Abstract:
Techniques are provided for improving the speed and accuracy of analytics on big data using theta sketches, by converting fixed-size sketches to theta sketches, and by performing set operations on sketches. In a technique for performing a set operation, two sketches are analyzed to identify the maximum value of each sketch. The maximum values of the two sketches are compared. Based the comparison, one or more values are removed from the sketch whose maximum value is greater. After the removal, a set operation (e.g., union, intersection, or difference) is performed based on the modified sketch and the unmodified sketch. A result of the set operation is a third sketch, which may be used to estimate a cardinality of the larger data sets that are represented by the two input sketches.

Description:
PRIORITY CLAIM 
       [0001]    This application claims benefit under 35 U.S.C. §120 as a Continuation of Ser. No. 14/448,487, filed Jul. 31, 2014, which claims the benefit under 35 U.S.C. §120 as a Continuation of Ser. No. 14/078,301, filed Nov. 12, 2013, which claims the benefit under 35 U.S.C. §119(e) of Provisional Appln. 61/887,375, filed Oct. 6, 2013, and Provisional Appln. 61/887,594, filed Oct. 7, 2013. The entire contents of each of the above listed applications are hereby incorporated by reference for all purposes as if fully set forth herein. The applicant hereby rescinds any disclaimer of claim scope in the parent applications or the prosecution history thereof and advises the USPTO that the claims in this application may be broader than any claim in the parent applications. 
     
    
     FIELD OF THE DISCLOSURE 
       [0002]    The present disclosure generally relates to statistical analysis of large data sets and, more specifically, to generating, storing, and performing set operations on statistical representations of large data sets. 
       BACKGROUND 
       [0003]    Methods and systems designed for analyzing smaller data sets begin to break or become non-functional as the size increases. The analysis of larger amounts of data (colloquially called “big data”) using conventional methods can require extensive computing resources, including processors and memory. Conventional methods may require data to be loaded into locally accessible memory, such as system memory or memory cache, where it can be processed to obtain results. However, as the amount of data increases, this can become impossible. In such situations, there is a need for generating results for complete analysis of the big data that do not require as much computational cost and latency. To achieve such an outcome, the accuracy of the results may be traded for less computational resources, such as memory. 
         [0004]    Big data analysis also frequently involves processing of data along multiple dimensions. Such dimensions could be time or data type. For example, big data may contain a log of user IDs and timestamps for users who have requested a particular web application. The big data may also contain user IDs of blacklisted users for each month. The data analysis may require a monthly count report of all unique non-blacklisted users that have visited the web application. To achieve such a result, not only does the large log of user IDs need to be extracted from the big data and processed but also set operations of difference need to be performed on the log data with the blacklisted user data. Such requirements for set operations further complicate the data analysis for big data performed with limited computational resources. 
         [0005]    Big data analysis presents a significant problem, in particular, for large website operators, such as Yahoo! Inc. A large website operator may generate terabytes of data per day describing the traffic to its website, or the content and advertisements displayed. While this vast pipeline of data can be mined for insights into the characteristics and behavior of its users, those insights are simply not available unless the pipeline of data can be analyzed, thereby permitting questions to be asked and answered within a relatively short period of time. For example, if the answer to a question about how many users visited a given website today takes until tomorrow to answer, then the answer may be of little use. Providing faster and more accurate answers to questions such as the unique number of visitors to a given website, or the number of clicks on a given item of content or advertisement, are technical problems of the utmost importance to website operators. 
         [0006]    More specifically, many big data analysis scenarios, such as user segment analysis, require set operations (e.g., intersection, union and difference) on sets of unique identifiers. When the data is larger than can be normally handled in memory, the unique counting as well as the set operations can be very expensive to compute exactly. If approximate answers are acceptable, then sketching technology can significantly reduce both the computational cost and the latency of obtaining results. 
         [0007]    A sketch can be more than just a mechanism to approximate unique counts. It can be thought of as a data structure that approximates a larger set of values. A sketch, in fact, may be a substantially uniform and random reservoir sample of all the unique values presented to it. It is then reasonable to ask: given two sketches can one determine, approximately, the number of unique values that form the intersection of the two large data sets represented by the sketches? Or, perhaps, could sketches represent other set operations such as difference—the number of unique values that are present in only one of two large data sets? 
         [0008]    For systems that generate millions of sketches or where query latency is critical, to be able to perform set operations on the same sketches that do the unique counting is a huge benefit and eliminates the need for separate processes. Example applications where set operations are intrinsic include segment overlap analysis (intersection), segment rollup analysis (union), retention analysis (intersection), and blacklist removal (set difference). Examples of segments for a website operator might be users (defined by login), impressions on items of content, clicks on advertisements, or other large-scale sets of data relating to website traffic. 
