PCMIgr: a fast packet classification method based on information gain ratio

To solve the problem of ambiguous attribute selection in existing decision tree classification algorithms, a decision tree construction method based on information entropy, PCMIgr, is proposed. PCMIgr is a heuristic method based on greedy strategy. At each decision tree node, when it is necessary to select classification attributes for division, the attribute with the highest information gain ratio is selected. The main innovation of this method is that the attribute selection in the traditional classification method based on decision tree is optimized, and the classification efficiency of the constructed decision tree is improved compared with that before optimization. At the same time, the decision tree ensures that each leaf node is only associated with one rule, which avoids the common problem of "rule replication" in the process of traditional decision tree construction, and effectively saves memory and calculation time. The experimental results show that the application of this method to the construction of classification decision tree can further improve the efficiency of packet classification method based on decision tree, and can be applied to high-speed real-time packet classification.


Introduction
With the development of a series of cutting-edge network technologies, such as network function virtualization (NFV) and software-defined network (SDN), the Internet carries more and more application services, the scale of network traffic, backbone network routing table and firewall access control rules have increased 1 3 PCMIgr: a fast packet classification method based on information… explosively, which puts forward higher requirements for packet classification and processing capacity of network devices.
Packet classification is the core technology to Internet devices and network services implementation. According to a series of information carried by the specified packet (such as source address, destination address, source port, destination port and protocol, etc.), it searches for the operation or task to be performed by the packet in a set of rules based on the principle of highest priority matching [1]. The rules in the classifier are generally expressed in the form of prefix or address range, and the two forms are semantically equivalent. In order to represent the structure of decision tree more intuitively, the classification rules are expressed in the form of address range in this paper, as shown in Fig. 1.
According to the basic principle of packet classification technology, it is well known that the most intuitive packet classification method is sequence matching. However, the time complexity of sequence matching algorithm is linearly related to the number of rules. With the increase in the scale and dimension of classification rules, sequence matching takes more time, which leads to the decrease in packet classification efficiency and becomes the bottleneck of network performance. Fortunately, the rules in the actual classifier have some inherent characteristics, which can be used to reduce the complexity of packet classification. Under this background, researchers have proposed many excellent packet classification algorithms, including the method based on dimension decomposition, the method based on tuple space search and the method based on decision tree, etc. [2].
Taking HiCuts [3], a method based on decision tree, as an example. It recursively divides the rule search space into multiple subspaces with equal size by using the equal-scale cutting method, until the number of rules in each subspace is less than the predefined threshold τ (τ is equal to 6 in the following example). According to HiCuts method, it first maps all rules to a two-dimensional space according to the address range. As shown in Fig. 2a, the area r 1 marked with backslashes is one rule mapping area, while areas with multiple mappings are marked with cross lines, such as areas r 1 ˄r 5 , r 4 ˄r 6 ˄r 9 , et al.
After all the rules are mapped to the two-dimensional space, the decision tree can be constructed. First, divide all rule spaces into two subspaces on the F 1 dimension equally. In this example, the address range of each dimension is [0,9], so the address ranges of these two subspaces are [0,4] and [5,9], respectively. Then a first-level decision tree with two leaf nodes can be constructed from the root node. However, according to the idea of HiCuts method, it is required that the number of rules associated with each leaf node should not exceed the specified threshold τ. Therefore, on the basis of the first-level decision tree division, it is necessary to continue to divide the subspace equally in the F 2 dimension, as shown in Fig. 2b, the subspace F 1 ∈ [0, 4] F 2 ∈ [0, 4] is only associated with three rules r 3 、r 5 and r 9 , which meet the threshold requirements. Continue to divide the remaining leaf nodes according to the same process until each leaf node meets the requirement that the number of association rules is not greater than the threshold value of 6, and the decision tree construction process ends.
It can be seen that HiCuts method can effectively improve the efficiency of packet classification by preprocessing the original classification rules and transforming the classification operation from sequence matching to search based on decision tree. This method provides a good reference for accelerating packet classification, but with the increase in link bandwidth, packet classification speed gradually becomes the bottleneck of network performance, and the existing packet classification algorithms still have room to further improve the classification speed.
