# Partition Selection for Large-Scale Data Management Using KNN Join Processing

Authors: Yue Hu; Ge Peng; Zehua Wang; Yanrong Cui; Hang Qin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

For the data processing with increasing avalanche under large datasets, thek nearest neighbors (KNN) algorithm is a particularly expensive operation for both classification and regression predictive problems. To predict the values of new data points, it can calculate the feature similarity between each object in the test dataset and each object in the training dataset. However, due to expensive computational cost, the single computer is out of work to deal with large-scale dataset. In this paper, we propose an adaptive vKNN algorithm, which adopts on the Voronoi diagram under the MapReduce parallel framework and makes full use of the advantages of parallel computing in processing large-scale data. In the process of partition selection, we design a new predictive strategy for sample point to find the optimal relevant partition. Then, we can effectively collect irrelevant data, reduce KNN join computation, and improve the operation efficiency. Finally, we use a large number of 54-dimensional datasets to conduct a large number of experiments on the cluster. The experimental results show that our proposed method is effective and scalable with ensuring accuracy.


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## Body

## 1. Introduction

In recent years, with the wide deployment of cloud computing, network, and radio, Internet of Things (IOT) products have been widely used in the natural sciences, such as industrial pollution areas [1], mobile devices [2], vehicle communication systems [3], and radar Systems [4]. In the face of complex and diverse signals, the traditional radars have limited performance in DOA estimating [5]. How to enhance useful signals and extract useful information is the focus of current research [6]. With the development of signal processing technology, researchers propose a variety of algorithms to achieve this purpose. Typical algorithms include the multiple signal classification (MUSIC) and its variations [3, 7], ULA-based method [8], reduced-complexity OGSBL [9], PARAFAC decomposition [10], Tensor-based subspace algorithm [11], and NNM [12].To date, searching for approximate objects from the vast amount of useful data is a very basic and critical operation. With the continuous expansion of network scale, the data scale presents explosive growth with large-volume, complex, and growing datasets for multiple and autonomous sources. As a result, thek nearest neighbors (KNN) algorithm with high accuracy, insensitive to outliers, and no data input assumptions represents an important paradigm shift in the evolution of partition selection with the trained metric. Existing KNN algorithm-based method assume the classification can be improved by learning a distance metric from labeled examples.In the classification process, the KNN algorithm calculates the distance (similarity) between the data sample to be classified and the entire known dataset. This method is simple, convenient, and inexpensive for small datasets. When dealing with such large-scale data, the complexity of similarity calculation increases dramatically with the intolerable calculation cost, which directly affects the classification efficiency and accuracy. When the dataset has more attributes, the impact is more evident, and the dimension catastrophes tend to occur to make distances very far when calculated in high-dimensional space. At this point, a natural idea is to introduce the idea of distribution with implementation of data parallel support.In particular, MapReduce is a distributed parallel programming framework proposed by Google for processing large datasets on large-scale clusters, in terms of a streamlined computational framework to assemble sequential and parallel computation. The programming model via MapReduce was originally designed to simplify large-scale data calculations [13, 14]. In recent years, MapReduce has been studied in many ways [15–18] for large-scale data-intensive computing under data-intensive [19], CPU-intensive, and memory-intensive applications, such as in the fields of smart cities [20], biological data management [21], spatial geometry calculation [22], and distributed computing over a wireless interference network [23].The goal of this paper is to propose an effective data partitioning strategy. The proposed KNN algorithm designed based on MapReduce framework is mainly used to solve the problem of too much computation and low classification efficiency. With this programming framework, we can divide KNN’s computing tasks into several small tasks and assign them to several computing nodes to calculate at the same time for the speedup the operation. Only by dividing the data reasonably and actually reducing the calculation cost practically can the running efficiency of the algorithm be effectively improved.The contributions of this paper are as follows:(1)
We introduce the idea of the Voronoi diagram to partition the sample objects and design the partition selection strategy to find the optimal relevant partition for the sample to be tested, thereby avoiding the extracalculation brought by irrelevant data.(2)
We address the MapReduce framework and propose a vKNN algorithm, which is implemented on the Hadoop cluster with KNN join processing, nearest center points selecting, relevant-partition selecting, and vKNN processing.(3)
We conduct many experiments using real datasets to study the effects of various parameters on the algorithm. The results show that our proposed algorithm is effective and scalable with the accuracy in relevant-partition selection.The contents of the paper are structured as follows. Section2 reviews the related work, Section 3 formally defines the problems to be solved in this paper, Section 4 describes the related technologies involved in this paper, Section 5 details the implementation and improvement of the KNN algorithm based on MapReduce, Section 6 reports the experimental results, and finally Section 7 summarizes the entire study.


## 2. Related Work

Existing KNN method assumed that the classification algorithm can be widely applied in the field of machine learning and large-scale data analysis. In order to better apply the traditional KNN algorithm, previous studies mainly used two kinds of methods,i.e., speeding up the process of finding k nearest neighbors and eliminating irrelevant data to reduce the overall computation. For instance, Cui et al. [24] introduced a B+-tree method that maps high-dimensional data points in one dimension. The one-dimensional distance computed in the principal component space and the first principal components of the sample points were indexed using a B+-tree. At the same time, the principal components were adopted to filter neighboring query points to improve query efficiency. When working with high-dimensional data, most indexing methods cannot scale up well and perform worse than sequential scanning. Xia et al. [25] designed and implemented KNN join algorithm based on block nested join Gorder using a grid-based sorting method, which can effectively assign similar objects to the same grid. Amagata et al. [26] proposed a dynamic set KNN self-join algorithm to trim unnecessary computations using index technology.However, the processing power of the single processor greatly limits the development of the KNN algorithm, which also makes the application of parallel and distributed compute imperatively. In recent years, MapReduce has been fully practiced in the field of machine learning [27–29]. In order to solve those problems of KNN algorithm, Zhang et al. [30] proposed the HBNJ algorithm implemented by Hadoop and its improved algorithm H-BRJ in document. However, due to the large impact of data size on the efficiency of the algorithm, this research focused more on approximate queries. Moutafis et al. [31] proposed a four-stage algorithm, where three optimization strategies were used to trim distant points, balance the number of reducers, and halve the output, which significantly reduced the computation time. In most studies, people chose to use the first data partitioning to reduce data calculation, such as R-tree, Δ-tree, Quad-Tree, and KDB-Tree [32]. These spatial partitioning-based indexing techniques will dramatically reduce efficiency as dimensions increase. Zhang et al. [30] proposed a Z-value-based partitioning strategy. The result of the algorithm depends to a large extent on the quality of the z-curve, which may cause problems in the processing of high-dimensional data. Ji et al. [33] proposed a distance-based partitioning method. However, this grid-based partitioning method is considered valid only for low-dimensional datasets. We also use a partitioning strategy. In this paper, we introduce the concept of Voronoi diagram because it can be applied to any dimension of data [34]. We use Voronoi diagrams to aggregate similar data so that irrelevant data can be clipped. The Voronoi diagram was proposed with the famous structure of computational geometry. It is widely used in many fields such as geometry, architecture, and geography [35–38]. Voronoi diagrams can partition data into set spaces and are effective in the study of local neighborhoods for each partition [39]. At the same time, Voronoi diagrams can help improve the performance of distance join queries [40].


## 3. Problem Formulation

In this part, we give the definition of KNN Join with its formulation. Table1 lists some symbols and their corresponding meanings involved mainly in this paper.Table 1
Symbols and their meanings.
SymbolDefinitionR(S)Thed-dimensional datasetr(s)The data object inR(S)ℜdThed-dimensional metric spacedis(r, s)The distance fromr to skThe number of nearest neighborsknn(s, R, k)Thek nearest neighbors of s from RknnJ(R, S)The KNN Join ofR and Smax(knn(s, R, k))The maximum distance froms to its k nearest neighbors from RNThe number of mappersNThe number of center pointsPThe set of center pointspiThe point inPΡiThe partition corresponds topiΡxThe partition wherex is locatedΡisThe partition corresponding to theith center point close to sLetR and S be two different d-dimensional datasets, and r(s) is the data object in R(S). For the convenience of discussion, we introduce the geometric space to represent them. ℜd is a d-dimensional geometric space, R and S can be regarded as a sample point set in ℜd, and the data objects r and s can be viewed as a d-dimensional sample point; then, we have r∈Rs∈S. In order to avoid loss of generality, the distance measurement method adopted in this paper is Euclidean distance. Also, the distance between the data objects r and s, denoted as dis(r, s), can be calculated as follows:(1)disr,s=∑l=1drl−sl21/2,where dis(r, s) ≥ 0 and the necessary condition for dis(r, s) = 0 is r = s.The similarity between data objectsr and s, denoted as sim(r, s), is(2)simr,s=11+disr,s,where the greater the distance, the greater the difference between objects and the smaller the similarity.Definition 1.
(KNN). Given a sample setR, a newly input sample point s. The KNN operation of them, denoted as knns,R,k, involves the k nearest neighbors of s from R. The formal description is as follows:(3)knns,R,k=z1,z2,…,zkz1,z2,…,zk∈R,for ∀zj∈R−z1,z2,…,zkz1,z2,…,zk∈R, and we have(4)ds,zj≥ds,zk≥⋯≥ds,z2≥ds,z1.Definition 2.
(KNN Join). Given two sample setsR and S. The KNN join operation denoted as knnJR,S returns each object s∈S with its k nearest neighbors from R. The formal description is as follows:(5)knnJR,S=s,knns,R,kfor alls∈S.


