A Router Queuing Delay Detection Method and Device Based on Unsupervised Learning

Abstract

The present invention discloses a method and device for detecting router queuing delay based on unsupervised learning, which relates to the technical field of wired networks. The method for detecting router queuing delay based on unsupervised learning includes the following steps: generating UDP packet pairs and transmitting the UDP packet pairs from the source address to the target address through the router to be tested; collecting the time interval data of the UDP packet pairs; using the Kmeans clustering algorithm of unsupervised learning to learn the collected time interval data to obtain a clustering result and a centroid; calculating the queuing delay of the router to be tested. By using the Kmeans clustering algorithm to learn the collected time interval data, the present invention achieves the effect of accurately analyzing the router queuing delay according to different transmission delay situations when calculating the queuing delay of the router to be tested based on the clustering result and the centroid, and solves the problem of low measurement accuracy of the existing scheme for measuring the router queuing delay.

Description

Technical Field

[0001] The present invention relates to the technical field of wired networks, and in particular, to a method and device for detecting router queuing delay based on unsupervised learning. Background Art

[0002] Queuing delay refers to the waiting time of data packets inside the intermediate nodes (i.e., routers) of the end-to-end path. It is a dynamic network parameter that depends on the instantaneous state of competing traffic on the router (such as rate and quantity). Although the high transmission speed of the backbone network can minimize the queuing delay on the router, queuing delay on the Internet path is still common. Therefore, in traditional Internet (such as IP geolocation) and visual Internet (such as remote surgery) environments, accurately measuring queuing delay is crucial for ensuring the performance of various applications.

[0003] There are various schemes for measuring queuing delay through passive or active measurement. Passive schemes are popular among Internet service providers. They usually use the continuous data flow on the measured path to measure queuing delay, with strict requirements for infrastructure support (such as dedicated equipment or specific network architectures) and management access (such as direct access to test routers). Active schemes usually use commercial equipment (such as workstations) and synthetic probe packets to measure parameters. However, the limiting problems of these schemes are either due to their special probe data packets (such as ICMP protocol data packets) or the strict topological configuration of the measured path (such as tree network topology), resulting in a lack of accuracy.

[0004] However, during the process of implementing the inventive technical solution in the embodiments of the present application, the inventors of the present application found that the above technologies have at least the following technical problems:

[0005] The existing passive schemes' strict requirements for infrastructure support and management access make them less suitable for deployment on the Internet; the existing active schemes have no restrictions on widespread deployment on the Internet. The limiting problems of these schemes are mainly due to their special probe data packets (such as ICMP protocol data packets) or the strict topological configuration of the measured path (such as tree network topology), resulting in a lack of accuracy. In summary, the existing schemes for measuring router queuing delay have low measurement accuracy. Summary of the Invention

[0006] The embodiments of the present application provide a method and device for detecting router queuing delay based on unsupervised learning, which solve the technical problem of low measurement accuracy of the existing schemes for measuring router queuing delay, and achieve the ability to accurately measure the router queuing delay situation without network support.

[0007] The embodiments of the present application provide a method for detecting router queuing delay based on unsupervised learning, including the following steps:

[0008] S1, generate UDP packet pairs and transmit the UDP packet pairs from the source address to the destination address via the router under test;

[0009] S2, collect the time interval data of the UDP packet pairs;

[0010] S3, use the Kmeans clustering algorithm of unsupervised learning to learn the collected time interval data, and obtain the clustering result and the centroid;

[0011] S4, calculate the queuing delay of the router under test.

[0012] Furthermore, the number of packet pairs transmitted via the router under test in S1 is not less than four.

[0013] Furthermore, the router under test in S1 is a router with cross traffic.

[0014] Furthermore, the time interval data of the UPD packet pairs collected in S2 includes three cases: unchanged time interval, compressed time interval, and extended time interval, and satisfies G co <G nc <G de ;

[0015] Among them, G co is the time interval compression data, G nc is the time interval unchanged data, G de is the time interval extended data.