If the packet is classified according to the decision tree in Fig. 2b, the sequence matching needs to be continued after accessing the leaf node from the root node, which reduces the efficiency of packet classification to some extent. In addition, as shown in the cross line marked area in Fig. 2a, there are often a lot of mutual "entanglement" between rules. These entanglements form the rule replication in the decision tree [4]. As shown in Fig. 2b, rule r 3 is copied twice and rule r 9 is copied three times, thus increasing the storage space consumption. In addition, most decision trees do not deal with rule conflicts during the construction process. For example, the rules r 1 and r 4 match the region F 1 ∈ [5, 5] ∧ F 2 ∈ [9,9] concurrently, but their decisions are different, which may lead to the erroneous discarding of legitimate packets or the acceptance of malicious packets, thus bringing security vulnerabilities to the network. To solve the above problems, we put forward an improved package classification method based on decision tree, Uscuts [4]. First, the original rules are mapped to a multi-dimensional matrix M k (where M k stands for a k-dimensional matrix) in reverse order, and the unit spaces with the same semantics as the original rules but independent of each other in space are obtained [5]. Figure 3 shows the result of mapping rule r: "F 1 ∈ [1, 7] ∧ F 2 ∈ [2,6] → accept." to a two-dimensional matrix M 2 . Here M 2 contains a unit space, represented as [(1,2) (7,5)]. Because it is a twodimensional space, there are only two attributes: F 1 dimension and F 2 dimension.
Based on the mapping method described in reference [5], mapping the rules shown in Fig. 1 to a two-dimensional matrix in reverse order can obtain seven unit spaces named cs 1 -cs 7 , which correspond to seven rectangles in the two-dimensional space as shown in Fig. 4.
This method divides the rule space corresponding to cs 1 -cs 7 according to the attribute order of F 1 dimension first and then F 2 dimension, and constructs the classification decision tree as shown in Fig. 5. It can be seen that each branch of the decision tree is in one-to-one correspondence with the rule subspace, that is, each leaf node of the decision tree is just associated with a rule. Therefore, when a packet matches a leaf node, the Uscuts method can directly determine that the packet classification decision is "accept", which is different from the traditional packet classification method based on decision tree, which needs to perform sequence matching in the rule group associated with the leaf node. Therefore, the speed of packet classification is effectively improved.
However, further research shows that for the unit spaces shown in Fig. 4, if they are divided according to the attribute order of F 2 dimension first and then F 1 dimension, the construction of decision tree is shown in Fig. 6. Obviously, if the selection order of classification attributes is different, the constructed decision tree and classification efficiency will also be different. For example, when the decision tree in Fig. 5 is used for packet classification, we first perform packet matching on eight nodes of F 1 dimension. Because the range value of each node is strictly increasing,    The decision tree constructed according to attribute order selection the Binary Search algorithm can be adopted, and the time complexity is log 2 (8) = 3. However, when the decision tree shown in Fig. 6 is used for packet classification, the time complexity of matching on the first layer is log 2 (3)≈1.58. Combing the matching time on the second layer, it can be easily calculated that the efficiency of packet classification according to the decision tree shown in Fig. 6 is higher than that in Fig. 5.
Through the analysis of the above examples, it can be inferred that when constructing a decision tree according to the specified classification rules, the selection of attribute order will affect the efficiency of package classification. Therefore, the problem we are faced with is how to determine the order of attribute selection when constructing the decision tree, so as to make the package classification based on the decision tree more efficient.
Generally speaking, the most intuitive way to deal with this problem is to construct all decision trees according to different arrangement of attributes, and then carry out package classification and efficiency comparison. However, for k-dimensional classification rules, to determine the decision tree with the highest classification efficiency, it is necessary to construct k factorial (e.g., 5 factorial equals 120) decision trees, and then classify packets based on them, respectively, and compare the classification efficiency. This is a global optimized solution, but it is obviously too time-consuming and difficult to realize. Therefore, we designed a heuristic decision tree construction method, PCMIgr. Based on the idea of greedy strategy, this method selects the attribute with the highest "information gain ratio" at each decision tree node. According to Informatics theory, information entropy can be used to measure the uncertainty of data samples and categories with different characteristics. The greater the information entropy of the eigenvector, the greater the uncertainty, so we should give priority to dividing from the eigenvector.
