Patent Publication Number: US-9424076-B1

Title: Dynamic distribution for distributed arrays and related rules

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 13/955,152, filed Jul. 31, 2013 which is a continuation of U.S. patent application Ser. No. 13/163,169, filed Jun. 17, 2011, now U.S. Pat. No. 8,510,366, which is a continuation of U.S. patent application Ser. No. 12/778,746, filed May 12, 2010, now U.S. Pat. No. 7,987,227, which is a continuation of U.S. patent application Ser. No. 11/387,565, filed Mar. 22, 2006, now U.S. Pat. No. 7,743,087, the disclosures of which are incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention generally relates to parallel computing. More particularly, the present invention relates to dynamic distribution of distributed arrays in the parallel computing environment. 
     BACKGROUND INFORMATION 
     MATLAB® is a product of The MathWorks, Inc. of Natick, Mass., that provides an environment for technical computing to engineers, scientists, mathematicians, and educators across a diverse range of industries. MATLAB® is an intuitive high performance language and a technical computing application that provides mathematical and graphical tools for mathematical computation, data analysis, visualization and algorithm development. As a desktop application, MATLAB® typically runs on a single computer. However, there is a limit to the complexity and the size of a problem that a single computer can solve. For example, for a data intensive program utilizing a few 1000 by 1000 arrays, there may not be enough processing power on the single computer to manipulate these arrays, or the program may take days to complete execution. Furthermore, there may not be enough memory on a processor to store one or more of the arrays needed for a data intensive program. 
     With many engineering and scientific problems requiring larger and more complex modeling, many scientists and engineers are interested in using parallel computing to solve their problems. Arrays are an integral part of technical computing and with distributed arrays, a user can employ more computing resources than with a single computer. U.S. patent application Ser. No. 11/173,645 discusses implementations of distributed arrays focused on evenly distributing arrays across multiple parallel processors. Flexible partitioning removes the limitation of having to evenly distribute the arrays across the parallel processors but introduces additional complexity in determining how an array is to be distributed. 
     Therefore, it is desirable to provide a technique that reduces the complexity in determining how arrays are partitioned to distribute them across multiple parallel processors. 
     SUMMARY OF THE INVENTION 
     The present invention relates in general to parallel computing. Specifically, the present invention is related to dynamic distribution of arrays in parallel computing in a technical computing environment. Operations performed on such distributed arrays are handled using flexible partitioning of the distributed array based on one or more criteria. Based on the established criterion, a partitioning scheme is used to perform a given operation. In some embodiments, rule sets are used to determine how distributed arrays are partitioned in performing an operation. The present invention provides a methodology for partitioning distributed arrays based on one or more goals established by one or more criteria. 
     In one aspect of the present invention, a method is provided for dynamically distributing arrays in a parallel computing environment. The method includes the steps of obtaining a criterion for the processing of such a distributed array and performing flexible partitioning based on obtained criteria. In certain embodiments, the method further includes performing analysis based on the criterion obtained. The criterion, how the flexible portioning is performed, and the analysis performed based on the criterion may each be specified by a user. In one embodiment of the invention, there may be predefined rules for partitioning arrays based on the desired criteria. Such rules may address the functions and manipulations with the arrays and what should be done in particular classes or individual operations. Certain embodiments of the present invention deal with the desire to minimize the communications between the processors as one of the criteria in distributing the arrays. Minimizing the communications may improve the overall performance. 
     In another aspect of the present invention, a computer-readable medium for holding instructions for dynamically distributing arrays in a parallel computing environment is provided. The instructions include the steps of obtaining a criterion for the distribution of an array, and performing flexible partitioning based on obtained criterion. In certain embodiments, the instructions further include the step of performing analysis based on the obtained criterion. The criterion, how the flexible portioning is performed, and the analysis performed based on the criterion may each by specified by a user. 
     In still another embodiment of the present invention, a system is provided for dynamically distributing arrays in a parallel computing environment. The system includes an application providing a parallel computing environment, a first parallel computing unit executing in the parallel computing environment, a second parallel computing unit executing in the parallel computing environment, and a dynamic distribution scheme for partitioning an array for distribution across the first and second parallel computing units based on an obtained criterion. In certain embodiments the dynamic distribution scheme, or portions thereof may be specified by a user 
     The details of various embodiments of the invention are set forth in the accompanying drawings and the descriptions below. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing and other objects, aspects, features, and advantages of the invention will become more apparent from the following description and accompanying drawings, in which like reference characters refer to the same parts throughout the different views. 
         FIG. 1A  is an exemplary depiction of a prior art distributed array evenly distributed across multiple processing units. 
         FIG. 1B  is an example depiction of a distributed array wherein the distribution is not even across multiple processing units. 
         FIG. 2  is an exemplary computing device that is suitable for practicing one embodiment of the present invention. 
