Li {\it et al}. introduced coded distributed computing (CDC) scheme to reduce the communication load in general distributed computing frameworks such as MapReduce. They also proposed cascaded CDC schemes where each output function is computed multiple times, and proved that such schemes achieved the fundamental trade-off between computation load and communication load. However, these schemes require exponentially large numbers of input files and output functions when the number of computing nodes gets large. In this paper, by using the structure of placement delivery arrays (PDAs), we construct several infinite classes of cascaded CDC schemes. We also show that the numbers of output functions in all the new schemes are only a factor of the number of computing nodes, and the number of input files in our new schemes is much smaller than that of input files in CDC schemes derived by Li {\it et al}.
Coded distributed computing (CDC) introduced by Li et al. in 2015 offers an efficient approach to trade computing power to reduce the communication load in general distributed computing frameworks such as MapReduce. For the more general cascaded CDC, Map computations are repeated at $r$ nodes to significantly reduce the communication load among nodes tasked with computing $Q$ Reduce functions $s$ times. While an achievable cascaded CDC scheme was proposed, it only operates on homogeneous networks, where the storage, computation load and communication load of each computing node is the same. In…
Proactive content caching at user devices and coded delivery is studied considering a non-uniform file popularity distribution. A novel centralized uncoded caching and coded delivery scheme, which can be applied to large file libraries, is proposed. The proposed cross-level coded delivery (CLCD) scheme is shown to achieve a lower average delivery rate than the state of art. In the proposed CLCD scheme, the same subpacketization is used for all the files in the library in order to prevent additional zero-padding in the delivery phase, and unlike the existing schemes in the literature, two users…
Coded caching scheme is a promising technique to migrate the network burden in peak hours, which attains more prominent gains than the uncoded caching. The coded caching scheme can be classified into two types, namely, the centralized and the decentralized scheme, according to whether the placement procedures are carefully designed or operated at random. However, most of the previous analysis assumes that the connected links between server and users are error-free. In this paper, we explore the coded caching based delivery design in wireless networks, where all the connected wireless links are…
Optimal delivery scheme for coded caching problems with small buffer sizes and the number of users no less than the amount of files in the server was proposed by Chen, Fan and Letaief ["Fundamental limits of caching: improved bounds for users with small buffers," (IET Communications), 2016]. This scheme is referred to as the CFL scheme. In this paper, the link between the server and the users is assumed to be error prone only during the delivery phase. Closed form expressions for average rate and peak rate of error correcting delivery scheme for CFL prefetching scheme is obtained. An optimal e…
Coded caching schemes with low subpacketization and small transmission rate are desirable in practice due to the requirement of low implementation complexity and efficiency of the transmission. Placement delivery arrays (PDA in short) can be used to generate coded caching schemes. However, many known coded caching schemes have large memory ratios. In this paper, we realize that some schemes with low subpacketization generated by PDAs do not fully use the users' caching content to create multicasting opportunities and thus propose to overcome this drawback. As an application, we obtain two new …
Coded caching can create coded multicasting thus significantly accelerates content delivery in broadcast channels with receiver caches. While the original delivery scheme in coded caching multicasts each coded message sequentially, it is not optimal for multiple-input multiple-output (MIMO) broadcast channels. This work aims to investigate the full spatial multiplexing gain in multi-antenna coded caching by transmitting all coded messages concurrently. In specific, we propose to treat the content delivery as the transmission problem with general message sets where all possible messages are pre…
We consider a centralized caching network, where a server serves several groups of users, each having a common shared homogeneous fixed-size cache and requesting arbitrary multiple files. An existing coded prefetching scheme is employed where each file is broken into multiple fragments and each cache stores multiple coded packets each formed by XORing fragments from different files. For such a system, we propose an efficient file delivery scheme with explicit constructions by the server to meet the arbitrary multi-requests of all user-groups. Specifically, the stored coded packets of each cach…
