Optimal Placement Delivery Arrays - arXiv

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Nov 14, 2016 - In a caching system, there is a server with N files and K users each with a ... Delivery phase, which takes place during peak times: Requested ...
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Optimal Placement Delivery Arrays Minquan Cheng, Jing Jiang, Qifa Yan, Xiaohu Tang, Member, IEEE and Haitao Cao

arXiv:1611.04213v1 [cs.IT] 14 Nov 2016

Abstract In wireless networks, coded caching scheme is an effective technique to reduce network congestion during peak traffic times. A (K, F, Z, S) placement delivery array ((K, F, Z, S)PDA in short) can be used to design a coded caching scheme with the delivery rate S/F during the peak traffic times. Clearly in order to minimize delivery rate, we only need to consider a (K, F, Z, S)PDA with minimum S. For the fixed parameters K, F and Z, a (K, F, Z, S)PDA with the minimum S is called optimal. So it is meaningful to study optimal PDAs. There are some known PDAs constructed by combinatorial design theory, hypergraphs and so on. However there is only one class of optimal PDAs (IEEE Trans. Inf. Theory 60:2856-2867, 2014) constructed by Ali and Niesen. We will focus on constructing optimal PDAs. In this paper, two kinds of lower bounds on the value of S in a (K, F, Z, S)PDA are derived first. Then two kinds of recursive constructions are proposed. Consequently (i) optimal PDAs with Z = 1 and F − 1 for any positive integers K and F are obtained; (ii) several infinite classes of optimal PDAs for some K, F and Z are constructed. Finally more existence of optimal PDAs with Z = F − 2 are given.

I. I NTRODUCTION Recently, with the explosive increasing broadband services such as video-on-demand (VoD) and catch-up TV, there is a tremendous pressure on the data transmission over the wireless network. However, it is widely acknowledged that the limited spectrum resource can not effectively support the ever increasing traffic in the wireless network. To make matters worse, the high temporal variability of network traffic results in disequilibrium about utilization of spectrum resource. That is, such a disequilibrium communication systems are congested during peak-traffic times and under utilized during off-peak times. Caching system, which proactively caches some contents at the network edge during off-peak hours, is a promising solution to smooth out traffic variability and reducing congestion (see [10], [2], [7], [8], [9], and references therein). In a caching system, there is a server with N files and K users each with a cache of size M . It is called (K, M, N ) caching system. The users are connected to the server by a shared link (see Fig. 1). In each time slot each user requests one of the N files. There are two distinct phases in a caching system: • •

Placement phase, which takes place during off-peak times: Parts of content are placed in users’ cache memories. Delivery phase, which takes place during peak times: Requested content is delivered by exploiting the local cache storage of users.

Clearly, it is carried out without the knowledge of the particular user requests in placement phase. And our goal is to minimize the rate of transmission over the shared link since the delivery phase takes place during peak traffic period. This rate is also called the delivery rate. In order to minimize the delivery rate, Maddah-Ali and Niesen proposed a coded caching approach based on network coding theory in [10]. Following the seminal work of [10], most researches focus on the case N < K in the coded caching system. For more details, the reader can refer to [3], [4], [6], [12], [15], [19]. When N ≥ K, there exists the worst-case that each user requests different file. So the objective of the system designer is to provide a two-phase scheme such that given the cahce storage the delivery rate in the delivery phase for the worst-case requests is minimized. M. Cheng and J. Jiang are with Guangxi Key Lab of Multi-source Information Mining & Security, Guangxi Normal University, Guilin 541004, China (e-mail: {chengqinshi,jjiang2008}@hotmail.com). Q. Yan and X. Tang are with the Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, 610031, China (e-mail: [email protected], [email protected]). H. Cao is with the Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China (e-mail: [email protected]).