[0016] Furthermore, when learning the collected data in S3, different types of data need to be cluster-labeled: the compression type is labeled as the co cluster, the unchanged type is labeled as the nc cluster, and the extended type is labeled as the de cluster, and applied to the time interval compression G co 、the time interval unchanged G nc and the time interval extended G de , and the cluster center M co of the co cluster, the cluster center M nc of the nc cluster, and the cluster center M de of the nc cluster.

[0017] Furthermore, when learning the collected data in S3, a three-peak distribution simulation diagram of the transmission time interval will be established.

[0018] Furthermore, the specific steps of using the Kmeans clustering algorithm to learn the collected data in S3 are as follows:

[0019] S31, randomly select 3 time intervals as the centroids;

[0020] S32. Truncate the time intervals in each cluster where the frequency is less than 0.2s i , where s i is the total packet pair size of cluster i;

[0021] S33. Update the centroid of each cluster using the remaining time interval values;

[0022] S34. Loop through S31 to S33 for iterative calculation until a relatively stable centroid is obtained, getting the clustering result and ending the clustering.

[0023] Furthermore, when calculating the queuing delay of the router to be measured in S4, the delay range value needs to be determined based on the clustering result and centroid obtained in step S3.

[0024] Furthermore, the specific method for determining the delay range value is as follows:

[0025] Step 1: Calculate the queuing delay w according to the time interval compression co :

[0026] w co = M nc - M co ,

[0027] where M nc is the cluster center of cluster nc, and M co is the cluster center of cluster co;

[0028] Step 2: Calculate the queuing delay w according to the time interval expansion de :

[0029] w de = M de - M nc ,

[0030] where M de is the cluster center of cluster de;

[0031] Step 3: Estimate the total queuing delay w:

[0032]

[0033] where w i is the queuing delay value of the cluster corresponding to i, is the weighted cluster size of the cluster corresponding to i;

[0034] Step 4: Calculate the standard deviation σ:

[0035]

[0036] where x jIncluding co and de, where n′ is the sum of the total data packet pair sizes of cluster co and the total data packet pair sizes of cluster de;

[0037] Step Five: Calculate the estimated range of the queue delay on the router under test as w ± σ.

[0038] The embodiment of the present application provides a router queuing delay detection device based on unsupervised learning, including a preparation module, a collection module, a processing module, and a calculation module:

[0039] The preparation module: is used to generate UDP data packet pairs and transmit the UDP data packet pairs from the source address through the router under test to the target address;

[0040] The collection module: is used to collect the time interval data of the data packet pairs;

[0041] The processing module: is used to use the Kmeans clustering algorithm of unsupervised learning to learn the collected time interval data to obtain the clustering result and the centroid;

[0042] The calculation module: is used to calculate the queuing delay of the router under test.

[0043] One or more technical solutions provided in the embodiment of the present application have at least the following technical effects or advantages:

[0044] 1. Since the Kmeans clustering algorithm is used to learn the collected time interval data and calculate the queuing delay of the router under test according to the clustering result and the centroid, the interference caused by manual data annotation to the measurement result is avoided, effectively solving the problem of low measurement accuracy of the existing scheme for measuring the router queuing delay, and thus realizing the accurate analysis of the router queuing delay according to different transmission delay situations.

[0045] 2. Since the technology of establishing a three-peak distribution simulation diagram of the transmission time interval is used when learning the collected data, it can intuitively show the corresponding situations of different transmission delays, effectively solving the problem that it is impossible to accurately locate which transmission delay causes the routing queuing delay in the existing method for measuring the routing queuing delay, and thus realizing the ability to directly find the type of transmission delay that causes the router queuing delay.

[0046] 3. Since the technology of actively generating UDP data packet pairs and transmitting the UDP data packet pairs from the source address through the router under test to the target address is used, the situation of the router queuing delay can be detected at any time, effectively solving the problem that the existing method cannot detect the routing queuing delay situation at any time, and thus realizing the ability to actively detect the routing queuing delay situation at any time. Description of the Drawings

[0047] Figure 1Flowchart of the router queuing delay detection method based on unsupervised learning provided in the first embodiment of this application;

[0048] Figure 2 Structural diagram of the router queuing delay detection device based on unsupervised learning provided in the second embodiment of this application;

[0049] Figure 3 Process diagram of the packet pair passing through the router with cross traffic provided in the first embodiment of this application;

[0050] Figure 4 Simulation diagram of the three - peak distribution provided in the first embodiment of this application. Detailed implementation manners

[0051] By providing a router queuing delay detection method and device based on unsupervised learning in the embodiments of this application, the problem of low measurement accuracy in the existing solutions for measuring router queuing delay is solved. After collecting the time intervals of the transmitted packet pairs, the Kmeans clustering algorithm is used to perform cluster labeling and learning on the time interval data, realizing the accurate analysis of the router queuing delay according to different transmission delay situations.

