JP2013530477A - Method for quantifying and analyzing parallel processing of algorithms - Google Patents

Abstract

A method for quantifying and analyzing an algorithm's intrinsic parallelism, wherein the method is configured to be implemented by a computer, the method configured to represent the algorithm by a plurality of sets of operations. A computer is configured to obtain a Laplace matrix according to the plurality of sets of operations, a computer is configured to calculate eigenvalues and eigenvectors of the Laplace matrix, and the computer is configured to calculate the eigenvalues and the eigenvalues of the Laplace matrix. Configuring the method to obtain a set of information relating to the intrinsic parallel processing of the algorithm according to the eigenvectors.

Description

  The present invention relates to a method for quantifying and analyzing parallel processing of an algorithm, and more particularly to a method for quantifying and analyzing intrinsic parallel processing of an algorithm.

  G. M.M. Amdahl introduced a method for parallel processing of algorithms according to the ratio of the sequential execution part of the algorithm. ("Effectiveness of a single processor approach to achieve large-scale computing performance," AFIPS conference proceedings, pages 483 to 485, 1967).

  The disadvantage of Amdahl's method is that the degree of parallel processing of the algorithm obtained by this method depends on the target platform on which the method is executed and not necessarily on the algorithm itself. Thus, the degree of parallelism obtained using Amdahl's method is not intrinsic to the algorithm and is biased by the target platform.

  A. Prehozy et al. Proposed a method for evaluating parallelism of an algorithm based on the ratio of the critical path length to the complexity of the algorithm ("Evaluation of parallelism for efficient multimedia implementation: Dynamic evaluation of algorithm critical path, "IEEE paper on circuits and systems for video technology, pages 593 to 608, volume 15, No. 5, May 2005). Complexity is the total number of operations of the algorithm, and critical path is the maximum number of operations that must be executed sequentially due to dependency on operation data. This method may be able to determine the average degree of parallel processing embedded in the algorithm, but is not sufficient to exhaustively determine the various multigrain parallel processing embedded in the algorithm.

  Accordingly, it is an object of the present invention to provide a method for quantifying and analyzing the intrinsic parallelism of an algorithm that is not biased by the target hardware and / or software platform.

Thus, the method for quantifying and analyzing the intrinsic parallelism of the algorithm according to the invention is configured to be implemented by a computer and comprises the following steps:
a) configure the computer to represent algorithms with multiple sets of operations,
b) configuring the computer to obtain a Laplace matrix according to multiple sets of operations;
c) configuring the computer to compute eigenvalues and eigenvectors of the Laplace matrix;
d) Configure the computer to obtain a set of information about the intrinsic parallelism of the algorithm according to the eigenvalues and eigenvectors of the Laplace matrix.

  Other features and advantages of the present invention will become apparent in the following detailed description of the preferred embodiments with reference to the accompanying drawings. The attached drawings are as follows.

Fig. 6 is a flow chart illustrating a preferred embodiment of a method for quantifying and analyzing the intrinsic parallelism of the algorithm of the present invention. FIG. 6 is a schematic diagram illustrating data flow information regarding an example algorithm. It is the schematic which shows the example of the set of a data flow graph. It is the schematic which shows the calculation set of a 4x4 discrete cosine transform algorithm. 6 is a schematic diagram illustrating an example of a configuration of intrinsic parallel processing corresponding to a dependency depth equal to 6. FIG. 6 is a schematic diagram illustrating an example of a configuration of intrinsic parallel processing corresponding to a dependency depth equal to 5. FIG. 3 is a schematic diagram illustrating an example of a configuration of intrinsic parallel processing corresponding to a dependency depth equal to 3. FIG.

  Referring to FIG. 1, a preferred embodiment of a method for evaluating the intrinsic parallelism of an algorithm according to the present invention is configured to be implemented by a computer and has the following steps.

  The essential degree of parallel processing indicates the degree of parallel processing of the algorithm itself without considering the design and configuration of software and hardware. That is, the method of the present invention is not limited by software and hardware when analyzing the algorithm.