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Fig. 1: Caching system

By investigating placement phase and delivery pahse from a combinatorial viewpoint, a new concept of placement delivery array (PDA) was introduced to characterize the involved strategies in two phases in [18]. As a result, the problem of designing a coded caching scheme can be transformed into the problem of designing an appropriate PDA. So it will be meaningful to study PDAs. There are some known constructions on PDAs. In [18] the authors showed that the first deterministic coded caching scheme proposed by Maddah-Ali and Niesen is a PDA, and also constructed two infinite classes of PDAs by means of partitions. Based on these results, the generalized constructions are proposed in [13]. Unfortunately, no one has yet proved that one of these constructions is optimal. In this paper, we focus on the constructions of optimal PDAs from a combinatorial viewpoint. We first derive two kinds of   F lower bounds on S(K, F, Z). According to our lower bounds, optimal ( FZ , F, Z, Z+1 )PDAs are obtained for any positive integers K and Z. Then we propose two kinds of recursive constructions. Consequently (i) optimal PDAs with Z = 1 and F − 1 are constructed for any integers K and F by the first recursive construction; (ii) several infinite classes of optimal PDAs for general Z and some K, F are obtained by the second recursive construction. Finally further existence of optimal PDAs are given when Z = F − 2. The rest of this paper is organized as follows. Section II introduces some preliminaries about PDAs. In Section III, two kinds of lower bounds on the value of S are derived. In Section IV two kinds of recursive constructions are proposed first. Then several infinite classes of optimal PDAs are constructed. Conclusion is drawn in Section V. II. P RELIMINARIES For later reference, the following notations are useful. • •



We denote arrays by bold capital letters. We use [a, b] = {a, a + 1, . . . , b} and [a, b) = {a, a + 1, . . . , b − 1} for intervals of integers for any integers a and b. If a > b, let [a, b) = ∅. Given an F × F array A = (ai,j ) on [0, S) ∪ {∗}, set A + s = (ai,j + s) where s + ∗ = ∗ for any integer s.

Definition 1: ([18]) For positive integers K, F , Z and S, an F × K array P = (pi,j ), i ∈ [0, F ), j ∈ [0, K), composed of a specific symbol “ ∗ ” and S nonnegative integers 0, 1, · · · , S − 1, is called a (K, F, Z, S) placement delivery array (PDA) if it satisfies the following conditions: C1. The symbol “ ∗ ” called star appears Z times in each column; C2. For any two distinct entries pi1 ,j1 and pi2 ,j2 , pi1 ,j1 = pi2 ,j2 = s is an integer only if a. i1 6= i2 , j1 6= j2 , i.e., they lie in distinct rows and distinct columns; and b. pi1 ,j2 = pi2 ,j1 = ∗, i.e., the corresponding 2 × 2 subarray formed by rows i1 , i2 and columns j1 , j2 must be of the following form ! ! s ∗ ∗ s or . ∗ s s ∗

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Example 1: There exists a (4, 6, 3, 4)PDA as follows:  0   ∗   ∗ P6×4 =   1    ∗ 2

∗ ∗ 1 2 0 ∗ ∗ 3 ∗ 0 3 ∗

3 ∗ 2 ∗ 1 ∗

     .    

In a caching scheme, if each file is sub-divided into F packets, we refer to such a scheme as an F -division caching scheme. The authors in [18] showed that an F -division caching scheme for a (K, M, N ) caching system can be realized by a (K, F, Z, S) PDA P = [pj,k ]F ×K with Z/F = M/N . Precisely, each user is able to decode its requested file correctly for any request with delivery rate R = S/F . So we should try to construct a PDA with S as small as possible. Let S(K, F, Z) = min{S | there exists a (K, F, Z, S)PDA}. A (K, F, Z, S)PDA is said to be optimal if S = S(K, F, Z). Clearly it is much more meaningful to study optimal PDAs. There are some known constructions of PDAs. Lemma 1: ([18]) The coded caching scheme proposed by Maddah-Ali and Niesen equals a (K,  K KM KM/N +1 )PDA with KM/N stars in each row, where N is an integer.

K KM/N



,

K−1 KM/N −1



,

Lemma 2: ([18]) For any positive integers m and q ≥ 2, there exist a (q(m + 1), q m , q m−1 , q m+1 − q m )PDA and a (q(m + 1), (q − 1)q m , (q − 1)2 q m−1 , q m )PDA. Based on these results, the generalized constructions are proposed in [13].     n  Corollary 1: ([13]) There exist an ( nb , na , na − n−b a , a+b )PDA for any positive integers a, b and n with a + b ≤ n,  t m m m−t and an ( m (q − 1)t , q m (q − 1)t )PDA for any positive integers q ≥ 2, t and m with t ≤ m. t q ,q ,q − q In [19], the authors proved that Ali and Niesen’s construction has the minimum delivery rate. That is, FS is minimum. So   we hvae the minimum S for the fixed parameters K = k, F = kt and Z = k−1 t−1 . This implies that the PDAs in Lemma 1 are optimal. Unfortunately except for this class of PDAs, no one has yet proved that one of the other PDAs is optimal. As far as we know, there is no studies on optimal PDAs. We will focus on the constructions of optimal PDAs for any integers K, F and Z from a combinatorial viewpoint in this paper. III. L OWER BOUNDS ON S(K, F, Z) In this section, we will derive two kinds of lower bounds on S(K, F, Z). Now let us introduce our first lower bound. Given a (K, F, Z, S)PDA P for some positive integers K, F , Z and S, let us consider a simple but very useful property of P. By the Pigeonhole Principle, there must exist one row with at lest d (F −Z)K e integers. With respect to row/column permutation F (F −Z)K (F −Z)K of P, we can assume that these d F e integers, say 0, 1, . . ., d F e − 1, are in the first row, i.e.,   e − 1 ... 0 1 2 . . . d (F −Z)K F      . P= (1) ..  0 .   P Clearly P0 is an (d (F −Z)K e, F −1, Z, S 0 )PDA. Furthermore, we claim that each integer in P0 does not belong to [0, d (F −Z)K e). F F (F −Z)K (F −Z)K 0 Otherwise if an entry in (i, j) of P is s ∈ [0, d F e) for some integers 0 ≤ i < F − 1 and 0 ≤ j < d F e, then the entry in (0, j) of P must be “∗” from C2 of a PDA. Clearly this is impossible since the entry in (0, j) of P is integer j. So we have   (F − Z)K + S 0 ≤ S. (2) F From above investigation, the following result can be obtained.