## 4. MapReduce for Data Processing under Voronoi-Based KNN Processing

### 4.1. MapReduce for Flexible Data Processing

MapReduce, a distributed parallel programming framework, is a member of the core designs of Hadoop [41, 42]. It separates the users from the bottom layer of the system. When users write the corresponding programs, they only need to write the Map function and Reduce function to give what needs to be calculated and how to calculate automatically by the framework. Meanwhile, MapReduce has been extensively used due to its high fault tolerance and scalability.MapReduce is mainly used for parallel computation of large amounts of data. A MapReduce program contains only two functions:Map function and Reduce function. The corresponding processing of these two functions can be customized by the user. At the beginning of the calculation task, Hadoop divides the entire job into two sequential phases: Map phase and Reduce phase. The model first breaks down the computational tasks of large-scale data that need to be processed into many individual tasks. These individual tasks can be run in parallel on a Hadoop server cluster; then, the model combines the results calculated by the cluster and calculates the final result. In the Map/Reduce phase, there are multiple instance tasks, which can be executed in parallel on each node. The MapReduce programs save both input and output results in HDFS. They use migration to transfer data to the nearest available node only if the node does not have local data or cannot process local data [43].Figure1 shows the specific execution process of MapReduce. The Client program divides the file data according to the parameter (m). The ResourceManager picks the idle nodes in the cluster and assigns the corresponding compute resources to them. At the same time, the Job is assigned the same number of Map tasks with the number of file blocks. Before starting the Map task, the file on the node is read and parsed into <key1, value1> key-value pairs by line, and then the Map function converts it into new <key2, Value2> key-value pairs. Next, the model uses the Hash function to partition and sort <key2, value2> key-value pairs, and groups them to <key2, {value2, …}> according to the key2. Finally, the Reduce function accepts the data, generates new <key3, value3> pairs from the corresponding business logic, and saves it in HDFS. When the cluster resources are not sufficient to host all Map (Reduce) tasks at the same time, the corresponding tasks are started in batches. In addition, the first Reduce task can only be started after the last Map task has been executed [44].Figure 1
The workflow of the MapReduce.


### 4.2. Voronoi Diagram with Partition Selection

The Voronoi diagram, also known as Dirichlet diagram, plays an important role in computational geometry. In the field of mathematics, this diagram is a decomposition of a given space, the simplest form to decompose a plane. The division yields that all points in each area are closer to the center of the area than to other centers.To illustrate this, we can take a two-dimensional plane as an example. Given a datasetR, each object in R can be regarded as a point of ℜ2. Partitioning using a Voronoi diagram, means selecting n objects as the center points and assigning all objects in R to the partition corresponding to their nearest center points. This divides the entire data space into n partitions, as shown in Figure 2. The large blue circle represents the center points of the Voronoi diagram, and each point represents an object. The green points represent the k nearest neighbors of object x, where the grid points represent the false k nearest neighbors. The orange area is where the area of object x is located, and the region enclosed by the red line is the set of regions, where the true k nearest neighbors of object x are located. The dashed areas correspond to the spatial ranges of the true k nearest neighbors and the false k nearest neighbors, respectively.Figure 2
Voronoi diagram of the plane.For the sake of brevity, letP be the selected set of center points, where P=p1,…,pn. Given two center points pi and pj, PBpi,pj represents the hyperplane dividing the partitions, where pi and pj are located for the point o on the hyperplane PBpi,pj, and we have(6)∀o∈PBpi,pj,diso,pi=diso,pj.According to formula (1), for any sample points s, located in the corresponding subspace of pi, the distance between s and pi, denoted as diss,pi, is(7)diss,pi=∑l=1dsl−pil21/2,for ∀pj∈P−pi, and we have(8)diss,pi<diss,pj.The distance froms to PBpi,pj, denoted as diss,PBpi,pj, can be calculated as follows:(9)diss,PBpi,pj=diss,pj2−diss,pi2×dispi,pj2.Figure3 shows the distance diss,PBpi,pj. Based on the characteristics of the Voronoi diagram, we can transform the process of finding k nearest neighbors in R in Definition 1 into the process of finding k nearest neighbors in the partition. Now, our work only considers the partition where the sample point is located.Figure 3
The distance from (s) toPBpi,pj.Hypothesis 1.
Given a sample setR, perform a KNN operation on the newly input sample point x, that is, find k nearest sample points of x in the partition px where x is located. The formal description is as follows:(10)knnx,R,k=knnx,px,k.
In this case, the computational effort of the running process is significantly reduced. However, whenx is near the px boundary of the partition where it is located, it is easy to produce large errors by roughly limiting the size of the partition involved in the calculation. Suppose k = 5, as shown in Figure 2, the sample point x to be measured is located in the partition corresponding to p4. According to Hypothesis 1, the k nearest sample points are z1,z2,…,z5. And when we look through Figure 2, it is easy to find that the real k nearest sample points are z1,z2,z3′,z4′,z5′, whereas the sample points z3′,z4′,z5′ outside of the partition. This means that, in the actual operation process, only the samples in the partition where x is located may not necessarily yield true results. Consequently, we introduce a new concept: relevant partition, which is given in conjunction with Definition 1 as follows.Definition 3.
(relevant partition). Given the sample setR, the corresponding relevant partition Rx for the newly input sample point x is the partition set of k nearest neighbors. We have(11)Rx=pz1∪pz2∪⋯∪pzk.
Therefore, KNN operation onx and R can be converted to finding the k nearest neighbors of x in the relevant partition Rx, and the formal description is as follows:(12)knnx,R,k=knnx,Rx,k.


## 4.1. MapReduce for Flexible Data Processing

MapReduce, a distributed parallel programming framework, is a member of the core designs of Hadoop [41, 42]. It separates the users from the bottom layer of the system. When users write the corresponding programs, they only need to write the Map function and Reduce function to give what needs to be calculated and how to calculate automatically by the framework. Meanwhile, MapReduce has been extensively used due to its high fault tolerance and scalability.MapReduce is mainly used for parallel computation of large amounts of data. A MapReduce program contains only two functions:Map function and Reduce function. The corresponding processing of these two functions can be customized by the user. At the beginning of the calculation task, Hadoop divides the entire job into two sequential phases: Map phase and Reduce phase. The model first breaks down the computational tasks of large-scale data that need to be processed into many individual tasks. These individual tasks can be run in parallel on a Hadoop server cluster; then, the model combines the results calculated by the cluster and calculates the final result. In the Map/Reduce phase, there are multiple instance tasks, which can be executed in parallel on each node. The MapReduce programs save both input and output results in HDFS. They use migration to transfer data to the nearest available node only if the node does not have local data or cannot process local data [43].Figure1 shows the specific execution process of MapReduce. The Client program divides the file data according to the parameter (m). The ResourceManager picks the idle nodes in the cluster and assigns the corresponding compute resources to them. At the same time, the Job is assigned the same number of Map tasks with the number of file blocks. Before starting the Map task, the file on the node is read and parsed into <key1, value1> key-value pairs by line, and then the Map function converts it into new <key2, Value2> key-value pairs. Next, the model uses the Hash function to partition and sort <key2, value2> key-value pairs, and groups them to <key2, {value2, …}> according to the key2. Finally, the Reduce function accepts the data, generates new <key3, value3> pairs from the corresponding business logic, and saves it in HDFS. When the cluster resources are not sufficient to host all Map (Reduce) tasks at the same time, the corresponding tasks are started in batches. In addition, the first Reduce task can only be started after the last Map task has been executed [44].Figure 1
The workflow of the MapReduce.


