Piggyback design optimization method and system for MDS array code

Abstract

The invention provides a Piggyback design optimization method and a Piggyback design optimization system for an MDS array code, for an index set of each node in a distributed storage system, a plurality of Piggyback functions are redesigned, each index set has a plurality of nodes, and a symbol in a previous instance in each node participates in the generation of the Piggyback functions; for each node, the symbols in the previous instance are named as follows according to the sequence of the instances, and the definition of the new generation function is shown in the specification. According to the invention, the average repair bandwidth ratio of system nodes can be further reduced.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of distributed storage, and particularly relates to a Piggybacking design optimization method and system of MDS array code. BACKGROUND

[0002] The statements in this section merely provide background information related to the present application and do not necessarily constitute the prior art.

[0003] In a distributed storage system, information is stored in different independent storage nodes, each of which has a certain probability of damage. When a storage node is damaged, the storage system needs to use the remaining surviving nodes to repair the damaged node, and the distributed storage code plays an important role in this process.

[0004] When repairing a damaged storage node, the computational load and network bandwidth in the repair process are two important problems to be considered. Generally, repair complexity and repair bandwidth are used to measure them respectively, wherein the repair bandwidth is defined as the amount of data read (transmitted) by the remaining surviving nodes for repairing the node when the node is damaged. The traditional maximum distance separable (MDS) storage code has a lower repair complexity, but has a higher repair bandwidth. On the contrary, the minimum storage regenerating code has a lower repair bandwidth, but has a higher repair complexity.

[0005] Piggybacking can solve the above problems. Piggybacking is a design framework for constructing MDS storage code using multiple instances of traditional MDS code as base code, wherein the number of base codes is usually named as sub-packetization level. The Piggybacking design can effectively reduce the repair bandwidth of the MDS storage code, while having a lower repair complexity.

[0006] However, in the existing Piggybacking design framework of MDS storage code, there is always a part of symbols in the check nodes that are not fully utilized by the piggyback function, and the average repair bandwidth ratio is still too high. SUMMARY

[0007] In order to solve the above problems, the present application provides a Piggybacking design optimization method and system of MDS array code, which can further reduce the average repair bandwidth ratio of the system node.

[0008] According to some embodiments, the present application adopts the technical solutions as follows: The Piggybacking design optimization method of MDS array code comprises the following steps: For the index set of each node in the distributed storage system , redesign piggyback functions for a system MDS array code, the number of system nodes, the number of check nodes, let be the index set of the system nodes, which is evenly or nearly evenly divided into non-intersecting sets , let , each index set contains nodes, and the signs in the first instances of each node participate in the generation of the piggyback function; For each node , , the signs in the first instances are named in the order of instances, and the definition of the new generating function is , where .

[0009] As an optional implementation, when any node is damaged, for the signs in the last instances of the node, download all data signs in the last instances of all undamaged system nodes and all check signs in the last instances of the node, a total of signs, repair the damaged node using the MDS property, and calculate all MDS codes using all nodes in the last instances.

[0010] As an optional implementation, for data signs in the first instances, download all data signs that satisfy , where , and all check signs containing the new generating function , and then calculate the first The data symbols of the damaged nodes in the instances, where .

[0011] As an alternative embodiment, each are equal or approximately equal.

[0012] As an alternative embodiment, The value of or Round to the nearest integer.

[0013] As an alternative embodiment, The value of is determined by traversing the integer The value of , using the Lagrange multiplier method to obtain each The value of makes the average repair bandwidth ratio reach the minimum value. The value of is the final value.

[0014] A piggybacking design optimization system for MDS array codes, comprising: The piggyback function generation module is configured as a set of indicators for each node in the distributed storage system. , redesigned piggyback function , for a system MDS array code, is the number of system nodes, , is the number of check nodes, let for The indicator set of each system node is divided evenly or nearly evenly into disjoint sets ,make , and then for each , each indicator set The CCP has nodes, and the first The symbols in each instance participate in the generation of the piggyback function; The symbolic naming module is configured to , , among which the former The symbols in the instances are named in the order of the instances. , the new generating function is defined as ,in, .