In this paper, the information gain ratio is used to select attributes, which overcomes the disadvantages of selecting attributes with more values when using information gain [29]. The experimental results show that the decision tree based on information gain ratio has higher classification efficiency, and does not require additional memory. It is suitable for high-speed real-time packet classification.
The main innovation of this method is that the attribute selection in the traditional classification method based on decision tree is optimized, and the classification efficiency of the constructed decision tree is improved compared with that before optimization. At the same time, the decision tree ensures that each leaf node is associated with only one rule, which avoids the deficiency that the traditional decision tree needs to continue sequence matching after the packet matches the leaf node. More importantly, the decision tree avoids the common problem of "rule replication" in traditional decision tree construction, and effectively saves memory and calculation time.
The rest of this paper is organized as follows: the related work is introduced in Sect. 2; In the third section, the problem description is given, and the classification decision tree construction algorithm based on information gain ratio is elaborated in detail. The fourth section gives the classification results of PCMIgr, Uscuts, Hicuts and HyperSplit, and followed by the analysis of experimental results; Finally, the conclusion is drawn in Sect. 5.

Related work
Packet classification can be regarded as the problem of "point location in multidimensional space" in computational geometry: given some disjoint areas in multidimensional space, to locate the area containing the specified "point." A classifier is a hypercube set with priority, and the packet header represents a point in k-dimensional space. Assuming that these areas do not intersect, and the dimension k is greater than 3, it has been proved that for n non-overlapping hyper-rectangles in k-dimensional space, the best bounds is either O (log 2 n) time with O (n k ) space, or O ((log 2 n) k−1 ) time with O (n) space [6]. In addition, in packet classification, hyperrectangles may overlap, which makes packet classification more difficult than point location, it may require too much memory or take a long time.
For the rules in the classifier, with the increase in rule dimensions, the performance of packet classification algorithm will drop sharply. Moreover, with the rapid expansion of Internet services, the scale of classification rules is also increasing, which brings severe challenges to packet classification. Therefore, the massive packets to be classified and the large-scale classification rules bring more severe challenges to the packet classification problem. In a word, although packet classification has been studied for decades, with the emergence of new network services and new requirements, there are still many technical barriers to break through. Especially considering the real-time and rapid classification requirements of massive packets, how to further improve the speed of packet classification, so as to meet the requirements of high-speed packet classification in the new generation network environment still needs to be explored and studied.
At present, large routers and high-end classifiers mainly use hardware devices to classify packets based on exhaustive search. Its core idea is to directly traverse all the rules in the rule list and get the matching result. Typical hardware-based solutions include ternary content addressable memory (TCAM) [7], field programmable gate array (FPGA) [8], and dedicated network processor chip. The exhaustive search algorithm has simple data structure and high classification efficiency. For example, the packet classification algorithm based on TCAM adopts parallel search scheme, and the time complexity of the algorithm is O (1). However, dedicated hardware has some disadvantages, such as high price, long development time and high energy consumption, which limit their application and scalability to some extent. At present, in academia, researchers have put forward many general solutions based on software for packet classification.
The algorithm based on dimension decomposition decomposes each multidimensional rule into multiple dimensions in a certain number of bytes or bits. Each dimension is searched separately, and then the final search result is obtained by combination. Cross-Producting [9] is an early classical algorithm, which firstly matches in each dimension respectively, then combines the results of each dimension to form a cross-product, and finally maps it to a product table to get the best match. The algorithm makes full use of the idea of sacrificing memory for speed to achieve fast matching of k-dimensional classification rules. The cross product algorithm has a short search time, but in the worst case, the space complexity is O (n k ) (n is the number of rules).
ABV [10] reduces memory access and improves classification speed by aggregating bit vectors. However, its memory consumption is high because it needs to store additional information, such as aggregated bit vectors. Through the modular BV architecture, StradBV [11] eliminates the rule expansion caused by the conversion from range to prefix. Different from the Cross-Producting method, RFC (Recursive flow classification) [12] uses multilevel mapping to transform packet classification into table lookup process, which has good classification performance. However, due to the long preprocessing time, extral class tables need to be stored, which consumes a lot of memory. To sum up, these methods are fast, but as the scale of classification rules increases, the memory consumption will increase exponentially in the worst case.
