Coded polynomial aggregation (CPA) enables the master to directly recover a weighted aggregation of polynomial evaluations without individually decoding each term, thereby reducing the number of required worker responses. In this paper, we extend CPA to straggler-aware distributed computing systems and introduce a straggler-aware CPA framework with pre-specified non-straggler patterns, where exact recovery is required only for a given collection of admissible non-straggler sets. Our main result shows that exact recovery of the desired aggregation is achievable with fewer worker responses than required by polynomial coded computing based on individual decoding, and that feasibility is fundamentally characterized by the intersection structure of the non-straggler patterns. In particular, we establish necessary and sufficient conditions for exact recovery in straggler-aware CPA and identify an intersection-size threshold that is sufficient to guarantee exact recovery. We further prove that this threshold becomes both necessary and sufficient when the number of admissible non-straggler sets is sufficiently large. We also provide an explicit construction of feasible CPA schemes whenever the intersection size exceeds the derived threshold. Finally, simulations reveal a sharp feasibility transition at the predicted threshold, providing empirical evidence that the bound is tight in practice.

Coded elastic computing, introduced by Yang et al. in 2018, is a technique designed to mitigate the impact of elasticity in cloud computing systems, where machines can be preempted or be added during computing rounds. This approach utilizes maximum distance separable (MDS) coding for both storage and download in matrix-matrix multiplications. The proposed scheme is unable to tolerate stragglers and has high encoding complexity and upload cost. In 2023, we addressed these limitations by employing uncoded storage and Lagrange-coded download. However, it results in a large storage size. To address the challenges of storage size and upload cost, in this paper, we focus on Lagrange-coded elastic computing based on uncoded download. We propose a new class of elastic computing schemes, using Lagrange-coded storage with uncoded download (LCSUD). Our proposed schemes address both elasticity and straggler challenges while achieving lower storage size, reduced encoding complexity, and upload cost compared to existing methods.

Coded elastic computing enables virtual machines to be preempted for high-priority tasks while allowing new virtual machines to join ongoing computation seamlessly. This paper addresses coded elastic computing for matrix-matrix multiplications with straggler tolerance by encoding both storage and download using Lagrange codes. In 2018, Yang et al. introduced the first coded elastic computing scheme for matrix-matrix multiplications, achieving a lower computational load requirement. However, this scheme lacks straggler tolerance and suffers from high upload cost. Zhong et al. (2023) later tackled these shortcomings by employing uncoded storage and Lagrange-coded download. However, their approach requires each machine to store the entire dataset. This paper introduces a new class of elastic computing schemes that utilize Lagrange codes to encode both storage and download, achieving a reduced storage size. The proposed schemes efficiently mitigate both elasticity and straggler effects, with a storage size reduced to a fraction 1/L of Zhong et al.'s approach, at the expense of doubling the download cost. Moreover, we evaluate the proposed schemes on AWS EC2 by measuring computation time under two different tasks allocations: heterogeneous and cyclic assignments. Both assignments minimize computation redundancy of the system while distributing varying computation loads across machines.

In 2018, Yang et al. introduced a novel and effective approach, using maximum distance separable (MDS) codes, to mitigate the impact of elasticity in cloud computing systems. This approach is referred to as coded elastic computing. Some limitations of this approach include that it assumes all virtual machines have the same computing speeds and storage capacities, and it cannot tolerate stragglers for matrix-matrix multiplications. In order to resolve these limitations, in this paper, we introduce a new combinatorial optimization framework, named uncoded storage coded transmission elastic computing (USCTEC), for heterogeneous speeds and storage constraints, aiming to minimize the expected computation time for matrix-matrix multiplications, under the consideration of straggler tolerance. Within this framework, we propose optimal solutions with straggler tolerance under relaxed storage constraints. Moreover, we propose a heuristic algorithm that considers heterogeneous storage constraints. Our results demonstrate that the proposed algorithm outperforms baseline solutions utilizing cyclic storage placements, in terms of both expected computation time and storage size.