[0015] A computer readable storage medium for storing computer instructions, which when executed by a processor, complete the steps in the above method.

[0016] An electronic device comprising a memory and a processor, and computer instructions stored on the memory and running on the processor, which when executed by the processor, complete the steps in the above method.

[0017] A distributed storage system comprising the above system or constructed using the steps in the above method.

[0018] Compared with the prior art, the beneficial effects of the present application are: The present application can utilize all available check nodes for piggyback function by design optimization, and since The equality holds if and only if In terms of size, when each node in each is repaired, the number of symbols downloaded is less, and the average repair bandwidth ratio is also smaller.

[0019] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the following preferred embodiments are specifically described below, and the accompanying drawings are used for detailed description. BRIEF DESCRIPTION OF DRAWINGS

[0020] The drawings accompanying the specification of the present application form a part of the present application and serve to provide a further understanding of the present application, the illustrative embodiments of the present application and their descriptions serve to explain the present application and do not constitute an improper limitation of the present application.

[0021] Figure 1 is the MDS storage code of the prior art system ; Figure 2 is the MDS storage code of the system obtained after piggybacking of the prior art system Figure 3 is a Piggybacking design schematic diagram of the MDS storage code in the prior art ; Figure 4 is a schematic diagram of the construction idea in the prior art Figure 5 is a Piggybacking design schematic diagram of the optimized MDS storage code in an embodiment Figure 6 is a schematic diagram of the construction idea in an embodiment. DETAILED DESCRIPTION

[0022] The present application will be further described below in conjunction with the drawings and embodiments.

[0023] It should be noted that the following detailed descriptions are illustrative and intended to provide further explanation of the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which the present invention belongs.

[0024] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0025] In the absence of conflict, the embodiments and features in the embodiments of this application can be combined with each other.

[0026] Example 1 To ensure that those skilled in the art have a clearer understanding of the solution provided by the present invention, this embodiment first provides a prior art solution.

[0027] A traditional MDS code (Minimum Distance Separable code) is a length of The definition of A linear code that reaches Singleton Bound on Information symbols and Check symbols (where each symbol is a Elements in ), reaching Singleton Bound means that The entire codeword of symbols can be represented by any This property is also called the MDS property.

[0028] MDS Storage Code is an array code formed by taking multiple traditional MDS code instances as base codes. MDS storage code is nodes, each node stores the total file size Due to the MDS nature of the base code, it can be downloaded by Any node It is not difficult to see that such a code can tolerate any Damage to a node.

[0029] If a node stores uncoded partial raw data, then this node is called a systematic node. A systematic MDS storage code is a storage code in which raw information is stored in uncoded form in a systematic node. The remaining nodes are called parity nodes, which store parity data of the systematic nodes. From a practical point of view, having systematic features is better because in general, data can be directly read from systematic nodes without performing decoding operations. Therefore, the repair property of a failed systematic node is very important in the design of a practical storage system. The present work is an optimization of an existing Piggybacking design for this problem.

[0030] In the definition of parameters, the repair bandwidth of a failed node is defined as the minimum possible amount of data that needs to be downloaded from surviving nodes to repair the node. The average repair bandwidth is the average of the repair bandwidths of all storage nodes. We also define the average repair bandwidth ratio as the ratio of the average repair bandwidth and the amount of raw information, i.e. We use and to denote the average repair bandwidth ratios of systematic nodes and parity nodes, respectively, so , and we have when . Since in many practical applications, high-rate MDS storage codes are considered, we only consider the repair of systematic nodes in this work, so in the following description, let be the average repair bandwidth ratio of systematic nodes. Obviously, in traditional MDS storage codes, the amount of data that needs to be downloaded to repair a single failed node is the same as the amount of raw data, so its . In MDS storage codes, the repair process of a failed node only involves addition and multiplication in a finite field, and multiplication operations in a finite field are at least four times slower than addition operations, so we use repair complexity to measure the computational load, which is defined as the number of multiplication operations needed in the repair of a single node, where the repair complexity of repairing a single node using the MDS property is set to . Since we do not focus on the change of repair complexity in the optimization process, we do not involve this parameter in this work.