## 4.2. Voronoi Diagram with Partition Selection

The Voronoi diagram, also known as Dirichlet diagram, plays an important role in computational geometry. In the field of mathematics, this diagram is a decomposition of a given space, the simplest form to decompose a plane. The division yields that all points in each area are closer to the center of the area than to other centers.To illustrate this, we can take a two-dimensional plane as an example. Given a datasetR, each object in R can be regarded as a point of ℜ2. Partitioning using a Voronoi diagram, means selecting n objects as the center points and assigning all objects in R to the partition corresponding to their nearest center points. This divides the entire data space into n partitions, as shown in Figure 2. The large blue circle represents the center points of the Voronoi diagram, and each point represents an object. The green points represent the k nearest neighbors of object x, where the grid points represent the false k nearest neighbors. The orange area is where the area of object x is located, and the region enclosed by the red line is the set of regions, where the true k nearest neighbors of object x are located. The dashed areas correspond to the spatial ranges of the true k nearest neighbors and the false k nearest neighbors, respectively.Figure 2
Voronoi diagram of the plane.For the sake of brevity, letP be the selected set of center points, where P=p1,…,pn. Given two center points pi and pj, PBpi,pj represents the hyperplane dividing the partitions, where pi and pj are located for the point o on the hyperplane PBpi,pj, and we have(6)∀o∈PBpi,pj,diso,pi=diso,pj.According to formula (1), for any sample points s, located in the corresponding subspace of pi, the distance between s and pi, denoted as diss,pi, is(7)diss,pi=∑l=1dsl−pil21/2,for ∀pj∈P−pi, and we have(8)diss,pi<diss,pj.The distance froms to PBpi,pj, denoted as diss,PBpi,pj, can be calculated as follows:(9)diss,PBpi,pj=diss,pj2−diss,pi2×dispi,pj2.Figure3 shows the distance diss,PBpi,pj. Based on the characteristics of the Voronoi diagram, we can transform the process of finding k nearest neighbors in R in Definition 1 into the process of finding k nearest neighbors in the partition. Now, our work only considers the partition where the sample point is located.Figure 3
The distance from (s) toPBpi,pj.Hypothesis 1.
Given a sample setR, perform a KNN operation on the newly input sample point x, that is, find k nearest sample points of x in the partition px where x is located. The formal description is as follows:(10)knnx,R,k=knnx,px,k.
In this case, the computational effort of the running process is significantly reduced. However, whenx is near the px boundary of the partition where it is located, it is easy to produce large errors by roughly limiting the size of the partition involved in the calculation. Suppose k = 5, as shown in Figure 2, the sample point x to be measured is located in the partition corresponding to p4. According to Hypothesis 1, the k nearest sample points are z1,z2,…,z5. And when we look through Figure 2, it is easy to find that the real k nearest sample points are z1,z2,z3′,z4′,z5′, whereas the sample points z3′,z4′,z5′ outside of the partition. This means that, in the actual operation process, only the samples in the partition where x is located may not necessarily yield true results. Consequently, we introduce a new concept: relevant partition, which is given in conjunction with Definition 1 as follows.Definition 3.
(relevant partition). Given the sample setR, the corresponding relevant partition Rx for the newly input sample point x is the partition set of k nearest neighbors. We have(11)Rx=pz1∪pz2∪⋯∪pzk.
Therefore, KNN operation onx and R can be converted to finding the k nearest neighbors of x in the relevant partition Rx, and the formal description is as follows:(12)knnx,R,k=knnx,Rx,k.


## 5. KNN Algorithm with MapReduce Performance Improvement

### 5.1. KNN Join Processing

The basic idea of the KNN join algorithm based on MapReduce is in general agreement with that of the KNN algorithm. Firstly, the MapReduce program divides the input test dataset, each node calculates the distance between the test samples in the corresponding slice and each sample in the training dataset, finds out thek nearest neighbors, and selects the label with the largest proportion of these adjacent points. As shown in Algorithm 1, the KNN join processing can be formulated by the following.Algorithm 1: KNN join processing.
Input:k, R, SOutput:s.predictLabel//prediction label of smap: <row number, s>foreachr ∈ Rdodis = dis(r, s);//calculate the Euclidean distance between r and sfori = 0 tokdoif dis < distance[i] then//find the minimum k distancesdistance[i] = dis;trainLabel[i] = r.label;break;forj = 0 tokdooutput(s, trainLabel[i]);reduce: <s, Labels>hmp = new HashMap(); //create a HashMap object hmpforeach label ∈ Labels do//count the number of each labelif hmp.get(label) ! = NULL then//if the label exists in hmplabel.value ++; //take the value of the label and add 1hmp.put(label) = label.value;//update the value of the label in hmpelse//if the label does not exist in hmphmp.put(label) = 1; //set the value of the label to 1 and insert to hmppredictLabel = hmp.maxvalue; //the label with the largest value as the prediction labeloutput(s, predictLabel);In theMap function, it first sets the parameter k of the algorithm. Next, it calculates the Euclidean distance between each sample r of the training dataset R and the test sample s and stores the labels of the k nearest training samples into trainLabel. The form of the input data <key, value> is <row number, sample>; the form of the output data <key, values> is <sample, the label of adjacent sample>. The Hash function partitions, sorts, and groups these intermediate results by key values. The Reduce function then reads them. The design of the Reduce function is relatively simple. Its main task is to obtain the label with the maximum number of k labels and assign it to the test sample. First, iterate through the data passed by the Map function in turn, and if the current data exists in the HashMap, add 1 to its value. If the current data does not exist in the HashMap, mark its value as 1, and add it to the HashMap. Finally, the label with the largest value in HashMap is used as the prediction label. The form of input data <key, values> is <sample, set (the labels of adjacent k samples)>. The form of output data <key, value> is <sample, the prediction label>.Obviously, the method is too expensive. It simply assigns computing tasks to the computing nodes, and each mapper needs to connect a subset fromS to the entire dataset R. Considering only the distance calculation to be performed for each node, the amount of computation reaches R⋅S/n. When the dataset involved in the calculation is large, the amount of computation is still a huge amount in comparison. In addition, the comparison of similarity may exceed the computing capacity of the node, resulting in the task being killed. Therefore, it is unreliable to rely solely on MapReduce to slice computational tasks for efficient classification. A better idea is to reduce the number of samples in R that is involved in the calculation.


### 5.2. Relevant-Partition Selecting and vKNN Processing

We consider the Voronoi diagrams for KNN join within the MapReduce framework. The basic idea is to partition the data using the Voronoi diagram and clip the unqualified data to reduce the amount of calculation. There are three main steps as follows.


#### 5.2.1. Preprocessing Step

Input datasetR and partition R using Voronoi diagram. First, randomly select N samples from the dataset R as the initial center point. Then, we use k-means clustering method to analyze the dataset globally to obtain the center points set P and the corresponding data clusters.


#### 5.2.2. Nearest Center Point Selecting Step

In this step, we use the output of the above processes and datasetS as input objects. Find the k nearest center points of each sample in S and save its index and distance information to help clip the unqualified data.Algorithm2 shows the execution of the mappers at this stage. Before the program starts, we can load the preprocess center point data into the main memory of each mapper. After each mapper reads the sample object s, it traverses each center point and calculates the distance of s from all the center points. In order to reduce the cost of data transmission between nodes, we only save the index and distance information of the k center points nearest to s. We use TreeMap to store relevant information so that we can get the first k center points more quickly, where TreeMap itself is an ordered set of key values. All elements remain in a specific order and are sorted in ascending order by default by the value of the key. So, we can easily get information about the nearest k center points.Algorithm 2: Nearest center point selecting.
Input:S, POutput: <s, Ρ1s,Ρ2s,…,Ρks>//the partitions corresponding to the k nearest center points of sforeachs∈Sdotmp = TreeMap(); //create a TreeMap object tmpnearPointSet = [ ]; //create an empty set nearPointSetforeachp∈Pdo//insert the distance and index information tmp.put(dis(p, s), p.index);fori = 0 tokdo//read the first k center points’ information nearPointSet.append(tmp.next()); //put the information to nearPointSet