In cloud computing systems, elastic events and stragglers increase the uncertainty of the system, leading to computation delays. Coded elastic computing (CEC) introduced by Yang et al. in 2018 is a framework which mitigates the impact of elastic events using Maximum Distance Separable (MDS) coded storage. It proposed a CEC scheme for both matrix-vector multiplication and general matrix-matrix multiplication applications. However, in these applications, the proposed CEC scheme cannot tolerate stragglers due to the limitations imposed by MDS codes. In this paper we propose a new elastic computing scheme using uncoded storage and Lagrange coded computing approaches. The proposed scheme can effectively mitigate the effects of both elasticity and stragglers. Moreover, it produces a lower complexity and smaller recovery threshold compared to existing coded storage based schemes.

In coded caching system we prefer to design a coded caching scheme with low subpacketization and small transmission rate (i.e., the low implementation complexity and the efficient transmission during the peak traffic times). Placement delivery arrays (PDA) can be used to design code caching schemes. In this article we propose a framework of constructing PDAs via Hamming distance. As an application, two classes of coded caching schemes with linear subpacketizations and small transmission rates are obtained.

In caching system, it is desirable to design a coded caching scheme with the transmission load$R$and subpacketization$F$as small as possible, in order to improve efficiency of transmission in the peak traffic times and to decrease implementation complexity. Yan et al. reformulated the centralized coded caching scheme as designing a corresponding$F\times K$array called placement delivery array (PDA), where$F$is the subpacketization and$K$is the number of users. Motivated by several constructions of PDAs, we introduce a framework for constructing PDAs, where each row is indexed by a row vector of some matrix called row index matrix and each column’s index is labelled by an element of a direct product set. Using this framework, a new scheme is obtained, which can be regarded as a generalization of some previously known schemes. When$K$is equal to${\binom{m}{ t}}q^{t}$for positive integers$m$,$t$with$t < m$and$q\geq 2$, we show that the row index matrix must be an orthogonal array if all the users have the same memory size. Furthermore, the row index matrix must be a covering array if the coded gain is${\binom{m}{ t}}$, which is the maximal coded gain under our framework. Consequently the lower bounds on the transmission load and subpacketization of the schemes are derived under our framework. Finally, using orthogonal arrays as the row index matrix, we obtain two more explicit classes of schemes which have significantly advantages on the subpacketization while the transmission load is equal or close to that of the schemes constructed by Shangguan et al. for the same number of users and memory size.

In an (H, r) combination network, a single content library is serving for (rH)users through H relays, where each user has local cache memories and simultaneously accesses a subset of r relays on orthogonal non-interfering and error-free channels. The combinatorial placement delivery array (CPDA in short) can be used to realize a coded caching scheme for combination networks. In this paper, a new algorithm used to realize a scheme for combination networks based on a CPDA is proposed. Based on the fixed CPDA, the scheme realized by our algorithm has smaller subpacketization. Then we focus on directly constructing CPDAs for any positive integers H and r with r <; H and obtain two new classes of CPDAs. Compared with the previously known CPDAs, the schemes realized by our CPDAs have significant advantages on the subpacketization levels with some costing of transmission rates.

Price promotions as a marketing tool can effectively attract customers to buy products impulsively. However, when customers are in the face of price promotions, their self-control also emerges. Their desire to consume collides with their self-control, which will react differently to vice and virtue category products. In order to figure out the relationship among self-control, price promotions and consumer purchase behavior, this paper uses shopping path and POS data to study customers' shopping and buying behaviors in stores. Through logistic regression analysis, we verified how self-control and price promotions affected customers' purchase process. Then we found that customers would be more likely to purchase vice category products with price promotion when they stayed in a store for a certain period. This paper is the first one to empirically study the effect of both self-control and price promotions on customers' purchase behavior of vice category products. It can provide market managers with a better understanding on customers' purchase behavior and a more effective way to increase sales.