[0052] The overall idea of the technical solution in the embodiments of this application to solve the problem of low measurement accuracy of the above - mentioned solution for measuring router queuing delay is as follows:

[0053] The generated UDP packet pairs are transmitted from the source address to the destination address through a router with cross traffic, and at the same time, the time intervals of the transmitted packet pairs are collected as the input data of the Kmeans clustering algorithm. After receiving the input data, the Kmeans algorithm performs cluster labeling on three different types of time intervals, obtains the cluster centers, and establishes a three - peak distribution simulation diagram based on the time intervals and the cluster centers. The Kmeans algorithm iteratively calculates the received data until a relatively stable centroid is obtained, and the clustering result is obtained, then the clustering ends. Finally, the queuing delay on the to - be - measured route is determined according to the clustering result and the centroid.

[0054] The specific steps for the above - mentioned Kmeans clustering algorithm to learn the collected time interval data are as follows: First, randomly select 3 time intervals as the centroids, and then truncate the time intervals with the cluster center frequency less than 0.2s i and update the centroid of each cluster using the remaining time interval values. By repeating the above steps, a relatively stable centroid can be obtained, and the clustering result is obtained, then the clustering ends.

[0055] The three peaks on the three - peak distribution simulation diagram are respectively the time interval unchanged G nc 、the time interval compressed G co and the time interval extended G deStatistical mean of the corresponding local peak: M co , M nc and M de , the queuing delay on the router can be estimated based on the results and centroids obtained by Kmeans clustering from the magnitude of the time interval M nc -M co and M de -M nc .

[0056] To better understand the above technical solution, the above technical solution will be described in detail below in conjunction with the accompanying drawings of the specification and specific embodiments.

[0057] Embodiment 1

[0058] As Figure 1 shown, it is a flowchart of a method for detecting router queuing delay based on unsupervised learning provided by an embodiment of the present application. This method is applied to a device for detecting router queuing delay based on unsupervised learning, and this method includes the following steps:

[0059] S1. Generate UDP packet pairs and transmit the UDP packet pairs from the source address through the router to be measured to the target address;

[0060] S2. Collect the time interval data of the UDP packet pairs;

[0061] S3. Use the Kmeans clustering algorithm of unsupervised learning to learn the collected time interval data to obtain the clustering result and the centroid;

[0062] S4. Calculate the queuing delay of the router to be measured.

[0063] Furthermore, the method provided by the embodiment of the present application can measure the queuing delay of the route during the wired transmission process through the router. The specific steps are as follows:

[0064] The first step is to generate UDP packet pairs, send n packet pairs from the source address src to the destination address dst, and count the time interval t for the corresponding packet pairs to reach the destination address dst after passing through the router to be measured;

[0065] The second step is to form an input vector with all the collected time interval data and provide it as input to the Kmeans model;

[0066] The third step is to use the Kmeans algorithm to learn the time interval data of the packet pairs to obtain the clustering result and the centroid;

[0067] The fourth step is to determine the queuing delay on the router to be measured according to the clustering result and the centroid.

[0068] In this embodiment, each packet pair consists of a small header packet (Ph) and a large tail packet (Pt), with no time interval in between. Relevant packet pairs are sent to the router to be measured, i.e., cross traffic is introduced.

[0069] Furthermore, the number of packet pairs transmitted through the router to be measured in S1 is not less than four.

[0070] Furthermore, the router to be measured in S1 is a router with cross traffic.

[0071] Furthermore, the time interval data of the UPD packet pairs collected in S2 includes three cases: unchanged time interval, compressed time interval, and extended time interval, and satisfies G co <G nc <G de ;

[0072] Among them, G co is the time interval compression data, G nc is the time interval unchanged data, G de is the time interval extended data.