         FIG. 3A  is a block diagram of an exemplary distributed computing environment that is suitable to practice an embodiment of the present invention in a single computing device. 
         FIG. 3B  is a block diagram of another exemplary distributed computing environment that is suitable to practice one embodiment of the present invention via a network. 
         FIG. 4  is a flow diagram depicting an illustrative embodiment of a method of the present invention. 
         FIG. 5  is a flow diagram depicting another illustrative embodiment of a method of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention provides a system and method for dynamically distributing an array in a parallel computing environment. In one aspect, the present invention is related to a method to execute a computer program in multiple execution environments. For a computer program utilizing a distributed array, the distributed array may be dynamically distributed across the multiple execution environments and each execution environment handles execution and calculation only for a portion of the distributed array. This provides an easy method for parallel processing of a distributed array executing on multiple computers and/or multiple processors to leverage the computing power of multiple computing resources. The automatic distribution also helps eliminate the requirement that the user of the parallel computing environment understand how to program parallel programming paradigms as well as how to best distribute or partition the distributed array. 
     With the availability of parallel computers, researchers and scientists are interested in parallel computing in a technical computing environment, such as, for example, MATLAB®. Arrays are an integral part of technical computing and in modern-day computation a single array may be large enough that an improvement in speed and latency may be achieved by distributing the single array across multiple computing resources. As used herein, arrays include matrices and multidimensional grids such as look-up tables. 
     With distributed arrays, a user can employ more computing resources than with a single computer. However, the implementation of the distributed arrays can affect the complexity of a computer program and user-visible side effects and performance. With identical low level parallel programming paradigms, different top level programming syntaxes can result in a very different user experience. 
     In U.S. patent application Ser. No. 11/173,645, a distributed array implementation for a parallel version of MATLAB has been introduced. The distribution scheme considered there largely addresses evenly distributing or partitioning the distributed array across the parallel processing units. The processing units may also be referred to as “labs.” An example of even distribution can be seen in  FIG. 1A . Here, a distributed array  102  having dimensionality of N is distributed over four (4) processing units or labs  104  with each partitioned as having N/4 of the distributed array  102 . 
     Flexible partitioning is built upon this concept and allows non-even distribution partitioning. An example of this can be seen in  FIG. 1B . Here, the distributed array  102  having dimensionality N is again distributed across four (4) labs  104   a ,  104   b ,  104   c ,  104   d . But, in this example the distributed array  102  is partitioned so that Lab  1   104   a  handles N/2 of the distributed array  102 , Lab  2   104   b  handles N/8 of the distributed array  102 , Lab  3   104   c  handles N/4 of the distributed array  102 , and Lab  4   104   d  handles N/8 of the distributed array  102 . The advantage of such flexible partitioning is that Lab  1   104   a  may have more resources available for processing so it can handle a larger portion on distributed array  102  while Labs  2   104   b  and Lab  4   104   d  have fewer resources available for processing, which necessitates Lab  2   104   b  and Lab  4   104   d  handling smaller portions of the distributed array  102 . It will be understood that resource management is but one of several possible factors that flexible partitioning can be based upon. A disadvantage of flexible partitioning is that it now introduces nearly limitless possibilities for distribution of arrays and the processing of distributed arrays that compounds the complexity issue that distributed arrays were created to address. 
     The present invention addresses the above-mentioned problems by using one or more criteria to guide how flexible partitioning is to be handled in a parallel processing environment. In one aspect of the present invention, one or more criteria are considered for directing the handling of flexible partitioning. Thus, an array may be dynamically distributed based on the one or more criteria. 
     The illustrated embodiments will be described for illustrative purposes relative to a MATLAB® based application, however one of ordinary skill in the art will appreciate that the present invention may be applied to parallel execution of other applications such as LabVIEW® or MATRIXx from National Instruments, Inc., or Mathematica® from Wolfram Research, Inc., or Mathcad of Mathsoft Engineering &amp; Education Inc., or Maple™ from Maplesoft, a division of Waterloo Maple Inc. The illustrative embodiment may also be applied in graphical modeling environments such as Simulink® for MATLAB®. 
       FIG. 2  illustrates an exemplary computing device that is suitable for practicing an illustrative embodiment of the present invention. Computing device  200  includes storage  202 , such as a hard drive or CD-ROM, for storing application  204  and operating system  208 . Computing device  200  may be any computer system such as a desktop computer, laptop, workstation, server, handheld computer, or other form of computing or telecommunication device that is capable of communication and that has sufficient computing power to perform the operations described herein. Computing device  200  further includes a display  218  which a user may interact with computing device  200  using I/O devices such as a keyboard  216  and a pointing device  214  such as a mouse or stylus. A first processor  210  and optionally a second processor  212  are utilized by computing device  200  for running operating system  208 , application  204 , and other software in storage  202 . One of ordinary skill in the art will appreciate that first and second processors  210  and  212  may be multiple core processors, multiple processors, other hardware component with computational capability, such as a microcontroller, application specific integrated circuit, field-programmable gate arrays, virtual machines or the like or any combination thereof. 