The algorithm based on tuple space constructs a hash table for each different prefix length, and the subsets of rules with the same prefix length are stored in the same hash table. When classifying packets, all hash tables are accessed sequentially until the longest matching prefix is found.
A classic algorithm is TSS (Tuple space search) [13], which divides the classification rules into multiple rule subsets according to the prefix bits of each field and stores them in hash tables. When a packet is received, TSS first finds the corresponding rule subset through the hash key, and then searches the subset for the most matching rule. The main disadvantage of tuple space search method is that the number of hash tables will greatly increase with time, which leads to slow packet classification. The representative algorithms are TupleMerge, PartitionSort [14][15][16][17], and so on.
Algorithms based on decision tree can be divided into two categories: one is based on trie, whose basic idea is to establish a hierarchical binary tree according to the classification rules, divide each dimension of the rules into one layer, then recursively expand the one-dimensional tree structure, and finally generate a k-dimensional hierarchical tree. The advantages of the algorithm based on trie are simplicity, directness and easy hardware implementation. Its disadvantage is that it takes a long backtracking time, which is not conducive to the expansion of the rule dimension and cannot directly support range matching. SplitTrie [18] improved the basic Triebased algorithm, supporting multi-field search and avoiding backtracking, but the algorithm still does not support range matching.
Another algorithm based on decision tree is to build a decision tree by recursively decomposing multidimensional space. Typical classical algorithms are HiCuts, HyperCuts [19] and EffiCuts [20], etc. These algorithms divide the search space into several subspaces of equal size by using local optimization until the number of rules in each subspace is less than the predefined threshold τ. It shows excellent search performance, but equal-scale cutting will lead to huge storage requirements. H. Lim et al. reduced the memory consumption of the algorithm through boundary-based cutting [21]. Hybridcuts [22] divide rules on a single rule field instead of all fields, which reduces the number of subsets and the frequency of memory access. Bitcuts [23] and Uscuts cut rules based on bit and unit space, respectively, achieving a better balance between classification speed and space consumption. Bytecuts [24] intelligently divides classification rules into multiple trees through byte segmentation, thus reducing rule duplication. Mbitcuts [25] reduces the space consumption and memory access in the algorithm by changing the bit selection mode when cutting the geometric space model of each tree node.
Compared with the cutting-based method, the segmentation-based method divides the search space into multiple equal-density subsets. "Equal-density" means that the number of rules in each subset is almost the same. HyperSplit [26] is a classical segmentation method, which divides the search space into two equal dense subspaces. However, this method will lead to the increase in memory consumption as the number of rules increases. As an improved version of HyperSplit, ParaSplit [27] uses a new partitioning algorithm to reduce the complexity of classification rules and its memory consumption. CutSplit [28] combines the advantages of cutting and segmentation to improve the performance of packet classification. However, for different rules, their performance varies greatly, which is a common problem faced by most decision tree-based algorithms except "rule replication".
At present, compared with other software-based classification methods, the method based on decision tree has more advantages in classification speed. Therefore, this paper continues to discuss the classification method based on decision tree, and optimizes the construction process of decision tree on the basis of existing methods, so as to further improve the classification speed.

Problem description
According to the research and analysis of previous packet classification methods based on decision tree, the existing algorithms focus on how to transform the representation of classification rules from access control list to decision tree. Its core idea is to construct one or more decision trees covering all rules according to the characteristics of rules, including scale cutting, density splitting and boundary division. However, few methods consider the order of each layer when constructing decision trees.
Generally speaking, the key to construct a decision tree is to measure attribute selection, then construct different branches according to the different division of a certain attribute at a node, and determine the topology between each attribute. The attribute selection metric is to divide the data of the training set marked by a given category into the "best" category, which determines the choice of topology and split position. So we need to consider how to choose an attribute as the root node of the decision tree from the data set composed of multidimensional attributes. That is, how to choose the attribute with the largest disorder degree in the attribute set as the dividing node every time?
According to the previous analysis, when the dimension k is high (e.g., greater than 4), the optimal scheme will be very time-consuming. Therefore, our idea is to find a heuristic local optimization solution, that is, to choose the "best" attribute every time we choose the attribute. At this time, the key to the problem is what criteria are used to measure the "best" attribute?
