In large-scale and complex network environments, frequent security events make security audits increasingly important. Effective audits rely on maintaining a clearer memory of recent security events to support reliable identification and analysis. However, in high-volume data streams, existing methods struggle to ensure more accurate recording of recent time periods under a fixed memory budget while keeping the overall error controllable. To address this, we propose a novel data stream processing structure, the Stair Sketch. The key idea is to organize limited memory into a staircase of atomic sketches, where newer time periods hold more sketches. When memory is exhausted, one sketch is reclaimed from each existing period for the new one, efficiently allocating more memory to recent periods without updating all sketches. To record longer time periods and handle load imbalance, we introduce vertical sampling and horizontal sharing techniques to optimize our method. We theoretically derive the error bounds of Stair Sketch. Extensive experiments show that Stair Sketch improves accuracy by at least one order of magnitude compared with state-of-the-art methods, while reducing memory accesses by up to 50%. All related codes are open-sourced at GitHub.

Logging network traffic plays a crucial role as it serves as the foundation for various network applications. As network scale continues to expand, contemporary network traffic becomes increasingly high-speed, high-volume, and dynamic. This growth poses challenges to traditional server-based solutions. In this paper, we proposeTitanLog, ahierarchicalandelasticlogging system designed specifically for large-scale network traffic. TitanLog utilizes thehierarchical loggingmethodology, which aims to identify the importance of each packet in real-time and log packet data of different importance at different levels. To enhance efficiency, we propose a co-design of the emerging programmable switch and the server, incorporating sketches and RDMA to boost performance. To achieve elasticity, we design mechanisms for run-time adjustments and monitoring for resource insufficiency. TitanLog possesses the capability to switch between these modes at run-time. We fully implement TitanLog on a testbed and conduct extensive evaluations. The experimental results demonstrate that TitanLog supports logging of 100Gbps traffic with a zero packet loss rate and reduces the log volume by up to 96.28%.

Increasingly stringent service-level objectives demand fast and accurate congestion control (CC) in data center networks. We observe that the feedback loop used by existing data center sender-driven CC schemes is inherently limited as it suffers from an inevitable delay. Consequently, a portion of traffic, which we call disobedient traffic, could finish before the feedback loop, thereby escaping the control of these schemes and exerting a negative impact on their performance. Furthermore, as link speeds continue to climb, the proportion and impact of disobedient traffic are concurrently escalating, exacerbating these issues. In this paper, we propose Compass, a solution implemented entirely on switch data plane to mitigate the impact of disobedient traffic. Compass utilizes sketching techniques to efficiently estimate the sending rate of disobedient traffic, and seamlessly integrates into HPCC and PowerTCP through incorporating the sending rate of disobedient traffic into their high-precision rate control algorithms. Such integration enhances network performance without incurring additional bandwidth overhead or modifications to host-side logic. Additionally, Compass supports incremental brownfield deployment, and all these properties make it highly practical for production deployment. Extensive simulations show that Compass improves both throughput and latency. For instance, Compass reduces tail flow completion times of medium and large flows by up to 35% for PowerTCP and 17% for HPCC.

The Mixture-of-Experts (MoE) models have gained significant attention in deep learning due to their dynamic resource allocation and superior performance across diverse tasks. However, efficiently training these models remains challenging. The MoE upcycling technique has been proposed to reuse and improve existing model components, thereby minimizing training overhead. Despite this, simple routers, such as linear routers, often struggle with complex routing tasks within MoE upcycling. In response, we propose a novel routing technique called Router Upcycling to enhance the performance of MoE upcycling models. Our approach initializes multiple routers from the attention heads of preceding attention layers during upcycling. These routers collaboratively assign tokens to specialized experts in an attention-like manner. Each token is processed into diverse queries and aligned with the experts’ features (serving as keys). Experimental results demonstrate that our method achieves state-of-the-art (SOTA) performance, outperforming other upcycling baselines.

Perfect hashing is a special hashing function that maps each item to a unique location without collision, which enables the creation of a KV store with small and constant lookup time. Recent dynamic perfect hashing attains high load factor by increasing associativity, which impacts bandwidth and throughput. This paper proposes a novel dynamic perfect hashing index without sacrificing associativity, and uses it to devise an RDMA-based remote memory KV store called CuckooDuo. CuckooDuo simultaneously achieves high load factor, fast speed, minimal bandwidth, and efficient expansion without item movement. We theoretically analyze the properties of CuckooDuo, and implement it in an RDMA-network based testbed. The results show CuckooDuo achieves 1.9~17.6x smaller insertion latency and 9.0~18.5x smaller insertion bandwidth than prior works.