Coded caching can significantly reduce the communication bandwidth requirement for satisfying users' demands by utilizing the multicasting gain among multiple users. Most existing works assume that the users follow the prescriptions for content placement made by the system. However, users may prefer to decide what files to cache. To address this issue, we consider a network consisting of a file server connected through a shared link to $K$ users, each equipped with a cache which has been already filled arbitrarily. Given an arbitrary content placement, the goal is to find a delivery strategy f…
Immersive viewing is emerging as the next interface evolution for human-computer interaction. A truly wireless immersive application necessitates immense data delivery with ultra-low latency, raising stringent requirements for next-generation wireless networks. A potential solution for addressing these requirements is through the efficient usage of in-device storage and computation capabilities. This paper proposes a novel location-based coded cache placement and delivery scheme, which leverages the nested code modulation (NCM) to enable multi-rate multicasting transmission. To provide a unifo…
Content caching at small-cell base stations (SBSs) is a promising method to mitigate the excessive backhaul load and delay, particularly for on-demand video streaming applications. A cache-enabled heterogeneous cellular network architecture is considered in this paper, where mobile users connect to multiple SBSs during a video downloading session, and the SBSs request files, or fragments of files, from the macro-cell base station (MBS) according to the user requests they receive. A novel coded storage and delivery scheme is introduced to reduce the load on the backhaul link from the MBS to the…
Decentralized proactive caching and coded delivery is studied in a content delivery network, where each user is equipped with a cache memory, not necessarily of equal capacity. Cache memories are filled in advance during the off-peak traffic period in a decentralized manner, i.e., without the knowledge of the number of active users, their identities, or their particular demands. User demands are revealed during the peak traffic period, and are served simultaneously through an error-free shared link. The goal is to find the minimum delivery rate during the peak traffic period that is sufficient…
Coded caching is used to reduce network congestion during peak hours. A single server is connected to a set of users through a bottleneck link, which generally is assumed to be error-free. During non-peak hours, all the users have full access to the files and they fill their local cache with portions of the files available. During delivery phase, each user requests a file and the server delivers coded transmissions to meet the demands taking into consideration their cache contents. In this paper we assume that the shared link is error prone. A new delivery scheme is required to meet the demand…
Network based on distributed caching of content is a new architecture to alleviate the ongoing explosive demands for rate of multi-media traffic. In caching networks, coded caching is a recently proposed technique that achieves significant performance gains compared to uncoded caching schemes. In this paper, we derive a lower bound on the average rate with a memory constraint for a family of caching allocation placement and a family of XOR cooperative delivery. The lower bound inspires us how placement and delivery affect the rate memory tradeoff. Based on the clues, we design a new placement …
A centralized coded caching system, consisting of a server delivering N popular files, each of size F bits, to K users through an error-free shared link, is considered. It is assumed that each user is equipped with a local cache memory with capacity MF bits, and contents can be proactively cached into these caches over a low traffic period; however, without the knowledge of the user demands. During the peak traffic period each user requests a single file from the server. The goal is to minimize the number of bits delivered by the server over the shared link, known as the delivery rate, over al…
Classical coded caching setting avails each user to have one dedicated cache. This is generalized to a more general shared cache scheme and the exact expression for the worst case rate was derived in [E. Parrinello, A. Unsal, P. Elia, "Fundamental Limits of Caching in Heterogeneous Networks with Uncoded Prefetching," available on arXiv:1811.06247 [cs.IT], Nov. 2018]. For this case, an optimal linear error correcting delivery scheme is proposed and an expression for the peak rate is established for the same. Furthermore, a new delivery scheme is proposed, which gives an improved rate for the ca…
This paper analyzes the achievable tradeoff between cache~size and download~rate in decentralized caching systems with the uncoded cache placement originally proposed by Maddah-Ali and Niesen. It proposes two novel delivery schemes that take advantage of the multicasting opportunities that arise when a file is demanded by multiple users. These delivery schemes are extensions of known ones to the regime where the file size is finite. Numerical evaluations for the case of file uniform popularity show that the proposed schemes outperform previous ones for all value of the cache size.