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Theorem 1: For any positive integers K, F , Z and S,            (F − Z − 1) (F − Z)K 1 2 (F − Z)K (F − Z)K + + ... + ... ... . S(K, F, Z) ≥ F F −1 F Z +1 Z +2 F

(3)

Proof: Denote an optimal (K, F, Z, S)PDA by P. From above discussion P can be written as (1). Then we have a (d (F −Z)K e, F − 1, Z, S 0 )PDA P0 and d (F −Z)K e + S 0 ≤ S. By induction, repeatedly using (1) and (2), formula (3) can be F F obtained. When Z = F − 1 and F − 2, (3) can be written in the following respectively. l 2K m l 1 l 2K mm lK m S(K, F, F − 2) ≥ + . S(K, F, F − 1) ≥ F F F −1 F

(4)

−Z−1) (F −Z)K In fact the right of inequality in (3) is not very easy to compute. However, if d (F F ee = 1, from (3) we have −1 d F   (F − Z)K S(K, F, Z) ≥ + F − Z − 1. F

For some parameters K, F and Z this lower bound is tight. (8−5)·6 −Z−1) (F −Z)K ee = d (8−5−1) ee = d 27 d 18 Example 2: When K = 6, F = 8 and Z = 5, we have d (F F −1 d F 8−1 d 8 8 ee = 1. So we have S(6, 8, 5) ≥ 3 + 3 − 1 = 5. It is easy to check that the following array is an optimal (6, 8, 5, 5)PDA.   0 ∗ ∗ ∗ 3 ∗    1 3 ∗ ∗ ∗ 4     ∗ 0 1 ∗ ∗ ∗     2 ∗ 3 ∗ ∗ ∗    P8×6 =    ∗ 2 ∗ 1 ∗ ∗     ∗ ∗ 4 0 2 ∗       ∗ ∗ ∗ ∗ 1 0  ∗ ∗ ∗ 3 ∗ 2

We could further improve the lower bound in (3) by the second lower bound. Theorem 2: For any positive integers K, F and Z, l (F − Z)K m l (F − Z)K m l (F − Z)K m  S ,F − ,Z − + 1 ≤ S(K, F, Z) − 1. S(K, F, Z) S(K, F, Z) S(K, F, Z)

(5)

Proof: Assume that there exists a (K, F, Z, S(K, F, Z))PDA P. Then by the Pigeonhole Principle, there must exist an (F −Z)K integer s ∈ [0, S(K, F, Z)) occurring at least d S(K,F,Z) e times in P. Without loss of generality, we can write P as follows.   s ∗ ∗ ... ∗ ...  ∗ s ∗ ... ∗ ...     .  .. ..  .  .   . .    P=  ∗ ∗ ∗ ... s ...        .  ..    P0 (F −Z)K (F −Z)K (F −Z)K Clearly P0 is a (d S(K,F,Z) e, F − d S(K,F,Z) e, Z − d S(K,F,Z) e + 1, S 0 )PDA. Then

S

l (F − Z)K m l (F − Z)K m  l (F − Z)K m ,F − ,Z − + 1 ≤ S0. S(K, F, Z) S(K, F, Z) S(K, F, Z)

Since S 0 ≤ S(K, F, Z) − 1, our statement is true. Then the proof is completed. −Z−1) (F −Z)K When d (F F ee = 1, a tighter lower bound can be derived by Theorems 1 and 2. −1 d F

