 Research
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Gapprox: using Gallup approach for approximation in Big Data processing
Journal of Big Data volume 6, Article number: 20 (2019)
Abstract
As Big Data processing often takes a long time and needs a lot of resources, sampling and approximate computing techniques may be used to generate a desired Quality of Result. On the other hand, due to not considering data variety, available samplebased approximation approaches suffer from poor accuracy. Data variety is one of the key features of Big Data which causes various parts of data to have different impact on the final result. To address this problem, we develop a data variety aware approximation approach called Gapprox. Our idea is to use a kind of cluster sampling to improve the accuracy of estimation. Our approach can decrease the amount of data to be processed to achieve the desired Quality of Result with acceptable error bound and confidence interval. We divide the input data into some blocks considering the intra/inter cluster variance. The size of the block and the sample size are determined in such a way that by processing small amount of input data, an acceptable confidence interval and error bound is achieved. We compared our work with two wellknown state of the art. The experimental results show that our result surpasses the state of the art and improve processing time up to 17× compared to ApproxHadoop and 8× compared to Sapprox when the user can tolerate an error of 5% with 95% confidence.
Introduction
Despite the huge number of researches in Big Data area, approximate computing in this area still remains a challenge. The approximation is used for reduction of resources such as time, cost or energy. Applications that analyze the input data, logs and queries to generate aggregated results or dashboards can benefit from approximation techniques in Big Data. In these applications, the output is much smaller than the input. This fact indicates that approximation can be used for increasing the processing performance for this kind of computation. Data skew causes reduction of performance in approximation. Data skew has many causes [1,2,3]. In this paper, we focused on the challenge that stems from variety. Data variety can be created by aggregating input data from multiple sources with different statistical distribution and uneven distribution of input data. This uneven distribution causes the data skew.
There are many approaches that have addressed the samplebased approximation. In case of lack of resources, approximate computing is a suitable approach for generating acceptable Quality of Result/Service [4,5,6,7,8,9]. The approximation can be used in wide range of applications such as image, audio or video processing, data analytics, Monte Carlo computation and machine learning processing [10,11,12,13,14].
Unfortunately, some of them do not consider data variety/skew. Some other process a large number of samples to achieve the desired error bound and confidence interval.
ApproxHadoop [15] is a wellknown state of the art of our work. The error bound and confidence interval of its sampling approach is good when there is no variety in data. But in the real world and real datasets, a large volume of variety and skew exists. ApproxHadoop considers a large cluster size and produces very small samples. In the case of existing data variety in input data, this method may suffer from inefficient sampling and large variance.
We also have considered another wellknown work for our evaluation. In Sapprox [16], the authors have used the cluster sampling with unequal probability to consider the distribution of input dataset. Implementation of this approach is not simple and may cause some inefficiency.
Based on the description presented, we have focused on the weakness of these two approaches and presented an approach to cover their weakness. Like Approxhadoop and Sapprox, we also have used cluster sampling for our goal. We select a suitable size as the block size in a way that interblock variance decreased and intrablock variance increased. By obtaining a sample from each block, the error bound is decreased. By using this approach we can achieve the suitable Quality of Result by processing a small volume of the input dataset.
In our previous work [17], we have shown that data variety has a great impact on the performance of progressive processing. In this paper, we present a samplebased approximation method with high accuracy in the case of data skew/variety. For this goal, we present Gapprox to increase the accuracy of approximate processing. We have offered a kind of cluster sampling for increasing performance of samplebased approximation. As Fig. 1 shows, in our approach, we divide the input data into some blocks and then divide each block into some same sized frames. We select the block size in such a way that the inter block variance is increased and intra block variance is decreased. By this approach, the acceptable Quality of Result is achieved by processing a very small amount of input data. Block and frame design are shown in Fig. 1.
We show the difference between approaches in Fig. 2. Approxhadoop selects some blocks and some frames in it as the sample. Sapprox uses sample interval for getting samples. Our approach divides the input data into some same size blocks and each block into some same size frames. In each block samples are selected among frames with certain error bound and confidence interval.
Weaknesses of ApproxHadoop

The authors in ApproxHadoop have not considered the data variety/Skew. They have evaluated their approach by uniform datasets. In the case of existing an impressive amount of skew in data, the users get forced to process a large amount of input data to achieve acceptable Quality of Result.
Weaknesses of Sapprox

In the beginning, the amount of sampled data to achieve acceptable error bound and the confidence interval is not definite. This weakness forces the user to process more data that the needed.