We consider the coded caching problem with shared caches where several users share a cache, but each user has access to only a single cache. For this network, the fundamental limits of coded caching are known for centralized and decentralized settings under uncoded placement. In the centralized case, to achieve the gains offered by coded caching, one requires a sub-packetization which increases exponentially with the number of caches. The dedicated cache networks had a similar issue, and placement delivery arrays (PDAs) were introduced as a solution to it. Using the PDA framework, we propose a…
The multi-access variant of the coded caching problem in the presence of an external wiretapper is investigated . A multi-access coded caching scheme with $K$ users, $K$ caches and $N$ files, where each user has access to $L$ neighbouring caches in a cyclic wrap-around manner, is proposed, which is secure against the wiretappers. Each transmission in the conventional insecure scheme will be now encrypted by a random key. The proposed scheme uses a novel technique for the key placement in the caches. It is also shown that the proposed secure multi-access coded caching scheme is within a constan…
We address the use of maximum distance separable (MDS) codes for distributed storage (DS) to enable efficient content delivery in wireless networks. Content is stored in a number of the mobile devices and can be retrieved from them using device-to-device communication or, alternatively, from the base station (BS). We derive an analytical expression for the download delay in the hypothesis that the reliability state of the network is periodically restored. Our analysis shows that MDS-coded DS can dramatically reduce the download time with respect to the reference scenario where content is alway…
Coded caching is an emerging technique to reduce the data transmission load during the peak-traffic times. In such a scheme, each file in the data center or library is usually divided into a number of packets to pursue a low broadcasting rate based on the designed placements at each user's cache. However, the implementation complexity of this scheme increases as the number of packets increases. It is crucial to design a scheme with a small subpacketization level, while maintaining a relatively low transmission rate. It is known that the design of caches in users (i.e., the placement phase) and…
The coded caching scheme is an efficient technique as a solution to reduce the wireless network burden during the peak times in a Device-to-Device (D2D in short) communications. In a coded caching scheme, each file block should be divided into $F$ packets. It is meaningful to design a coded caching scheme with the rate and $F$ as small as possible, especially in the practice for D2D network. In this paper we first characterize coded caching scheme for D2D network by a simple array called D2D placement delivery array (DPDA in shot). Consequently some coded caching scheme for D2D network can be …
In private information delivery (PID) problem, there are $K$ messages stored across $N$ servers, each capable of storing $M$ messages and a user. Servers want to convey one of the $K$ messages to the user without revealing the identity (index) of the message conveyed. The capacity of PID problem is defined as maximum number of bits of the desired message that can be conveyed privately, per bit of total communication, to the user. For the restricted case of replicated systems, where coded messages or splitting one message into several servers is not allowed, the capacity of PID has been charact…
In this paper, we consider multiple cache-enabled end-users connected to multiple transmitters through a linear network. We also prevent a totally passive eavesdropper, who sniffs the packets in the delivery phase, from obtaining any information about the original files in cache-aided networks. Three different secure centralized multi-transmitter coded caching scenarios namely, secure multi-transmitter coded caching, secure multi-transmitter coded caching with reduced subpacketization, and secure multi-transmitter coded caching with reduced feedback, are considered and closed-form coding delay…
This paper addresses the problem of exponentially increasing sub-packetization with the number of users in a centralized coded caching system by introducing a new coded caching scheme inspired by the symmetric neighboring consecutive side information index coding problem. The scheme has a placement policy where the number of sub-packets required grows only linearly with the number of users, with no restriction on file size, and a delivery policy which is instantaneously decodable. Further, an application of the new delivery scheme in a multi-access coded caching set-up is studied and a few res…
The multi-antenna coded caching problem, where the server having $L$ transmit antennas communicating to $K$ users through a wireless broadcast link, is addressed. In the problem setting, the server has a library of $N$ files, and each user is equipped with a dedicated cache of capacity $M$. The idea of extended placement delivery array (EPDA), an array which consists of a special symbol $\star$ and integers in a set $\{1,2,\dots,S\}$, is proposed to obtain a novel solution for the aforementioned multi-antenna coded caching problem. From a $(K,L,F,Z,S)$ EPDA, a multi-antenna coded caching schem…
Fog networks benefit from content caching at edge nodes (ENs) as well as fronthaul connectivity to the cloud. In previous works, both dedicated and multicast fronthaul links has been considered under different assumptions with Normalized Delivery Time (NDT) as a standard to measure the performance. While dedicated links has been studied extensively in different caching scenarios, multicast links has been developed under assumption of 2-by-2 fog network where cloud is connected to two ENs serving two users. The extension to arbitrary number of ENs and users only provides the lower bound on the …