In Informatics, the concept of information entropy is introduced to measure the order (or disorder) of an object's attribute value. Information entropy is used to measure the expected value of random variables. The greater the information entropy of a variable, the more information it contains, that is, more information is needed to fully determine the value of the variable.
For the set of random variables X = {x 1 ,x 2 ,…,x m }, if the occurrence probability of any random variable x i (i = 1, 2,…, m) is p i , then the information entropy of X is expressed as: Considering that information entropy is used to measure the expected value of a random variable, information entropy can be used to measure the uncertainty of categories in decision trees. The greater the information entropy of the attribute, the greater the uncertainty of the vector. Therefore, it can be considered to divide based on this attribute vector first.
Dong. X et al. proposed an attribute selection method based on information entropy [29]. Although this method is only applicable to rules expressed in prefix form, it has been proved by a large number of experiments that using information gain to measure the priority of attribute selection is helpful to construct a decision tree with better classification performance.
In this paper, C4.5 algorithm is used to calculate the optimal eigenfunction. Compared with ID3 algorithm, it is easy to fall into the trap of selecting attributes with the most values. C4.5 algorithm uses information gain ratio instead of information gain to select attributes, which overcomes the deficiency that information gain tends to select attributes with more values. Next, we use a specific classification rule to illustrate the implementation steps of the algorithm. For simplicity and intuition, two-dimensional classification rules are adopted here.

Classification algorithm based on information gain ratio
The implementation process of the algorithm includes the following four steps: (1) Pre-processing the original rules, and mapping the rules into multidimensional matrix space by rule mapping method to form a series of independent unit spaces; (2) Constructing a data set according to the coordinate projection interval, wherein the attribute set C = {F 1 , F 2 , …, F k }, and calculating the information gain ratio of each attribute; (3) Using the top-down recursive divide-and-conquer method and the greedy strategy without backtracking, the attribute with the largest information gain ratio is selected as the partition node to construct the classification decision tree; (4) Classify data packets by decision tree.
Using the rule mapping method, the input k-dimensional classification rules are mapped to the k-dimensional matrix space M k in reverse order, forming a series of independent unit spaces. Generally speaking, classification rules can be expressed in the form of intervals, such as " represents the source address, destination address, source port and destination port, etc., D(F i ) indicates the corresponding domain value interval, and decision represents the action (accept or discard) performed by the rule.
According to the rule mapping idea based on multi-dimensional matrix [5], any k-dimensional classification rule can be mapped to k-dimensional matrix space M k . In the mapping process, we use unit space cs (corresponding to a k-dimensional rectangle in k-dimensional matrix space) to represent the area that is finally decided to accept: [(l 1 ,l 2 ,…,l k )(d 1 ,d 2 ,…,d k )], where l i and d i refer to the minimum boundary value and range of the area in each dimension, respectively.
(2) Calculating the Information gain ratio.
The purpose of this step is to calculate the information gain ratio of attributes according to the unit space obtained in the rule preprocessing stage. Firstly, the data set is constructed according to the definition of coordinate projection interval, as shown in Table 1. In this example, the dataset has two attributes, corresponding to the attribute set C = {F 1 , F 2 }; In addition, there are two category labels, which constitute the category set L = {accept, discard}. FPC [30] introduces the construction process of data set in detail. For any two unit spaces u and v, if they satisfy these two  (2) in any other dimension Fy, R(u, v, Fy) = included or R(u, v, Fy) = crossed, we let the coordinate value of u in dimension Fy to be the secant, then cut u into two or three sub-unit spaces. This operation on unit spaces is iteratively conducted in a certain dimension until all the k dimensions are completed. In the follows, we calculate the information gain ratio of each attribute based on the C4.5 algorithm (as described in Algorithm 1) and the data set described in Table 1.
Step 1: Using information entropy to measure the uncertainty of category label to the whole sample.
Let S be a set of data samples, and its category label C = {c 1 ,c 2 ,…,c m }. Class c divides the data sample set S into S c = {S c1 ,S c2 ,…,S cm }, where S ci = {s|s.label = c i , s ∈ S} and S ci ∩ S cj = Ø,1 ≤ i ≠ j ≤ m, s.label represents the label of sample s. The information entropy formula of sample classification is as follows: where p i = length(S ci )/length(S) is the probability that the sample belongs to category c i . length(S ci ) indicates the number of elements of category c i in sample set S; length(S) indicates the number of elements in the sample set S, that is, the total number of samples. Substituting the data set in Table 1 into Formula (2), the category information entropy can be calculated as follows: I (S c ) = -15/24 * log 2 (15/24)-9/24 * log 2 (9/24) = 0.955. (2) Step 2: Using information entropy to measure the uncertainty of different values of each attribute.