To approximate sums of values in key-value data streams, sketches are widely used in databases and networking systems.They offer high-confidence approximations for any given key while ensuring low time and space overhead.While existing sketches are proficient in estimating individual keys, they struggle to maintain this high confidence across all keys collectively, an objective that is critically important in both algorithm theory and its practical applications.We propose ReliableSketch, the first to control the error of all keys to less than Λ with a small failure probability Δ, requiring only 𝑂 (1 + Δ ln ln( 𝑁 Λ )) amortized time and 𝑂 ( 𝑁 Λ + ln( 1 Δ )) space.Furthermore, its simplicity makes it hardware-friendly, and we implement it on CPU servers, FPGAs, and programmable switches.Our experiments show that under the same small space, ReliableSketch not only keeps all keys' errors below Λ but also delivers competitive throughput among accuracy-oriented baselines, outperforming * Both authors contributed equally to this research.

Diffusion-based generative models have exhibited considerable success in conditional video synthesis and editing. Nevertheless, prevailing video diffusion models primarily rely on conditioning with specific input modalities, predominantly text, restricting their adaptability to alternative modalities without necessitating retraining of modality-specific components. In this work, we present EnergyViD, a universal spatio-temporal Energy-guided Video Diffusion model designed for zero-shot video synthesis and editing across diverse conditions. Specifically, we leverage off-the-shelf pre-trained networks to construct generic energy functions, guiding the generation process under specific conditions without the need for retraining. To precisely capture temporal dynamics related to motion conditions (e.g., pose sequences), we introduce a novel kernel Maximum Mean Discrepancy (MMD)-based energy function, which minimizes the global distribution discrepancy between the conditioning input and the generated video. Our extensive qualitative and quantitative experiments demonstrate that our algorithm consistently produces high-quality results across a wide range of motion and non-motion conditions, including text, face ID, style, poses, depths, sketches, canny edges, and segmentation maps, in the context of zero-shot video synthesis and editing. We will release source code upon acceptance of the paper.

Finding top-$K$frequent items has been a hot topic in data stream processing with wide-ranging applications. However, most existing sketch algorithms focus on finding local top-$K$in a single data stream. In this paper, we tackle finding global top-$K$across multiple data streams. We find that using prior sketch algorithms directly is often unfair in global scenarios, degrading global top-$K$accuracy. We define top-$K$-fairness and show its importance for finding global top-$K$. To achieve this, we propose the Double-Anonymous (DA) sketch, where double-anonymity ensures fairness. We also propose two techniques, hot-filtering and early-freezing, to improve accuracy further. We theoretically prove that the DA sketch achieves top-$K$-fairness while maintaining high accuracy. Extensive experiments verify top-$K$-fairness in disjoint data streams, showing that the DA sketch's error is up to 129 times (60 times on average) smaller than the state-of-the-art. To enhance the applicability and technical depth, we also investigate how to extend the DA sketch to general distributed data stream scenarios and how to provide a fairer and more accurate global ranking for top-$K$items. The experimental results show that the extended version of the DA sketch can indeed compute better rankings and still has significant advantages in general data streams.

The growing memory demands of embedding tables in Deep Learning Recommendation Models (DLRMs) pose great challenges for model training and deployment. Existing embedding compression solutions cannot simultaneously achieve memory efficiency, low latency, and adaptability to dynamic data distribution. This article presents CAFE+, a Compact, Adaptive, and Fast Embedding compression framework that meets the above requirements. The design philosophy of CAFE+ is to dynamically allocate more memory to important features and less to unimportant ones. We assign unique embedding to important feature and allow multiple unimportant features sharing one embedding. We propose a fast and lightweight feature monitor, to real-time capture feature importance and report important features. We theoretically analyze the accuracy of our feature monitor and prove the superiority of CAFE+ from the aspect of model convergence. Extensive experiments show CAFE+ outperforms existing embedding compression methods, yielding \(3.94\%\) and \(3.94\%\) superior testing AUC on Criteo Kaggle dataset and CriteoTB dataset at a compression ratio of \(10{,}000\times\) . Building on our conference version [ 114 ], this journal version introduces several novel designs (implicit importance attenuation, adaptive threshold adjustment, and ColdSifter) that enable CAFE+ to more effectively adapt to long-term online learning and achieve better model quality. All codes are available at GitHub [ 112 ].