         [0009]    Getting faster and more accurate answers to questions about website traffic is an important technical problem facing many website operators. 
         [0010]    The approaches described in this section are approaches that could be pursued, but not necessarily approaches that have been previously conceived or pursued. Therefore, unless otherwise indicated, it should not be assumed that any of the approaches described in this section qualify as prior art merely by virtue of their inclusion in this section. 
       SUMMARY 
       [0011]    By introduction of a defined threshold theta to the computation of sketches, as disclosed below, a system and method is provided that achieve improvements in the speed and efficiency with which set operations may be performed (and questions thereby answered) on big data. More specifically, advantages over prior art techniques are achieved for computing and performing set operations with sketches, including one or more of the following: 
         [0012]    The same sketch used for unique counting can also be used for set operations with very good accuracy. This enables significant reduction in process complexity that would otherwise require separate processes for these computations. 
         [0013]    The result of a set operation is another sketch, not just a number. This enables asynchronous or out-of-order computations that are very frequent in batch pipeline operations. 
         [0014]    The size-controlling role of K (which refers to a pre-defined maximum sketch sample set size) can be disabled when performing union operations. Such disabling allows the union of two sets to be larger than either of two initial sketches with improved accuracy. This feature may be critical for data query operations where the query engine constructs a set expression with multiple terms. Disabling K also enables set expression evaluation order independence. 
         [0015]    Sketches with different values of K (which implies different configured accuracy or size) can now be targets of any of the set operations. 
         [0016]    These and other advantages are provided by the method and system further disclosed below. These systems and methods are operable to provide, among other things, faster and more accurate unique counts of visitors to a given website, faster and more accurate answers to questions about particular demographic segments that are represented in visits, impressions, click-through rates, or other data signals collected by website operators. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0017]    In the drawings of certain embodiments in which like reference numerals refer to corresponding parts throughout the figures: 
           [0018]      FIG. 1  is a system diagram that depicts program logic for generating a result for big data using a sketch data structure, in an embodiment. 
           [0019]      FIG. 2  is a block diagram that depicts a sketch sample set within memory. 
           [0020]      FIG. 3  is a block diagram that depicts a fixed-size sketch with K maximum sample set size, in an embodiment. 
           [0021]      FIG. 4  is a block diagram that depicts a threshold sketch with Theta, an upper threshold, and a target sample set size, in an embodiment. 
           [0022]      FIG. 5  is a flow diagram that depicts a process for generating a fixed-size sketch with K maximum size sample set, in an embodiment. 
           [0023]      FIG. 6  is a flow diagram that depicts a process for generating a theta sketch, in an embodiment. 
           [0024]      FIG. 7  is a flow diagram that depicts a process for generating a theta sketch with an asynchronous cleanup of data, in an embodiment. 
           [0025]      FIG. 8A  is a flow diagram that depicts a process for converting a fixed-size sketch into a theta sketch, in an embodiment. 
           [0026]      FIG. 8B  is a flow diagram that depicts a process for converting a theta sketch into a fixed-size sketch, in an embodiment. 
           [0027]      FIG. 9  is a block diagram that depicts a SketchMart database with a SketchMart comprising of aggregation of sketches along a dimension, in an embodiment. 
           [0028]      FIG. 10  is a block diagram that depicts the generation and usage of sketches in a SketchMart, in an embodiment. 
           [0029]      FIG. 11  is a flow diagram that depicts a process for performing a union operation on two fixed-size sketches to yield a union fixed-size sketch, in an embodiment. 
           [0030]      FIG. 12  is a flow diagram that depicts an alternative process for performing a union operation on two fixed-size sketches to yield a union fixed-size sketch, in an embodiment. 
           [0031]      FIG. 13  is a block diagram that depicts resulting theta sketches from set operations on theta sketches, in an embodiment. 
           [0032]      FIG. 14  is a flow diagram that depicts a process for performing a set operation on theta sketches, in an embodiment. 
           [0033]      FIG. 15  is a flow diagram that depicts a process for performing an intersection operation on two theta sketches to yield an intersection threshold sketch, in an embodiment. 
           [0034]      FIG. 16  is a flow diagram that depicts a process for performing a union operation on two theta sketches to yield a union threshold sketch, in an embodiment. 
           [0035]      FIG. 17  is a system diagram that illustrates a sketch system, in an embodiment. 
           [0036]      FIG. 18  is a block diagram that depicts various infrastructure components through which a sketch system may be implemented, in an embodiment. 
           [0037]      FIG. 19  is a block diagram that illustrates a computer system upon which an embodiment of the invention may be implemented. 
       