[0031] Piggybacking is a design framework that uses multiple instances of traditional MDS codes as base codes to construct MDS storage codes. The number of base codes is often referred to as the sub-packetization level. Piggybacking can effectively reduce the repair bandwidth of MDS storage codes while also maintaining low repair complexity. Next, we will introduce the system's MDS storage code piggybacking design framework: make for The original information is encoded into a traditional system The MDS code is .Pick Put together the examples of such MDS codes as base codes, and let The nodes are system nodes, and then The nodes are check nodes, and we can get Figure 1 The system shown MDS storage code.

[0032] In order to maintain its system properties, only piggyback design operations can be performed on its check nodes. For the On the node The piggyback function of each instance, that is, each piggyback function is a function of the original information in the previous instance. The MDS storage code of the system obtained after piggyback is as follows: Figure 2 shown.

[0033] Its MDS property is reflected in that when any When a node is damaged, the MDS property of the first instance's MDS code can repair all its damaged data, and then the piggyback function of the second instance can be calculated and subtracted to obtain the original MDS code of the second instance. The surviving data can be restored by its MDS property, and the data in the damaged node can be repaired recursively in subsequent instances. The above is the content of the Piggybacking design framework of the system's MDS storage code.

[0034] The present invention aims at an existing method in which the number of sub-packets is a redundant value. The optimization of the piggybacking design of the MDS storage code of the system is to further reduce the average repair bandwidth ratio of the system nodes. The existing design is described as follows: make for The index set of a system node is divided into disjoint sets For convenience of calculation, let , and for each , Let be the piggyback function generated by the data of the index in the original information of the th instance in the th node, that is . Add to the th node of the th instance, where , and the number of packages after piggybacking is The MDS storage code of a system of , the construction result is shown in Figure 3 , and the parameter Figure 3 in is not the same as above. The construction idea can be shown by Figure 4 .

[0035] If it cannot be evenly divided, only a few sets of nodes have different numbers, but the calculation method is the same as other nodes, so it is not repeated here.

[0036] In the existing design scheme, when a node is damaged, let the th node be damaged, then for the symbols in the last instances in the node, all data symbols in the last instances in all undamaged system nodes and all check symbols in the last instances in the th node need to be downloaded, a total of symbols, using the MDS property to repair the damaged data node, and using all data nodes in the last instances to calculate all . For the data symbols in the first instances, all data nodes in the first instances in all nodes and the th to th symbols in the th instance need to be downloaded, a total of symbols, and then the data symbols of the damaged node in the first instances can be calculated.

[0037] ​​In this design, the average repair bandwidth ratio of the system nodes can be calculated as:

[0038] when hour, Get the minimum value .

[0039] Through the description of this design and the idea of ​​piggybacking design, it can be seen that the shortcoming of its design is that there are some symbols in the check nodes that are not utilized by adding a piggyback function. This shows that the design can be further optimized to further reduce the average repair bandwidth ratio.

[0040] To address the problem in existing designs where piggybacking doesn't fully utilize all symbols in the check nodes, the present invention further reduces the average repair bandwidth ratio by utilizing these symbols. Specifically, for certain storage code constructions with specific parameters, the average repair bandwidth ratio can be reduced by more than 5% or even 6%.

[0041] This paper provides a piggybacking design optimization method for MDS array codes. Its innovation lies in redesigning the piggyback function added to each check symbol, fully utilizing all available piggyback check symbols, and ensuring that different data in each node is added to a different piggyback function. The specific design optimization is as follows: For each node's index set , redesigned piggyback function Same as the previous design, assuming , still take , then each The CCP has nodes, and the first The symbols in each instance participate in the generation of the piggyback function. The nodes in the , and for each node , , the front of its node The symbols in the instances are named in the order of the instances. , the new generating function is defined as ,in , the optimized results are as follows Figure 5 As shown, Figure 6 Visually demonstrate its construction ideas.