#### 5.2.3. vKNN Processing Step

In this step, we use the output from the previous two processes as input. On the basis of the distance, we can filter out the relevant partitionRs corresponding to each sample s in S to find out the labels of the k nearest neighbors. Finally, reducer counts the labels and outputs the label that appears most often.In order to ensure the accuracy of the prediction results, the selected relevant partitionRs contains all knns,R,k as minimum as possible. How to determine the relevant partition Rs that will participate in the final calculation is an issue we need to consider now.Theorem 1.
Given a sample points and a center point pi, s is located in the partition pi corresponding to pi, and we have(13)pi∈Rs.Proof.
Ass is located in the partition pi, s is very similar to the sample points in pi. This means that the probability of having the nearest neighbors in pi is greater than that in any other partition. In other words, pi are more likely to have the nearest neighbors.
Based on the analysis of Hypothesis1, we may have errors in predicting using only the sample points within pi. We can design a selection strategy to determine whether other partitions meet the criteria.Definition 4.
Given two center pointspi and pj, PBpi,pj addresses the hyperplane dividing the subspace, where pi and pj are located, s is located in the partition pi corresponding to pi, and the maximum distance from s to it in its k nearest neighbor samples, denoted as θ, is(14)θ=maxknns,pi,k.Theorem 2.
Given two center pointspi and pj, PBpi,pj is the hyperplane dividing the subspace where pi and pj are located, s is in the partition pi corresponding to pi, and the necessary condition for pj∈Rs is(15)diss,PBpi,pj<θ.
Figure4(a) shows the case of diss,PBpi,pj>θ, and Figure 4(b) shows the case of diss,PBpi,pj<θ, respectively. When diss,PBpi,pj>θ, there is no intersection between the hypersphere with a radius of θ and the hyperplane PBpi,pj. That is to say, ∀x∈pj,diss,x>θ all hold. At this time, we can directly discard the partition pj. When diss,PBpi,pj<θ, the hypersphere with the radius of θ intersect with the hyperplane PBpi,pj. It means that there is probably a sample point x in the partition pj, making diss,x<θ, which also means that the calculated knns,Rs,k are not the real nearest neighbors. Therefore, we need to add the partition pj to the relevant partition Rs.In (Algorithm3), due to the center point information sorted using TreeMap before, according to Theorem 1, we can get that the initial relevant partition Rs is p1s. We calculated k nearest neighbors of s on Rs and saved them in knnDisSet. Next, we judge the subsequent partitions in sequence according to Theorem 2. If the partition pisi<k makes diss,PBp1,pi<θ, then the partition pis may contain the actual k nearest neighbors of s. We need to include the partition pis in Rs and calculate the k nearest neighbors of s in the latest correlation partition. When a certain partition pisi<k appears where diss,PBp1,pi>θ, it means that the partition pis does not contain the actual k nearest neighbors of s, and we can discard the partition directly. Meanwhile, since the center point of the subsequent partitions are farther away from s, we also believe that they do not contain the actual k nearest neighbors of s, so we do not continue to make judgments and discard them directly. Finally, we can assume that the knnDisSet stores actual k nearest neighbors of s.Algorithm4 describes the specific details of vKNN. Before running the Map function, the program loads the center point data. In the Map function, call the selectTrainSet() function to calculate the k nearest neighbors. After the selectTrainSet() function finishes executing, knnSet[k] accepts the k nearest neighbor samples returned. Meanwhile, the Map function creates an array trainLable[k] to hold their labels. In the Reduce function, we use the HashMap to count the labels, find out the label that occur most frequently, and merge the output with s.In addition, we partition the data, and each partition is roughly the same size, aboutR/N. The provided partition selection policy limits the number of filtered partitions. This means that even in the worst case, each sample only needs to be compared with k×R/N training samples to find k nearest neighbors, which greatly reduces the computational effort of the entire KNN join process. Meanwhile, we use k-means clustering in the partitioning process, which makes the sample points in each partition highly similar, and the partition to be filtered is also the most similar to the test sample. Therefore, in theory, we can still maintain a high degree of accuracy while drastically reducing the number of training samples involved in the calculation.Figure 4
Partition selection. (a) DiscardΡj. (b) Reserve Ρj.
(a)(b)Algorithm 3: Relevant partition selecting.
Input:s, p1s,p2s,…,pksOutput:knnx,Rs,kselectTrainSet() {Rs=p1s; //determine the initial Relevant-partitionfori = 2 tokdoknnx,Rs,k; //calculate the k nearest neighbors of s in Rsθ=maxknns,Rs,k; //calculate the maximum distance from s to its k nearest neighbors from Rdiss,PBp1,pi=diss,pi2−diss,p12/2×disp1,pi; //calculate the distance from s to PBp1,pi,if (diss,PBp1,pi>θ) then//pιs does not belong to Rsbreak;else//pis belongs to RsRs=Rs∪pis; //add the partition pis to Rs}Algorithm 4: vKNN processing.
Input:p1s,p2s,…,pks, S, kOutput:s.predictLabel//prediction label of smap: <row number, s>knnSet[k] ← selectTrainSet(); //calculate the k nearest neighborsfori = 0 tokdo//fetch the k nearest neighbors labeltrainLable[i] = knnSet[i].label;forj = 0 tokdooutput(s, trainLabel[j]);reduce: <s, Labels>hmp = new HashMap(); //create a HashMap object hmpforeach label ∈ Labels do//count the number of each labelif hmp.get(label) ! = NULL then//if the label exists in hmplabel.value ++; //take the value of the label and add 1hmp.put(label) = label.value; //update the value of the label in hmpelse//if the label does not exist in hmphmp.put(label) = 1; //set the value of the label to 1 and insert to hmppredictLabel = hmp.maxvalue;//the label with the largest value as the prediction labeloutput(s, predictLabel);


## 5.1. KNN Join Processing

The basic idea of the KNN join algorithm based on MapReduce is in general agreement with that of the KNN algorithm. Firstly, the MapReduce program divides the input test dataset, each node calculates the distance between the test samples in the corresponding slice and each sample in the training dataset, finds out thek nearest neighbors, and selects the label with the largest proportion of these adjacent points. As shown in Algorithm 1, the KNN join processing can be formulated by the following.Algorithm 1: KNN join processing.
Input:k, R, SOutput:s.predictLabel//prediction label of smap: <row number, s>foreachr ∈ Rdodis = dis(r, s);//calculate the Euclidean distance between r and sfori = 0 tokdoif dis < distance[i] then//find the minimum k distancesdistance[i] = dis;trainLabel[i] = r.label;break;forj = 0 tokdooutput(s, trainLabel[i]);reduce: <s, Labels>hmp = new HashMap(); //create a HashMap object hmpforeach label ∈ Labels do//count the number of each labelif hmp.get(label) ! = NULL then//if the label exists in hmplabel.value ++; //take the value of the label and add 1hmp.put(label) = label.value;//update the value of the label in hmpelse//if the label does not exist in hmphmp.put(label) = 1; //set the value of the label to 1 and insert to hmppredictLabel = hmp.maxvalue; //the label with the largest value as the prediction labeloutput(s, predictLabel);In theMap function, it first sets the parameter k of the algorithm. Next, it calculates the Euclidean distance between each sample r of the training dataset R and the test sample s and stores the labels of the k nearest training samples into trainLabel. The form of the input data <key, value> is <row number, sample>; the form of the output data <key, values> is <sample, the label of adjacent sample>. The Hash function partitions, sorts, and groups these intermediate results by key values. The Reduce function then reads them. The design of the Reduce function is relatively simple. Its main task is to obtain the label with the maximum number of k labels and assign it to the test sample. First, iterate through the data passed by the Map function in turn, and if the current data exists in the HashMap, add 1 to its value. If the current data does not exist in the HashMap, mark its value as 1, and add it to the HashMap. Finally, the label with the largest value in HashMap is used as the prediction label. The form of input data <key, values> is <sample, set (the labels of adjacent k samples)>. The form of output data <key, value> is <sample, the prediction label>.Obviously, the method is too expensive. It simply assigns computing tasks to the computing nodes, and each mapper needs to connect a subset fromS to the entire dataset R. Considering only the distance calculation to be performed for each node, the amount of computation reaches R⋅S/n. When the dataset involved in the calculation is large, the amount of computation is still a huge amount in comparison. In addition, the comparison of similarity may exceed the computing capacity of the node, resulting in the task being killed. Therefore, it is unreliable to rely solely on MapReduce to slice computational tasks for efficient classification. A better idea is to reduce the number of samples in R that is involved in the calculation.


## 5.2. Relevant-Partition Selecting and vKNN Processing

We consider the Voronoi diagrams for KNN join within the MapReduce framework. The basic idea is to partition the data using the Voronoi diagram and clip the unqualified data to reduce the amount of calculation. There are three main steps as follows.


### 5.2.1. Preprocessing Step

Input datasetR and partition R using Voronoi diagram. First, randomly select N samples from the dataset R as the initial center point. Then, we use k-means clustering method to analyze the dataset globally to obtain the center points set P and the corresponding data clusters.


### 5.2.2. Nearest Center Point Selecting Step

In this step, we use the output of the above processes and datasetS as input objects. Find the k nearest center points of each sample in S and save its index and distance information to help clip the unqualified data.Algorithm2 shows the execution of the mappers at this stage. Before the program starts, we can load the preprocess center point data into the main memory of each mapper. After each mapper reads the sample object s, it traverses each center point and calculates the distance of s from all the center points. In order to reduce the cost of data transmission between nodes, we only save the index and distance information of the k center points nearest to s. We use TreeMap to store relevant information so that we can get the first k center points more quickly, where TreeMap itself is an ordered set of key values. All elements remain in a specific order and are sorted in ascending order by default by the value of the key. So, we can easily get information about the nearest k center points.Algorithm 2: Nearest center point selecting.
Input:S, POutput: <s, Ρ1s,Ρ2s,…,Ρks>//the partitions corresponding to the k nearest center points of sforeachs∈Sdotmp = TreeMap(); //create a TreeMap object tmpnearPointSet = [ ]; //create an empty set nearPointSetforeachp∈Pdo//insert the distance and index information tmp.put(dis(p, s), p.index);fori = 0 tokdo//read the first k center points’ information nearPointSet.append(tmp.next()); //put the information to nearPointSet