In quantile estimation within a stream of key-value pairs, recent work has made significant progress in query flexibility, supporting quantile estimation for any key using a unified statistical structure. However, despite this flexibility, their query speed falls behind, unable to match the high speed of online data insertion. This “offline query + online insertion” model is not ideal for online quantile estimation. Our goal is to online detect keys whose quantiles exceed a user-queried threshold in real-time, such as identifying the user whose 95 % latency exceeds 200ms in network data. These keys, termed “Quantile-Outstanding Keys,” are vital for anomaly detection in streaming data. In this paper, we propose QuantileFilter, the first approximate algorithm specifically designed for detecting quantile-outstanding keys. QuantileFilter overcomes existing limitations by 1) enabling fast online computation, capable of handling streaming data in real-time with a constant processing time for each data item, accelerating the state-of-the-art (SOTA) by 10 ~ 100 times, and 2) maintaining high space efficiency, saving 50 ~ 500 times storage space compared to the SOTA while maintaining the same accuracy. All associated code is available on GitHub.

In-memory database in the data center plays an indispensable role in many fields. B+tree is the most recognized index in in-memory database, but its indexing latency has become the bottleneck that prevents the database from achieving higher performance. The existing work reduces latency by caching B+tree nodes using slow DRAM on computing clients, or directly caching data using fast SRAM on programmable switch. However, no existing work can meet all three key requirements: (1) Efficiency: providing enough fast memory to accelerate indexing. (2) Compatibility: compatible with multiple architectures and query types. (3) Adaptability: adapting to various workloads and database scales. Inspired by structrual similarity between the network topology of modern data centers and the B+tree structure, we propose Fat-B+Tree, which embeds the B+tree into the in-network fast memory provided by the hierarchical connected programmable switches, thus meeting all design requirements. We have fully implemented the Fat-B+Tree prototype, and the experimental results show that compared with the baseline system, it reduces query latency by up to 76% and improves throughput by up to 3.93 times. The source codes of Fat-B+Tree are open-sourced at GitHub.

Recently, the growing memory demands of embedding tables in Deep Learning Recommendation Models (DLRMs) pose great challenges for model training and deployment. Existing embedding compression solutions cannot simultaneously meet three key design requirements: memory efficiency, low latency, and adaptability to dynamic data distribution. This paper presents CAFE, a Compact, Adaptive, and Fast Embedding compression framework that addresses the above requirements. The design philosophy of CAFE is to dynamically allocate more memory resources to important features (called hot features), and allocate less memory to unimportant ones. In CAFE, we propose a fast and lightweight sketch data structure, named HotSketch, to capture feature importance and report hot features in real time. For each reported hot feature, we assign it a unique embedding. For the non-hot features, we allow multiple features to share one embedding by using hash embedding technique. Guided by our design philosophy, we further propose a multi-level hash embedding framework to optimize the embedding tables of non-hot features. We theoretically analyze the accuracy of HotSketch, and analyze the model convergence against deviation. Extensive experiments show that CAFE significantly outperforms existing embedding compression methods, yielding 3.92% and 3.68% superior testing AUC on Criteo Kaggle dataset and CriteoTB dataset at a compression ratio of 10000x. The source codes of CAFE are available at GitHub.

Layer-3 load balancing is a key topic in the networking field. It is well acknowledged that flowlet is the most promising solution because of its good trade-off between load balance and packet reordering. However we find its one significant limitation: it makes the forwarding paths of flows unpredictable. To address this limitation this paper presents BurstBalancer a simple yet efficient load balancing system with a sketch named BalanceSketch. Our design philosophy isdoing less changesto keep the forwarding path of most flows fixed which guides the design of BalanceSketch and our balance operations. We have fully implemented BurstBalancer in a small-scale testbed built with Tofino switches and conducted both large-scale event-level (NS-2) and ESL (electronic system level) simulations. Our results show that BurstBalancer achieves 5%$\sim$35% smaller FCT than LetFlow in symmetric topology and up to 30× smaller FCT in asymmetric topology while 58× fewer flows suffer from path changing. All related codes are open-sourced at GitHub