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m l m l m l (F −Z−1) (F −Z)K (F −Z)K . If d e = 1 and F < Lemma 3: For any positive integers K, F and Z, let h = (F −Z)K F F −1 F F l m (F −Z)K h+F −Z−1 (h + F − Z − 1) we have S(K, F, Z) ≥ h + F − Z. Proof: Assume S(K, F, Z) = d (F −Z)K e + F − Z − 1. From (5) we have F l (F − Z)K m l (F − Z)K m l (F − Z)K m   (F − Z)K  S ,F − ,Z − +1 ≤ + F − Z − 1 − 1. S(K, F, Z) S(K, F, Z) S(K, F, Z) F From (3), we have l (F − Z)K m l (F − Z)K m l (F − Z)K m  S ,F − ,Z − +1 S(K, F, Z) S(K, F, Z) S(K, F, Z) l m l m l m  (F −Z)K (F −Z)K (F −Z)K l m l m (F − S(K,F,Z) − (Z − S(K,F,Z) + 1)) S(K,F,Z)  + F − (F − Z)K − (Z − (F − Z)K + 1) − 1 l m ≥   (F −Z)K S(K, F, Z) S(K, F, Z)   F − S(K,F,Z) l m  (F −Z)K (F − Z − 1) S(K,F,Z) +F −Z −2 m l ≥   (F −Z)K   F − S(K,F,Z) From (6) and (7), we have l m  (F −Z)K   (F − Z − 1) S(K,F,Z)   + F − Z − 2 ≤ (F − Z)K + F − Z − 2 l m   (F −Z)K F   F − S(K,F,Z) l m l m (F −Z)K Let r = (F −Z)K F − (F − Z)K and r = 1 F S(K,F,Z) S(K, F, Z) − (F − Z)K. From (8) we have

(6)

(7)

(8)

(F − Z)K + r (F − Z − 1)((F − Z)K + r1 ) ≤ S(K, F, Z)F − ((F − Z)K + r1 ) F From above formula we have (F − Z)K + r1 (F − Z)K + r1 r1 − r + ≤ + F − Z − 1. (F − Z)K + r F F l m l m (F −Z)K r1 −r This implies that (F −Z)K+r ≤ 0. This is impossible since (F −Z)K F < h+F F −Z−1 (h + F − Z − 1), i.e., r1 > r. So we have S(K, F, Z) ≥ h + F − Z. Then the proof is completed. F −Z −1+

−Z−1) (F −Z)K When Z = F − 2, it is easy to check that d (F F ee = 1 holds if and only if K ≤ F (F2−1) holds. From Lemma −1 d F 3, the following tighter lower bound on S(K, F, F − 2) can be obtained. l m 2K Corollary 2: For any positive integers K, F with K ≤ F (F2−1) , if hF < h+1 (h + 1), then S(K, F, F − 2) ≥ h + 2

where h = d 2K F e. l mm l l mm l −Z−1) (F −Z)K −Z−1) (F −Z)K = 1, we can also study the case (F F > 1. Similar to the discussion of the case (F F −1 F −1 F However this case is more complex than the first case. IV. C ONSTRUCTIONS OF OPTIMAL PDA S In this section, several infinite classes of optimal PDAs will be constructed. According to Theorem 1, we first show that the permutation of PDA in Lemma 1 is also optimal. Then two classes recursive constructions are proposed. Underlying these recursive constructions, several optimal PDAs are obtained. That is, (i) optimal PDAs with Z = 1 and F − 1 are constructed for any integers K and F ; (ii) several infinite classes of optimal PDAs for general Z and some K, F are obtained. Finally further existence of optimal PDAs are given when Z = F − 2.   k  For any positive integers k and t with 0 ≤ t ≤ k, we have an optimal (k, kt , k−1 )PDA P with t stars in each row. t−1 , t+1   k k 0 0 Denote the permutation of P by P . It is easy to check P is also a PDA with K = t , F = k, Z = t and S = t+1 . So the following statement holds.      k k k Lemma 4: The permutation of (k, kt , k−1 t−1 , t+1 )PDA in Lemma 1 is a ( t , k, t, t+1 )PDA.