In step 11, the computer is configured to represent an algorithm with a plurality of operation sets. Each set of operations may take any form that represents an equation, program code, flowchart, or other algorithm. In the example below, the algorithm includes three operation sets O1, O2, and O3, represented as follows:
O1 = A ₁ + B ₁ + C ₁ + D ₁ ,
O2 = A ₂ + B ₂ + C ₂ , and
O3 = A ₃ + B ₃ + C ₃

Step 12 is to configure the computer to obtain a Laplace matrix L _d according to the operation set and includes the following sub-steps.

In sub-step 121, according to the operation set, the computer is configured to obtain data flow information regarding the algorithm. As shown in FIG. 2, the data flow information corresponding to the operation set of this example may be expressed as follows.
Data 1 = A ₁ + B ₁
Data 2 = A ₂ + B ₂
Data 3 = A ₃ + B ₃
Data 4 = Data 1 + Data 7
Data 5 = Data 2 + C ₂
Data 6 = Data 3 + C ₃
Data 7 = C ₁ + D ₁

In sub-step 122, the computer is configured to obtain a data flow graph according to the data flow information. The data flow graph is composed of a plurality of vertices representing operations in the algorithm and a plurality of directional edges indicating the interconnection of two corresponding vertices of the vertices and indicating the source and destination of the data in the algorithm . For the data flow information shown in FIG. 2, operational symbols V ₁ through V ₇ (ie, vertices) are used instead of addition symbols, and arrows (ie, directional edges) indicate the data source and destination. As a result, the data flow graph of FIG. 3 is obtained. In particular, the operation symbol V ₁ indicates an addition operation A ₁ + B ₁ , the operation symbol V ₂ indicates an addition operation A ₂ + B ₂ , the operation symbol V ₃ indicates an addition operation A ₃ + B ₃ , and the operation symbol V ₄ adds. Operation data 1 + data 7 is shown, operation symbol V ₅ is addition operation data 2 + C ₂ , operation symbol V ₆ is addition operation data 3 + C ₃ , and operation symbol V ₇ is addition operation D ₁ + C ₁ .

From the data flow graph shown in FIG. 3, it can be seen that the operation symbol V ₄ depends on the operation symbols V ₁ and V ₇ . Similarly, the operation symbol V ₅ depends on the operation symbol V ₂ , the operation symbol V ₆ depends on the operation symbol V _3, and the operation symbols V ₄ , V _5, and V ₆ do not depend on each other.

In sub-step 123, the computer is configured to obtain a Laplace matrix L _d according to the data flow graph. In the Laplace matrix L _d , the i-th diagonal element indicates the number of arithmetic symbols connected to the arithmetic symbol V _i, and the element outside the diagonal indicates whether two arithmetic symbols are connected. . Therefore, the Laplace matrix L _d can clearly show a data flow graph in linear algebra form. The set of data flow graphs shown in FIG. 3 may be shown as follows:

The Laplace matrix L _d indicates connectivity between the operation symbols V ₁ to V ₇ , and the first to seventh columns indicate the operation symbols V ₁ to V ₇ , respectively. For example, in the first column, the operation symbol V ₁ is connected to the operation symbol V ₄ , and thus the matrix element (1, 4) is −1.

In step 13, the computer is configured to calculate the eigenvalue λ and eigenvector X _d of the Laplace matrix L _d . For the Laplace matrix L _d acquired in the above example, the eigenvalue λ and the eigenvector X _d are as follows.

In step 14, the computer, in accordance with eigenvalue λ and eigenvectors X _d Laplace matrix L _d, configured to obtain a set of information about the intrinsic parallelism of the algorithm. The set of information regarding the essential parallel processing is strictly defined so as to recognize the independent operation sets that are independent from each other and can be executed in parallel processing. The set of information related to strict parallel processing includes the degree of strict parallel processing that indicates the number of independent operation sets, and the set of strict parallel processing configurations corresponding to the operation sets. Including. F. R. K. Based on the spectral graph theory introduced by Chang (Mathematical Regional Conference Series, No. 92, 1997), the number of connected components of the graph is equal to the number of eigenvalues of the Laplace matrix equal to zero. The degree of strict parallel processing embedded in the algorithm is therefore equal to the number of eigenvalues λ equal to zero. Moreover, based on spectral graph theory, the exact parallel processing configuration may be specified according to the eigenvector X _d associated with the eigenvalue λ equal to zero.