In global data analysis, the central server needs the global statistic of the user data stored in local clients. In such cases, an Honest-but-Curious central server might put user privacy at risk in trying to collect individual statistics of each user. In response, the secure aggregation provides a solution for calculating global statistics without revealing users’ privacy data. However, existing secure aggregation protocols only focus on the data in the form of vectors or common sets, which limits their application scope. We formalize a general problem—key-value set secure aggregation—that not only includes secure vector aggregation and private set union but also supports more applications. To address the proposed problem, we devise our solution (called the KVSAgg framework) that promises satisfactory performance in security, efficiency, and accuracy. Our key technique is a homomorphic transform algorithm (called HyperIBLT) that is not only capable of bidirectionally transforming data between key-value sets and vectors, but also able to transform sum operation of sets to addition of vectors. We implement KVSAgg on both CPU and GPU platforms and perform the evaluation on three use cases including federated learning, distributed data counting, and finding global hot items. Compared with our baselines, KVSAgg simultaneously achieves the best security, efficiency higher by orders of magnitude, and zero-error in nearly all cases. All codes are open-source anonymously.

In this paper, we propose a new type of item in data streams, called simplex items. Simplex items have frequencies in consecutive p windows that can be approximated by a polynomial of degree at most k, where k = 0, 1, 2. These low-order representable simplex items have a wide range of potential applications. For example, when k = 1, we can leverage these items whose frequency has obvious linear increase or decrease to speed up the running time of a class of machine learning models and detect network attacks such as distributed denial-of-service (DDoS), etc. To find k-degree simplex items in real time, we propose a novel sketch, namely X-Sketch, to accurately record simplex items in a compact space. The key idea of X-Sketch is to effectively filter out non-simplex items with less memory overhead, and then monitor the remaining potential simplex items and keep those items with more consecutive windows. We conduct extensive experiments, and the experimental results show that the F1 Score of X-Sketch is on average 68.6%, 57.9%, and 42.2% higher than the baseline solution for k = 0, 1, 2, respectively. Finally, we also provide a case study that applies X-Sketch to "accelerate" the two machine learning models through end-to-end experiments. We have released our source code at GitHub.

Batch is an important pattern in data streams, which refers to a group of identical items that arrive closely. We find that some special batches that arrive periodically are of great value. In this paper, we formally define a new pattern, namely periodic batches. A group of periodic batches refers to several batches of the same item, where these batches arrive periodically. Studying periodic batches is important in many applications, such as caches, financial markets, online advertisements, networks, etc. We propose a one-pass sketching algorithm, namely the HyperCalm sketch, which takes two phases to detect periodic batches in real time. In phase 1, we propose a time-aware Bloom filter, namely HyperBloomFilter (HyperBF), to detect the start of batches. In phase 2, we propose an enhanced top-k algorithm, called Calm Space-Saving (CalmSS), to report top-k periodic batches. We theoretically derive the error bounds for HyperBF and CalmSS. Extensive experiments show HyperCalm outperforms the strawman solutions 4× in term of average relative error and 13.2× in term of speed. We also apply HyperCalm to a cache system and integrate HyperCalm into Apache Flink. All related codes are open-sourced.

Sketches have risen as promising solutions for frequency estimation, which is one of the most fundamental tasks in approximate data stream processing. In many scenarios, users have a strong demand to apply sketches under the expected error constraints. In this paper, we explore how to configure sketch parameters to satisfy user-defined error constraints. We propose SketchConf, an automatic sketch configuration framework, which efficiently generates memory-optimal configurations for the first time. We show that SketchConf can be applied to order-independent sketches, including CM, Count, Tower, and Nitro sketches. We further discuss how to deal with the unknown and changeable workloads when applying SketchConf to the real scenarios of streaming data processing. Experimental results show that SketchConf can be up to 715.51 times faster than the baseline algorithm, and the outputted configurations save up to 99.99% memory and achieve up to 27.44 times throughput, compared with the theory-based configurations. The code is open sourced at Github.