[0073] In this embodiment, as Figure 3 shown, it is the process diagram of the packet pair provided by this embodiment passing through the router with cross traffic. Each packet pair has a small header packet (Ph) and a large tail packet (Pt). In the case of cross traffic in the router, there are different time intervals for packet transmission, i.e., G co 、G nc and G de .

[0074] Furthermore, when learning the collected data in S3, different types of data need to be cluster - labeled: the compression class is labeled as co - cluster, the unchanged class is labeled as nc - cluster, and the extended class is labeled as de - cluster, and applied to the time interval compression G co 、the time interval unchanged G nc and the time interval extended G de , and the cluster centers M co 、the cluster center M nc of the nc - cluster and the cluster center M de of the nc - cluster.

[0075] In this embodiment, the cluster centers M co 、M nc and M de when learning the collected data satisfy M co <M nc <M de .

[0076] Further, when learning the collected data in S3, a simulation graph of the three-peak distribution of the transmission time interval will be established.

[0077] In this embodiment, as Figure 4 shown, it is the three-peak distribution simulation graph provided by the embodiment of the present application. When there is cross traffic on the router, in the three-peak distribution of the time interval of packet pair transmission, it respectively includes: time interval compression, time interval unchanged, and time interval extension, and the three cluster centers M co , M nc and M de ;

[0078] When learning the collected data, according to the time interval unchanged G nc , the time interval compression G co and the time interval extension G de obtain the statistical mean of the corresponding local peaks, that is, the cluster centers: M co , M nc and M de , and draw a three-peak distribution simulation graph.

[0079] Further, the specific steps of using the Kmeans clustering algorithm to learn the collected data in S3 are as follows:

[0080] S31, randomly select 3 time intervals as the centroids;

[0081] S32, truncate the time intervals with a frequency less than 0.2s i in each cluster, where s i is the total packet pair size of cluster i;

[0082] S33, update the centroid of each cluster using the remaining time interval values;

[0083] S34, loop S31 to S33 for iterative calculation until a relatively stable centroid is obtained, obtain the clustering result, and end the clustering.

[0084] In this embodiment, when using a fixed number of clusters in Kmeans and the fixed number k of the cluster types is 3, there will be no ambiguity when the subsequent Kmeans algorithm optimizes the data to generate a consistent metric, where cluster i includes clusters co, nc, and de.

[0085] Further, calculating the queuing delay of the router to be measured in S4 requires determining the delay range value according to the clustering result and the centroid obtained in step S3.

[0086] In this embodiment, according to the clustering result and the centroid, from the size of the time interval M nc -M co and M de-M nc to estimate the queuing delay on the router.

[0087] Furthermore, the specific method for determining the delay range value is as follows:

[0088] Step 1: Calculate the queuing delay w according to the time interval compression co :

[0089] w co = M nc - M co ,

[0090] where M nc is the cluster center of cluster nc, and M co is the cluster center of cluster co;

[0091] Step 2: Calculate the queuing delay w according to the time interval expansion de :

[0092] w de = M de - M nc ,

[0093] where M de is the cluster center of cluster de;

[0094] Step 3: Estimate the total queuing delay w:

[0095]

[0096] where w i is the queuing delay value of the cluster corresponding to i, is the weighted cluster size of the cluster corresponding to i;

[0097] Step 4: Calculate the standard deviation σ:

[0098]

[0099] where x j includes co and de, and n′ is the sum of the total packet pair sizes of cluster co and the total packet pair sizes of cluster de;

[0100] Step 5: Calculate the queue delay estimation range on the router to be measured as w ± σ.

[0101] In this embodiment, the calculation process of the weighted size of cluster co is The calculation process of the weighted size of cluster de is x j ∈ {co, de}, n′ = s co + s de .

[0102] The technical solutions in the embodiments of the present application have at least the following technical effects or advantages: The unsupervised clustering algorithm Kmeans is used to analyze the changes in UDP packet pairs, namely the three cases of unchanged time interval, compressed time interval, and extended time interval. It uses the unsupervised algorithm to learn the corresponding delay situations for the above three cases, so as to better measure the queuing delay situation of the route; at the same time, the cluster centers of the corresponding clusters are obtained according to the above three cases, supporting the drawing of a three-peak distribution simulation diagram to more intuitively show the changes generated when the packet pair passes through a router with cross traffic.