From the above example, it can be seen that since there are three Laplace eigenvalues equal to 0, the dataflow graph set is composed of three independent sets of operations. Therefore, the degree of strict parallel processing embedded in the illustrated algorithm is equal to 3. Next, the first, second, and third eigenvectors of eigenvector X _d are associated with an eigenvalue λ equal to zero. By observing the first eigenvector of the eigenvector X _d , it is clear that the values corresponding to the operation symbols V ₁ , V ₄ and V ₇ are not 0, that is, the operation symbols V ₁ , V ₄ and V ₇ depends and forms a connected (V ₁ -V ₄ -V ₇ ) of the data flow graph. Similarly, the second and third eigenvectors of the eigenvector X _d associated with the eigenvalue λ equal to 0 are formed, the operation symbols V ₂ , V ₅ and the operation symbols V ₃ , V ₆ are dependent, and the data flow graph the remaining two connected ones (V ₂ -V ₅ and V ₃ -V ₆₎ is understood to form respectively. Thus, the computer obtains a degree of strict parallel processing equal to 3 and a strict parallel processing configuration that can be expressed in the form of a graph, table, equation, or program code (as shown in FIG. 3). Configured to do.

  In step 15, the computer is configured to obtain a plurality of sets of information regarding multi-grain parallelism of the algorithm according to at least one of the plurality of dependency depths of the algorithm and a set of information regarding strict parallelism. The The set of information regarding multi-grain parallel processing includes a set of information regarding broad parallel processing of algorithms that characterize all possible parallel processing embedded in an independent set of operations.

  Note that the depth of dependency of an algorithm represents an associated sequential step that is important in the processing of the algorithm, that is, complementary to the possible parallel processing of the algorithm. Thus, information regarding the intrinsic parallel processing of different algorithms may be obtained based on different dependency depths. In particular, information on strict parallel processing is information on the intrinsic parallel processing of the algorithm corresponding to the largest of the algorithm dependency depths, and information on parallel processing in a broad sense is information on the dependency depths. Information on the essential parallel processing of the algorithm corresponding to the smallest one.

For example, the above algorithm includes two different configurations of strictly parallel processing: V ₁ -V ₄ -V ₇ and V ₂ -V ₅ (V ₃ -V ₆ is similar to V ₂ -V _5. And is considered the same configuration). The configuration of strict parallelism V ₁ -V ₄ -V _7, an independent operation symbol V ₁ and V ₇ are mutually, i.e., it can be seen that the operation symbol V ₁ and V ₇ can be processed in parallel . Therefore, the set of information regarding the parallel processing in the broad sense of the algorithm includes the degree of parallel processing in the broad sense equal to 4, and the configuration of the parallel processing in the broad sense is similar to the configuration of strict parallel processing.

According to the method of this embodiment, the degree of parallel processing in the broad sense of the above algorithm is equal to 4. The processing element requires seven processing cycles to implement the algorithm. This is because the algorithm includes seven operational symbols V ₁ -V ₇ . According to the degree of strict parallel processing equal to 3, to implement an algorithm using three processing elements, three processing cycles are used. According to the degree of parallel processing in a broad sense equal to 4, two processing cycles are used to implement an algorithm using four processing elements. Furthermore, although more processing elements are used, it can be seen that at least two processing cycles are required to implement the algorithm. Thus, the optimal number of processing elements used to implement the algorithm can be obtained according to the method of this embodiment.

