d=1 Let u now chooe )) 1/ m d = m d, θ) = d logρ, d)) + log for d = 1,...,, 6) where ρ, d) = e max{1, / log e)d)}) 1/. 7) 1

13 An elementary analyi how m 1 m m. ow we can etimate n in term of, and θ: n = m d log ρ, d)) + 1 )) 1/ d=1 d=1 d d log + 1) ) 1/ = log ρ, d)) + 1 )) 1/ ) d! d=1 d log + < π) 1/4 e / log ρ, )) + 1 )) ) 1/ log + 1. Then the particular choice δ := 1 )) 1/ logρ, )) + log 8) and 4) imply the following verion of Theorem 3. Theorem 4 Let the condition of Theorem 3 hold. Then, with the choice of m 1,..., m and ρ, ) a in 6) and 7), we obtain with probability > θ D P ) log ρ, )) + 1 )) 1/ log + D P 1). 9) One can prove the following corollary in a imilar manner to Corollary. Corollary 3 For any 1 and the equence m d ) d 1 of poitive integer defined a above, there exit a point et P = {p 0,...,p } [0, 1) with ai,1 + 1 p i =, a ) i, + 1,..., m 1 m where a i,d {0,..., m d 1} for all d 1 and 0 i < uch that for all 1 for the point et P, coniting of the projection of the point from P to the firt component, we have DP 3/ ) log ρ, )) + 1 ) 1/ log). 30) We now derandomize the above contruction, that i, we tranform it into a determinitic algorithm that compute point et having a dicrepancy imilar to the one which the randomized algorithm ha with poitive probability. In order to be able to ue exiting derandomization, we reformulate our problem of adding one dimenion a a rounding problem with hard contraint. We briefly recall our problem: Let P 1 = {y 0,..., y } G 1 with mall tar dicrepancy D P 1). We aim at finding X 0,...,X G uch that P = {y j, X j ) : 0 j < } ha mall dicrepancy. For convenience, let u write m = m. Our problem become a rounding problem a follow. Let X be the et of all familie x jk ),...,;k=1,...m uch that x jk [0, 1] for all j and k and m k=1 x jk = 1 for all j. Let 13

14 t 1,..., t n be the enumeration of G choen above. Define a linear function A : R m R n by Ax) i = m x jk 1 [0,ti )y j, ˆk) for x = x 01,...,x )m ) and i = 1,..., n, k=1 where we ued the horthand ˆk = k 1. m If x X i integral put X j = ˆk j for j = 0, 1,..., 1 if x jkj = 1. Then Px) := {y j, X j ) : 0 j < } i an point et in [0, 1] and Ax) i i Px) [0, t i ). If in addition x X i defined by x jk = 1 for all j, k, then we have the following. Let B B m and t i = ΦB). Then Ax) i = Due to 1) we have m 1 m 1 [0,t i )y j, ˆk)) m ) 1 = 1 [0,t i )y j ) m 1 [0,t i, )ˆk) k=1 = t i, 1 [0,t i )y j ) = t i, Ax) i Ax) i = 1 B y j ). k=1 Ξ i y j, X j ), 31) thu ) implie D Px)) 1 Ax) Ax) +D P 1)+ 1. Hence low-dicrepancy m point-et Px) correpond to rounding x X of x with mall rounding error Ax) Ax). Generating and derandomizing randomized rounding atifying certain equalitie hard contraint ; here m k=1 x jk = 1 for all j) without violation i highly non-trivial a recent reult how, ee e.g. [, 3, 6]. Fortunately, in the cae that the hard contraint require variable-dijoint um to equal one, the claical method of Raghavan can be ued. We briefly outline thi method. Let u aume that we have an arbitrary x X, which we want to round to an integral x X uch that Ax x)) i i mall for all i. The randomized contruction would be to chooe for each j independently a k j [m] at random uch that P[k j = k] = x jk for all j, k. Then for all j we define random variable X jkj = 1 and X jk = 0, k k j. ote that by contruction any outcome of X lie in X. Let λ R uch that P := i P[ AX x)) i λ] < 1 mall initial failure probability ). Then there exit an x X uch that Ax x)) i λ for all i. We can actually compute uch rounding x by derandomizing the probabilitic contruction above. Let u confine ourelve to the pecial cae x jk = 1/m for all j, k. Due to 31) and 3) the initial failure probability i at mot P n exp λ /), which i maller than one if λ = δ, δ choen a in 8). For k = 1,..., m, let e k denote the kth m dimenional unit vector and conider the conditional probability P k := i P[ AX x)) i λ X 01,..., X 0m ) = e k ]. Since P = m 1 k=1 P m k, there i a 1 k0 m uch that P k0 P < 1 decreaing failure probability ). ext, let P k 0 k := i P[ AX x)) i λ X 01,...,X 0m ) = e k 0, X 11,...,X 1m ) = e k ]. Again, P k 0 = m 1 k=1 P m k0 k, and there i a 1 k1 m uch that P k0 P k 1 k0 < 1. Proceeding like thi we end up with k0,...,k uch that P k0 := P[ AX x)),...,k i 14