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From Theorem 1 we claim that this PDA is optimal.  F , F, Z, Z+1 )PDA.   F Proof: For any positive integers F and Z with 0 ≤ Z ≤ F , from Lemma 4, there exists an ( FZ , F, Z, Z+1 )PDA. So it   F F is sufficient to show that S( Z , F, Z) = Z+1 . From (3) we have Theorem 3: For any positive integers F and Z with 0 ≤ Z ≤ F , there exists an optimal (

F Z +1

! ≥ = = = =

F Z



! ! F , F, Z Z & & & & & &  ''  ' ''' ' & F (F − Z) F (F − Z) F (F − Z − 1) (F − Z) Z 1 2 Z Z + + ... + ... ... F F −1 F Z +1 Z +2 F !' & & & ! ''' ! & (F − Z − 1) F −1 1 2 F −1 F −1 + ... + ... ... + F −1 F −Z −1 Z +1 Z +2 F −Z −1 F −Z −1 ! ! ! ! F −1 F −2 F − (F − Z − 1) F − (F − Z) + + ... + + F −Z −1 F −Z −2 F − Z − (F − Z − 1) F − Z − (F − Z) ! ! F F = . F −Z −1 Z +1 S

The last equation holds since         n+r+1 n+r n+r−1 n = + + ... + r r r−1 0 for any positive integers n and r. Then the proof is completed. In the following subsections, we will propose two kinds of recursive constructions of PDAs which can be used to construct several classes of optimal PDAs.

A. The first recursive construction Construction 1: Let P be a (K, F, Z, S)PDA with integers [0, S). For any positive integer m, we can define an F × mK array   P0 = P, P + S, . . . , P + (m − 1)S . (9) From the definition of a PDA it is not difficult to check that P0 is an (mK, F, Z, mS)PDA.     F F Let P be an optimal ( FZ , F, Z, Z+1 )PDA with integers [0, Z+1 ). When K = m FZ , the P0 generated in Construction   F 1 is an (m FZ , F, Z, m Z+1 )PDA. Similar to the proof of Theorem 3, we could also show P0 is optimal. Furthermore using Construction 1, we could obtain an optimal PDA with F − 1 and Z = 1 for any positive integers K and F respectively. K  0 • When Z = F − 1, we have K = mF . From (4) we have S(K, F, F − 1) ≥ F . Clearly if F |K, P generated by (9) is an optimal PDA with Z = F − 1. When F 6 |K, let h = d K F e. Then the PDA obtained by deleting last hF − K columns of P0 is also optimal. K • When Z = 1, we also have K = mF . Let w = b F c, u = K − F w for any positive integers K and F . From (3) we have l (F − Z)K m l (F − Z − 1) l (F − Z)K mm l 1 l 2 l l (F − Z)K m mmm S(K, F, 1) ≥ + + ... + ... ... F F −1 F Z +1 Z +2 F l (F − 2) m l 1 l 2 l mmm = ((F − 1)w + u) + ((F − 1)w + u) + . . . + . . . ((F − 1)w + u) . . . F −1 Z +1 Z +2 l 1 l 2 l mmm = ((F − 1)w + u) + ((F − 2)w + u) + · · · + . . . ((F − 2)w + u) . . . Z +1 Z +2 .. . F (F − 1) u(u − 1) w + u(F − 1) − 2 2 F (F − 1) u(u + 1) = w + uF − 2 2

=

7

Similarly if F |K, P0 generated by (9) is an optimal PDA with Z = 1. When F 6 |K, let h = d K F e. Then the PDA obtained by deleting the last hF − K columns of P0 is also optimal.

B. The second recursive construction Construction 2: Suppose that Pi is an optimal (Ki , Fi , Zi , Si )PDA with integer set [0, Si ), i = 0, 1. If F0 −Z0 = F1 −Z1 , we can define an (F0 + F1 ) × (K0 + K1 ) array ! P0 ∗ P= . (10) ∗ P1 From the definition of a PDA it is not difficult to check that P is an (K0 +K1 , F0 +F1 , F0 +Z1 , S)PDA with S = max{S0 , S1 }. −Z−1) (F −Z)K ee = 1 in the following. Now let us study Construction 2 for the case d (F F −1 d F i )Ki Theorem 4: For any two optimal (Ki , Fi , Zi , Si )PDAs with Si = d (Fi −Z e + (Fi − Zi ) − 1 i = 0, 1, there exists an Fi 1 )K1 0 )K0 optimal (K0 + K1 , F0 + F1 , F0 + Z1 , S1 )PDA if F0 − Z0 = F1 − Z1 and 0 ≤ d (F1 −Z e − d (F0 −Z e ≤ 1. F1 F0