  Taking a 4 × 4 discrete cosine transform (DCT) as an example, the operation set of the DCT algorithm is represented by the data flow graph shown in FIG. Since 4x4 DCT is well known by those skilled in the art, further details thereof are omitted here for the sake of brevity. FIG. 4 shows that the maximum dependency depth of the 4 × 4 DCT algorithm is equal to 6. For the maximum dependency depth (ie 6), the exact parallelism configuration of this algorithm may be obtained as shown in FIG. According to the method of this embodiment, it is equal to 4. When analyzing the intrinsic parallelism of the 4 × 4 DCT algorithm with one of the dependency depths equal to 5, the intrinsic parallelism configuration of this algorithm can be obtained as shown in FIG. Well, the degree of intrinsic parallelism is equal to 8. Furthermore, when analyzing the intrinsic parallelism of the 4 × 4 DCT algorithm with one of the dependency depths equal to 3, the intrinsic parallelism configuration of this algorithm is obtained as shown in FIG. The degree of intrinsic parallel processing is equal to 16.

  That is, the method according to the invention may be used to evaluate the intrinsic parallelism of the algorithm.

  Although the present invention has been described in connection with what are considered to be the most practical and preferred embodiments, the invention is not limited to the described embodiments and is included in the spirit and scope of the broadest interpretation. It is intended to cover various arrangements that may be considered, and should be considered to include all such modifications and equivalent arrangements.

Claims (11)

A method for quantifying and analyzing the intrinsic parallelism of an algorithm, said method being configured to be implemented by a computer, said method comprising:
a) configuring the computer to represent the algorithm with a plurality of sets of operations;
b) configuring the computer to obtain a Laplace matrix according to the plurality of operation sets;
c) configuring the computer to calculate eigenvalues and eigenvectors of the Laplace matrix; and d) setting the computer with information about intrinsic parallel processing of the algorithm according to the eigenvalues and eigenvectors of the Laplace matrix. Comprising the steps of: obtaining the method. Step b)
b1) Substep for configuring the computer to obtain data flow information regarding the algorithm according to a set of operations;
b2) The computer is configured to acquire a dataflow graph composed of a plurality of vertices indicating operations in the algorithm according to the dataflow information, and indicates an interconnection of two corresponding vertices among the vertices. A sub-step configured to obtain a plurality of directional edges indicating the source and destination of the data, and b3) a sub-step configured to obtain the Laplace matrix according to the data flow graph ,
The method of claim 1 comprising: Step d)
d1) a sub-step for configuring the computer to obtain a set of information relating to strict parallel processing of the algorithm according to the eigenvalues and eigenvectors of the Laplace matrix; and d2) a plurality of dependencies of the algorithm. A sub-step configured to obtain a set of information relating to multi-grain parallelism of the algorithm according to at least one of the probabilities and a set of information relating to the strict parallelism;
The method of claim 1 comprising:   The strict parallel processing information set includes a strict parallel processing degree representing the number of independent operation sets of the algorithm, and a strict parallel processing configuration set corresponding to the operation set, respectively. The method according to claim 3.   In sub-step d2), the computer is configured to respectively obtain a plurality of sets of information regarding multi-grain parallel processing of the algorithm according to the dependency depth and the set of information regarding the strict parallel processing. The method according to claim 3.   6. The method of claim 5, wherein each of the plurality of sets of information regarding multi-grain parallel processing of the algorithm includes a degree of multi-grain parallel processing and a set of configurations of multi-grain parallel processing.   The set of information regarding multi-grain parallel processing includes a set of information regarding parallel processing of the algorithm in a broad sense obtained according to the minimum of the dependency depth and the set of information regarding strict parallel processing. The method according to claim 3.   The set of information regarding the parallel processing in the broad sense includes a degree of parallel processing in the broad sense characterizing all possible parallel processing embedded in the independent operation set of the algorithm, and a set of configurations of the broad sense of parallel processing. The method of claim 7, comprising:   4. The method according to claim 3, wherein in sub-step d1), the degree of strict parallel processing is equal to the number of eigenvalues equal to 0 based on spectral graph theory.   4. The method of claim 3, wherein the information regarding multi-grain parallel processing includes a degree of multi-grain parallel processing and a set of configurations for multi-grain parallel processing.   A computer program product comprising a machine-readable storage medium having recorded thereon program instructions for causing a computer to perform the method for quantifying and analyzing the intrinsic parallel processing of the algorithm of claim 1.

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