[0103] Embodiment 2

[0104] As Figure 2 shown, it is a structural diagram of a router queuing delay detection device based on unsupervised learning provided by the embodiments of the present application. The router queuing delay detection device based on unsupervised learning provided by the embodiments of the present application includes a preparation module, a collection module, a processing module, and a calculation module:

[0105] Preparation module: used to generate UDP packet pairs and transmit the UDP packet pairs from the source address through the router to be tested to the target address;

[0106] Collection module: used to collect the time interval data of the packet pairs;

[0107] Processing module: used to learn the collected time interval data by using the Kmeans clustering algorithm of unsupervised learning to obtain the clustering result and the centroid;

[0108] Calculation module: used to calculate the queuing delay of the router to be tested.

[0109] In this embodiment, first, the preparation module generates UDP packet pairs and transmits the UDP packet pairs from the source address through the router to be tested with cross traffic to the target address. Then, the collection module collects the time interval data of the packet pairs. Next, the Kmeans clustering algorithm in the processing module learns the collected time interval data to obtain the clustering result and the centroid. Finally, the calculation module calculates the queuing delay of the router to be tested.

[0110] The technical solutions in the embodiments of the present application have at least the following technical effects or advantages: Since the time interval data is one-dimensional and not easy to be processed by complex machine learning and deep learning methods, and the Kmeans algorithm is simple, the Kmeans clustering algorithm is used to process the time interval data, and there is no need for manual annotation of the data.

[0111] Those skilled in the art should understand that the embodiments of the present invention can be provided as a method, a system, or a computer program product. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present invention can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code.

[0112] The present invention is described with reference to the flowcharts and / or block diagrams of methods, apparatuses (systems), and computer program products according to embodiments of the present invention. It should be understood that each flow and / or block in the flowchart and / or block diagram, and the combination of flows and / or blocks in the flowchart and / or block diagram, can be realized by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing devices generate a device for realizing the functions specified in Figure 1 one flow or multiple flows and / or blocks Figure 1 one block or multiple blocks.

[0113] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing devices to work in a specific manner, such that the instructions stored in the computer-readable memory generate a manufactured article including an instruction device that realizes the functions specified in Figure 1 one flow or multiple flows and / or blocks Figure 1 one block or multiple blocks.

[0114] These computer program instructions can also be loaded onto a computer or other programmable data processing devices, such that a series of operation steps are executed on the computer or other programmable devices to generate a computer-implemented process, so that the instructions executed on the computer or other programmable devices provide steps for realizing the functions specified in Figure 1 one flow or multiple flows and / or blocks Figure 1 one block or multiple blocks.

[0115] Although the preferred embodiments of the present invention have been described, those skilled in the art can make additional changes and modifications to these embodiments once they know the basic creative concepts. Therefore, the appended claims are intended to be construed as including the preferred embodiments and all changes and modifications falling within the scope of the present invention.

[0116] Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalent technologies, the present invention is also intended to include these modifications and variations.