Assuming that the attribute A = {a 1 ,a 2 ,…,a v } has v different values, then the sample set S can be divided into v disjoint subsets {S 1 A ,S 2 A ,…,S v A } by using attribute A, where S j A = {s|s ∈ S,s.A = a j } and j = 1,2,…,v. If attribute A is selected as the optimal partition feature, then the partitioned subset is the branch of the decision tree that grows out the S node of the sample set. The information entropy of the subset divided by attribute A is shown in the following formula.
Step 3: Using information gain to determine the division basis of decision tree branches.
The formula for calculating the difference between the information entropy of the whole data set on a branch of the decision tree and the information entropy of the current node is: Thus, the information gain can be calculated: Step 4: Calculating the split information Split info (S).
Split information is defined as: PCMIgr: a fast packet classification method based on information… is the division of sample set S on attribute A, and it is assumed here that attribute A has v different values. Thereby, the attribute splitting information metric can be calculated: Split info (F 1 ) = 3.0 Split info (F 2 ) = 1.585. STEP 5: Calculating the information entropy gain ratio IGR(S).
(2) Constructing a classification decision tree based on the information gain ratio.
The purpose of this step is to determine the priority of attribute selection and build a classification decision tree according to the information gain ratio calculated in the STEP 5. In this example, it can be seen that IGR (F 2 ) > IGR (F 1 ), so when constructing the decision tree, the F 2 dimension with larger information gain ratio is selected as the preferred attribute. Assuming that there are k attributes in the data set, the information entropy gain ratio of each attribute is calculated, which is recorded as IGR [k] = {IGR (F 1 ), IGR (F 2 ),…, IGR (F k )}. Initially, the decision tree T only contains the root node 'root'. Next, we briefly describe the general process of building a classification decision tree based on the information gain ratio, as described in Algorithm 2.
Taking Fig. 6 as an example. In the initial case, T is a decision tree containing only the root node 'root'. Because IGR (F 2 ) > IGR (F 1 ), we choose the attributes in the order of F 2 before F 1 to construct the decision tree. In Fig. 4, seven unit spaces (us 1~u s 7 ) in the two-dimensional matrix space form three coordinate projection intervals { [4,4], [5,6], [7,8]} in the F 2 dimension. These three intervals are added to the decision tree T as child nodes Node i (i: = 1 to 3) of the root, respectively. Each child node constitutes a subtree T i (i = 1 to 3) of T, and each node Node i is the root node of the corresponding subtree T i .
The root node of the subtree T 1 corresponds to the interval [4,4], and all the associated unit spaces us 1 and us 5 form three coordinate projection intervals [1,1], [3,3] and [7,8] on the F 1 dimension, and then these three projection intervals are added to the subtree T 1 as child nodes, respectively. Similarly, projection intervals [0,4], [6,9] are added to the subtree T 2 as child nodes; [0,2] and [7,8] are added to the subtree T 3 , and finally form the decision tree T, as shown in Fig. 6.

(4) Classifying packets
In the packet classification method based on decision tree, packet classification is essentially a query operation. For the decision tree or any of its sub-trees, the interval coordinate values corresponding to the child nodes of the root are strictly increasing, so the binary search method can be directly applied to search. As shown in Fig. 6, the root node of the decision tree has three sub-nodes, and the corresponding interval coordinate values are [4,4], [5,6] and [7,8], which meet the strict increasing relationship.
Consider the classification process of k-tuple packet P:(e 1 , e 2 ,…, e k ), the classification starts from the root node of the decision tree. Because the first layer of the decision tree is divided based on the attribute F 2 , first, binary search is performed on all the child nodes of the root to determine whether the second metadata e 2 of the packet is included in the corresponding interval of a child node. If any interval cannot be matched, it can be directly determined that the packet P cannot match any node, and the packet is determined to be discarded; Otherwise, continue searching on the subtree with the node as the root. If each tuple e i (i = 1 to k) of the packet P matches the corresponding interval of each layer node of a subtree branch, it can be determined that the packet P matches the decision tree, and the decision of P is accept.