Sapprox is very difficult for implementation in case of NoSQL datasets. In this kind of datasets determination of sampling interval is very hard for users. This fact causes inefficiency in case of NoSQL datasets.
Advantages of our Approach

We have considered different skew conditions in Gapprox. In the case of uniform distribution, a negligible amount of input data must be processed to achieve the acceptable error bound and confidence interval. In case of uneven distribution, our approach is able to manage this issue by dividing input data into some blocks and using sampling with a guarantee. By this technique, we can tolerate the weakness of ApproxHadoop.

Our approach can achieve sampling targets by processing a low amount of data (less than 1%) in each condition of skew. Our approach can estimate the amount of data that must be processed before starting the process. Due to the techniques that we have used, our approach can achieve an acceptable Quality of Result by processing less data than Sapprox.

Our approach can produce targeted Quality of Result in any condition of data skew by processing less than 1% of input data. But based on the experimental results, in case of uneven distribution, a large and undefined amount of data must be processed in ApproxHadoop.

Our approach is applicable for SQL and NoSQL datasets and efficiently generate acceptable Quality of Result in case of both.

We have used some techniques and statistical methods to help the user to implement the approach. We have used Simple Random Sample to determine the primary condition of data skew. We also divide data into some equal parts and present a simple approach to implement, in compare to Sapprox.
We proposed three techniques in our approach

1.
Defining suitable size for the block. Block size should be defined in a way that the intra block variance is minimized.

2.
Input data sampling. Only a subset of input data is processed. We use a sampling technique with a 95% confidence interval and 5% error bound.

3.
Userdefined approximation. In our work user can decide the level of approximation which the generated outcome will have.
We evaluate our work by using some wellknown datasets. We also used the TPC Benchmark and Amazon review dataset for our evaluations [18, 19]. We compared our work with two wellknown previous works as the state of the art: ApproxHadoop [15] and Sapprox [16].
Contributions In this paper, we have the following contributions:

1.
We proposed a variety/skew aware approach for using an approximation in Big Data processing. That is practical and easy to use for users.

2.
We use some statistical theories for sampling with an acceptable confidence interval and error bound in case of existing skew/variety in data.

3.
We show how sampling theories can be used to estimate sample size and compute error bounds for approximations in MapReducelike systems.