Claims

1. A router queuing delay detection method based on unsupervised learning, characterized in that It includes the following steps: S1. Generate UDP packet pairs and transmit the UDP packet pairs from the source address to the target address via the router under test; S2. Collect the time interval data of the UDP packet pairs; S3. Use the Kmeans clustering algorithm of unsupervised learning to learn the collected time interval data, and obtain the clustering result and the centroid; S4. Calculate the queuing delay of the router under test; Among them, the time interval data of the UPD data packet pairs collected in S2 includes three cases: the time interval remains unchanged, the time interval is compressed, and the time interval is extended, and it satisfies G co <G nc <G de ; among them, G co is the time interval compression data, G nc is the time interval unchanged data, G de is the time interval extension data; Among them, when calculating the queuing delay of the router under test in S4, it is necessary to determine the delay range value according to the clustering result and the centroid obtained in step S3; Among them, the specific method for determining the delay range value is: Step 1: Calculate the queuing delay w according to the time interval compression co : w co = M nc - M co , Among them, M nc is the cluster center of cluster nc, and M co is the cluster center of cluster co; Step 2: Calculate the queuing delay w according to the time interval expansion de : w de = M de -M nc , Among them, M de is the cluster center of the cluster; Step three: Estimate the total queuing delay w: where, w i is the queuing delay value of the cluster corresponding to i, is the weighted cluster size of the cluster corresponding to i; Step four: Calculate the standard deviation σ: where x j includes co and de, and n' is the sum of the total number of data packet pairs of cluster co and the total number of data packet pairs of cluster de; Step five: Calculate the estimated range of the queue delay on the router under test as w±σ; Among them, when learning the collected data in S3, different types of data need to be cluster-labeled: compression type is labeled as co-cluster, invariant type is labeled as nc-cluster, and expansion type is labeled as de-cluster, and applied to time interval compression G co , time interval invariant G nc and time interval expansion G de , and the cluster center M of co-cluster co , the cluster center M of nc-cluster nc and the cluster center M of nc-cluster de .

2. The method for detecting router queuing delay based on unsupervised learning according to claim 1, characterized in that: The number of packet pairs transmitted via the router under test in S1 is not less than four.

3. The method for detecting the queuing delay of a router based on unsupervised learning according to claim 1, characterized in that: The router under test in S1 is a router with cross traffic.

4. The method for detecting router queuing delay based on unsupervised learning according to claim 1, characterized in that: When learning the collected data in S3, a three-peak distribution simulation diagram of the transmission time interval will be established.

5. The method for detecting the queuing delay of a router based on unsupervised learning according to claim 1, wherein The specific steps for using the Kmeans clustering algorithm to learn the collected data in S3 are: S31. Randomly select 3 time intervals as the centroids; S32, truncate the time intervals in each cluster where the frequency is less than 0.2s i where s i is the total packet pair size of cluster i; S33. Use the remaining time interval values to update the centroid of each cluster; S34. Loop S31 to S33 for iterative calculation until a relatively stable centroid is obtained, obtain the clustering result, and end the clustering.

6. A router queuing delay detection device based on unsupervised learning, characterized in that, It includes a preparation module, a collection module, a processing module, and a calculation module: Preparation module: Used to generate UDP packet pairs and transmit the UDP packet pairs from the source address to the target address via the router under test; Collection module: Used to collect the time interval data of the packet pairs; Processing module: Used to use the Kmeans clustering algorithm of unsupervised learning to learn the collected time interval data, and obtain the clustering result and the centroid; Calculation module: Used to calculate the queuing delay of the router under test; Among them, the time interval data of the collected UPD data packet pairs includes three cases: the time interval remains unchanged, the time interval is compressed, and the time interval is extended, and it satisfies G co <G nc <G de ; among them, G co is the time interval compression data, G nc is the time interval unchanged data, G de is the time interval extension data; Among them, when calculating the queuing delay of the router under test, it is necessary to determine the delay range value according to the obtained clustering result and the centroid; Among them, the specific method for determining the delay range value is: Step 1: Calculate the queuing delay w according to the time interval compression co : w co = M nc - M co , Among them, M nc is the cluster center of cluster nc, and M co is the cluster center of cluster co; Step 2: Calculate the queuing delay w according to the time interval expansion de : w de = M de -M nc , Among them, M de is the cluster center of the cluster; Step three: Estimate the total queuing delay w: where, w i is the queuing delay value of the cluster corresponding to i, is the weighted cluster size of the cluster corresponding to i; Step four: Calculate the standard deviation σ: where x j includes co and de, and n' is the sum of the total data packet pair size of cluster co and the total data packet pair size of cluster de; Step five: Calculate the estimated range of the queue delay on the router under test as w±σ; Among them, when the processing module learns the collected data, different types of data need to be cluster - labeled: the compression type is labeled as the co - cluster, the invariant type is labeled as the nc - cluster, and the expansion type is labeled as the de - cluster, and they are applied to the time - interval compression G co , the time - interval invariant G nc and the time - interval expansion G de , and the cluster center M of the co - cluster co , the cluster center M of the nc - cluster nc and the cluster center M of the nc - cluster de .

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