    
    
     DETAILED DESCRIPTION 
       [0038]    In the following description, for the purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be apparent, however, that the present invention may be practiced without these specific details. In other instances, well-known structures and devices are shown in block diagram form in order to avoid unnecessarily obscuring the present invention. 
       General Overview 
       [0039]    Analyzing big data can be an extremely computationally intensive task with results obtained with significant latency. Sketching techniques can significantly reduce both the computational cost and the latency in obtaining results for the big data. Sketching techniques involve transforming big data into a mini data set, or sample set, that is representative of big data&#39;s particular aspects (or attributes) and are used, in various embodiments, to obtain a particular data analysis result estimation, such as unique entry counts, within big data. These mini sample sets, and in some embodiments associated metadata, are generally known as “sketches” or sketch data structures. 
         [0040]    Sketches are useful, for example, for counting unique values (i.e. for estimating cardinality) for big data. This is because sketches can be order insensitive, duplicate insensitive, and have well-defined and configurable bounds of size and error distribution. The order insensitive property means that the input data stream from big data can be processed as is; no ordering of the incoming data is required. The duplicate insensitive property means that all duplicates in the input stream will be ignored, which is important in estimating cardinality. The importance of having a bounded size is that one can plan for a maximum memory or disk storage capacity independent of how large the input data stream from the big data becomes. With a bounded, well defined and queryable error distribution, it is possible to report to the user not only any estimated result but the upper and lower bounds of that estimate, based on a confidence interval, as well. 
       Overview of Generating and Using a Sketch 
       [0041]      FIG. 1  is a system diagram that depicts program logic for generating a sketch and estimating cardinality of a large data set that the sketch represents, in an embodiment. A data stream is generated from Big Data  100  and fed into transformer  110 . Transformer  110  transforms the data stream into a representative set of values in such a way where only a portion of the transformed set needs to be analyzed to yield a desired estimation of a result. For example, to estimate cardinality within the data stream, transformer  110  may be a hash function that produces the same hash for the same data value from the data stream. In some instances, a hash function may generate the same hash value based on two or more unique input data values. But taking into consideration the kind of data in the data stream, the choice of hash algorithm may be made to reduce or avoid such collisions. 
         [0042]    Transformer  110  also produces close to a uniform, random distribution of transformed values. Due to this property, only a particular subset of the transformed values needs to be examined to yield an estimation for cardinality for the whole set. Thus, only this subset of transformed values needs to be stored in memory  120 , as part of sketch data structure (or simply “sketch”)  130 . Similarly for other transformations, only a subset of transformed data values is collected in sketch  130  in memory  120 . The selected subset of transformed values and, thus, the size of sketch  130  may be limited by the available memory  120 . Estimator  140  processes sketch  130  and its subset of values to estimate result  150 . For example, in some embodiments, when cardinality is requested as result  150 , estimator  140  would use the size of sketch  130  (i.e., the number of elements) and/or values within sketch  130  to estimate the cardinality of the large data set that sketch  130  represents and output the estimated cardinality as a result for Result  150 . 
         [0043]    Each of transformer  110  and estimator  140  are part of computer system and may be implemented in software, hardware, or a combination of software and hardware. For example, one or more of transformer  110  and estimator  140  may be implemented using stored program logic, in a non-transitory medium such as the memory of a general-purpose computer or in hardware logic. 
         [0044]    Generally, a sketch is generated to estimate a single result. Such a sketch may represent only a particular attribute of big data relevant to the result. For example, a sketch may be generated to estimate a number of unique user IDs for users who have used a web application for a particular month. However, estimations for other months may also be requested for latter comparisons and, thus, a separate sketch would need to be generated for each month requested. These generated sketches then can be stored in a persistent storage, as “SketchMart,” along with the dimension for which the sketches were generated. A “SketchMart” is a set of sketches, where each sketch is related to at least one other sketch in the set and where combining two related sketches in the SketchMart yields a meaningful result. In this example, the dimension is a time dimension of months. Once stored in in a SketchMart, the sketches may be queried based on the time dimension and used for an estimation of a result, but even more importantly, for set operations to yield new sketches. 
         [0045]    In much of data analysis for big data, multiple large data sets from big data need to be processed to yield a particular result. For example, the big data may contain a log of user IDs and timestamps for users who have visited a particular web application. The big data may also contain user IDs of blacklisted users for each month. The data analysis may require a monthly count report of all unique non-blacklisted users that have visited the web application. 
         [0046]    An estimate of such results could be achieved by proper set operations on sketches. Set operations may be performed on sketches to produce another sketch. The produced sketch can then be used for an estimation of a result that would be an estimation of the result that would have been produced by an actual intersection of large data sets from big data. Thus, sketches that are stored in a SketchMart may be queried for performing various set operations to obtain a desired result estimation. 
       Sketch Data Structure 
       [0047]    A sketch data structure (or “sketch”) includes a sample set of transformed values and metadata associated with the sketch. In an embodiment, a sketch does not contain metadata, such as metadata that indicates the size of (or the number of transformed values in) the sample set. Instead, such metadata may be derived from the sample set itself. A sketch contains values resulting from transformation of big data. The sketch contains only a subset of all transformed values generated from big data, where the size of the sketch may be based on available computational resources, such as memory. 