Cache-aided coded multicast leverages side information at wireless edge caches to efficiently serve multiple groupcast demands via common multicast transmissions, leading to load reductions that are proportional to the aggregate cache size. However, the increasingly unpredictable and personalized nature of the content that users consume challenges the efficiency of existing caching-based solutions in which only exact content reuse is explored. This paper generalizes the cache-aided coded multicast problem to a source compression with distributed side information problem that specifically accou…
Cache-aided content delivery is studied in a multi-server system with $P$ servers and $K$ users, each equipped with a local cache memory. In the delivery phase, each user connects randomly to any $ρ$ out of $P$ servers. Thanks to the availability of multiple servers, which model small-cell base stations (SBSs), demands can be satisfied with reduced storage capacity at each server and reduced delivery rate per server; however, this also leads to reduced multicasting opportunities compared to the single-server scenario. A joint storage and proactive caching scheme is proposed, which exploits cod…
This paper considers a cache-aided device-to-device (D2D) system where the users are equipped with cache memories of different size. During low traffic hours, a server places content in the users' cache memories, knowing that the files requested by the users during peak traffic hours will have to be delivered by D2D transmissions only. The worst-case D2D delivery load is minimized by jointly designing the uncoded cache placement and linear coded D2D delivery. Next, a novel lower bound on the D2D delivery load with uncoded placement is proposed and used in explicitly characterizing the minimum …
Cache-aided coded content delivery is studied for devices with diverse quality-of-service (QoS) requirements, specified by a different average distortion target. The network consists of a server holding a database of independent contents, and users equipped with local caches of different capacities. User caches are filled by the server during a low traffic period without the knowledge of particular user demands. As opposed to the current literature, which assumes that the users request files in their entirety, it is assumed that the users in the system have distinct distortion requirements; an…
We consider the content delivery problem in a fading multi-input single-output channel with cache-aided users. We are interested in the scalability of the equivalent content delivery rate when the number of users, $K$, is large. Analytical results show that, using coded caching and wireless multicasting, without channel state information at the transmitter (CSIT), linear scaling of the content delivery rate with respect to $K$ can be achieved in some different ways. First, if the multicast transmission spans over $L$ independent sub-channels, e.g., in quasi-static fading if $L = 1$, and in blo…
This work considers the coded caching problem with shared caches, where users share the caches, and each user gets access only to one cache. The user-to-cache association is assumed to be known at the server during the placement phase. We focus on the schemes derived using placement delivery arrays (PDAs). The PDAs were originally designed to address the sub-packetization bottleneck of coded caching in a dedicated cache setup. We observe that in the setup of this paper permuting the columns of the PDA results in schemes with different performance for the same problem, but the sub-packetization…
Coded caching is an effective technique to reduce the redundant traffic in wireless networks. The existing coded caching schemes require the splitting of files into a possibly large number of subfiles, i.e., they perform coded subfile caching. Keeping the files intact during the caching process would actually be appealing, broadly speaking because of its simpler implementation. However, little is known about the effectiveness of this coded file caching in reducing the data delivery rate. In this paper, we propose such a file caching scheme which uses a decentralized algorithm for content place…
Coded caching utilizes proper file subpacketization and coded delivery to make full use of the multicast opportunities in content delivery, to alleviate file transfer load in massive content delivery scenarios. Most existing work considers deterministic environments. An important practical topic is to characterize the impact of the uncertainty from user inactivity on coded caching. We consider a one server cache-enabled network under homogeneous file and network settings in presence of user inactivity. Unlike random or probabilistic caching studied in the literature, deterministic coded cachin…
Coded caching scheme provides us an effective framework to realize additional coded multicasting gain by exploiting coding into multiple transmitted signals. The goal of coded caching design is to jointly optimize the placement and delivery scheme so as to minimize the amount of transmitted coded signals. However, few research efforts consider multiple-round coded caching design problem, in which fixed and mobile users may coexist in one network different number of active users present in successive rounds. Obviously, such dynamic network configurations may lead to undesired frequent placement…
We investigate the use of maximum distance separable (MDS) codes to cache popular content to reduce the download delay of wireless content delivery. In particular, we consider a cellular system where devices roam in an out of a cell according to a Poisson random process. Popular content is cached in a limited number of the mobile devices using an MDS code and can be downloaded from the mobile devices using device-to-device communication. We derive an analytical expression for the delay incurred in downloading content from the wireless network and show that distributed caching using MDS codes c…