     Application  204  provides a parallel computing environment  206  that allows parallel processing of application  204  on the first and second processors  210  and  212 . Application  204  is illustrated here as Parallel MATLAB® or a MATLAB® based application that is suitable for parallel or distributed processing. However, one of ordinary skill in the art will appreciate that the present invention may apply to other technical computing environments suitable for parallel or distributed processing. Computing device  200  may further include a network interface  220  to interface to a Local Area Network (LAN), Wide Area Network (WAN), or the Internet through a variety of connections including, but not limited to, standard telephone lines, LAN or WAN links, broadband connections, wireless connections, or some combination of any or all of the above. The network interface  220  allows computing device  200  to interface with another computing device that is capable of executing in the parallel computing environment to allow parallel processing on multiple computing devices. 
       FIG. 3A  illustrates a tightly coupled environment that is suitable for practicing the present invention. Computing device  200  includes a first parallel computing unit  222  and a second parallel computing unit  224 . In this embodiment, a parallel computing unit may be a processor, a multiple core processor, multiple processors, or other hardware component with computational capability, such as a microcontroller, application specific integrated circuit, or field-programmable gate arrays. 
       FIG. 3B  illustrates another embodiment that is suitable to practice another exemplary parallel computing environment of the present invention. First parallel computing unit  222  communicates with second parallel computing unit  224  via network  228  such as a LAN, WAN, or wireless network. In this embodiment, a parallel computing unit may be a computing device such as computing device  200 , a cluster of workstations or the like. One of ordinary skill in the art will appreciate that a distributed computing environment may comprise a combination of both embodiments illustrated in  FIG. 3A  and  FIG. 3B  as well as any number of parallel computing units. 
     In either of the embodiments shown in  FIG. 3A  and  FIG. 3B , the first parallel computing unit  222  manages a first execution environment or lab and the second parallel computing unit  224  manages a second execution environment or lab. Each execution environment has a local copy of a computer program that is executed in the corresponding parallel computing unit, so there is no main parallel computing unit that distributes executions to the other parallel computing units. 
     The first execution environment and second execution environment are capable of communicating with each other via an interface. In an exemplary embodiment, the first execution environment and the second execution environment can communicate with each other by sending messages via a Message Passing Interface (MPI). MPI is a de facto standard for communication among the nodes running a parallel program on a distributed memory system. Implementation of the MPI specification permits programs with separate address spaces to synchronize with one another and move data from the address space of one process to that of another by sending and receiving messages. In another embodiment, the first and second execution environments may interface via socket based communications over TCP/IP implementing a custom message specification or other available message communications projects that allow the execution environments to send and receive messages. 
     A distributed array is implemented such that the executions of the distributed array are distributed or partitioned among the multiple execution environments or labs. Each of the parallel computing units stores locally a partitioned portion of the data contents of the distributed array. The multiple execution environments know which portion of the distributed array is stored on which of the multiple parallel computing units. A parallel computing unit performs execution only on the portion of the distributed array that is stored locally and if the parallel computing unit needs some portion of the distributed array that is not stored locally, it can send a message to other parallel computing units to request for the information needed. 
     In some embodiments, platform independent technology may be leveraged to achieve the distributed computations. For example, dynamic distribution may be performed on top of any general purpose scheduler. As such, no specific tailoring to the hardware is required at the user level. This enables the distribution to be achieved on heterogeneous clusters that may include different types of hardware and operating systems, as long as the scheduler and the application support them. 
     Flexible partitioning allows a distributed array to be partitioned across multiple execution environments or labs in any number of ways. In one embodiment of the present invention, a method is provided for dynamically distributing arrays in a parallel computing environment as shown in the flow diagram  400  of  FIG. 4 . In this example, the method comprises obtaining a criterion for the processing of a distributed array (step  410 ) and performing flexible partitioning based on obtained criterion (step  430 ). In certain embodiments, the method further comprises performing analysis based on the obtained criteria. 
     The criterion acts as a guideline for how flexible partitioning should be handled. The distribution scheme is selected based on the criterion. In one exemplary embodiment the criterion comprises minimizing communication in the parallel computing environment. In order to achieve this goal, flexible distribution schemes are used. It should be understood that this is just one of the possible criteria. Other examples include, but are not limited to, maximizing use of execution environment resources, minimizing processing time, and profiling history. In certain embodiments multiple criteria may be obtained. In some embodiments where multiple criteria are obtained, a weighting scheme may be employed giving some criteria priority over others. Other possible criteria and embodiments will be apparent to one skilled in the art given the benefit of this disclosure. 