### 5.2.3. vKNN Processing Step

In this step, we use the output from the previous two processes as input. On the basis of the distance, we can filter out the relevant partitionRs corresponding to each sample s in S to find out the labels of the k nearest neighbors. Finally, reducer counts the labels and outputs the label that appears most often.In order to ensure the accuracy of the prediction results, the selected relevant partitionRs contains all knns,R,k as minimum as possible. How to determine the relevant partition Rs that will participate in the final calculation is an issue we need to consider now.Theorem 1.
Given a sample points and a center point pi, s is located in the partition pi corresponding to pi, and we have(13)pi∈Rs.Proof.
Ass is located in the partition pi, s is very similar to the sample points in pi. This means that the probability of having the nearest neighbors in pi is greater than that in any other partition. In other words, pi are more likely to have the nearest neighbors.
Based on the analysis of Hypothesis1, we may have errors in predicting using only the sample points within pi. We can design a selection strategy to determine whether other partitions meet the criteria.Definition 4.
Given two center pointspi and pj, PBpi,pj addresses the hyperplane dividing the subspace, where pi and pj are located, s is located in the partition pi corresponding to pi, and the maximum distance from s to it in its k nearest neighbor samples, denoted as θ, is(14)θ=maxknns,pi,k.Theorem 2.
Given two center pointspi and pj, PBpi,pj is the hyperplane dividing the subspace where pi and pj are located, s is in the partition pi corresponding to pi, and the necessary condition for pj∈Rs is(15)diss,PBpi,pj<θ.
Figure4(a) shows the case of diss,PBpi,pj>θ, and Figure 4(b) shows the case of diss,PBpi,pj<θ, respectively. When diss,PBpi,pj>θ, there is no intersection between the hypersphere with a radius of θ and the hyperplane PBpi,pj. That is to say, ∀x∈pj,diss,x>θ all hold. At this time, we can directly discard the partition pj. When diss,PBpi,pj<θ, the hypersphere with the radius of θ intersect with the hyperplane PBpi,pj. It means that there is probably a sample point x in the partition pj, making diss,x<θ, which also means that the calculated knns,Rs,k are not the real nearest neighbors. Therefore, we need to add the partition pj to the relevant partition Rs.In (Algorithm3), due to the center point information sorted using TreeMap before, according to Theorem 1, we can get that the initial relevant partition Rs is p1s. We calculated k nearest neighbors of s on Rs and saved them in knnDisSet. Next, we judge the subsequent partitions in sequence according to Theorem 2. If the partition pisi<k makes diss,PBp1,pi<θ, then the partition pis may contain the actual k nearest neighbors of s. We need to include the partition pis in Rs and calculate the k nearest neighbors of s in the latest correlation partition. When a certain partition pisi<k appears where diss,PBp1,pi>θ, it means that the partition pis does not contain the actual k nearest neighbors of s, and we can discard the partition directly. Meanwhile, since the center point of the subsequent partitions are farther away from s, we also believe that they do not contain the actual k nearest neighbors of s, so we do not continue to make judgments and discard them directly. Finally, we can assume that the knnDisSet stores actual k nearest neighbors of s.Algorithm4 describes the specific details of vKNN. Before running the Map function, the program loads the center point data. In the Map function, call the selectTrainSet() function to calculate the k nearest neighbors. After the selectTrainSet() function finishes executing, knnSet[k] accepts the k nearest neighbor samples returned. Meanwhile, the Map function creates an array trainLable[k] to hold their labels. In the Reduce function, we use the HashMap to count the labels, find out the label that occur most frequently, and merge the output with s.In addition, we partition the data, and each partition is roughly the same size, aboutR/N. The provided partition selection policy limits the number of filtered partitions. This means that even in the worst case, each sample only needs to be compared with k×R/N training samples to find k nearest neighbors, which greatly reduces the computational effort of the entire KNN join process. Meanwhile, we use k-means clustering in the partitioning process, which makes the sample points in each partition highly similar, and the partition to be filtered is also the most similar to the test sample. Therefore, in theory, we can still maintain a high degree of accuracy while drastically reducing the number of training samples involved in the calculation.Figure 4
Partition selection. (a) DiscardΡj. (b) Reserve Ρj.
(a)(b)Algorithm 3: Relevant partition selecting.
Input:s, p1s,p2s,…,pksOutput:knnx,Rs,kselectTrainSet() {Rs=p1s; //determine the initial Relevant-partitionfori = 2 tokdoknnx,Rs,k; //calculate the k nearest neighbors of s in Rsθ=maxknns,Rs,k; //calculate the maximum distance from s to its k nearest neighbors from Rdiss,PBp1,pi=diss,pi2−diss,p12/2×disp1,pi; //calculate the distance from s to PBp1,pi,if (diss,PBp1,pi>θ) then//pιs does not belong to Rsbreak;else//pis belongs to RsRs=Rs∪pis; //add the partition pis to Rs}Algorithm 4: vKNN processing.
Input:p1s,p2s,…,pks, S, kOutput:s.predictLabel//prediction label of smap: <row number, s>knnSet[k] ← selectTrainSet(); //calculate the k nearest neighborsfori = 0 tokdo//fetch the k nearest neighbors labeltrainLable[i] = knnSet[i].label;forj = 0 tokdooutput(s, trainLabel[j]);reduce: <s, Labels>hmp = new HashMap(); //create a HashMap object hmpforeach label ∈ Labels do//count the number of each labelif hmp.get(label) ! = NULL then//if the label exists in hmplabel.value ++; //take the value of the label and add 1hmp.put(label) = label.value; //update the value of the label in hmpelse//if the label does not exist in hmphmp.put(label) = 1; //set the value of the label to 1 and insert to hmppredictLabel = hmp.maxvalue;//the label with the largest value as the prediction labeloutput(s, predictLabel);


## 5.2.1. Preprocessing Step

Input datasetR and partition R using Voronoi diagram. First, randomly select N samples from the dataset R as the initial center point. Then, we use k-means clustering method to analyze the dataset globally to obtain the center points set P and the corresponding data clusters.


## 5.2.2. Nearest Center Point Selecting Step

In this step, we use the output of the above processes and datasetS as input objects. Find the k nearest center points of each sample in S and save its index and distance information to help clip the unqualified data.Algorithm2 shows the execution of the mappers at this stage. Before the program starts, we can load the preprocess center point data into the main memory of each mapper. After each mapper reads the sample object s, it traverses each center point and calculates the distance of s from all the center points. In order to reduce the cost of data transmission between nodes, we only save the index and distance information of the k center points nearest to s. We use TreeMap to store relevant information so that we can get the first k center points more quickly, where TreeMap itself is an ordered set of key values. All elements remain in a specific order and are sorted in ascending order by default by the value of the key. So, we can easily get information about the nearest k center points.Algorithm 2: Nearest center point selecting.
Input:S, POutput: <s, Ρ1s,Ρ2s,…,Ρks>//the partitions corresponding to the k nearest center points of sforeachs∈Sdotmp = TreeMap(); //create a TreeMap object tmpnearPointSet = [ ]; //create an empty set nearPointSetforeachp∈Pdo//insert the distance and index information tmp.put(dis(p, s), p.index);fori = 0 tokdo//read the first k center points’ information nearPointSet.append(tmp.next()); //put the information to nearPointSet


## 5.2.3. vKNN Processing Step

In this step, we use the output from the previous two processes as input. On the basis of the distance, we can filter out the relevant partitionRs corresponding to each sample s in S to find out the labels of the k nearest neighbors. Finally, reducer counts the labels and outputs the label that appears most often.In order to ensure the accuracy of the prediction results, the selected relevant partitionRs contains all knns,R,k as minimum as possible. How to determine the relevant partition Rs that will participate in the final calculation is an issue we need to consider now.Theorem 1.
Given a sample points and a center point pi, s is located in the partition pi corresponding to pi, and we have(13)pi∈Rs.Proof.
Ass is located in the partition pi, s is very similar to the sample points in pi. This means that the probability of having the nearest neighbors in pi is greater than that in any other partition. In other words, pi are more likely to have the nearest neighbors.
Based on the analysis of Hypothesis1, we may have errors in predicting using only the sample points within pi. We can design a selection strategy to determine whether other partitions meet the criteria.Definition 4.
Given two center pointspi and pj, PBpi,pj addresses the hyperplane dividing the subspace, where pi and pj are located, s is located in the partition pi corresponding to pi, and the maximum distance from s to it in its k nearest neighbor samples, denoted as θ, is(14)θ=maxknns,pi,k.Theorem 2.
Given two center pointspi and pj, PBpi,pj is the hyperplane dividing the subspace where pi and pj are located, s is in the partition pi corresponding to pi, and the necessary condition for pj∈Rs is(15)diss,PBpi,pj<θ.
Figure4(a) shows the case of diss,PBpi,pj>θ, and Figure 4(b) shows the case of diss,PBpi,pj<θ, respectively. When diss,PBpi,pj>θ, there is no intersection between the hypersphere with a radius of θ and the hyperplane PBpi,pj. That is to say, ∀x∈pj,diss,x>θ all hold. At this time, we can directly discard the partition pj. When diss,PBpi,pj<θ, the hypersphere with the radius of θ intersect with the hyperplane PBpi,pj. It means that there is probably a sample point x in the partition pj, making diss,x<θ, which also means that the calculated knns,Rs,k are not the real nearest neighbors. Therefore, we need to add the partition pj to the relevant partition Rs.In (Algorithm3), due to the center point information sorted using TreeMap before, according to Theorem 1, we can get that the initial relevant partition Rs is p1s. We calculated k nearest neighbors of s on Rs and saved them in knnDisSet. Next, we judge the subsequent partitions in sequence according to Theorem 2. If the partition pisi<k makes diss,PBp1,pi<θ, then the partition pis may contain the actual k nearest neighbors of s. We need to include the partition pis in Rs and calculate the k nearest neighbors of s in the latest correlation partition. When a certain partition pisi<k appears where diss,PBp1,pi>θ, it means that the partition pis does not contain the actual k nearest neighbors of s, and we can discard the partition directly. Meanwhile, since the center point of the subsequent partitions are farther away from s, we also believe that they do not contain the actual k nearest neighbors of s, so we do not continue to make judgments and discard them directly. Finally, we can assume that the knnDisSet stores actual k nearest neighbors of s.Algorithm4 describes the specific details of vKNN. Before running the Map function, the program loads the center point data. In the Map function, call the selectTrainSet() function to calculate the k nearest neighbors. After the selectTrainSet() function finishes executing, knnSet[k] accepts the k nearest neighbor samples returned. Meanwhile, the Map function creates an array trainLable[k] to hold their labels. In the Reduce function, we use the HashMap to count the labels, find out the label that occur most frequently, and merge the output with s.In addition, we partition the data, and each partition is roughly the same size, aboutR/N. The provided partition selection policy limits the number of filtered partitions. This means that even in the worst case, each sample only needs to be compared with k×R/N training samples to find k nearest neighbors, which greatly reduces the computational effort of the entire KNN join process. Meanwhile, we use k-means clustering in the partitioning process, which makes the sample points in each partition highly similar, and the partition to be filtered is also the most similar to the test sample. Therefore, in theory, we can still maintain a high degree of accuracy while drastically reducing the number of training samples involved in the calculation.Figure 4
Partition selection. (a) DiscardΡj. (b) Reserve Ρj.
(a)(b)Algorithm 3: Relevant partition selecting.
Input:s, p1s,p2s,…,pksOutput:knnx,Rs,kselectTrainSet() {Rs=p1s; //determine the initial Relevant-partitionfori = 2 tokdoknnx,Rs,k; //calculate the k nearest neighbors of s in Rsθ=maxknns,Rs,k; //calculate the maximum distance from s to its k nearest neighbors from Rdiss,PBp1,pi=diss,pi2−diss,p12/2×disp1,pi; //calculate the distance from s to PBp1,pi,if (diss,PBp1,pi>θ) then//pιs does not belong to Rsbreak;else//pis belongs to RsRs=Rs∪pis; //add the partition pis to Rs}Algorithm 4: vKNN processing.
Input:p1s,p2s,…,pks, S, kOutput:s.predictLabel//prediction label of smap: <row number, s>knnSet[k] ← selectTrainSet(); //calculate the k nearest neighborsfori = 0 tokdo//fetch the k nearest neighbors labeltrainLable[i] = knnSet[i].label;forj = 0 tokdooutput(s, trainLabel[j]);reduce: <s, Labels>hmp = new HashMap(); //create a HashMap object hmpforeach label ∈ Labels do//count the number of each labelif hmp.get(label) ! = NULL then//if the label exists in hmplabel.value ++; //take the value of the label and add 1hmp.put(label) = label.value; //update the value of the label in hmpelse//if the label does not exist in hmphmp.put(label) = 1; //set the value of the label to 1 and insert to hmppredictLabel = hmp.maxvalue;//the label with the largest value as the prediction labeloutput(s, predictLabel);