A centralized coded caching scheme has been proposed by Maddah-Ali and Niesen to reduce the worst-case load of a network consisting of a server with access to N files and connected through a shared link to K users, each equipped with a cache of size M. However, this centralized coded caching scheme is not able to take advantage of a non-uniform, possibly very skewed, file popularity distribution. In this work, we consider the same network setting but aim to reduce the average load under an arbitrary (known) file popularity distribution. First, we consider a class of centralized coded caching s…
A cache-aided $K$-user Gaussian broadcast channel (BC) is considered. The transmitter has a library of $N$ equal-rate files, from which each user demands one. The impact of the equal-capacity receiver cache memories on the minimum required transmit power to satisfy all user demands is studied. Considering uniformly random demands across the library, both the minimum average power (averaged over all demand combinations) and the minimum peak power (minimum power required to satisfy all demand combinations) are studied. Upper bounds are presented on the minimum required average and peak transmit …
Coded caching and delivery is studied taking into account the correlations among the contents in the library. Correlations are modeled as common parts shared by multiple contents; that is, each file in the database is composed of a group of subfiles, where each subfile is shared by a different subset of files. The number of files that include a certain subfile is defined as the level of commonness of this subfile. First, a correlation-aware uncoded caching scheme is proposed, and it is shown that the optimal placement for this scheme gives priority to the subfiles with the highest levels of co…
Most results on coded caching focus on a static scenario, in which a fixed number of users synchronously place their requests from a content library, and the performance is measured in terms of the latency in satisfying all of these demands. In practice, however, users start watching an online video content asynchronously over time, and often abort watching a video before it is completed. The latter behaviour is captured by the notion of audience retention rate, which measures the portion of a video content watched on average. In order to bring coded caching one step closer to practice, asynch…
Coded caching scheme is a technique which reduce the load during peak traffic times in a wireless network system. Placement delivery array (PDA in short) was first introduced by Yan et al.. It can be used to design coded caching scheme. In this paper, we prove some lower bounds of PDA on the element and some lower bounds of PDA on the column. We also give some constructions for optimal PDA.
Information theoretic analysis of a coded caching system is considered, in which a server with a database of N equal-size files, each F bits long, serves K users. Each user is assumed to have a local cache that can store M files, i.e., capacity of MF bits. Proactive caching to user terminals is considered, in which the caches are filled by the server in advance during the placement phase, without knowing the user requests. Each user requests a single file, and all the requests are satisfied simultaneously through a shared error-free link during the delivery phase. First, centralized coded cach…
Coded-caching delivery is considered over a symmetric noisy broadcast channel whose state is unknown at the transmitter during the cache placement phase. In particular, the delivery phase is modeled by a state-dependent broadcast channel where the state remains constant over each transmission block and is learned by the transmitter (and the receivers) only at the beginning of each block. A state-adaptive coded caching scheme is proposed that improves either on rate or decoding latency over two baseline schemes that are based on standard coded caching.
We consider a cache-aided wireless device-to-device (D2D) network of the type introduced by Ji, Caire, and Molisch [1], where the placement phase is orchestrated by a central server. We assume that the devices' caches are filled with uncoded data, and the whole content database is contained in the collection of caches. After the cache placement phase, the files requested by the users are serviced by inter-device multicast communication. For such a system setting, we provide the exact characterization of the optimal load-memory trade-off under the assumptions of uncoded placement and one-shot d…
Coding theoretic approached have been developed to significantly reduce the communication load in modern distributed computing system. In particular, coded distributed computing (CDC) introduced by Li et al. can efficiently trade computation resources to reduce the communication load in MapReduce like computing systems. For the more general cascaded CDC, Map computations are repeated at r nodes to significantly reduce the communication load among nodes tasked with computing Q Reduce functions s times. In this paper, we propose a novel low-complexity combinatorial design for cascaded CDC which …
A decentralized coded caching scheme has been proposed by Maddah-Ali and Niesen, and has been shown to alleviate the load of networks. Recently, placement delivery array (PDA) was proposed to characterize the coded caching scheme. In this paper, a neural architecture is first proposed to learn the construction of PDAs. Our model solves the problem of variable size PDAs using mechanism of neural attention and reinforcement learning. It differs from the previous attempts in that, instead of using combined optimization algorithms to get PDAs, it uses sequence-to-sequence model to learn construct …