         [0048]      FIG. 2  illustrates an embodiment of sketch  130  residing within memory  120 . In an embodiment, data values from big data are transformed using a hash function. The distribution of hash function values are illustratively denoted as being all real values from Transformation Range Minimum Value  220  of 0.0 to Transformation Range Maximum Value  210  1.0. As the transformed values are received, only the values within Retained Value Range  230  are retained within sample set  200 . Metadata  240  includes metadata for sketch  130 , such as the maximum size of sample set  200  and Retained Value Range  230  thresholds. In another embodiment, the retained value range may be adjacent to the Transformation Range Maximum Value  210 . In still another embodiment, the retained value range may be between the Transformation Range Minimum Value  220  and the Transformation Range Maximum Value  210 . 
       Fixed-Size Sketch Data Structure 
       [0049]    A fixed-size sketch data structure, also referred to as “fixed-size sketch,” is a type of sketch data structure where the sample set retains values based on a predefined size. If the sample set has reached its predefined size (e.g., maximum allowed entries) and a new transformed value is received, then the new value would either replace an existing value in the sketch sample set, or will be discarded and not stored within the sample set. The predefined size for the sample set may be stored as part of Metadata  240  or for already generated fixed-size sketches, may be computed from the sample set itself. The predefined size is denoted as K, maximum sample set size.  FIG. 3  illustrates an embodiment of Fixed-Size Sketch  300  that has Sample Set  200  and K, Maximum Sample Set Size  310 . 
         [0050]      FIG. 5  illustrates an embodiment to generate a fixed-size sketch for cardinality computation. As an example of a fixed-size sketch, the following description refers to Fixed-Size Sketch  300  of  FIG. 3 . In block  500 , a transformed value is received. If it is determined at block  510  that the transformed value is already within Sample Set  200  of  FIG. 3 , then the value is discarded at block  505  and the process proceeds to block  500 . If the received transformed value is not within Sample Set  200 , then Sample Set  200  is evaluated for available space at block  520 . 
         [0051]    If it is determined at block  520  that Sample Set  200  has already reached K, Maximum Sample Set Size  310  of  FIG. 3 , then, at block  525 , the transformed value is compared with the maximum value in Sample Set  200 . If the transformed value is less than the maximum value, then, at block  530 , the maximum value is discarded from Sample Set  200  to make space for the transformed value. However, if the transformed value is greater than the maximum value in Sample Set  200 , then the value is discarded at block  505  and the process proceeds to block  500 . 
         [0052]    At block  540 , the received transformed value is inserted into Sample Set  200 . In an embodiment, Sample Set  200  is an ordered list of values, where the smallest value in Sample Set  200  is the first element and the greatest value is the K-th element in Sample Set  200 . Thus, according to such embodiment, the K-th value in Sample Set  200  would be evaluated at block  525  and would be removed at block  530 . 
       Threshold (Theta) Sketch Data Structure 
       [0053]    A threshold sketch data structure (or “theta sketch”) is a type of sketch where the sample set retains values based on one or more threshold values and a target sample set size. Determination of whether to insert a received transformed value into the sample set is based on whether the received value is greater or less than one or more threshold values for the sketch. The threshold values may be adjusted to accommodate a bound size for the sample set. The bound size for the sample set may be computed based on a target sample set size for the sketch that is stored as part of metadata for the sketch. Any values that are outside of the adjusted threshold values may be discarded from the sketch. In an embodiment, the discarding of sketch data values that are outside of the adjusted threshold values may be performed asynchronously from the receipt of a transformed value. In a related embodiment, the threshold value adjustment itself may be performed asynchronously from the receipt of a transformed value. 
         [0054]      FIG. 4  is a block diagram that depicts a theta sketch  400 , in an embodiment. Theta sketch  400  contains sample set  200  with K′, Target Sample Set Size  420  and Theta, Upper Threshold  410  as metadata for the sketch. Theta sketch  400  is generated by receipt of transformed values from big data. Theta sketch  400  has a minimum threshold value equal to the transformation range minimum value, and thus the minimum threshold value need not stored as part of the metadata for Theta Sketch  400 . 
         [0055]      FIG. 6  is a flow diagram that depicts a process for generating a theta sketch, in an embodiment.  FIG. 6  is described using theta sketch  400  and sample set  200  of  FIG. 4 . At block  600 , Theta, Upper Threshold  410  is initialized to Transformation Range Maximum Value  210  of  FIG. 2 . At block  605 , a transformed value is received. At block  610 , if the value is greater than or equal to Theta  410 , the transformed value is discarded at block  615  and the process returns to block  605  to process the next transformed value. 
         [0056]    At block  620 , it is determined whether the transformed value is unique relative to the other transformed values in the sample set. If not, then the transformed value is discarded at block  615  and the process returns to block  605 . However, if it is determined that the transformed value is unique, then, at block  630 , the received transformed value is inserted into Sample Set  200 . 
         [0057]    At block  640 , the current size of Sample Set  200  is compared with K′, Target Sample Set Size  420 . If the size of Sample Set  200  has already reached K′ plus one value, then at block  650 , the maximum value is removed from Sample Set  200 , and, at block  660 , Theta is assigned to the removed maximum value. Otherwise, if the size of Sample Set  200  has not reached K′ plus one value, then Theta  410  stays unchanged, and the process returns to block  605 . 
         [0058]    In an embodiment, Sample Set  200  is an ordered list of values, where the smallest value in Sample Set  200  is the first element and the greatest value is the last element in Sample Set  200 . Thus, in such an embodiment, the last value is discarded from Sample Set  200  at block  650 , and Theta  410  is assigned to the last value in Sample Set  200  at block  660 . 
         [0059]    While the above example indicates that the range of sample set  200  is from 0 to a value (theta) that represents a value less than 1, embodiments are applicable to the scenario where the range of a sketch&#39;s sample set is from a value that represents ‘1’ to a value that represents a value greater than ‘0’. In such an embodiment, a received threshold value would be compared to a lower threshold (not depicted) instead of upper threshold  410 . Cardinality estimations and set operation for such embodiments would change accordingly. 