     The obtained criterion may be a default criterion or specified. In the current example, the criterion of minimizing communication in the parallel computing environment is a default criterion provided for handling flexible distribution of arrays provided by a parallel processing environment. In some embodiments of the invention, a criterion may be specified by a user, for example via a graphical user interface. In some such embodiments, the user may be prompted to select a criterion from pre-existing criterion choices. In other embodiments the criterion is specified by the parallel processing environment, the processing unit, other applications running on the processing unit, or other applications of devices in communication with the processing unit and environment. In some embodiments, a new criterion may be specified or provided. This allows for the extensibility of the present invention for as new situations or conditions occur new criterion may be created to address them. In certain embodiments, criteria may be provided via an Application Programming Interface (API). Other possible embodiments and configurations will be apparent to one skilled in the art given the benefit of this disclosure. 
     Once a criterion is obtained (step  410 ) then flexible partitioning may be performed based on the obtained criterion (step  430 ). In certain embodiments the flexible partitioning is performed according to instructions determined the criterion. In some instances the instructions may be the commands used to achieve the desired partitioning. In still further examples, the instructions may comprise the code for implementing a distribution scheme. 
     As with the criterion, the instructions may be default or specified by an operator such as user, device, or application. That is, once a criterion is obtained, a default set of instructions associated with the selected criterion may then be used or a new set of instruction may be provided for the selected criterion. In other embodiments where a new criterion is specified, instruction for implementing flexible partitioning in accordance with the new criterion may also be specified. In certain embodiments, criteria and corresponding instructions may be provided via an Application Programming Interface (API). Other embodiments and implementations will be apparent to one skilled in the art given the benefit of this disclosure. 
     In one embodiment of the invention, the method shown in  FIG. 4  further comprises the step of performing analysis based the criterion to determine how to perform flexible partitioning (step  420 ). In certain embodiments the analysis may also be performed as flexible partitioning is performed to allow for further adjustment to the dynamic distribution. As with the criterion and instructions, the analysis performed may be default or specified by an operator such as user, device, or application. That is, once a criterion is obtained a default analysis associated with the selected criterion may then be performed or a new analysis to be performed may be provided for the selected criterion. In other embodiments where a new criterion is specified, an analysis to be performed in accordance with the new criterion may also be specified. In certain embodiments, the analysis to be provided may be provided via an Application Programming Interface (API). Other embodiments and implementations will be apparent to one skilled in the art given the benefit of this disclosure. The flexible partitioning performed (step  430 ) may then be based on the results of the analysis. In some embodiments this may include determining the instructions to be used for implementing the flexible partitioning. 
     In some embodiments, the analysis (step  420 ) is performed using a rule set. The rules set may be rules for performing the analysis or rules performed as part of the analysis. The rule set may be default or specified by an operator such as user, device, or application. That is, once a criterion is obtained a default rule set associated with the selected criterion or performed analysis may then be utilized or a new rule set may be provided for the selected criterion or analysis. In other embodiments where a new criterion or analysis is specified, a new rule set in accordance with the new criterion or analysis may also be specified. In certain embodiments, the rule set may be provided via an Application Programming Interface (API). Other embodiments and implementations will be apparent to one skilled in the art given the benefit of this disclosure. The flexible partitioning performed (step  430 ) may then be based on the rule set. In some embodiments this may include determining the instructions to be used for implementing the flexible partitioning. 
     In some embodiments, the step of performing an analysis (step  420 ) comprises determining the type of operation to be performed on a distributed array. In some further embodiments, the determination of the type of operation to be performed on a distributed array determines a rule set to be used to perform flexible partitioning. An exemplary embodiment of this methodology can be seen in the flow diagram  500  of  FIG. 5 . The first step involves obtaining a criterion (step  510 ). Then the type operation being performed is determined (step  520 ). Based on the type of operation being performed a rule set is then applied (step  525 ). Then flexible partitioning may be performed based on an applied rule set (step  530 ). 
     In one exemplary embodiment, wherein the criterion is to minimize communication in the parallel computing environment, some of the operations, such as math functions, performed on a distributed array may be grouped into different categories for the purposes of determining how arrays should be distributed before, during and/or after the execution of a particular function. Some examples of possible categories include: Functions with Scalar or Small Vector Results, Unary Element-wise and Other Size Preserving Functions, Reduction Functions of One Data Input, Binary Element-wise Functions, Transpose Functions, and Size/Value-Dependent Functions. Each of these category groups will be discussed in more detail below. It should be understood that this is but one possible category grouping and does not cover all possible functions. Other possible functions, categories and groupings will be apparent to one skilled in the art given the benefit of the disclosure. 
     The category of Functions with Scalar or Small Vector Results includes the functions that return scalar or small vector (length of the output is the same as the number of dimensions of the input) outputs. Examples of functions in this category include:
         det, length, isempty, isequal, isreal, ndims, issparse, isscalar, issorted, isvector, norm, numel, rcond, size, isequalwithequalnans.
 