## 6. Experimental Evaluation

### 6.1. Experimental Environment and Dataset

The experimental platform used to evaluate the performance of the proposed algorithm is mainly configured as Intel (R) Core (TM) i5-8300H 2.3 GHz processor, 16G memory, and 500G NVMe hard disk. The Hadoop cluster consists of six virtual machines, and each allocating 2G of memory and 40G of hard disk. On each node, we install CentOS 8.0 operating system with Java 1.8.0 and Hadoop 2.10.0. We select one of them as the Master node and the other five as Slave node to be managed through VMware® Workstation 15 Pro. The development environment used in the experiment is Eclipse-jee-2019-12-R-linux-gtk-x86_64.The experimental data in this study uses the Forest CoverType dataset, a standard dataset in the UCI database. The dataset has 581012 records, each comprising 54-dimensional features (10 quantitative variables, 4 binary wilderness areas, and 40 binary soil type variables) and seven labels: Spruce/Fir, Lodgepole Pine, Ponderosa Pine, Cottonwood/Willow, Aspen, Douglas-fir, and Krummholz, each represented by a number from 1–7. For simplicity, we first randomly selected 200,000 data from the dataset as the train setR, and the rest as the test set S.Consequently, we evaluated the methods mentioned in our experiments. For the two methods described in Section5, we set the number of reducers to 1 by default.


### 6.2. Experimental Evaluation Indicators

There are many available evaluation indicators in the classification tasks of machine learning. The two most common types are accuracy and error rate. For a given test setS, the classification error rate is defined as(16)ES=1S∑s∈SIstype≠snew_type,and the accuracy is defined as(17)AccS=1−ES=1S∑s∈SIstype=snew_type,where stype refers to the actual label of s and snew_type refers to the label of s predicted by the model.In addition, we will evaluate the proposed method in terms of the elapsed time and the acceleration ratio. The elapsed time involves the global time for the MapReduce program to run. The acceleration ratio is the ratio of the elapsed time of the original version to the improved version of the relevant parameters:(18)Speedup=original_timeimproved_time,where original_time is the time when the original version runs and improved_time is the time when the program runs after the improved parameter.


### 6.3. Evaluation of Experimental Results

#### 6.3.1. Effect of Different Center Point Sizes

For our first experiment, we analyze the effect of the number of center points on the performance of vKNN. To further illustrate the situation, the number of mappers is given (n = 4). Then, we randomly select 600, 800, 1000, 1200, and 1400 pieces of data from the training set in 5 times as the initial center point set. In Figure 5(a), the execution time of Algorithm 2 increases approximately linearly as N increases. This is because when finding the nearest k center points for each element of S, the distance between each element and each center point needs to be calculated with a time complexity of OS×N. When the center point increases linearly, it means that the computation time will also increase linearly. We adopt the TreeMap for sorting distances, so we need not spend any extra time on it. In Figure 5(b), we can see that vKNN execution time decreases as N increases. This is because, as the number of center points increases, the training set is divided more finely when dividing the partitions, i.e., the number of distance calculations needed for each sample decreases accordingly. Also, we notice that the actual reduction is getting smaller as the center point increases. This is because when the partitions are divided more finely, the probability of occurrence of the scenario shown in Figure 4(b) increases as the set of corresponding partitions is determined. It is shown that some of the samples correspond to a larger set of partitions than before, which also results in increased computation time. Figure 5(c) shows how the accuracy of the algorithm varies from 600 to 1400. As N increases, there is little change in accuracy, which also indicates that the choice of N has no effect on the accuracy of the proposed method.Figure 5
Effect of different center point sizes. (a) Running time of nearest center point selecting. (b) Running time of vKNN. (c) Accuracy of vKNN.
(a)(b)(c)


#### 6.3.2. Effect of the Number of Nearest Neighbors

Next, we study the effect ofk on the performance of the two algorithms. Similarly, given the number of mappers (n = 4). Figure 6 shows the experimental results of k increasing from 3 to 20 gradually.Figure 6
Effect of the number of nearest neighbors. (a) Running time of nearest center point selecting. (b) Running time of vKNN. (c) Accuracy of vKNN. (d) Running time comparison of different methods.
(a)(b)(c)(d)Figures6(a) and 6(b) address the operation of two programs of vKNN algorithm. The running time of the algorithm increases approximately with the increase of k value, which means that the vKNN algorithm is not sensitive to the change of the k value. In Algorithm 2, the effect of k is mainly reflected in the following aspects: we added relevant information about k nearest center points of each sample to the training set. The greater the k value, the more information will be added, and the communication cost of data will increase. When the vKNN algorithm is run at the end, the communication cost of the file on HDFS increases accordingly. The increase of k value shows that θ will be larger in the same sample, and the hypersphere with this radius will be larger and easier to cross the hyperplane, as shown in Figure 4(b). At this point, more sample points will be added to the relevant partition, resulting in more computational effort.Figure6(c) shows the change of vKNN algorithm accuracy in the process of k increasing from 3 to 20. When k goes from 3 to 5, the algorithm accuracy is improved. As k continued to increase, the accuracy begins to decrease slightly. Explain that the k value of vKNN algorithm is not as large as possible, and we need to select the appropriate k value for the specific situation.Figure6(d) shows that vKNN performs better than KNN for the results of the two methods. The execution time of KNN increases linearly with the increase of k. The influence of k value on KNN algorithm is mainly reflected in the selection of the nearest k sample points. However, because the KNN algorithm itself is too computationally intensive when dealing with large datasets, the increase in the amount of computations caused by the increase in k value is less obvious than the amount of computations itself.


#### 6.3.3. Effect of Speedup

Now, we measure the effect of the number of mappers. Given the number of center points (N = 1000) and the number of nearest neighbors (k = 5), Figure 7 shows the running time and acceleration ratio of vKNN as the number of mappers gradually increases from 1 to 4. Figure 7(a) shows that the run time decreases as the number of mappers increases. However, the scale is shrinking. This is also reflected in Figure 7(b), where the acceleration ratio gradually stabilizes. It is because increasing the number of mappers means increasing the number of nodes participating in the operation, at which point the amount of computation allocated to each node will also be reduced, so the execution time of the map function on each node will be shortened accordingly. With the increasing number of nodes, the computing resources are larger than the required resources actually needed. At this time, the computing resources are not fully utilized, the execution time decreases less and less obvious, and cluster computing resources are also wasted. On the contrary, each node needs to read files from the HDFS, which increases the corresponding communication costs. When the acceleration effect of computing node growth is insufficient to offset the pressure of increased communication costs, the algorithm execution efficiency will decrease. Therefore, the number of mappers needs to be selected appropriately when facing datasets of different sizes.Figure 7
(a) Effect of the number of mappers. (b) Speedup.
(a)(b)


#### 6.3.4. Effect of Accuracy

Now, we study the accuracy of the two algorithms. Given the number of center points (N = 1000) and the number of nearest neighbors (k = 5). Figure 8 shows the accuracy of the two methods when k is gradually increased from 3 to 20. In general, vKNN is more accurate than KNN. This is because the partitioned data we use is the result of k-means clustering during the preprocessing phase, in which samples with the same characteristics are divided into the same cluster, i.e., the data in the same partition are similar to each other. Subsequently, we search for the center point closest to the sample to be classified. Distance closest represents less difference and more similarity. Therefore, when the vKNN algorithm is executed and compared with other samples, the higher the degree of similarity with the sample to be classified, the smaller the calculation error will be.Figure 8
Accuracy of different methods.