4.
Via experimental results, we show that our approach can significantly reduce processing time compared to our state of the arts.
Organization The rest of the paper is organized as follows: “Related works” section presents an overview of the state of the arts and previous works including a classification of methods and tools. In “Background” section we introduced the background of our work. “Motivation” section presents the overall motivation of our work. “Methods” section describes the system design and experiments. The experimental result and discussion is presented in “Results and discussion” section, and finally, “Conclusions and future work” section extracts the main conclusions of the paper and proposes future work.
Related works
Approximation Approximate computing is used in largescale computing in case of lack of processing resources [4,5,6,7,8,9]. This kind of processing can be used for a wide range of application types including data analytics, machine learning, Monte Carlo computations, and image, audio or video processing [10,11,12,13,14]. The approximation can be used in case of the lack of resources such as time, cost or energy [9, 15, 20]. In [17] we also have considered data variety and have the benefit from it to develop efficient progressive processing. The acceptable quality of approximation results is achieved by using our approach.
The authors in [9] have presented a survey of techniques for approximate computing. They discussed strategies and approaches for approximation and monitoring output quality. They have considered different processing units (e.g., CPU, GPU and FPGA), processor components memory techniques and different framework for approximation computing. The authors in [20] have improved the performance of Big Data processing by using a kind of MapReduce that supports the early return from a job as it is being computed. ApproxHadoop [15] is wellknown research in Big Data approximate computing. The authors have presented a framework for approximate computing by using sampling. Their framework has good performance in case of uniform distribution. But our analysis shows that ApproxHadoop does not have acceptable performance in case of existing data variety or input data with real distribution. The authors have paid attention to the weakness of Approxhadoop in [16]. They have considered data skew in their research. They present an approach with unequal probability sampling. Their solution has better performance in compare to ApproxHadoop. But this approach also has some weaknesses about NoSQL datasets and complexity of implementation. The authors have presented an approximate computing technique in [21] by learning how an approximation can be applied in code regions. They have used Neural Network techniques to learn how to reduce energy consumption by replacing the code regions.
We also have presented an approximation framework for Big Data processing. Our approach is more efficient than ApproxHadoop and more flexible than Sapprox. The authors in ApproxHadoop do not consider the data skew/variety. The plan that presented in Sapprox is hard to implement for NoSQL datasets and achieved to the acceptable Quality of Result by processing more data than Gapprox.
Samplebased approximation approaches Approaches and researches in approximate query processing are presented in [22]. BlinkDB [10] offers a solution to select samples for each query column set and uses samples to answer online queries in the distributed file system. Offline sampling can use preprocessing and a priori knowledge of dataset. Traditional sampling approach like [10, 23,24,25], that present uniformbasedsampling are unreliable in case of data skew. These methods select more tuples for big groups and enough tuples in rare groups. But in the case of data variety, the size of the sample group and the values in each sample group may be highly skewed.
Agarwal et al. [26], Pol and Jermaine [27], Zeng et al. [28], Zeng et al. [29] used closedform estimation and Bootstrap method for providing high confidence interval. They repaired the results by using resampling. The authors in [30] with the knowledge of data distributions use errorbounded stratified sampling and can reduce the sample size. The authors in [31] offer a solution for approximate computing for more kinds of data besides relational data. Researchers in [32] use online uniform HDFS samples and by using Bootstrap method incrementally evaluate the accuracy. This method can be used efficiently for complex queries. But the researchers unlike us did not consider the data variety and skew. Authors in ApproxHadoop [15] consider the datasets with uniform distributions. But real datasets are not uniformly distributed. Random sampling in this kind of datasets generate biased sampling and does not work very well. The authors in Sapprox [16] have noted the weakness of ApproxHadoop [15] and tried to offer a way to overcome it. They have used cluster sampling with unequal probability theory. It is very hard for users to define suitable Sampling Interval for the various condition of skew. Our approach can achieve acceptable Quality of Result with processing fewer amount data than Sapprox. A clusterbased sampling approach is presented in [33]. This approach combines three kinds of sampling: uniform sampler, distinct sampler, and universal sampler. This method uses Horvitz–Thompson (HT) to estimate the exact answer and uses the centrallimit theorem to compute the confidence interval. It also uses “dominance transitivity” before using the estimator. The authors in [34] have applied sampling to MapReduce facilitate analysis of various film details, review comments and users profile.