We abstract a MIMO scenario in distributed data stream mining, where a stream of multiple items is mined by multiple nodes. We design a framework named MimoSketch for the MIMO-specific scenario, which improves the fundamental mining task of item frequency estimation. MimoSketch consists of an algorithm design and a policy to schedule items to nodes. MimoSketch's algorithm applies random counting to preserve a mathematically proven unbiasedness property, which makes it friendly to the aggregate query on multiple nodes; its memory layout is dynamically adaptive to the runtime item size distribution, which maximizes the estimation accuracy by storing more items. MimoSketch's scheduling policy balances items among nodes, avoiding nodes being overloaded or underloaded, which improves the overall mining accuracy. Our prototype and evaluation show that our algorithm can improve the item frequency estimation accuracy by an order of magnitude compared with the state-of-the-art solutions, and the scheduling policy further promotes the performance in MIMO scenarios.

Network measurement is critical to many network applications. There are mainly two kinds of flow-level measurement tasks: 1) packet accumulation tasks and 2) packet loss tasks. In practice, the two kinds of tasks are often required at the same time, but existing works seldom handle both. In this paper, we design ChameleMon to support the two kinds of tasks simultaneously. The key design of ChameleMon is to shift measurement attention as network state changes, through two dimensions of dynamics: 1) dynamically allocating memory between the two kinds of tasks; 2) dynamically monitoring the flows of importance. To realize the key design, we propose a key technique, leveraging Fermat's little theorem to devise a flexible data structure, namely FermatSketch. FermatSketch is dividable, additive, and subtractive, supporting the two kinds of tasks. We have implemented a ChameleMon prototype on a testbed with a Fat-tree topology. We conduct extensive experiments and the results show ChameleMon supports the two kinds of tasks with low memory/bandwidth overhead, and more importantly, it can automatically shift measurement attention as network state changes.

The data plane cache, a critical functionality found in numerous network devices, such as programmable switches, intelligent NICs, and DPUs, is often subject to limitations in its programmability and memory access capacity. As a result, the majority of existing data plane caches rely on simple and inefficient replacement policies. This paper is set to introduce LRU, a near-optimal replacement policy, into the programmable data plane. We first explore the reasons why the traditional implementation of LRU is not suitable for deployment on the data plane. Consequently, we propose P4LRU, a pipeline-optimized version of the LRU implementation. Building on P4LRU, we conceive three distinct in-network systems - LruTable, LruIndex, and LruMon, and successfully bring them to life on Tofino switches. Our thorough experimental trials establish that P4LRU provides a significant performance boost over existing data plane caches in these three systems. We have open-sourced the source codes for the three systems on GitHub [1].

Membership (membership query/membership testing) is a fundamental problem across databases, networks and security. However, previous research has primarily focused on either approximate solutions, such as Bloom Filters, or exact methods, like perfect hashing and dictionaries, without attempting to develop an integral theory. In this paper, we propose a unified and complete theory, namely chain rule, for general membership problems, which encompasses both approximate and exact membership as extreme cases. Building upon the chain rule, we introduce a straightforward yet versatile algorithm framework, namely ChainedFilter, to combine different elementary filters without losing information. Our evaluation results demonstrate that ChainedFilter improves performance of many applications including static dictionary, lossless data compression, Cuckoo Hashing, LSM-Tree and Learned Filters.

Finding top-K frequent items has been a hot topic in data stream processing in recent years, which has a wide range of applications. However, most of existing sketch algorithms focuses on finding local top-K in a single data stream. In this paper, we work on finding global top-K in multiple disjoint data streams. We find that directly deploying prior sketch algorithms is often unfair under global scenarios, which will degrade the accuracy of global top-K. We define top-K-fairness and show that it is important for finding global top-K. To achieve top-K-fairness, we propose a new sketch framework, called the Double-Anonymous sketch. The process of finding global top-K items is similar to that of paper reviewing and democratic elections. In these scenarios, double-anonymity is often an effective strategy to achieve top-K-fairness. We also propose two techniques, hot panning, and early freezing, to further improve the accuracy. We theoretically prove that the Double-Anonymous sketch achieves top-K-fairnesswhile keeping high accuracy. We perform extensive experiments to verify top-K-fairness in the scenario of disjoint data streams. The experimental results show that the Double-Anonymous sketch's error is up to 129 times (60 times on average) smaller than the state-of-the-art. All the related source code is open-sourced and available at Github.