[0042] In this scheme, the node repair process is as follows: When damaged, for the back of the node The symbols in each instance need to be downloaded from all undamaged system nodes. All data symbols in the instance and The back of the node All check symbols in instances, total symbols, using the MDS property to repair the damaged data nodes, and using the known back All data nodes in the instance are calculated to obtain all . And for the former Data symbols in instances, all data symbols need to be downloaded , which satisfies , and all the included of check symbols, a total of approximately symbols (not considering decimals for now), and then we can calculate the previous Data symbols for the damaged nodes in each instance.

[0043] In the above design, we make each are equal or approximately equal, and The value of is also fixed to (Actually, we need to take the nearest integer.) However, without considering that each value should be an integer, we can also use an algorithm to traverse the integers based on the size of the value. The value of , and then use the Lagrange multiplier method to get each The value of makes the average repair bandwidth ratio reach the minimum value, and this algorithm can give this minimum value.

[0044] After the above design optimization, all the check nodes available for adding piggyback function are utilized. , if and only if When the equal sign is established. Compared with the size, this design repairs each When each node in the network is smaller, the number of symbols required to be downloaded is smaller, and the average repair bandwidth ratio will be smaller.

[0045] Specifically for the optimization of a real parameter code, we let , number of subcontracts , using the piggyback framework to construct an MDS storage code. From the parameters before optimization, we can see that when When , the average repair bandwidth ratio is .

[0046] When using the optimized framework, let Let be the data symbols in the 5th instance, Let be the check symbols in the 5th instance, Let be the check symbols in the 5th instance, where all the check functions are designed as:

[0047]

[0048]

[0049]

[0050]

[0051]

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[0055]

[0056]

[0057]

[0058] Suppose the set of all data symbols that generate the check functions is .

[0059] The final construction form is as follows:

[0060] Suppose the node is damaged, i.e. the data symbol is lost, the process of recovering them is as follows: (1) To repair , download all the data symbols except and the first check symbol in the 5th instance, using the MDS property can repair , a total of 30 symbols need to be downloaded. At the same time, using all the data symbols in the 5th instance including the just repaired , it can also calculate Used for recovery of other nodes.

[0061] (2) For repair , download symbol , and using the calculated value in (1) It can be calculated , then download the symbol , that is, the symbol , then you can Subtract these signs to get , a total of 10 symbols need to be downloaded.

[0062] (3) For repair , download symbol , and using the calculated value in (1) It can be calculated , then download the symbol , that is, the symbol , then you can Subtract these signs to get , a total of 10 symbols need to be downloaded.

[0063] (4) For repair , download symbol , and using the calculated value in (1) It can be calculated , then download the symbol , that is, the symbol , then you can Subtract these signs to get , a total of 10 symbols need to be downloaded.

[0064] (5) For repair , download symbol , and using the calculated value in (1) It can be calculated , then download the symbol , that is, the symbol , then you can Subtract these signs to get , a total of 10 symbols need to be downloaded.

[0065] Hypothetical Node Corruption, i.e. data symbols If lost, the process to recover them is as follows: (1) For repair , download the 5th instance except All data symbols except and the first check symbol , we can repair it by using MDS properties , a total of 30 symbols need to be downloaded.

[0066] (2) For repair , download the 4th instance except All data symbols except and the first check symbol , we can repair it by using MDS properties , a total of 30 symbols need to be downloaded. At the same time, using the newly repaired All data symbols in the fourth instance including Used for recovery of other nodes.

[0067] (3) For repair , download symbol , and using the calculated value in (2) It can be calculated , then download the symbol , that is, the symbol , then you can Subtract these signs to get , a total of 7 symbols need to be downloaded.

[0068] (4) For repair , download symbol , and using the calculated value in (2) It can be calculated , then download the symbol , that is, the symbol , then you can Subtract these signs to get , a total of 8 symbols need to be downloaded.

[0069] (5) For repair , download symbol , and using the calculated value in (2) It can be calculated , then download the symbol , that is, the symbol , then you can Subtract these signs to get , a total of 8 symbols need to be downloaded.

[0070] Hypothetical Node Corruption, i.e. data symbols If lost, the process to recover them is as follows: (1) For repair , download the 5th instance except All data symbols except and the first check symbol , we can repair it by using MDS properties , a total of 30 symbols need to be downloaded.