We have also presented a sample based framework for Big Data processing that can be used in case of data skew/variety. Previous approaches that are presented in this section do not consider generating acceptable Quality of Result in any condition of data skew and any types of dataset. In some approaches, we must have some information about input data distribution. In some other, the authors consider uniform distribution. There is no sensitivity about NoSQL datasets. So, Gapprox is generally different from previous approaches.
Distributed approximation systems The authors in [10] present an approximate query processing system that provides an efficient strategy to get a sample from the dataset. Sapprox [16] presents a more flexible approach than BlinkDB and efficient than ApproxHadoop [15]. We also present an approach that is more efficient than ApproxHadoop and more flexible than Sapprox. Quickr [33] is also designed presented adhoc queries on bigdata clusters like BlinkDB [10], ApproxHadoop and Sapprox. Quickr [33] does not need to discover all dataset to achieve a certain confidence interval. It has been used by Microsoft’s search engine, i.e., Bing. Microsoft also develop and use a framework for progressive analytics “Now!” and use it for Windows Azure [35]. SnappyData [36] is a Sparkbased system for delivering interactive analytics. FluoDB [37] is another Sparkbased approach to support general (online analytical processing) OLAP queries. XDB [38] is another system that can support online aggregation. This approach support complex queries including join operator. We also have presented an approach that can be used in any condition of data skew. Our approach can be used in case of uniform or uneven distribution. None of the above solutions can perform acceptable Quality of Result in any condition of skew and input data types. So, our approach is different from them.
Variation We present a samplebased approximation in this paper. Our approach improved the Quality of Result by processing a small amount of input data. The authors in [39] have shown how approximation techniques can be used to address network and processing variations. They have used an approximation to reduce computation requirement without a reduction in Quality of Result. The authors in [40] investigated the problem of imbalanced subdatasets and inefficient sampling on subdatasets over a Hadoop cluster. They have mentioned that uneven subdatasets distribution leads to lowerperformance in parallel data analysis. They have offered a framework to balance workload among computational nodes. The authors in [41] consider the data skew to reduce the finishing time and computational resources. They have modeled different degrees of skew in the input data and provided experimental results under different conditions. The authors in [42] have presented a framework to increase dataaware scheduling for jobs with combinatorial choices. They have considered intermediate data transfer for locality and balancing. The authors in [43] have presented a survey of techniques for approximate computing. They have discuss different approaches to find program portions that are suitable for approximation and monitor Quality of Results. The authors in [44] have presented a framework to mitigate the skew in Big Data processing applications. They have mitigated the skew of two sources: (1) Skew due to an uneven distribution of input data, and (2) the skew caused by some subset of data taking longer to process than other. The authors in [45] have surveyed Big Data processing platforms. They have divided the Big Data processing platforms into 2 main categories: (1) Horizontal Scaling that also called Scale out and (2) Vertical scaling that also called Scaleup. Using more distributed servers is considered in Scaleout strategy. In Scale up approach, current processing element such as CPU and memory is replaced by more powerful ones. We have also presented an approach to scale up for big data processing in [17] considering the data variety.
None of the above researches have not yet presented a framework to increase the accuracy and reduce the amount of data to be processed to achieve acceptable Quality of Result. ApproxHadoop and Sapprox are two works that can be compared with our approaches. We show our work’s advantage in the case of efficiency and flexibility in compared with them.
Background
In this section, we present some tools, topics and statistical methods that will be noted in this paper.
Spark Apache Spark is an opensource clustercomputing framework [46]. Originally developed at the University of California, Berkeley’s AMPLab, the Spark codebase was later donated to the Apache Software Foundation, which has maintained it since then. Spark provides an interface for programming entire clusters with implicit data parallelism and fault tolerance.