As shown in Fig. 7, assuming that the packets to be classified is E = {p 1 ,p 2 } = {(1,4), (3,7)}, we first analyze the matching result of packet p 1 : (1,4), where e 1 = 1 and e 2 = 4. Starting from the root node, through binary search on all child nodes, it can be seen that e 2 = 4 matches the interval [4,4] of the first node, and then continue searching on the subtree with this node as the root. Since e 1 = 1 can match one of the branch intervals [1,1] in the subtree, it can be determined that the classification decision of p 1 is accept.
Next, continue to classify the packet p 2 : (3,7), where e 1 = 3 and e 2 = 7. Because e 2 matches the coordinate interval [7,8] of the third child node, the search continues on the subtree with the child node as the root. However, the first dimension of p 2 :(3,7) is '3', which does not match any interval [0,2] or [7,8] corresponding to the two branches, therefore it can be determined that the classification decision of packet p 2 is discard. The specific process of packet classification is described in Algorithm 3.

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PCMIgr: a fast packet classification method based on information…

Effectiveness
In order to test the effectiveness of PCMIgr method in constructing decision tree using C4.5 algorithm, according to the classification rules shown in Fig. 1, we use PCMIgr and Uscuts methods to classify packets with different sizes (from 10 KB to 200 MB). Unlike the PCMIgr method, the Uscuts method directly constructs the decision tree from the F 1 dimension to the F k dimension. The experimental results of classification are shown in Fig. 8. It can be seen that the classification efficiency of PCMIgr method is higher than that of Uscuts method, which indicates that reconstructing the decision tree based on the information gain ratio can further improve the classification efficiency.
To further verify the classification efficiency of PCMIgr, we have generated eight classification rules of different number (ranging from 25 to 1800, respectively) and six, data packets of different sizes (10 KB, 50 KB, 500 KB, 1 MB, 100 MB, and 200 MB respectively) to test the time required to classify packets using PCMIgr algorithm. We generated these rules by ClassBench [31], which is a well-known benchmark that provides classifiers similar to real classifiers used in Internet routers and inputs traces corresponding to the classifiers. These algorithms were implemented in Java jdk1.7, and our experiment is conducted on a desktop PC running Windows10, which has 16G memory and 1.80 GHz Intel (R) core (TM) i7-10510u Processor. The test results are shown in Table 2.
It can be seen that for the classification rules with the scale of 1800, when the data set reaches 200 MB, the time required to classify these packets is no more than 3500 ms. Moreover, the running time of the system also includes the process of rule preprocessing and decision tree construction, which can be carried out offline in advance in actual classification.

Efficiency
According to the decision tree constructed by PCMIgr method, only one rule is associated with each leaf node. Therefore, in the process of packet classification, when visiting the leaf nodes of the decision tree, it is not necessary to continue the sequential search in the rule grouping as the traditional packet classification method based on the decision tree, but to directly determine the classification decisions of packets. At the same time, for the decision tree or any of its subtrees, the interval coordinate values of each child node corresponding to the root of the tree are strictly increasing, so the efficient binary search method can be adopted.