Proof: Suppose that Pi is an optimal (Ki , Fi , Zi , Si )PDA with integer set [0, Si ), i = 0, 1. By Theorem 1 we have S0 ≤ S1 −Z−1) (F −Z)K 0 )K0 1 )K1 since d (F F ee = 1 and (F0 −Z ≤ (F1 −Z . So P generated by Construction 2 is a (K0 + K1 , F0 + F1 , F0 + −1 d F F0 F1 Z1 , S1 )PDA. Clearly it is sufficient to show S(K0 + K1 , F0 + F1 , F0 + Z1 ) = S1 . From (3), S1 ≥S(K0 + K1 , F0 + F1 , F0 + Z1 )      (F0 + F1 − (F0 + Z1 ) − 1) (F0 + F1 − (F0 + Z1 ))(K0 + K1 ) (F0 + F1 − (F0 + Z1 ))(K0 + K1 ) + + ... ≥ F0 + F 1 F0 + F 1 − 1 F0 + F1       (F0 + F1 − (F0 + Z1 ))(K0 + K1 ) 1 2 + ... ... (F0 + Z1 ) + 1 (F0 + Z1 ) + 2 F0 + F 1      (F1 − Z1 )(K0 + K1 ) (F1 − Z1 − 1) (F1 − Z1 )(K0 + K1 ) = + + ... F0 + F1 F 0 + F1 − 1 F0 + F1       (F1 − Z1 )(K0 + K1 ) 1 2 + ... . ... (F0 + Z1 ) + 1 (F0 + Z1 ) + 2 F0 + F1

Since

(F0 −Z0 )K0 F0



(F1 −Z1 )K1 F1

(11)

we have (F0 − Z0 )K0 (F0 − Z0 )(K0 + K1 ) (F1 − Z1 )K1 ≤ ≤ . F0 F0 + F1 F1

1 )K1 0 )K0 e = d (F1 −Z e we have S1 = S0 . Then (11) can be written as When d (F0 −Z F0 F1   (F0 − Z0 )K0 S1 ≥ S(K0 + K1 , F0 + F1 , F0 + Z1 ) ≥ + (F0 − Z0 ) − 1 = S0 . F0

(12)

0 )K0 1 )K1 e < d (F1 −Z e we have S1 = S0 + 1 and When d (F0 −Z F0 F1

(F0 − Z0 )K0 (F0 − Z0 )(K0 + K1 ) (F1 − Z1 )K1 < < . F0 F0 + F1 F1 Then (11) can be written as  S1 ≥ S(K0 + K1 , F0 + F1 , F0 + Z1 ) >

 (F0 − Z0 )K0 + (F0 − Z0 ) − 1 = S0 . F0

(13)

From (12) and (13), P is optimal. Then the proof is completed. For instance from Example 2 and Theorem 4, there exists an optimal (6m, 8m, 8m − 3, 5)PDA for any positive integer m by induction. In fact, we can construct more infinite classes of optimal PDAs by Construction 2. Since all the optimal PDAs with Z = F − 1 and 1 are constructed in Subsection IV-A, let us consider the case 1 < Z < F − 1.

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−Z−1) (F −Z)K e. When d (F F ee = 1 we have 1) The general case 1 < Z < F − 1: Let h = d (F −Z)K F −1 d F

F ≥ h(F − Z − 1) + 1

and

K=

hF − r F −Z

(14)

where r = hF − (F − Z)K, 0 ≤ r < F . Suppose that there exists an optimal (K0 , F0 , Z0 , h + F0 − Z0 − 1)PDA P1 with the minimum F0 ≥ h(F0 − Z − 1) + 1 0 −r and maximum K0 = hF F0 −Z . Then by Theorem 4 we have an optimal (mK0 , mF0 , (m − 1)F0 + Z0 , h + F0 − Z0 − 1)PDA Pm for any positive integer m. For any integer x, if h−1
1. In the following we will obtain several infinite classes optimal PDAs with Z = F − 2 by means of Lemma 1 and Theorem 3. Firstly, it is not difficult to obtain the following result by using Theorem 3. Theorem 6: For any positive integers K and F with F |2K and (h+1)|F , there exists an optimal (K, F, F −2, h+1)PDA Proof: From Theorem 3, we can obtain an optimal ( h(h+1) , h + 1, h − 1, h + 1)PDA P0 . Let F = g(h + 1), then 2 h(h+1) hF K = 2 = g 2 . By applying Theorem 4 g − 1 times with P0 , we can derive an optimal (g h(h+1) , g(h + 1), (g − 1)(h + 2 1) + h − 1, h + 1)PDA which is indeed an optimal (K, F, F − 2, h + 1)PDA.