RDD RDDs are faulttolerant, parallel data structures that let users explicitly persist intermediate results in memory, control their partitioning to optimize data placement, and manipulate them using a rich set of operators [47,48,49].
We use a command in Scala language for getting suitable size sample from RDDs.
Multistage sampling We leverage multistage sampling [50] to compute error bounds for approximate MapReduce applications that compute aggregations (e.g., counting accesses to Web pages from a log file). The set of supported aggregation functions includes sum, count, average, and ratio. For simplicity, we next discuss twostage sampling. Depending on the computation, it may be necessary to use additional sampling stages as discussed at the end of the section.
Suppose we have T units in N clusters. Each cluster contains M_{i} units. Suppose further that each unit j in cluster i has an associated value V_{ij}, and we want to compute the sum of these values.
To compute an approximate sum, we can create a sample by randomly choosing n clusters, and then randomly choosing m_{i} units from each chosen cluster i. Twostage sampling then allows us to estimate the sum from this sample as:
where (s _{ u} ^{2} ) is the intercluster variance (computed using the sum and average of the values associated with units from each cluster in the sample), (s _{ i} ^{2} ) is the intracluster variance for cluster i, and t_{n−1,1−α/2} is the value of the Student tdistribution with n−1 degrees of freedom at the desired confidence 1−α.
Cochran sampling The Cochran formula allows you to calculate an ideal sample size given a desired level of precision, desired confidence level, and the estimated proportion of the attribute presented in the population [51]. Cochran’s formula is considered especially appropriate in situations with large populations. A sample of any given size provides more information about a smaller population than a larger one, so there’s a ‘correction’ through which the number given by Cochran’s formula can be reduced if the whole population is relatively small.
Aggregation Applications Many companies are used Aggregation Applications widely in transaction operations, Business calculations and Analytical computations. An aggregate function is a function that computes multiple values grouped together and forms final value. In this paper, we present an approach to increase the accuracy of approximation in aggregation applications. This kind of applications can benefit from approximation. As a real and concrete example, the popularity of individual Web pages is important for the Web site operators. They use logs of their Websites for this information. In this kind of computation, the relative popularity is more important than the exact access.
Motivation
Motivation We consider Data Variety in our work and show the difference of results when different amount of skew exists. We have used bootstrapping method to generate 12 GB data without Data Variety and with uniform distribution [52]. We use this uniform input data for the first experiment. We also have used an input with 12 GB size for our second experiment that is collected from three sources. We consider IMDB, Gutenberg, and Quotes and get 4 GB from each of them [53,54,55]. We show the result of this experiment in Fig. 3. The vertical axis represents the “amount of data processed “to meet the sampling requirements, and the horizontal axis represents the “Approaches”.
ApproxHadoop is a wellknown state of the art. We study the functionality of ApproxHadoop in two conditions: existing and not existing data variety. As discussed before multiple resources increase data variety and uneven distribution. Figure 3 shows the amount of data processed to meet the acceptable confidence interval (95%) and error bound (5%). In case of not existing data variety, ApproxHadoop has acceptable performance and by processing less than 1% of input data the sampling requirement is achieved. But ApproxHadoop has achieved expected Quality of Result by processing a large amount of input data as sample. In this case, more than a half of the input data must be processed. The authors in ApproxHadoop have assumed that input data is uniformly distributed. Some reasons such as aggregating data from multiple sources may cause an impressive amount of variety in most real datasets. We have considered this fact in our research.
Our approach is a kind of multistage sampling method. Multistage designs are used in many practical cases [56]. For example in Gallup method, first approximately 300 election districts used in sampling. At the second stage, they select 5 households per district. We employ the Gallup idea in our work and consider Data Variety in our twostage sampling.
Methods
Based on the motivation presented in “Motivation” section, in the case of Data Variety, previous approaches do not have acceptable performance. So, we offer our approach and discuss our system design in this section. As Fig. 4 shows, we divide the input data into some same sized blocks and then divide each block into some same sized frames. As discussed in “Results and discussion” section, we used multistage sampling in this paper. We consider variance of blocks as interblock variance (S_{u}) and variance of frames as intrablock variance (S_{i}).
In our approach, based on formula 3, there are 4 approaches to reduce sample size for achieving the acceptable result:

1.
Minimizing S_{u}: For minimizing S_{u}, we should consider bigger size as the Block Size. This decision causes a bigger value for S_{i}.

2.
Minimizing S_{i}: For minimizing S_{i}, we should consider lower size as the Block Size, this decision causes bigger value for S_{u}.

3.
Processing all blocks (n = N): In case of selecting all blocks, the first term of Eq. (3) will be zero.

4.
Processing all frames: In case of selecting all frames in selected Blocks, the second term of Eq. (3) will be zero.
Decision 1 If we select 1 and try to minimize the S_{u}, the Block Size would have big value and it increases S_{i}. So, based on formula 3, for minimizing the data variance we should try to select all frames in selected Blocks for processing. Based on the big volume of Blocks in this approach, processing all frames have great overheads.
Decision 2 If we select 2 and try to minimize the S_{i} the Block Size would have a small value and increase S_{u}. So, based on formula 3, for minimizing data variance we should try to select all Blocks for processing. Based on the low volume of frames in our approach, processing all samples have low overheads.
Determining Frame Size
The frame size should be determined small enough that intraframe has unique characteristics and behavior. Also, the Frame Size should be big enough to be a good representation of Block. In this paper, we consider 1 KB as the frame size.
Determining Sample Size
Sample Size We use Cochran Sampling technique for our work. Based on this approach if we process limited samples, we achieve acceptable error bound and confidence interval. Based on the Frame Size we can consider each frame as one unique sample.
Cochran Sampling For populations that are large, Cochran yields a representative sample for proportions.
Which is valid where n_{0} is the sample size, Z^{2} is the abscissa of the normal curve that cuts off an area at the tails (1 − α equals the desired confidence level, e.g., 95%), e is the desired level of precision, p is the estimated proportion of an attribute that is present in the population, and q is 1 − p. The value for Z is found in statistical tables which contain the area under the normal curve.
Determining Block Size
Problem definition As discussed in previous sections data skew reduces the efficiency of approximation. We must design an approach to reduce the impact of skew on approximation efficiency. We must design an approach that can present acceptable performance in any condition of data skew and in case of SQL and NoSQL data sets. For this goal, we present 2 algorithms to solve the problem. Algorithm 1 presents our approach for approximation in case of data skew to minimize the processing time. Algorithm 2 presents our approach for determining the Suitable block size for approximation.
Notation We use some notation for the presentation of our algorithm. We show this notation in Table 1.
Problem statement Based on Eqs. 1 to 3, error bound and confidence interval depend on inter and intra block variance. For reduction of the error bound the block size must be determined in a way that the amount of sample is reduced.
Problem formulation The objective function, to be minimized, is the total processing time of blocks and the constraint is the error bound. PT, EB, and PEB presents Processing Time, Error Bound and Preferred Error Bound. Thus, the problem can be formulated as below:
Subject to:
Equation (5) is the objective function which should be minimized. Equation (6) is the constraint of our work. Based on the Spark and RDD limitation these constraints exist in our work. Default RDD size in Spark is 64 MB. We can merge RDDs for generating a bigger RDD.
Lines 1–2 of the algorithm is initializing the variables. Note that DV, IDS, BS and PT show the Data variety, Input Data Size, Block Size and Processing Time. Line 3 presents algorithm 2 to select SBSA. Line 4 estimate the S_{u}, S_{i}, PD, PT, and EB.
The loop in lines 5 to 9 shows the multistage sampling from which the Preferred Error Bound and minimum Processing Time is achieved.
Complexity Analysis The algorithm time complexity is of O (1). Because the algorithm selects all blocks and sample intra them based on the Cochran method.
Figure 5 shows our approach in general (Algorithm 1). Samples derived from primary RDDs form new RDDs and intermediate RDDs join each other and make the final RDD.
Simple Random Sample We have used Simple Random Sample to determine the skew condition. Using a cluster sample generally requires either a larger sample size than a SRS or using a wider confidence interval. To determine segment size, we have used the design effect formula [57]. The design effect is used to determine how much larger sample size or confidence interval need to be. Based on Eq. 7 the design effect increases as cluster sizes increases, and as interclass correlation increases.
DEFF is design effect, \(\rho\) is the intraclass correlation, n is the average size of the cluster.
Based on Eq. 8 the estimated variance decreases when the variance of Simple Random Sample decreases and the DEFF decreases.
We present algorithm 2 to determine the Suitable Block Size for Approximation. We have used DEFF to estimate the amount of data skew. To overcome this issue, we divide the Block Size and increase the amount of samples by this approach.
Lines 1–2 of the algorithm is initializing the variables. Note that DV, IDS, and BS show the Data variety, Input Data Size and Block Size. The loop in Lines 3 to 10 selects SBSA as BS. Line 4 select all data as Block Size. Line 5 estimates data skew by simple random sampling. In line 6, 7 and 8 if the variance of SRS is bigger than Expected Variance the RDDs will be divided.
Complexity Analysis Line 6–9 of the algorithm is a loop procedure and takes O (log2 (n)) time. Where n represents the volume of input data.
Figure 6 shows the overview of algorithm 2. In the case of existing low skew in input data, the block size can be big. In the case of existing negligible skew in data, the input data can be on the block. In this case, our algorithm works like ApproxHadoop.
By using these two algorithms, we take Decision 1 when low skew exists and take Decision 2 when there is a high skew in data.
Implementation in Spark
We have used commands and abilities of Spark and Scala language to implement our approach. We get some samples from RDDs as Fig. 7 shows. For this purpose, we consider frames within RDDs. In our approach, the frame is the smallest unit of our sampling. Based on the Cochran sampling technique, when we get a certain number of frames as a sample, the acceptable confidence interval is achieved. For example in Fig. 7, we get 384 samples with a volume ratio of 0.0001 (per each sample) of all data. This samples make a new RDD and will be processed.
Results and discussion
We evaluate our approach using applications from a wide area, multiple applications, and datasets.
Datasets We evaluated seven benchmarks WordCount, Grep, and AverageLength from BigDataBench suite [58]. The datasets comprised of 100 GB of data from three different sources [53,54,55]. We also used wellknown TPC benchmark [18] and Amazon review for our evaluations [19, 59]. We have used a bootstrapping method for generating 100 GB data as input datasets [52].
Application

WordCount: This application Counts the number of words in the file.