#### 6.3.5. Effect of Scalability

In this section, we randomly extract 100,000 pieces of data from the original dataset as a training set. Then, randomly extract 100,000, 200,000, 300,000, and 400,000 pieces of data from the rest of the set four times as a test set. Based on this, the scalability of the proposed method is evaluated. In the experiment, given the number of center points (N = 1000) and the number of nearest neighbors (k = 5). Figure 9(a) is the time-consuming result of Algorithm 2, and Figure 9(b) is the time-consuming result of two methods for performing KNN join.Figure 9
Scalability. (a) Running time of nearest center point selecting. (b) Running time of different methods.
(a)(b)On the whole, the execution time of both methods increases approximately linearly with the increase of training set data size. The reason is that, as the data size increases, the data allocated to each computing node will also increase proportionally, and the computing time of each node will increase in the same proportion. However, when using vKNN algorithm to perform KNN join, the growth is gentler. The average growth time of Algorithm2 is only 5 seconds, which indicates that the scalability of vKNN algorithm is better than that of KNN algorithm. The time complexity of KNN algorithm is OR⋅S, which is a Cartesian product of the sample set R and S. Obviously, no matter which sample set is increased in size, the effect is enormous. However, vKNN algorithm adopts the idea of partition which alleviates the computational changes caused by the increase of S. Therefore, as the dataset increases, the difference in execution time between the two methods increases.


## 6.1. Experimental Environment and Dataset

The experimental platform used to evaluate the performance of the proposed algorithm is mainly configured as Intel (R) Core (TM) i5-8300H 2.3 GHz processor, 16G memory, and 500G NVMe hard disk. The Hadoop cluster consists of six virtual machines, and each allocating 2G of memory and 40G of hard disk. On each node, we install CentOS 8.0 operating system with Java 1.8.0 and Hadoop 2.10.0. We select one of them as the Master node and the other five as Slave node to be managed through VMware® Workstation 15 Pro. The development environment used in the experiment is Eclipse-jee-2019-12-R-linux-gtk-x86_64.The experimental data in this study uses the Forest CoverType dataset, a standard dataset in the UCI database. The dataset has 581012 records, each comprising 54-dimensional features (10 quantitative variables, 4 binary wilderness areas, and 40 binary soil type variables) and seven labels: Spruce/Fir, Lodgepole Pine, Ponderosa Pine, Cottonwood/Willow, Aspen, Douglas-fir, and Krummholz, each represented by a number from 1–7. For simplicity, we first randomly selected 200,000 data from the dataset as the train setR, and the rest as the test set S.Consequently, we evaluated the methods mentioned in our experiments. For the two methods described in Section5, we set the number of reducers to 1 by default.


## 6.2. Experimental Evaluation Indicators

There are many available evaluation indicators in the classification tasks of machine learning. The two most common types are accuracy and error rate. For a given test setS, the classification error rate is defined as(16)ES=1S∑s∈SIstype≠snew_type,and the accuracy is defined as(17)AccS=1−ES=1S∑s∈SIstype=snew_type,where stype refers to the actual label of s and snew_type refers to the label of s predicted by the model.In addition, we will evaluate the proposed method in terms of the elapsed time and the acceleration ratio. The elapsed time involves the global time for the MapReduce program to run. The acceleration ratio is the ratio of the elapsed time of the original version to the improved version of the relevant parameters:(18)Speedup=original_timeimproved_time,where original_time is the time when the original version runs and improved_time is the time when the program runs after the improved parameter.


## 6.3. Evaluation of Experimental Results

### 6.3.1. Effect of Different Center Point Sizes

For our first experiment, we analyze the effect of the number of center points on the performance of vKNN. To further illustrate the situation, the number of mappers is given (n = 4). Then, we randomly select 600, 800, 1000, 1200, and 1400 pieces of data from the training set in 5 times as the initial center point set. In Figure 5(a), the execution time of Algorithm 2 increases approximately linearly as N increases. This is because when finding the nearest k center points for each element of S, the distance between each element and each center point needs to be calculated with a time complexity of OS×N. When the center point increases linearly, it means that the computation time will also increase linearly. We adopt the TreeMap for sorting distances, so we need not spend any extra time on it. In Figure 5(b), we can see that vKNN execution time decreases as N increases. This is because, as the number of center points increases, the training set is divided more finely when dividing the partitions, i.e., the number of distance calculations needed for each sample decreases accordingly. Also, we notice that the actual reduction is getting smaller as the center point increases. This is because when the partitions are divided more finely, the probability of occurrence of the scenario shown in Figure 4(b) increases as the set of corresponding partitions is determined. It is shown that some of the samples correspond to a larger set of partitions than before, which also results in increased computation time. Figure 5(c) shows how the accuracy of the algorithm varies from 600 to 1400. As N increases, there is little change in accuracy, which also indicates that the choice of N has no effect on the accuracy of the proposed method.Figure 5
Effect of different center point sizes. (a) Running time of nearest center point selecting. (b) Running time of vKNN. (c) Accuracy of vKNN.
(a)(b)(c)


### 6.3.2. Effect of the Number of Nearest Neighbors

Next, we study the effect ofk on the performance of the two algorithms. Similarly, given the number of mappers (n = 4). Figure 6 shows the experimental results of k increasing from 3 to 20 gradually.Figure 6
Effect of the number of nearest neighbors. (a) Running time of nearest center point selecting. (b) Running time of vKNN. (c) Accuracy of vKNN. (d) Running time comparison of different methods.
(a)(b)(c)(d)Figures6(a) and 6(b) address the operation of two programs of vKNN algorithm. The running time of the algorithm increases approximately with the increase of k value, which means that the vKNN algorithm is not sensitive to the change of the k value. In Algorithm 2, the effect of k is mainly reflected in the following aspects: we added relevant information about k nearest center points of each sample to the training set. The greater the k value, the more information will be added, and the communication cost of data will increase. When the vKNN algorithm is run at the end, the communication cost of the file on HDFS increases accordingly. The increase of k value shows that θ will be larger in the same sample, and the hypersphere with this radius will be larger and easier to cross the hyperplane, as shown in Figure 4(b). At this point, more sample points will be added to the relevant partition, resulting in more computational effort.Figure6(c) shows the change of vKNN algorithm accuracy in the process of k increasing from 3 to 20. When k goes from 3 to 5, the algorithm accuracy is improved. As k continued to increase, the accuracy begins to decrease slightly. Explain that the k value of vKNN algorithm is not as large as possible, and we need to select the appropriate k value for the specific situation.Figure6(d) shows that vKNN performs better than KNN for the results of the two methods. The execution time of KNN increases linearly with the increase of k. The influence of k value on KNN algorithm is mainly reflected in the selection of the nearest k sample points. However, because the KNN algorithm itself is too computationally intensive when dealing with large datasets, the increase in the amount of computations caused by the increase in k value is less obvious than the amount of computations itself.


### 6.3.3. Effect of Speedup

Now, we measure the effect of the number of mappers. Given the number of center points (N = 1000) and the number of nearest neighbors (k = 5), Figure 7 shows the running time and acceleration ratio of vKNN as the number of mappers gradually increases from 1 to 4. Figure 7(a) shows that the run time decreases as the number of mappers increases. However, the scale is shrinking. This is also reflected in Figure 7(b), where the acceleration ratio gradually stabilizes. It is because increasing the number of mappers means increasing the number of nodes participating in the operation, at which point the amount of computation allocated to each node will also be reduced, so the execution time of the map function on each node will be shortened accordingly. With the increasing number of nodes, the computing resources are larger than the required resources actually needed. At this time, the computing resources are not fully utilized, the execution time decreases less and less obvious, and cluster computing resources are also wasted. On the contrary, each node needs to read files from the HDFS, which increases the corresponding communication costs. When the acceleration effect of computing node growth is insufficient to offset the pressure of increased communication costs, the algorithm execution efficiency will decrease. Therefore, the number of mappers needs to be selected appropriately when facing datasets of different sizes.Figure 7
(a) Effect of the number of mappers. (b) Speedup.
(a)(b)


### 6.3.4. Effect of Accuracy

Now, we study the accuracy of the two algorithms. Given the number of center points (N = 1000) and the number of nearest neighbors (k = 5). Figure 8 shows the accuracy of the two methods when k is gradually increased from 3 to 20. In general, vKNN is more accurate than KNN. This is because the partitioned data we use is the result of k-means clustering during the preprocessing phase, in which samples with the same characteristics are divided into the same cluster, i.e., the data in the same partition are similar to each other. Subsequently, we search for the center point closest to the sample to be classified. Distance closest represents less difference and more similarity. Therefore, when the vKNN algorithm is executed and compared with other samples, the higher the degree of similarity with the sample to be classified, the smaller the calculation error will be.Figure 8
Accuracy of different methods.