9

Example 4: From [18], when F = 3 the following optimal (3, 3, 1, 3)PDA can be obtained.   ∗ 0 1   P3×3 =  0 ∗ 2  1 2 ∗ From Theorem 6, there exists an optimal (3m, 3m, 3m − 2, 3)PDA for any positive integer m. The following PDAs are optimal (6, 6, 4, 3)PDA and optimal (9, 9, 7, 3)PDA,  ∗ 0 1  0 ∗ 2    ∗ 0 1 ∗ ∗ ∗   1 2 ∗     0 ∗ 2 ∗ ∗ ∗   ∗ ∗ ∗     1 2 ∗ ∗ ∗ ∗    P = P6×6 =   ∗ ∗ ∗ 9×9  ∗ ∗ ∗ ∗ 0 1      ∗ ∗ ∗     ∗ ∗ ∗ 0 ∗ 2   ∗ ∗ ∗  ∗ ∗ ∗ 1 2 ∗   ∗ ∗ ∗ ∗ ∗ ∗

respectively. ∗ ∗ ∗ ∗ 0 1 ∗ ∗ ∗

∗ ∗ ∗ 0 ∗ 2 ∗ ∗ ∗

∗ ∗ ∗ 1 2 ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1

∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ 2

∗ ∗ ∗ ∗ ∗ ∗ 1 2 ∗

                

Now we will consider the case F |K, that is h = 2K F is even. In this case, optimal PDAs can be obtained if the parameter F is proper large. Firstly, the following two lemmas which are very useful for the main result will be given. , n + 2, n, n + 2)PDA for any positive even integer n. Lemma 6: There exists an optimal ( n(n+2) 2 Proof: By using Corollary 2, we have S( n(n+2) , n+2, n) ≥ n+ 2 since d 2(n(n+2)/2) e(h+ 1) = n2 + 2n+ 1 > n(n+ 2) = 2 h+1 (n+2)(n+1) h(n + 2). We can also obtain a ( , n + 2, n, n + 2)PDA P0 from Theorem 3. By deleting (h+2) columns from P0 , 2 2 we can derive a ( n(n+2) , n + 2, n, n + 2)PDA. So this PDA is optimal. The the proof is completed. 2 2

Lemma 7: Then there exists an optimal ( n2 , n, n − 2, n + 2)PDA for any positive even integer n. 2

2

/2) 2 2 Proof: By using Corollary 2, we can drive that S( n2 , n, n − 2) ≥ n + 2 since d 2(n h+1 e(h + 1) = n + n > n = hn. From n(n−1) Theorem 3 and Example 3-(i), there exists a ( 2 , n, n − 2, n)PDA P0 and ( n2 , n, n − 2, 2)PDA P1 , respectively. Suppose that the integer set of array P0 and array P1 are {0, 1, · · · , n − 1} and {n, n + 1}, respectively. Let P = (P0 , P1 ). Then it is 2 not difficult to check that P is a ( n2 , n, n − 2, n + 2)PDA. From Corollary 2, P is optimal. Then the proof is completed.

Example 5: There exist a (6, 4, 3, 4)PDA and a  0 ∗ ∗  ∗ ∗ 0  P4×6 =   ∗ 2 ∗ 3 ∗ 2

(2, 4, 2, 2)PDA as following.   4 1 ∗ 2  5 ∗ ∗ 3    P4×2 =    ∗ 3 0 ∗  ∗ ∗ 1 ∗

According to Lemma 7, we can construct an optimal (8, 4, 2, 6)PDA  0 ∗ ∗ 1 ∗ 2  ∗ ∗ 0 ∗ ∗ 3  P4×8 =   ∗ 2 ∗ 3 0 ∗ 3 ∗ 2 ∗ 1 ∗

∗ ∗ 4 5

    

as follows.  4 ∗ 5 ∗    ∗ 4  ∗

5

By using Theorem 3, Lemmas 6 and 7, we can construct several infinite classes of optimal PDAs which are listed in Lemmas 8 and 9. Lemma 8: Suppose that a and n are positive integers and 2|n. Then there exists an optimal ( n2 (an + i), an + i, an + i − 2, n + 2)PDA for any integer i ∈ (a, 2a]. Proof: By using Corollary 2, we can drive that S( n2 (an + i), an + i, an + i − 2) ≥ n + 2 since d

2(n(an + i)/2) n(an + i) e(h + 1) = d e(n + 1) = an2 + in + a > an2 + in = n(an + i). h+1 n+1