Grep: It searches and counts a pattern in a file.

AverageLength: This application calculates the average length of words in the file.

We also consider AVG for TPCH datasets and SUM for Amazon datasets.
Methodology
Hardware (Servers) We ran Experiments on 5 machines, Intel Corei7 4 core CPU at 2.8 GHz with 4 GB of RAM.
Software We have used Spark version 2.0 on Ubuntu12.04 as the framework of experiments. For our experiments.
Experimental results
The following figures presented the impact of Data variety on the volume of data that must be processed to achieve a 95% confidence interval and 5% error bound. In this experiment, we consider 0.5 GB for Block size.
In this section, we used two wellknown datasets for our evaluations:

Amazon datasets (Mc: Music, Bs: Books, Ms: Movies, Cg: Clothing, PS: Phones), and

TPCH datasets (M: MAIL, S: SHIP, A: AIR, R: RAIL, T: TRUCK).
Amazon product data contains product reviews and metadata from Amazon, including 142.8 million reviews spanning May 1996–July 2014.
(TPCH) is a decision support benchmark. It consists of a suite of business oriented adhoc queries and concurrent data modifications. The queries and the data populating the database have been chosen to have broad industrywide relevance. This benchmark illustrates decision support systems that examine large volumes of data, execute queries with a high degree of complexity, and give answers to critical business questions.
We compared our work with other approaches in two factors: the amount of data processed and the processing time. In Figs. 8, 9, and 10. Vertical axis shows the relative amount of data to be processed and the horizontal axis presents the datasets. As discussed in previous sections, our approach can surpass Approxhadoop and Sapprox. Based on the uneven distribution of real datasets, ApproxHadoop processed a large amount of data to achieve processing requirements. Sapprox cannot perform efficient estimation of the amount of data to be processed. So, our approach can surpass other approaches.
In Figs. 11, 12, and 13, processing time (in seconds) is presented as the vertical axis and the horizontal axis shows the datasets. Our approach can reduce the time of processing as these figures show.
Our approach improved the processing time up to 17× and 8× compared to ApproxHadoop and Sapprox, Respectively. We also reduce the amount of data processed to achieve the sampling requirement up to 58× and 32× compared to ApproxHadoop and Sapprox. In NoSQL datasets, Sapprox has lower performance and we achieve up to 9× speed up in compared with it.
Sensitivity analysis
Impact of Block Size on Inter/Intra Block Variance Figures 14 and 15 show the impact of BS on Inter/Intra block Variance in SQL and NoSQL datasets. Vertical axis presents the variance (normalized to the 0.5 GB variance) and horizontal axis presents different block size.

1.
In the case of increasing the Block Size, the intra Block variance is increasing and the interblock variance is decreasing.

2.
In the case of decreasing the Block Size, the inter Block variance is increasing and the intrablock variance is decreasing.
Based on these observations we decide to select 0.5 GB as Block Size. By this decision, we are able to achieve desirable confidence interval and error bound.
Impact of data skew on volume of sample We discover the impact of data skew on volume of sampled data. We used a statistical method to model skewed data.
Modeling data skew We have used Zipfian [41, 60], distribution to generate skewed data. Zipf’s law states that out of a population on N elements, the frequency of elements of rank k, f(k; z, N) is:
Following the Zipfian distribution, the frequency of occurrence of an element is inversely proportional to its rank.
In the current context, let:

1.
N = total number of input partitions;

2.
k be their rank; partitions are ranked as per the number of records in the partition that satisfy the given predicate;