[0071] (2) To repair , all the data symbols in the 4th example except and the first check symbol are downloaded, and the repaired can be obtained using the MDS property. A total of 30 symbols need to be downloaded.

[0072] (3) To repair , all the data symbols in the 3rd example except and the first check symbol are downloaded, and the repaired can be obtained using the MDS property. A total of 30 symbols need to be downloaded. Meanwhile, using all the data symbols in the 3rd example including the just repaired , the for the remaining nodes can also be calculated.

[0073] (4) To repair , the symbol is downloaded, and the calculated in (3) can be used to calculate , and the symbol is downloaded, i.e. the symbol , which can be obtained by subtracting these symbols. A total of 5 symbols need to be downloaded.

[0074] (5) To repair , the symbol is downloaded, and the calculated in (3) can be used to calculate , and the symbol is downloaded, i.e. the symbol , which can be obtained by subtracting these symbols. A total of 5 symbols need to be downloaded.

[0075] By combining the above repair methods for nodes in different index sets , the sum of the repair bandwidths of all system nodes can be calculated as . The average repair bandwidth of all system nodes is , and the average repair bandwidth ratio is , which is lower than the average repair bandwidth ratio of the previous construction.

[0076] ​​Similarly, for other general parameters, the constructed and calculated optimized design can reduce the average repair bandwidth of the system nodes by more than Even .

[0077] A piggybacking design optimization method of MDS array code, comprising the following steps: For each node in the distributed storage system , redesign piggyback function For a system of MDS array code, The number of system nodes is The number of check nodes is Let The index set of The number of system nodes is evenly or nearly evenly divided into Non-intersecting sets For convenience of calculation, let , and for each , there are Nodes in each index set , and the signs in the first Instance participate in the generation of the piggyback function; For each node , The signs in the first Instance are named in the order of the instance , and the new generating function is defined as .

[0078] As an optional implementation, when any node is damaged, for the signs in the last Instance in the node, download all data symbols in the last Instance in all undamaged system nodes and all check symbols in the last Instance in the first Node, a total of Symbols, repair the damaged node using the MDS property, and calculate all MDS codes Using all nodes in the last Instance.

[0079] As an optional implementation, for the data symbols in the first Instance, download all data symbols , which satisfy ,in , and all containing new generating functions The check symbol of The data symbols of the damaged nodes in the instances, where .

[0080] As an alternative embodiment, each are equal or approximately equal.

[0081] As an alternative embodiment, The value of or Round to the nearest integer.

[0082] As an alternative embodiment, The value of is determined by traversing the integer The value of , using the Lagrange multiplier method to obtain each The value of makes the average repair bandwidth ratio reach the minimum value. The value of is the final value.

[0083] Example 2 A piggybacking design optimization system for MDS array codes, comprising: The piggyback function generation module is configured as a set of indicators for each node in the distributed storage system. , redesigned piggyback function , for a system MDS array code, is the number of system nodes, , is the number of check nodes, let for The indicator set of each system node is divided evenly or nearly evenly into disjoint sets , for the convenience of calculation, let , and then for each , each indicator set The CCP has nodes, and the first The symbols in each instance participate in the generation of the piggyback function; The symbolic naming module is configured to , , among which the former The symbols in the instances are named in the order of the instances. , the new generating function is defined as wherein, .

[0084] Embodiment Three A computer readable storage medium storing computer instructions, which when executed by a processor, complete the steps in the above method.

[0085] Embodiment Four An electronic device comprising a memory and a processor, and computer instructions stored on the memory and running on the processor, which when executed by the processor, complete the steps in the above method.

[0086] Embodiment Five A distributed storage system comprising the above system or constructed using the steps in the above method.