         [0060]      FIG. 7  illustrates another embodiment to generate Theta Sketch  400  with Sample Set  200  of  FIG. 4  for cardinality computation. Similar to block  600 , at block  700 , Theta, Upper Threshold  410  is initialized to Transformation Range Maximum Value  210  of  FIG. 2 . Similar to block  605 , at block  705 , a transformed value is received. Similar to block  610 , block  710  involves determining whether the transformed value is less than Theta  410 . If not, then the transformed value is discarded and the process proceeds to block  705  to receive the next transformed value, if any. If the transformed value is less than Theta  410 , then the process proceeds to block  720 , where it is determines whether the transformed value is unique relative to other transformed values in Sample Set  200  of  FIG. 4 . 
         [0061]    If it is determined that the transformed value is not unique (i.e., the transformed value is already within Sample Set  200 ), then the transformed value is discarded. However, unlike block  630 , if the transformed value is not within Sample Set  200 , then the size of Sample Set  200  is not compared with K′, Target Sample Set Size  420 . Rather, Theta  410  is decreased at block  740 , and the transformed value is inserted into Sample Set  200  at block  760 . At block  750 , which may be performed asynchronous from the receipt of transformed values, values in Sample Set  200  that are greater than or equal to Theta  410  are discarded from Sample Set  200 . 
         [0062]    According to an embodiment, at block  740 , Theta  410  is decreased based on K′, Target Sample Set Size  420 . Theta  410  may be decreased based on K′ in multiple ways. Embodiments are not limited to any particular technique. For example, Theta (θ)  410  may be decreased according to the following equation, where K is the actual sample set size: 
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         [0063]    As another example, Theta ( 6 )  410  may be decreased according to the following equation: 
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         [0064]    As another example, Theta  410  may be decreased according to the following equation: 
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         [0065]    As yet another example, Theta  410  may be decreased according to the following equation: 
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       Sketch Conversion 
       [0066]    In an embodiment, a fixed-size sketch is converted to a theta sketch and/or vice versa. Converting a fixed-size sketch to a theta sketch involves removing minimum and/or maximum values from the fixed-size sketch and storing the values in the sketch metadata as thresholds. Also, if K (i.e., the maximum sample set size) is stored in the metadata of a fixed-size sketch, then K may be removed from the metadata. The resulting sketch would constitute a theta sketch rather than a fixed-size sketch. 
         [0067]    Similarly, converting a theta sketch to a fixed-size sketch involves discarding the one or more threshold values from theta sketch metadata, while the size of the current sample set may be stored in the metadata. The resulting sketch would constitute a fixed-size sketch. 
         [0068]      FIG. 8A  illustrates a conversion of a fixed-size sketch into a theta sketch, using the example of the fixed-size sketch in  FIG. 3 , in an embodiment. At block  800 A, the K-th element is selected from the Sample Set  200  of Fixed-Size Sketch  300 . At block  810 A, the K-th element stored in the metadata as the theta upper threshold. At block  820 A, the K-th element is discarded from the Sample Set  200 . At block  830 A, in the metadata, K, Maximum Sample Set Size  310 , is renamed to K′, Target Sample Set Size at block  830 A. The resulting sketch is then considered a theta sketch. 
         [0069]      FIG. 8B  illustrates a conversion of a theta sketch into a fixed-size sketch, using the example in  FIG. 4 , in an embodiment. At block  800 B, Theta, Upper Threshold  410  is discarded from the metadata. At block  810 B, K′, Target Sample Set Size  420  is also discarded from the metadata. At block  820 B, the size of the sample set is stored in metadata as K, maximum sample set size. The resulting sketch is then considered a fixed-size sketch. Although  FIGS. 8A and 8B  are depicted and described in a particular order, embodiments are not limited to that particular order. For example, block  830 A may be performed before block  800 A and block  810 B may be performed before  800 B. 
         [0070]    In an embodiment, sketch conversion is performed on multiple sketches. For example, a set of multiple fixed-size sketches are converted to a set of theta sketches as part of a single operation. Such a single operation may be performed in response to receiving a single command from a user. The single command may specify individual sketches or may specify a physical or logical container that stores the set of sketches that are to be converted. Thus, individual input is not required after one sketch is converted and before another sketch is converted. Instead, multiple sketches may be converted from one type (e.g., fixed) to another type (e.g., theta) in parallel. 
       Cardinality Estimation 
       [0071]    Cardinality for a large data set from big data can be estimated based on a sketch for the large data set. In an embodiment, large data set values are transformed using a hash function. Such transformation uniformly randomizes the large data set values without losing one-to-one correspondence between the large data set values and transformed values. The transformed values are uniformly distributed within a transformation range maximum value and a transformation range minimum value, and a sample set is generated by capturing only a contiguous subset of the transformed values. Thus, the cardinality of the captured sample set is proportional to the cardinality of the transformed value data set and to the cardinality of the large data set. The proportionality can be represented by the following equation, where est (|M i |) is estimated cardinality of the large data set, |S i | is the cardinality of S i , a sketch of M i , F(M i ) max  is a transformation range maximum value and F(M i ) min  is a transformation range minimum value, x max  is the sample set maximum value in the retained value range and x min  is the sample set minimum value in the retained value range: 
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         [0072]    In an embodiment, the above equation (5) can be further simplified for theta sketches. For a theta sketch, |S i |−2 is the cardinality of the theta sketch, x max  is the upper threshold, θ hi , and x min  is the lower threshold, θ io . Thus, for theta sketch, the equation (5) can be further simplified to the following: 
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         [0073]    According to an embodiment, the transformation value range can be normalized from 0 to 1, where the sketch retained value range has minimum value of 0. Thus, the equation (5) can be further simplified for this embodiment: 
         [0000]    
       