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, n + 1, n − 1, n + 1)PDA P0 and ( n(n+2) , n + 2, n, n + 2)PDA P1 , respectively. From Theorems 3 and 6, there exist an ( n(n+1) 2 2 Suppose that the integer set of array P0 and array P1 are {0, 1, · · · , n} and {0, 1, · · · , n + 1}, respectively. We can construct an (an + i) × ( n2 (an + i)) array by using (2a − i) P0 s and (i − a) P1 s as followings.   P0 ∗ · · · ∗ ∗ ∗ ··· ∗    ∗ P0 ∗ ∗ ∗ ∗ ··· ∗     .. .. .. .. .. ..  ..   . . . . . . .    ∗ ∗ · · · P0 ∗ ∗ ··· ∗    (15) P= .   ∗ ∗ · · · ∗ P ∗ · · · ∗ 1    ∗ ∗ ··· ∗ ∗ P1 ∗ ∗     . .. .. .. .. ..  ..  .. . . . . . .    ∗ ∗ ··· ∗ ∗ ∗ · · · P1 It is not difficult to check that P is a PDA. Furthermore, P is optimal since S( n2 (an + i), an + i, an + i − 2) ≥ n + 2. The proof is completed. Lemma 9: Suppose that a and n are positive integers and 2|n. Then there exists an optimal ( n2 (an + i), an + i, an + i − 2, n + 2)PDA for any integer i ∈ [0, a). Proof: Similar to the proof of Lemma 8, we can construct an optimal ( n2 (an + i), an + i, an + i − 2, n + 2)PDA P by 2 using i P0 s and a − i P1 s, where P0 is a ( n(n+1) , n + 1, n − 1, n + 1)PDA, and P1 is a ( n2 , n, n − 2, n + 2)PDA which can 2 be obtained from Lemma 7. The proof is completed. Notice that the case i = a is not covered by Lemmas 8 and 9. In fact, such a case can be found in Theorem 6. So we can obtain the following result from Theorem 6, Lemmas 8 and 9. Corollary 3: Suppose that a and n are positive integers with 2|n. Then there exists an optimal ( n2 (an + i), an + i, an + i − 2, S)PDA for any integer i ∈ [0, 2a], S = n + 1 when i = a or S = n + 2 when i 6= a. Example 6: We take n = 2, 4, 6 as examples. S∞ S∞ • The case n = 2. Then a=1 [na, na + 2a] = a=1 [2a, 4a] = [2, ∞). That is, there exists an optimal (F, F, F − 2, S)PDA for any integer F ≥ 2, where S = 3 when 3|F or S = 4 when 3 6 |F . S∞ S∞ S • The case n = 4. Then a=1 [na, na + 2a] = a=1 {4a, 6a} = [4, 6] [8, ∞). That is, there exists an optimal (2F, F, F − S 2, S)PDA for any integer F ∈ [4, 6] [8, ∞), where S = 5 when 5|F or S = 6 when 5 6 |F . S∞ S∞ S S • The case n = 6. Then a=1 [na, na + 2a] = a=1 [6a, 8a] = [6, 8] [12, 16] [18, ∞). That is, there exists an optimal S S (3F, F, F − 2, S)PDA for any integer F ∈ [6, 8] [12, 16] [18, ∞), where S = 7 when 6|F or S = 8 when 6 6 |F . From Example 6, it is not difficult to see that Corollary 3 in fact give optimal PDAs with F |K if F is large enough. Theorem 7: Suppose that F and K are positive integers with F |K. Then there exists an optimal (K, F, F − 2, S)PDA 2 for any integer F ≥ h2 , where h = 2K F , and S = h + 1 when (h + 1)|F or S = h + 2 when (h + 1) 6 |F . 2

h Proof: Obviously, h = 2K F is even since F |K. Since F ≥ 2 , we can write F = ah + i, where 0 ≤ i ≤ h − 1, a ≥ Then i < 2a. From Corollary 3, we can obtain an optimal (K, F, F − 2, S)PDA. This completes the proof.

h 2.

V. C ONCLUSION  In this paper, two kinds of lower bounds on S(K, F, Z) were derived. According to our lower bounds, an optimal ( FZ , F ,  F Z, Z+1 )PDA was obtained for any positive integers K and Z. Then two kinds of recursive constructions were proposed. Consequently (i) optimal PDAs with Z = 1 and F − 1 were constructed for any integers K and F by the first recursive construction; (ii) several infinite classes of optimal PDAs for general Z and some K, F were obtained by the second recursive construction. Finally when Z = F − 2, further constructions of optimal PDAs were proposed. It would be of interest if we could find a tighter lower bound on S(K, F, Z) for some Z and more constructions of the corresponding optimal PDAs.

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