Assuming that the original rule number is n, then according to the rule mapping method based on multidimensional matrix [5], the number of unit spaces formed will be less than n, and the projection interval of n unit spaces in any dimension will not be greater than 2n-1. As described in the decision tree construction process of  PCMIgr method, the number of child nodes of the root of the decision tree or any of its subtrees will not exceed 2n-1, so in the worst case, the corresponding search time is log 2 (2n-1) and the time complexity is O(log 2 (n)). Therefore, when searching on a k-level decision tree, the time complexity in the worst case is O(T worst ) = O(k·log 2 n), where n and k refer to the number and dimension of rules, respectively. In practical network classification applications, the data packets and rules are dynamic. Two hypotheses are put forward and verified by experiments: (1) About 50% of the m data packets to be classified are classified as discard; (2) The packet classified as discard has about 1/k probability of not finding a matching node in the F i (i: = 1 to k) dimension [4]. The experimental results show that if the number of data packets to be classified is m, about m/2 data packets will be classified as accept, and these m/2 data packets will need up to log 2 (2n) access time in each dimension, so the total time required is (m/2)·[k·log 2 (2n)]; For the other m/2 packets whose decision is discard, there is a probability of 1/k in each dimension that no matching coordinate interval can be found. Therefore, when the matching coordinate interval cannot be found in the i(i: = 1 to k) dimension, the execution time is (m/2 k)·i·log 2 (2n), and the time complexity of PCMIgr in average case, T avg , is: Considering the searching process of the k-layer decision tree, the time complexity of PCMIgr method in average case is O((0.75 k + 1)·log 2 n). According to the classification results shown in Table 2, with the number of classification rules n as the abscissa and the classification time of packet as the ordinate, we randomly select the classification time of two packets with different sizes (50 KB and 100 MB, respectively) under different classification rules, and map them on the coordinate chart. The mapping results are shown in Fig. 9. It can be seen that the classification time trend approximately conforms to the logarithmic function curve, and basically consistent with the average time complexity analysis results of the PCMIgr method. Table 3 lists the time complexity of PCMIgr method and several classical packet classification methods in the worst case. In Table 3, n and k refers to the number and dimension of rules, respectively, and w is the length of rule domain (in IPv4 protocol: w = 104); τ is the threshold of the number of rules associated with leaf nodes, and RuleSize is the number of bytes occupied by a single rule (generally RuleSize = 24.5Byte [21]). C is the number of bytes of CacheLine, and its size generally ranges from 16 to 256 bytes. Currently, the mainstream CacheLine is 64 Byte.
As shown in Table 3, in the worst case, the time complexity of TSS algorithm is O(n k ), Grid-of-Tries algorithm is O(2w) and HiCuts algorithm is O(w + τ·Rulesize/C). Even if the number of classification rules n reaches 100,000, k·log 2 n is still less than w (typical rule dimension k = 5). Therefore, compared with Grid-of-Tries, HiCuts and other algorithms, the execution efficiency of PCMIgr classification algorithm has certain advantages, and it does not need extra space to store the classification rules, which reduces the requirement of storage space to some extent. For example, HyperSplit is close to PCMIgr in time complexity, but it requires more memory storage.
For the sake of intuition, we comprehensively compare the classification speed of Hicuts, Uscuts, PCMIgr and HyperSplit. The difference among the first three methods is that the order of attribute selection is different when constructing decision tree. The Hicuts and Uscuts methods are in the order from F 1 dimension to F k dimension, while the PCMIgr method selects the corresponding dimensions to construct the decision tree according to the information gain ratio of each attribute from the largest to the smallest. Four data packet sets with different sizes, namely 50 KB, 1 MB, 100 MB and 200 MB, were selected in the experiment. The classification rules included five different sizes, ranging from 100 to 2000. The experimental results are shown in Fig. 10.
As shown in Fig. 10, when classifying data packets of the same size, the classification speed of the PCMIgr method is slightly higher than that of the HyperSplit method, and it has a certain improvement compared with the Uscuts method, while it has obvious advantages compared with the Hicuts method. The experimental results show that the PCMIgr method preprocesses the original rules and avoids the problem of "rule replication", which is of great significance to improve the classification speed. In particular, compared with the Uscuts method, PCMIgr chooses to (8) i⋅ log 2 n) m = 3k + 1 4 ⋅ log 2 n construct the decision tree based on the information gain ratio, and the comparison of classification results further proves that this idea has certain reference value for the improvement of classification efficiency based on decision tree.

Conclusions
With the development of network application, higher requirements are put forward for the speed of network packet classification. In this paper, a heuristic decision tree construction method, PCMIgr, is proposed. It is based on the greedy strategy. When each decision tree node needs to select the classification attribute, it selects the attribute with the highest information gain ratio for classification. This method optimizes the attribute selection in the traditional decision tree construction process, and the classification efficiency is greatly improved compared with that before optimization. This method also avoids the common problem of "rule replication" in traditional classification methods based on decision tree, and effectively saves storage space. The experimental results show that applying PCMIgr method to the construction of classification decision tree can further improve the efficiency of packet classification method based on decision tree. This idea also provides a new way for the research of packet classification.