         
           
             
               
                 
                   
                     est 
                      
                     
                       ( 
                       
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                         max 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   7 
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         [0074]    Based on the above equation (7), the cardinality estimation for both fixed-size sketches and theta sketches can be easily derived. For a fixed-size sketch, |S i | is the cardinality of the fixed-size sketch, and x max  is the K-th value of the fixed-size sketch. For a theta sketch, |S i |−1 is the cardinality of the theta sketch, and x max  is the upper threshold, θ. Thus, for theta sketch, the cardinality equation can be further simplified to the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     est 
                      
                     
                       ( 
                       
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                       ) 
                     
                   
                   = 
                   
                     
                       
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                     . 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0075]    In another embodiment, a theta sketch, with K′, Target Sample Set Size, is constructed from a large data set using the process in  FIG. 7 , where θ, Upper Threshold, is decreased at block  740  using equation (4). In such embodiment, the following equation provides est (|M i |), estimated cardinality of the large data set: 
         [0000]    
       
         
           
             
               
                 
                   
                     est 
                      
                     
                       ( 
                       
                          
                         
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                           i 
                         
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                       ) 
                     
                   
                   = 
                   
                     
                       
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                     . 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
       Sketchmart 
       [0076]    Sketches may be stored in a database for later use. The database may be any queryable persistent storage, such as a relational database or a distributed file system. 
         [0077]    Sketches may be aggregated along a particular dimension into a set of sketches, referred herein as a “SketchMart.” Each sketch in a SketchMart is related to at least one other sketch in the set. Also, combining two related sketches in the SketchMart yields a meaningful result, such as the number of unique users who have visited both a financial website and a sports website during a particular month. 
         [0078]      FIG. 9  is a block diagram that depicts a SketchMart  910  that is stored in SketchMart Database  900 , in an embodiment. Although only one SketchMart is depicted in  FIG. 9 , SketchMart Database  900  may include multiple SketchMarts. Sketch (S′ j )  912  and Sketch (S′ k )  914  are part of SketchMart  910  and have been collected and/or generated along dimension  915 . The number and type of dimensions that may be used to group Sketches into SketchMarts are numerous. Example dimensions include time (e.g., day, month, or year), web portal, web application, geographical location of client device submitting requests, type of those client devices (e.g., tablet, desktop, smartphone), type of OS of the client devices (e.g., Windows, Android, iOS). 
         [0079]    SketchMart Database  900  may be queried based on a SketchMart identifier and a dimension value to retrieve a particular sketch or set of sketches from the identified SketchMart. For example, a query to retrieve one or more sketches from SketchMart  910  may specify a range of dimension values (e.g., “month=January &amp;&amp; February”) and might retrieve Sketches  912  and  914  from SketchMart Database  900 . 
         [0080]    In an embodiment, a SketchMart is either a “fixed-size SketchMart” or a “Theta SketchMart.” A fixed-size SketchMart is a SketchMart that comprises fixed-size sketches. Each sketch in a fixed-size SketchMart has the same number of transformed values as each other sketch in the fixed-size SketchMart. 
       Sketch Set Operations 
       [0081]    Sketches may be combined to estimate a result for combination of larger data that the sketches represent. Since a sketch consists of a sample set and metadata, two or more sketches may be combined using a desired set operation on sample set and an adjustment to resulting sketch metadata, if any exists. Set operations include union, intersection, and difference. There are numerous scenarios in which it would be desirable to combine two or more sketches. One example scenario is determining a number of users that visited a certain webpage or website at least once each month of a particular year. Such a scenario is reflected in  FIG. 10 . 
         [0082]      FIG. 10  is a block diagram that depicts combining multiple sketches to generate another sketch, in an embodiment. Specifically,  FIG. 10  depicts a database (M i )  1050  of user IDs. The user IDs from database  1050 , for each month i, are transformed into Sketch S′(M i )  1055 , which represents user ID data for each month i. The generated sketches are stored in SketchMart (S′ i )  1060  along month dimension i. Therefore, Sketch (S′ j )  912  and Sketch (S′ k )  914  may represent sketches for the j month and k month, respectively. Thus, query to SketchMart Database  900  for a user ID sketch for the j month would return Sketch (S′ j )  912 . See further description of  FIG. 10  below. 
       Fixed-Size Sketch Set Operations 
       [0083]    Set operations may be performed on fixed-size sketches to yield a resulting fixed-size sketch. The resulting fixed-sized sketch may depend on the number of values produced from the set operation. However, the resulting sample set size may be reduced to the minimum sample set size of the operated sketches to accommodate limited computational resources. 
         [0084]    In an embodiment, a fixed-size union sketch is constructed by performing a union set operation on the sample sets of operated sketches. The resulting union sample set size is matched with the size of the smallest operated sketch size sample set by removing the greater value. In other words, if K is the sample set size of the smallest operated fixed-size sketch, then only the first K smallest values in the union sample set are preserved; the rest of the values are discarded. Thus, the fixed-size union sketch would contain the resulting union sample set and have maximum sample set size equal to the smallest K, maximum sample set size, of the operated fixed-size sketches. 
         [0085]    For example, Sketch S′ j  may contain sample set with values 0.1, 0.3, 0.4, 0.5, 0.6 and 0.7 with K, maximum sample set size set to 6. Sketch S′ k  may contain sample set values 0.2, 0.4 and 0.6 with K, maximum sample set size set to 3. The union set operation performed on the two sample sets would yield a union sample set of values: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7. Since the union sample set size is 7 that is greater than the smallest K, which is 3, the greater values are removed from the union sample set to reduce the size to the smallest K, 3. Thus, the resulting union Sketch S″ u  contains sample set values: 0.1, 0.2 and 0.3 and has K, maximum sample set size of 3. 