An example of an associated rule regarding the distribution of the result of a function included in this category is:
   The output of the function is replicated.       

     The category of Unary Element-wise and Other Size Preserving Functions includes functions that have the characteristic such that if:
         y=fun(x)       

     then:
         the size of the output y is the same as the size of the input x.
 
Examples of functions in this category include the one input forms of:
   abs, acos, acos h, asin, asin h, atan, atan h, ceil, char, conj, cos, cos h, double, exp, fft, fftn, fix, floor, full, ifft, ifftn, imag, int8, int16, int32, int64, isfinite, isinf, isnan, log, log 2, logical, not, pow2, real, real log, realsqrt, round, sign, sin, single, sin h, sort, sparse, sqrt, tan, tan h, tril, triu, uint8, uint16, uint32, uint64, uminus, uplus.
 
An example of an associated rule regarding the distribution of the result of a function included in this category is:
   The distribution scheme of the output is the same as that of the input.       

     The category Reduction Functions of One Data Input includes functions that have the characteristic such that if:
         y=fun(x, . . . , dim, . . . )       

     then:
         The size of the output y has been “reduced” from that of the input x such that the size of the output yin the dimension dim is 1. In some cases, this shrinks the dimensionality of the output y to less than that of the input x.
 
Examples of functions in this category include:
   all, any, cumprod, cumsum, max, min, prod, sum.
 
An example of an associated rule regarding the distribution of the result of a function in this category is:
   The distribution dimension of the output is the same as that of the data input, even if the reduction function operates down the distribution dimension and the entirety of the output y ends up stored locally on one lab.       

     The category Binary Element-wise Functions includes functions that have the characteristic such that if:
         z=fun(x,y)       

     and
         the size of the input x is the same as the size of the input y, or one of x and y is a scalar (the scalar expansion case)       

     then:
         the size of the output z is the same as that of x if the inputs are the same size, otherwise the size of the output z is the size of the non-scalar input.
 
Examples of functions in this category include the two input forms of:
   and, bitand, bitcmp, bitget, bitor, bitset, bitshift, bitxor, eq, ge, gt, le, lt, ldivide, max, min, minus, mod, ne, or, plus, pow2, power, rdivide, rem, times, xor.
 