Data stream processing has become fundamental in computer science, with a wide range of applications, such as in databases, data mining, and security. Memorizing when an item appears in the data stream is one important task in stream processing. Because the older data is, the less value it has, memorizing recent events with higher accuracy is desirable. To achieve this, we propose a novel data stream processing structure named the Stair sketch. Our key idea is to organize the memory used by different time periods in the shape of stairs. We deploy the Stair sketch on Bloom filters, CM sketches, and CU sketches as case studies. Experiment results show that our approach outperforms state-of-the-art algorithms by more than 5× in accuracy while providing comparable efficiency. The source code of the Stair sketch is available at GitHub.

Layer-3 load balancing is a key topic in the networking field. It is well acknowledged that flowlet is the most promising solution because of its good trade-off between load balance and packet reordering. However, we find its one significant limitation: it makes the forwarding paths of flows unpredictable. To address this limitation, this paper presents BurstBalancer, a simple yet efficient load balancing system with a sketch, named BalanceSketch. Our design philosophy is doing less changes to keep the forwarding path of most flows fixed, which guides the design of BalanceSketch and balance operations. We have fully implemented BurstBalancer in a small-scale testbed built with Tofino switches, and conducted large-scale NS-2 simulations. Our results show that BurstBalancer achieves 5%∼35% smaller FCT than LetFlow in symmetric topology and up to 30× smaller FCT in asymmetric topology, while 58× fewer flows suffer from path changing. All related codes are open-sourced at Github22https://github.com/BurstBalancer/Burst-Balancer.

Nowadays, wide-area data analyses are pervasive with emerging geo-distributed systems. These analyses often need to do the global aggregation in the wide area. Since scarce and variable WAN bandwidth may degrade the aggregation performance, it is highly desired to design a communication scheme for global aggregation in WAN. Unfortunately, no existing algorithm can meet the three design requirements of communication schemes: fast computation, adaptive transmission, and accurate aggregation. In this paper, we propose MinMax Sampling, a fast, adaptive, and accurate communication scheme for global aggregation in WAN. We first focus on the accuracy and design a scheme, namely MinMaxopt, to achieve optimal accuracy. However, MinMaxopt does not meet the other two requirements: fast computation and adaptive transmission. Based on MinMaxopt, we propose MinMaxadp, which trades little accuracy for the other two requirements. We evaluate MinMaxadp with three applications: federated learning, distributed state aggregation, and hierarchical aggregation. Our experimental results show that MinMaxadp is superior to existing algorithms (8.44× better accuracy on average) in all three applications. The source codes of MinMax Sampling are available at Github [1].

Flow scheduling is crucial in data centers, as it directly influences user experience of applications. According to different assumptions and design goals, there are four typical flow scheduling problems/solutions: SRPT, LAS, Fair Queueing, and Deadline-Aware scheduling. When implementing these solutions in commodity switches with limited number of queues, they need to set static parameters by measuring traffic in advance, while optimal parameters vary across time and space. This paper proposes a generic framework, namely QCluster, to adapt all scheduling problems for limited number of queues. The key idea of QCluster is to cluster packets with similar weights/properties into the same queue. QCluster is implemented in Tofino switches, and can cluster packets at a speed of 3.2 Tbps. To the best of our knowledge, QCluster is the fastest clustering algorithm. Experimental results in testbed with programmable switches and ns-2 show that QCluster reduces the average flow completion time (FCT) for short flows up to 56.6%, and reduces the overall average FCT up to 21.7% over state-of-the-art. All the source code in ns-2 is available in Github [45].

Sketches, a type of probabilistic algorithms, have been widely accepted as the approximate summary of data streams. Compressing sketches is the best choice in distributed data streams to reduce communication overhead. The ideal compression algorithm should meet the following three requirements: high efficiency of compression procedure, support of direct query without decompression, and high accuracy of compressed sketches. However, no prior work can meet these requirements at the same time. Especially, the accuracy is poor after compression using existing methods. In this paper, we propose Cluster-Reduce, a framework for compressing sketches, which can meet all three requirements. Our key technique nearness clustering rearranges the adjacent counters with similar values in the sketch to significantly improve the accuracy. We use Cluster-Reduce to compress four kinds of sketches in two use-cases: distributed data streams and distributed machine learning. Extensive experimental results show that Cluster-Reduce can achieve up to 60 times smaller error than prior works. The source codes of Cluster-Reduce are available at Github anonymously[1].