3.
z be the value of the exponent characterizing the distribution.
We have considered z = 0 for uniform distribution, z = 1 for moderate skew and z = 2 for high skew.
Figures 16, 17 and 18 show the impact of data skew on the amount of data to be processed to achieve processing requirements. The vertical axis shows the relative amount of processed data and the horizontal axis shows the datasets. As above figures present, Sapprox has lower efficiency in case of NoSQL data. The reason for this is that the determination of Sampling Interval is complex for NoSQL datasets.
Impact of Block Size on Volume of Sample In Fig. 19, we show the effect of block size on the volume of the sample. The vertical axis shows the volume of sample and the horizontal axis shows the block size. Aggregating input data from multiple resources is one of the factors that cause data skew. We have shown it’s effect in previous sections. Figure 19, shows that in case of aggregating input data from the single source, the big block is better than a small block and in case of input data from multiple sources, the small block size is better than big block sizes. We have used Wikipedia for single source and four different sources [53,54,55] and Wikipedia as multiple sources. The same behavior is observed for all applications.
Qualitative and quantitative analysis comparisons
In this section, we present a qualitative and quantitative comparison of approaches. Table 2. Presents the comparisons in summary. More detail descriptions are provided below in Table 2.
The advantage of Gapprox to ApproxHadoop

Determined sized sample

Considering data variety and skew

Processing lower amount of input data to achieve acceptable confidence interval and error bound.
The advantage of Gapprox to Sapprox

Processing lower amount of input data to achieve acceptable confidence interval and error bound.

No need to specify the sampling interval.

Applicable for SQL and NoSQL applications.

More simple than Sapprox for implementation
Our novelties in the presented approach

We offer an approach that has acceptable performance in various conditions of data variety/skew.

Our approach is simple for users to be implemented. The users can easily divide the input data into some same size blocks and apply the Gapprox to them.

Our approach has low overhead.

Our approach can be used for SQL and NoSQL simply and presents acceptable efficiency in any type of data.

We have used abilities in Spark and Scala language to present a practical simple approach. So, the users can easily implement this approach in Spark.
ApproxHadoop have high performance in case of uniform distribution. But in case of skewed data does not have acceptable performance. Sapprox has better performance in case of data variety. Our algorithm also can surpass Sapprox in SQL and NoSQL datasets. Sapprox has another weakness in the case of NoSQL datasets. In this kind of dataset determining Sample Interval is complex.
Conclusions and future work
In this paper, we offer a solution for reducing processing time in case of existing data variety in input data. We consider data variety/skew and use multistage sampling to discover the input data. We select fixed block (cluster) and frame size. We use Cochran sampling for intra block sampling.
Based on the experimental results our approach can surpass other approaches, improve processing time and reduce the amount of data to be processed to achieve the desired error bound and confidence interval. Our approach processes a certain amount of input data to achieve processing requirements. For evaluation, we have used Amazon and TPC datasets as SQL datasets and some other as NoSQL datasets. Our approach surpasses two other approaches in SQL and NoSQL datasets. We also have modeled data skew and show that our approach is more efficient than other approaches in various skew conditions. We have shown that in uniform data sets large block size and in skewed datasets, small block size is suitable in case of sampling overheads.
There are some directions for future works. One of them is the key management in intermediate data to reduce data transportation among the network. In aggregation applications, the input and intermediate data are converted to a certain value of a key. By intermediate processing and key management, we are able to reduce data transportation.
Abbreviations
 Gapprox:

using Gallup approach for approximation in Big Data processing
 BS:

block size
 DV:

data variety
 EB:

error bound
 IDS:

input data size
 UBS:

upper bound of block size
 LBS:

lower bound of block size
 PEB:

preferred error bound
 PT:

processing time
 PPT:

preferred processing time
 PD:

processed data
 SABA:

suitable block size for approximation
 VAR_{SRS} :

variance of simple random sample
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Authors’ contributions
HA is the primary researcher for this study. His contributions include the original idea, literature review, implementation and initial drafting of the article. MG guided the initial research concept and played a crucial role in the design of the analytics approaches presented. FF discussed the results with the primary author to aid writing of the evaluation and conclusion sections and played an essential role in editing the paper. All authors read and approved the final manuscript.
Acknowledgements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
BigDataBench: http://prof.ict.ac.cn/.
TPC Benchmark: http://www.tpc.org/information/benchmarks.asp.
Amazon product data: http://jmcauley.ucsd.edu/data/amazon/.
IMDB data files: https://datasets.imdbws.com/.
Gutenberg datasets: https://www.gutenberg.org/.
Quotesdataset: https://www.kaggle.com/akmittal/quotesdataset.
Funding
Not applicable.
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Keywords
 Data variety
 Quality of Result
 Approximation
 Cluster sampling