         [0086]    In a related embodiment, a fixed-size union sketch is similarly constructed by performing a union set operation on the sample sets of operated sketches. Then, the resulting union sample set is reduced based on the K-th values, the greatest values in each of the operated sample sets. The K-th values of the operated sample sets are compared, and all the values in the union sample set that are greater than the smallest of K-th values are removed from the union sample set. Subsequently, the fixed-size union sketch would contain the resulting union sample set and have maximum sample set size equal to the resulting sample set size. 
         [0087]    For example, Sketch S′ j  may contain sample set with values 0.1, 0.3, 0.4, 0.5, 0.6 and 0.7 with K, maximum sample set size set to 6. Sketch S′ k  may contain sample set with values 0.2, 0.4 and 0.6 with K, maximum sample set size set to 3. The union set operation performed on the two sample sets would yield a union sample set of values: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7. S′ j  has the greatest value, K-th value, of 0.7, and S′ k  has the greatest value, K-th value, of 0.6. Thus, the smallest K-th value would be 0.6, and all the values in the union sample set that are larger than 0.6 would be removed. Thus, the resulting union Sketch S″ u  would contain sample set values: 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 and have K, maximum sample set size of 6. 
         [0088]      FIG. 11  is a flow diagram that depicts a process for performing a union set operation on fixed-size sketches, in an embodiment. The fixed-sized sketches are referred to as Sketch S′ j  and Sketch S′ k , respectively, the resulting sketch is referred to as Sketch S″ u . At block  1100 , all the values from Sketch S′ k  are copied into (initially empty) Sketch S″ u . Then, a value from Sketch S′ j  is selected at block  1110  and compared to one or more values in Sketch S′ k  at block  1120 . The values that are not found in are then inserted into the Sketch S″ u  sample set at block  1130 . Once all the values in the Sketch S′ j  sample set have been selected, the process eventually proceeds from block  1120  to block  1150 . At block  1150 , K (i.e., the maximum size of Sketch S″ u ) is set to the minimum of the Ks for Sketch S′ k  and Sketch S′ j . At block  1160 , K is compared with the actual sample set size of Sketch S″ u . If the actual sample set size of Sketch S″ u  is lower than or equal to K, then, at block  1170 , all the values in the Sketch S″ u  sample set size that are larger than K&#39;th value in Sketch S″ u  are removed. 
         [0089]      FIG. 12  is a flow diagram that depicts a process for performing a union set operation on fixed-size sketches, in an embodiment. Again, the fixed-sized sketches are referred to as Sketch S′ j  and Sketch S′ k , respectively, the resulting sketch is referred to as Sketch S″ u . Similar to  1100  block in  FIG. 11 , at  1200  block, all the values from Sketch S′ k  are copied into Sketch S″ u . Then, a value in Sketch S′ j  is selected at block  1210  and compared to the values in Sketch S′ k  at block  1220 . The values that are not found in Sketch S′ k  are inserted into Sketch S″ u  at block  1230 . Once all the values in the Sketch S′ j  sample set have been selected (as determined in block  1240 ), the process proceeds to block  1270 . However, unlike block  1170  in  FIG. 11 , at block  1270 , only those values in Sketch S″ u  are removed that are either larger than the K-th value in Sketch S′ j  or larger than the K-th value in Sketch S′ k . 
       Theta Sketch Set Operations 
       [0090]    In an embodiment, set operations are performed on theta sketches to yield a resulting theta sketch. The size of a resulting theta sketch may depend on the respective thresholds of the input theta sketches. 
         [0091]      FIG. 13  is a block diagram that depicts theta sketches that result from different set operations on two input theta sketches, in an embodiment. Sketch A consists of a sample set, Sample Set A  1300 , and an upper threshold, θ A    1305 , where values in Sample Set A  1300  are depicted in descending order: value V 1  is greater than value V 2 , and value V 2  is greater than value V 3 . Sketch B consists of a sample set, Sample Set B  1310 , where values in Sample Set B  1310  are depicted in descending order: value V 3  is greater than value V 4 . Sketch B further consists of an upper threshold, θ B    1315 , where θ B    1315  is less than value V 1  and thus, is further less than θ A    1305 . 
         [0092]    To yield a resulting theta sketch for any sketch operation, A, such as union, difference or intersection, a resulting sketch upper threshold value is calculated, in an embodiment. The resulting sketch upper threshold, θ Δ   1325 , is determined by taking the minimum of the upper thresholds of the operated sketches: θ A    1305  and θ B    1315 . In this embodiment, since θ B    1315  is less than θ A    1305 , θ Δ   1325  is equal to θ B    1315 . 
         [0093]    Next, a resulting sketch sample set for an operation on Sketch A and Sketch B is determined. First, the set operation is performed on Sample Set A  1300  and Sample Set B  1310 , and second, any value that is greater than θ Δ   1325  is removed. 
         [0094]    For a union operation on Sketch A and Sketch B, the union operation is performed on Sample Set A  1300  values V 1 , V 2 , V 3  and Sample Set B  1310  values V 3 , V 4 . The union operation yields a super set of the values: V 1 , V 2 , V 3  and V 4 , as depicted in Sample Set A∪B  1320 . However, since value V 1  is greater than the resulting upper threshold, θ Δ   1325 , V 1  is removed from Sample Set A∪B  1320 , as depicted by the parenthesis around V 1  in  FIG. 13 . 
         [0095]    Similarly, a difference operation of Sketch A and Sketch B first yields Sample Set A\B  1330 . Any value in Sample Set B  1310  that also exists in Sample Set A  1300  is removed from Sample Set A  1300  to produce values V 1  and V 2  for Sample Set A\B  1330 . However, since V 1  is greater than the resulting upper threshold, θ Δ   1325 , value V 1  is removed from Sample Set A\B  1330 , as depicted by the parenthesis around V 1  in  FIG. 13 , while value V 2  remains in Sample Set A\B  1330 . 
         [0096]    A difference operation of Sketch B and Sketch A yields Sample Set B\A  1340 . Any value in Sample Set A  1300  that also exists in Sample Set B  1310  is removed from Sample Set B  1310  to produce value V 4  for Sample Set B\A  1340 . Since value V 4  is less than θ Δ   1325 , value V 4  remains in Sample Set B\A  1340 . 
         [0097]    An intersection operation of Sketch A and Sketch B yields Sample Set A∩B  1350 . Values V 1 , V 2 , V 3  from Sample Set A  1300  are intersected with values V 3 , V 4  from Sample Set B  1310  to yield common value V 3  for Sample Set A∩B  1350 . Since value V 3  is less than θ Δ   1325 , value V 3  remains in Sample Set A∩B  1350 . 
         [0098]      FIG. 14  is a flow diagram that depicts a process for performing a set operation using two theta sketches as input. At blocks  1400  and  1410  thresholds for a resulting sketch is set to the minimum and maximum of thresholds of the respective input sketches. Then, at block  1420 , a desired set operation is performed on the sample sets of the operated sketches. The resulting sample set is then stored into the resulting theta sketch at block  1430 . All the values in the resulting theta sketch sample set that are not within the thresholds of the resulting sketch are then removed at block  1440 . The resulting theta sketch is a product of the desired set operation on the operated theta sketches. 