### 6.3.5. Effect of Scalability

In this section, we randomly extract 100,000 pieces of data from the original dataset as a training set. Then, randomly extract 100,000, 200,000, 300,000, and 400,000 pieces of data from the rest of the set four times as a test set. Based on this, the scalability of the proposed method is evaluated. In the experiment, given the number of center points (N = 1000) and the number of nearest neighbors (k = 5). Figure 9(a) is the time-consuming result of Algorithm 2, and Figure 9(b) is the time-consuming result of two methods for performing KNN join.Figure 9
Scalability. (a) Running time of nearest center point selecting. (b) Running time of different methods.
(a)(b)On the whole, the execution time of both methods increases approximately linearly with the increase of training set data size. The reason is that, as the data size increases, the data allocated to each computing node will also increase proportionally, and the computing time of each node will increase in the same proportion. However, when using vKNN algorithm to perform KNN join, the growth is gentler. The average growth time of Algorithm2 is only 5 seconds, which indicates that the scalability of vKNN algorithm is better than that of KNN algorithm. The time complexity of KNN algorithm is OR⋅S, which is a Cartesian product of the sample set R and S. Obviously, no matter which sample set is increased in size, the effect is enormous. However, vKNN algorithm adopts the idea of partition which alleviates the computational changes caused by the increase of S. Therefore, as the dataset increases, the difference in execution time between the two methods increases.


## 6.3.1. Effect of Different Center Point Sizes

For our first experiment, we analyze the effect of the number of center points on the performance of vKNN. To further illustrate the situation, the number of mappers is given (n = 4). Then, we randomly select 600, 800, 1000, 1200, and 1400 pieces of data from the training set in 5 times as the initial center point set. In Figure 5(a), the execution time of Algorithm 2 increases approximately linearly as N increases. This is because when finding the nearest k center points for each element of S, the distance between each element and each center point needs to be calculated with a time complexity of OS×N. When the center point increases linearly, it means that the computation time will also increase linearly. We adopt the TreeMap for sorting distances, so we need not spend any extra time on it. In Figure 5(b), we can see that vKNN execution time decreases as N increases. This is because, as the number of center points increases, the training set is divided more finely when dividing the partitions, i.e., the number of distance calculations needed for each sample decreases accordingly. Also, we notice that the actual reduction is getting smaller as the center point increases. This is because when the partitions are divided more finely, the probability of occurrence of the scenario shown in Figure 4(b) increases as the set of corresponding partitions is determined. It is shown that some of the samples correspond to a larger set of partitions than before, which also results in increased computation time. Figure 5(c) shows how the accuracy of the algorithm varies from 600 to 1400. As N increases, there is little change in accuracy, which also indicates that the choice of N has no effect on the accuracy of the proposed method.Figure 5
Effect of different center point sizes. (a) Running time of nearest center point selecting. (b) Running time of vKNN. (c) Accuracy of vKNN.
(a)(b)(c)


## 6.3.2. Effect of the Number of Nearest Neighbors

Next, we study the effect ofk on the performance of the two algorithms. Similarly, given the number of mappers (n = 4). Figure 6 shows the experimental results of k increasing from 3 to 20 gradually.Figure 6
Effect of the number of nearest neighbors. (a) Running time of nearest center point selecting. (b) Running time of vKNN. (c) Accuracy of vKNN. (d) Running time comparison of different methods.
(a)(b)(c)(d)Figures6(a) and 6(b) address the operation of two programs of vKNN algorithm. The running time of the algorithm increases approximately with the increase of k value, which means that the vKNN algorithm is not sensitive to the change of the k value. In Algorithm 2, the effect of k is mainly reflected in the following aspects: we added relevant information about k nearest center points of each sample to the training set. The greater the k value, the more information will be added, and the communication cost of data will increase. When the vKNN algorithm is run at the end, the communication cost of the file on HDFS increases accordingly. The increase of k value shows that θ will be larger in the same sample, and the hypersphere with this radius will be larger and easier to cross the hyperplane, as shown in Figure 4(b). At this point, more sample points will be added to the relevant partition, resulting in more computational effort.Figure6(c) shows the change of vKNN algorithm accuracy in the process of k increasing from 3 to 20. When k goes from 3 to 5, the algorithm accuracy is improved. As k continued to increase, the accuracy begins to decrease slightly. Explain that the k value of vKNN algorithm is not as large as possible, and we need to select the appropriate k value for the specific situation.Figure6(d) shows that vKNN performs better than KNN for the results of the two methods. The execution time of KNN increases linearly with the increase of k. The influence of k value on KNN algorithm is mainly reflected in the selection of the nearest k sample points. However, because the KNN algorithm itself is too computationally intensive when dealing with large datasets, the increase in the amount of computations caused by the increase in k value is less obvious than the amount of computations itself.


## 6.3.3. Effect of Speedup

Now, we measure the effect of the number of mappers. Given the number of center points (N = 1000) and the number of nearest neighbors (k = 5), Figure 7 shows the running time and acceleration ratio of vKNN as the number of mappers gradually increases from 1 to 4. Figure 7(a) shows that the run time decreases as the number of mappers increases. However, the scale is shrinking. This is also reflected in Figure 7(b), where the acceleration ratio gradually stabilizes. It is because increasing the number of mappers means increasing the number of nodes participating in the operation, at which point the amount of computation allocated to each node will also be reduced, so the execution time of the map function on each node will be shortened accordingly. With the increasing number of nodes, the computing resources are larger than the required resources actually needed. At this time, the computing resources are not fully utilized, the execution time decreases less and less obvious, and cluster computing resources are also wasted. On the contrary, each node needs to read files from the HDFS, which increases the corresponding communication costs. When the acceleration effect of computing node growth is insufficient to offset the pressure of increased communication costs, the algorithm execution efficiency will decrease. Therefore, the number of mappers needs to be selected appropriately when facing datasets of different sizes.Figure 7
(a) Effect of the number of mappers. (b) Speedup.
(a)(b)


## 6.3.4. Effect of Accuracy

Now, we study the accuracy of the two algorithms. Given the number of center points (N = 1000) and the number of nearest neighbors (k = 5). Figure 8 shows the accuracy of the two methods when k is gradually increased from 3 to 20. In general, vKNN is more accurate than KNN. This is because the partitioned data we use is the result of k-means clustering during the preprocessing phase, in which samples with the same characteristics are divided into the same cluster, i.e., the data in the same partition are similar to each other. Subsequently, we search for the center point closest to the sample to be classified. Distance closest represents less difference and more similarity. Therefore, when the vKNN algorithm is executed and compared with other samples, the higher the degree of similarity with the sample to be classified, the smaller the calculation error will be.Figure 8
Accuracy of different methods.


## 6.3.5. Effect of Scalability

In this section, we randomly extract 100,000 pieces of data from the original dataset as a training set. Then, randomly extract 100,000, 200,000, 300,000, and 400,000 pieces of data from the rest of the set four times as a test set. Based on this, the scalability of the proposed method is evaluated. In the experiment, given the number of center points (N = 1000) and the number of nearest neighbors (k = 5). Figure 9(a) is the time-consuming result of Algorithm 2, and Figure 9(b) is the time-consuming result of two methods for performing KNN join.Figure 9
Scalability. (a) Running time of nearest center point selecting. (b) Running time of different methods.
(a)(b)On the whole, the execution time of both methods increases approximately linearly with the increase of training set data size. The reason is that, as the data size increases, the data allocated to each computing node will also increase proportionally, and the computing time of each node will increase in the same proportion. However, when using vKNN algorithm to perform KNN join, the growth is gentler. The average growth time of Algorithm2 is only 5 seconds, which indicates that the scalability of vKNN algorithm is better than that of KNN algorithm. The time complexity of KNN algorithm is OR⋅S, which is a Cartesian product of the sample set R and S. Obviously, no matter which sample set is increased in size, the effect is enormous. However, vKNN algorithm adopts the idea of partition which alleviates the computational changes caused by the increase of S. Therefore, as the dataset increases, the difference in execution time between the two methods increases.


## 7. Conclusion

In this paper, we propose a vKNN algorithm based on Voronoi diagram concerning MapReduce-based KNN join scheme. Our algorithm can partition the training set using the idea of Voronoi diagram. Then, we design a partition selection strategy to find the optimal relevant partition for the sample to be tested, which effectively avoids the enormous amount of computation caused by irrelevant data. This strategy takes full advantage of the parallel processing capabilities of the MapReduce framework and is suitable for large-scale data. A large number of experiments based on real datasets show that our proposed algorithm can accelerate the calculation with good scalability while ensuring accuracy.



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*Source: 7898230-2020-09-08.xml*