An example of associated rules regarding the distribution of the result of a function in this category is:
   In the case of two same-sized distributed inputs, the distribution scheme of the result y is that of the left-most input x.   In the case of two same-sized inputs, if one is distributed and the other is replicated, then the result is replicated.   For the scalar expansion case, the non-scalar input provides the distribution scheme of the output. If the non-scalar operand is replicated, then the output will also be replicated.       

     The category of Transpose Functions includes functions that have the characteristic such that if:
         y=fun (x)       

     and:
         the size of the input x is m-by-n       

     then:
         the size of the output y is n-by-m
 
Examples of functions in this category include:
   transpose, ctranspose.
 
An example of associated rules regarding the distribution of the result of a function in this category is:
   The distribution dimension of the output y is the dimension of the input x that is not distributed. The partition of the output y is the same as that of the input x.       

     The category Size/Value-Dependent Functions includes functions that have the characteristic such that:
         The size of the output cannot be deduced simply from the size of the inputs. Instead, it relies on the value of the inputs. For example:   A (I1, I2, . . . , IN)=RHS   can have any number of impacts on the resulting size of A. Some choices include:   Reduce the size of A in any one dimension.   Increase the size of A in an existing dimension or dimensions.   Increase the number of dimensions of A.
 
Examples of functions in this category include:
   diag, fft, find, permute, reshape, subsasgn, subsref.
 
An example of associated rules regarding the distribution of the result of a function in this category is the following subset of cases for the function subsref.
   B=A(:)   For the indexing expression B=A(:), where the size of A is szA, the distribution dimension of A is dim and the partition of A is par.   If dim is the last dimension, then perform the A(:) indexing operation locally.   If dim is not the last dimension, then the indexing operation A(:) may not be performed locally and communication may be required.   In both cases, the distribution dimension of the output B is 1, and the partition of B is given by the MATLAB expression   prod(szA([1:dim−1, dim+1:end]))*par.   B=A(:,q)   If A is a matrix distributed by its second dimension (by columns) and B=A(:,q), where q is a vector of columns selected from A to create B.   If q is a sorted vector in increasing order, then we may perform the indexing operation locally and thereby eliminate communication. Otherwise, communication may be required to preserve the monotonicity of the columns in the output B.   If q is a sorted vector in increasing order, then the columns of A are left on the labs where they are and B is formed using the distribution scheme that falls out. For example, if there are four labs and A is 10-by-16 with four columns on each lab (the default partition), then B=A(:, 12:16) is a 10-by-5 matrix and has a partition of [0 0 1 4]. Forcing B into the default partition of [2 1 1 1] would require moving four of the resulting five columns of B to other labs.   If q is not sorted in increasing order, then the columns of B may move to different labs according to the following algorithm:   First, sort the indices into a new vector sortedQ.   Next, determine how many of those columns of A in sortedQ are on each lab. This determines the partition of B, partitionB.   For example, consider our 10-by-16 matrix A distributed by columns with the default partition from above.   If q=[10 11 16 3 1] then sortedQ=[1 3 10 11 16], defining the partition for B (:,q) as partitionB=[2 0 2 1]       

     One of ordinary skill in the art will appreciate that the present invention has several advantages. The use of criterion allows for a flexible methodology for guiding the handling of operations on distributed arrays. Providing a pre-defined guiding criterion and relates analysis, rules, and instructions reduces the need for a user to have to specify how to implement flexible partitioning for each instance an operation is preformed. The ability to specify new criterion as well as the underlying analysis, rules, and instructions for flexible partitioning provide a flexible, robust, extensible system for implementing dynamic distribution as new functions, conditions, and situations arise. 
     It should be understood that the provided embodiments are exemplary in nature. Many alterations and modifications may be made to the present invention, especially the implementations of different criterion and rules, by those having ordinary skill in the art without departing from the spirit and scope of the invention. For example, interpolation and/or extrapolation functionality, such as used for lookup tables, could be implemented using the teachings of the present invention. Therefore, it must be expressly understood that the illustrated embodiments and listed functions have been shown only for the purposes of example and should not be taken as limiting the invention, which is defined by the following claims. These claims are to be read as including what they set forth literally and also those equivalent elements which are insubstantially different, even though not identical in other respects to what is shown and described in the above illustrations.