15 λ 0 j < : X j1,...,x jm ) = e k j ] < 1. Since P k 0,...,k involve no randomne all variable are bound in the conditional tatement), we actually have P k,...,k = 0. We define x a follow: For each 1 j <, we et x jk j := 1 and x jk := 0 for all other k. Thi yield an integral vector x X uch that Ax x)) i λ for all i. The only problem with the above derandomization i that we uually cannot compute the conditional probabilitie P k 0 k1... in time polynomially bounded in, m and n. However, it would uffice if we can compute in polynomial time) upper bound U k 0 k1... for the exact conditional probabilitie P k 0 k1... uch that the following key propertie are maintained: Small initial etimated) failure probability: U < 1. Decreaing etimated) failure probability: For all 0 l < and k 0,...,k l 1 {1,..., m} there i a 1 k m uch that U k 0 k 1...k l 1 k U k 0 k 1...k l 1. The U k 0 k1... are called peimitic etimator for the conditional probabilitie P k 0 k1.... Thi notion wa introduced by Raghavan [0], who alo howed that uch peimitic etimator exit for the conditional probabilitie that occur in our derandomization. They are the um of n expreion, two for each 1 i n, etimating the probability that the box [0, t i ) receive too many or too few point. Both cae lead to imilar expreion. Hence we ketch only the one for the cae of too many point. ote that we do not want give a precie decription of how to implement the derandomization, but only prove bound on the amortized time needed to compute the etimator. The probability that the box [0, t i ) receive too many point i at mot exp c i ) m x jk exp1 [0,ti )y j, ˆk)c i ). ) k=1 Here, x jk hall alway denote the expected value of the random variable X jk, if it ha not been rounded, and the outcome of the rounding thereafter. Thi i the reaon why we can compute the peimitic etimator efficiently uing the previou computation: When determining the rounded value for ome x j ), we only need to replace the term involving thee variable with the m poible choice for x j ). Hence thi can be done quite efficiently in time Onm). Thu both computing the initial value of the etimator and computing all ubequent value take time Onm). ote that for the final rounding x, the peimitic etimator implicitly tell u the number of point in the box [0, t i ): The expreion m k=1 x jk exp1 [0,ti )y j, ˆk)c i ) i expc i ), if the jth point i in [0, t i ), and one, if not. Hence we can extract the number of point in each box eaily from the computation o far, and thu alo compute the precie dicrepancy eaily by computing for each box the deviation of the actual and the aimed at number of point). The contant c i and c i depend only on the aimed at rounding error λ in particular, they do not change during the algorithm). Alo, the c i, c i are uch that the reulting exponential function can be computed in the RAM model. A a conequence ince we choe all component of x to be rational number we can compute the initial value of the peimitic etimator in time Onm) in the RAM model. Unfortunately, Raghavan peimitic etimator do not admit the Hoeffding bound given in 4), but rather one that, in the etting of 4) and u i = 0, v i = 1, implie that P [ A A r rη] e 1/3)rη. 15 3)

16 In conequence, to achieve that the initial etimated failure probability U i le than one, we have to chooe the δ in Equation 3) to 5) to be 6 time larger than there. Chooing m 1... m = m a in 6) and θ = 1/ in the choice of δ any mall poitive contant i fine), thi lead to a dicrepancy bound analoguou to 9) of D P ) ) log ρ, )) + 1 log4) ) 1/ + D P 1). 33) ote that thi etimate i certainly trivial for /3 alo, in thi cae we have m = m = 1, i.e., our random experiment i completely determinitic and X 0 =... = X = 1/). Hence in the following run-time etimate, we may aume /3. Then, the run-time of our derandomized algorithm i Onm) = O c log ) +1 where c i ome contant independent of and. We ummarize the dicuion above in following theorem. Theorem 5 Let,. Let θ = 1/, and m 1,...,m a in 6). Let P 1 = {y 0,...,y } be a ubet of G 1. Then there i a determinitic algorithm that compute in time OnM) = O c log c a contant independent of and, X 0,...,X G uch that the point et P = {y 0, X 0 ),...,y, X )} atifie ) +1 ) 3 D P 1 ) + log ρ, )) + 1 ) 1/ log 4) + D P 1). Obviouly, we can ue the derandomized algorithm from Theorem 5 to contruct point et P atifying the dicrepancy etimate 30) multiplied by the factor )/ ) component by component. 5 Concluion We preented a determinitic algorithm that generate in time c +3 O an -point et P [0, 1) atifying D P ) O 1 4 log 3/ 1/ ) +1,, log 1 + )) ) 1/. 34) 16