[0087] Those skilled in the art should understand that embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including, but not limited to, disk memory, optical storage, etc.) containing computer-usable program code. CD ROM - Figure 1

[0088] The present application is described in reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, and the combination of flows and / or blocks in the flowcharts and / or block diagrams can be implemented by computer program instructions. These computer program instructions can be provided to a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing apparatus to produce a machine, so that the instructions executed by the computer or other programmable data processing apparatus produce the functions specified in the flowcharts and / or block diagrams of the flows and / or blocks. Figure 1 The flowcharts and / or block diagrams of the flows and / or blocks Figure 1 The flowcharts and / or block diagrams of the flows and / or blocks

[0089] These computer program instructions can also be stored in a computer-readable memory that can direct the computer or other programmable data processing apparatus to work in a specific manner, so that the instructions stored in the computer-readable memory produce a manufactured product comprising instruction means that implement the functions specified in the flowcharts and / or block diagrams of the flows and / or blocks. Figure 1 The flowcharts and / or block diagrams of the flows and / or blocks Figure 1 The flowcharts and / or block diagrams of the flows and / or blocks

[0090] ​These computer program instructions can also be loaded into a computer or other programmable data processing devices, so that a series of operational steps are generated to realize the computer-implemented processes in the computer or other programmable devices, and the instructions executed in the computer or other programmable devices provide processes for implementing the functions specified in the flowchart Figure 1 one flow or a plurality of flows and / or the functions specified in the block ​ one flow or a plurality of flows and / or the functions specified in the block

[0091] The above only describes the preferred embodiments of the present application and is not intended to limit the present application. Various modifications and changes can be made to the present application by those skilled in the art without departing from the spirit and principles of the present application. Any modifications, equivalent replacements, improvements, etc. made by those skilled in the art without creative efforts should be included in the protection scope of the present application.

Claims

1. A Piggybacking design optimization method for MDS array codes, characterized by: The following steps are involved: For each node in the distributed storage system, the index set , redesigned piggyback function , for a system MDS array code, is the number of system nodes, , is the number of check nodes, let for The indicator set of each system node is divided evenly or nearly evenly into disjoint sets , each indicator set The CCP has nodes, and the first The symbols in each instance participate in the generation of the piggyback function; For each node , , among which the former The symbols in the instances are named in the order of the instances. , the new generating function is defined as ,in, .

2. The Piggybacking design optimization method for an MDS array code according to claim 1, wherein: When any node is damaged, the following symbols in each instance, download the following from all undamaged system nodes All data symbols in the instance and The back of the node All check symbols in instances, total symbols, using the MDS property to repair the damaged nodes, and using the known back All nodes in the instance calculate all MDS codes .

3. The Piggybacking design optimization method for an MDS array code according to claim 2, wherein: For the former Data symbols in instances, download all data symbols , which satisfies ,in , and all containing new generating functions The check symbol of The data symbols of the damaged nodes in the instances, where .

4. A Piggybacking design optimization method for an MDS array code according to any one of claims 1 to 3, characterized in that: Each are equal or approximately equal.

5. A Piggybacking design optimization method for an MDS array code according to any one of claims 1 to 3, characterized in that: The value of or Round to the nearest integer.

6. A Piggybacking design optimization method for an MDS array code according to any one of claims 1 to 3, characterized in that: The value of is determined by traversing the integer The value of , using the Lagrange multiplier method to obtain each The value of makes the average repair bandwidth ratio reach the minimum value. The value of is the final value.

7. A Piggybacking design optimization system for MDS array codes, characterized by: include: The piggyback function generation module is configured as a set of indicators for each node in the distributed storage system. , redesigned piggyback function , for a system MDS array code, is the number of system nodes, , is the number of check nodes, let for The indicator set of each system node is divided evenly or nearly evenly into disjoint sets , each indicator set The CCP has nodes, and the first The symbols in each instance participate in the generation of the piggyback function; The symbolic naming module is configured to , , among which the former The symbols in the instances are named in the order of the instances. , the new generating function is defined as ,in, .

8. A computer-readable storage medium, characterized in that: Used to store computer instructions, which, when executed by a processor, complete the steps of the method according to any one of claims 1 to 6.

9. An electronic device, characterized in that: The method comprises a memory and a processor, and computer instructions stored in the memory and executed on the processor, wherein the steps of the method according to any one of claims 1 to 6 are completed when the computer instructions are executed by the processor.

10. A distributed storage system, characterized in that: The system comprises the system of claim 7 or is constructed using the steps of the method of any one of claims 1 to 6.

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