We study downlink beamforming in a single-cell network with a multi-antenna base station serving cache-enabled users. Assuming a library of files with a common rate, we formulate the minimum transmit power with proactive caching and coded delivery as a non-convex optimization problem. While this multiple multicast problem can be efficiently solved by successive convex approximation (SCA), the complexity of the problem grows exponentially with the number of subfiles delivered to each user in each time slot, which itself grows exponentially with the number of users. We introduce a low-complexity…
This paper studies device to device (D2D) coded-caching with information theoretic security guarantees. A broadcast network consisting of a server, which has a library of files, and end users equipped with cache memories, is considered. Information theoretic security guarantees for confidentiality are imposed upon the files. The server populates the end user caches, after which D2D communications enable the delivery of the requested files. Accordingly, we require that a user must not have access to files it did not request, i.e., secure caching. First, a centralized coded caching scheme is pro…
Content delivery networks often employ caching to reduce transmission rates from the central server to the end users. Recently, the technique of coded caching was introduced whereby coding in the caches and coded transmission signals from the central server are considered. Prior results in this area demonstrate that carefully designing the placement of content in the caches and designing appropriate coded delivery signals from the server allow for a system where the delivery rates can be significantly smaller than conventional schemes. However, matching upper and lower bounds on the transmissi…
We consider the classical coded caching problem as defined by Maddah-Ali and Niesen, where a server with a library of $N$ files of equal size is connected to $K$ users via a shared error-free link. Each user is equipped with a cache with capacity of $M$ files. The goal is to design a static content placement and delivery scheme such that the average load over the shared link is minimized. We first present a class of centralized coded caching schemes consisting of a general content placement strategy specified by a file partition parameter, enabling efficient and flexible content placement, and…
Cache-aided content delivery is studied in a multi-server system with $P$ servers and $K$ users, each equipped with a local cache memory. In the delivery phase, each user connects randomly to any $ρ$ out of $P$ servers. Thanks to the availability of multiple servers, which model small base stations with limited storage capacity, user demands can be satisfied with reduced storage capacity at each server and reduced delivery rate per server; however, this also leads to reduced multicasting opportunities compared to a single server serving all the users simultaneously. A joint storage and proacti…
The performance of existing coded caching schemes is sensitive to the worst channel quality, a problem which is exacerbated when communicating over fading channels. In this paper, we address this limitation in the following manner: in short-term, we allow transmissions to subsets of users with good channel quality, avoiding users with fades, while in long-term we ensure fairness among users. Our online scheme combines (i) the classical decentralized coded caching scheme \cite{maddah2013decentralized} with (ii) joint scheduling and power control for the fading broadcast channel, as well as (iii…
Caching is a promising solution to satisfy the ever increasing demands for the multi-media traffics. In caching networks, coded caching is a recently proposed technique that achieves significant performance gains over the uncoded caching schemes. However, to implement the coded caching schemes, each file has to be split into $F$ packets, which usually increases exponentially with the number of users $K$. Thus, designing caching schemes that decrease the order of $F$ is meaningful for practical implementations. In this paper, by reviewing the Ali-Niesen caching scheme, the placement delivery ar…
This paper studies the decentralized coded caching for a Fog Radio Access Network (F-RAN), whereby two edge-nodes (ENs) connected to a cloud server via fronthaul links with limited capacity are serving the requests of $K_r$ users. We consider all ENs and users are equipped with caches. A decentralized content placement is proposed to independently store contents at each network node during the off-peak hours. After that, we design a coded delivery scheme in order to deliver the user demands during the peak-hours under the objective of minimizing the normalized delivery time (NDT), which refers…
In this paper we provide theoretical and simulation-based study of the delivery delay performance of a number of existing throughput optimal coding schemes and use the results to design a new dynamic rate adaptation scheme that achieves improved overall throughput-delay performance. Under a baseline rate control scheme, the receivers' delay performance is examined. Based on their Markov states, the knowledge difference between the sender and receiver, three distinct methods for packet delivery are identified: zero state, leader state and coefficient-based delivery. We provide analyses of each …
In wireless networks, coded caching is an effective technique to reduce network congestion during peak traffic times. Recently, a new concept called placement delivery array (PDA) was proposed to characterize the coded caching scheme. So far, only one class of PDAs by Maddah-Ali and Niesen is known to be optimal. In this paper, we mainly focus on constructing optimal PDAs. Firstly, we derive some lower bounds. Next, we present several infinite classes of PDAs, which are shown to be optimal with respect to the new bounds.