17 In [4, 5] determinitic algorithm were preented that contruct -point et P [0, 1) atifying ) D 1/ P ) O log 1 + ))1/ 35) 1/ in run-time OC + log) log) 1 ), C a uitable contant, and O log)σ) ), σ = σ) = Θlog) / log log)), repectively. The comparion of the algorithm how that the dicrepancy guaranteed by our new algorithm ha a lightly wore dependence on than the dicrepancy guaranteed by the other two algorithm. Converely, the run-time of our new algorithm improve coniderably on the other two algorithm, epecially with regard to the dependence on the number of point. Another advantage i that the new algorithm calculate along the way the exact number of point in each box from a ditinguihed et of half-open boxe. Thi allow to eaily compute the precie dicrepancy of the output et P. We cloe thi paper by relating the problem of contructing low-dicrepancy et of mall ize via derandomization to the problem of approximating the dicrepancy of a given et. Intead of trying to derandomize the random experiment to contruct low-dicrepancy et, one may think of a emi-contruction by performing a random experiment, calculating the actual dicrepancy of the received et, and accept it if bound like 34) or 35) are atified or tart a new random experiment otherwie. Large deviation bound like Hoeffding inequality guarantee that with high probability we only have to perform the random experiment a few time to end up with a low-dicrepancy point et. Beide the need of true?) random bit, thi overlook the difficulty of calculating or approximating) the tar dicrepancy of a given et. Indeed, all algorithm that have been preented for thi problem o far have a run-time exponential in or no run-time guarantee at all, ee e.g. [7, 1,, 3] and the literature mentioned therein. In fact, it ha been hown recently in [3] that the deciion problem whether an arbitrary point et ha dicrepancy maller than ε i if uitably poed) Pcomplete. Thi indicate that it may be not poible to perform emi-contruction a decribed above in polynomial time, a long a the poible out-put et do not exhibit a pecial tructure that make the approximation of their tar dicrepancy particularly imple. In the light of thee reult it i not too urpriing that the run-time of the derandomized algorithm in thi paper and in [4, 5] are exponentially in, ince we cannot expect to do the determinitic) derandomized contruction with le effort than the probabilitic) emi-contruction. Reference [1] Dick, J.: A note on the exitence of equence with mall tar dicrepancy. J. Complexity 3: , 007. [] Doerr, B.: Generating randomized rounding with cardinality contraint and derandomization. In Durand, B., Thoma, W. ed) Proceeding of the 3rd Annual Sympoium on Theoretical Apect of Computer Science STACS 06), Lecture ote in Comput. Sci. 3884: ,

18 [3] Doerr, B.: Randomly rounding rational with cardinality contraint and derandomization. In Thoma, W., Weil, P. ed) Proceeding of the 4rd Annual Sympoium on Theoretical Apect of Computer Science STACS 07), Lecture ote in Comput. Sci. 4393: , 007. [4] Doerr, B., Gnewuch, M.: Contruction of low-dicrepancy point et of mall ize by bracketing cover and dependent randomized rounding. In Keller, A., Heinrich, S., iederreiter, H. ed) Monte Carlo and Quai-Monte Carlo Method 006: 99 31, Springer, 007. [5] Doerr, B., Gnewuch, M. and Srivatav, A.: Bound and contruction for the tar dicrepancy via δ-cover. J. Complexity 1: , 005. [6] Gandhi, R., Khuller, S., Parthaarathy, S., and Srinivaan, A.: Dependent rounding and it application to approximation algorithm. J. ACM 53: , 006. [7] Gnewuch, M.: Bracketing number for d-dimenional boxe and application to geometric dicrepancy. J. Complexity 007, doi: /j.jco [8] Gnewuch, M.: Contruction of minimal bracketing cover for rectangle. Berichtreihe de Mathematichen Seminar der Univerität Kiel, Report 07-14, Kiel, 007. [9] Heinrich, S., ovak, E., Wailkowki, G.W., and Woźniakowki, H.: The invere of the tar dicrepancy depend linearly on the dimenion. Acta Arith. 96: 79 30, 001. [10] Heinrich, S.: Some open problem concerning the tar dicrepancy. J. Complexity 19: , 003. [11] Hinrich, A.: Covering number, Vapnik Červonenki clae and bound for the tar-dicrepancy. J. Complexity 0: , 004. [1] Hlawka, E.: Funktionen von bechränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura Appl., 54: , [13] Hoeffding, W.: Probability inequalitie for um of bounded random variable. J. Amer. Statit. Aoc. 58: 13 30, [14] Kritzer, P.: Improved upper bound on the tar direpancy of t, m, )-net and t, )-equence. J. Complexity : , 006. [15] Kuiper, L. and iederreiter, H.: Uniform Ditribution of Sequence. Wiley, ew York, [16] iederreiter, H.: Point et and equence with mall dicrepancy. Monath. Math. 104: , [17] iederreiter, H.: Random umber Generation and Quai-Monte Carlo Method. CBMS-SF Serie in Applied Mathematic, 63, SIAM, Philadelphia, 199. [18] Petrov, V.V.: Sum of Independent Random Variable. Ergebnie der Mathematik und ihrer Grenzgebiete, Band 8, Springer, Berlin-Heidelberg-ew York,

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