         [0099]    In a related embodiment, a similar process is performed to that of  FIG. 14 . One difference is that values in the resulting sketch are not removed synchronously to the process. In other words, a separate process may be responsible for removing the values that are to be discarded. 
         [0100]      FIG. 15  is a flow diagram that depicts a process for performing an intersection operation on theta sketches, in an embodiment. The theta sketches are referred to as Sketch S′ j  and Sketch S′ k , respectively, and the resulting sketch is referred to as Sketch S″ j,k . At block  1500 , theta for the resulting sketch, S″ j,k , is set to the minimum of the theta&#39;s for operated sketches, S′ k  and S′ j . At block  1510 , a value in Sketch S′ j  is selected and, at block  1520 , compared to the values in Sketch S′ k . If the selected value is found in Sketch S′ k , then the value is inserted into Sketch S″ j,k  at block  1530 . Once all the values in Sketch S′ j  are selected and compared and there are no more values from Sketch S′ j  left to process (as determined in block  1540 ), then the process proceeds to block  1550 . 
         [0101]    At block  1550 , all the values in the Sketch S″ j,k  sample set that are greater than Theta for Sketch S″ j,k  are removed. Thus, Sketch S″ j,k  is a result of the intersection operation on Sketch S′ k  and Sketch S′ j . 
         [0102]      FIG. 16  is a flow diagram that depicts a process for performing a union operation on theta sketches, in an embodiment. The theta sketches are referred to as Sketch S′ j  and Sketch S′ k , respectively, and the resulting union sketch is referred to as Sketch S″ u . At block  1600 , Theta for the resulting sketch, S″ u , is set to the minimum of Theta&#39;s for operated sketches, S′ k  and S′ j . At block  1610 , the values from Sketch S′ k  are copied into Sketch S″ u . Then, at block  1620 , a value from Sketch S′ j  is selected and, at block  1640 , compared to the values in Sketch S′ k . If the selected value does not exist in Sketch S′ k , then the value is inserted into Sketch S″ j,k  at block  1650 . Once all the values in Sketch S′ j  have been selected and compared and it is determined (at block  1630 ) that there are no more values from Sketch S′ j  left to process, the process proceeds to block  1660 . At block  1660 , all the values in Sketch S″ u  that are greater than Theta for Sketch S″ u  are removed. 
       Sketch Set Operation Cardinality Estimation 
       [0103]    Set operations on sketches allow estimated results for combinations of large data sets to be obtained. For example, a large data set may have a log of users who have used a particular web application or who have visited a particular website. Such data may contain user IDs with timestamps. To determine retention amongst users of the web application for each month, a data set of user IDs for one month needs to be queried from the log and intersected with a data set of user IDs for another month. The unique count of the intersection would yield the retention number of users for the web application for those months. An estimation of this result may be obtained by generating sketches from large data set, performing set operations on those sketches, and estimating results based on the sketches produced by performing the set operations. 
         [0104]      FIG. 10  (a portion of which was described previously) illustrates such a scenario. Database (M i )  1050  represents a log of user IDs with timestamps for each month i. To obtain an estimation of unique users per month, a sketch, S′(M i ) is generated at block  1055  for each month i. A number of S′(Mi) sketches (e.g.,  12 , corresponding to each month of a particular year) are stored in SketchMart (S′ i )  1060 . For generating a retention results for users on monthly basis, sketches, Sketch S′ j  and Sketch S′ k  may be queried from SketchMart S′ i  for j and k month respectively. At block  1065 , an intersection operation may be performed on sketches, Sketch and Sketch S′ k  to produce Sketch S″ j,k . The intersection operation may be performed according to any of the processes described in the set operation sections. Also, since set operations, like intersection, may be performed on sketches retrieved from SketchMart database, such set operations may be performed completely asynchronous from sketch generation described in block  1055 . The resulting sketch, Sketch S″ j,k , at block  1070  would represent the common user ID data for months j and k. Sketch S″ j,k  may be evaluated for cardinality at block  1075  for estimation of cardinality Result j,k . The cardinality, Result j,k , would represent the unique count of common users that have visited the site both in month j and k. Result j,k  may be stored in a database at block  1080 . From the data base of results, at block  1080 , a bar chart,  990 , can be generated, where each bar represents a count of retention of users who have visited the web application each month. 
         [0105]    Set operation produced sketch cardinality can be estimated by the same equations used for cardinality estimation for data generated sketches. The following equation may be used to estimate cardinality based on intersection operation: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where |C U | is the cardinality of union sketch of operated sketches, x u  is the maximum value in the union sample set, and |C I | is the cardinality of the intersection sketch of the operated sketches. 
         [0106]    For set operation resulting in a theta sketch, the equation may be further simplified to: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0107]    Thus, for theta sketches, cardinality estimation using a resulting theta sketch for any set operation can be accurately estimated as: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where Δ is any set operation, |C Δ | is a cardinality of a resulting sketch sample set from Δ set operation on operated sketches, and θ Δ  is a resulting Theta from Δ operation on the operated sketches. In an embodiment, θ Δ  may be the minimum of θ&#39;s for the operated sketches. 
       Sketch System 
       [0108]      FIG. 17  is a block diagram that depicts a sketch system  1700 , in an embodiment. 
         [0109]    Sketch system  1700  may be used for transforming values of large data sets from big data, generating sketches, storing and retrieving sketches from SketchMart database, performing set operations on sketches and estimating cardinalities of large data sets that sketches represent. In an embodiment, a data stream is generated from Big Data  1710  and fed into sketch system  1700 . Value Transformer  1720 , a component of sketch system  1700 , receives the data stream from Big Data  1710 . Value Transformer  1720  transforms the data stream into a representative set of values based on which sketches can be generated. 
         [0110]    Sketch Generator  1730 , a component of sketch system  1700 , receives the transformed data set from Value Transformer  1720  and based on the transformed data set, generates a sketch. Sketch Generator  1730  may feed sketches into Cardinality Estimator  1740  or Sketch Operator  1750 , or store sketches into SketchMart database  1760  for later retrieval. In an embodiment, Sketch Generator  1730  may receive multiple transformed data sets at once, generate sketches in parallel and feed the sketches to other components of sketch system  1700 . 
         [0111]    Sketch Operator  1750 , a component of sketch system  1700 , may receive sketches as input from Sketch Generator  1730  or retrieve sketches from SketchMart database  1760 . Sketch Operator  1750  performs set operations on input sketches producing a resulting sketch. Sketch Operator  1750  may store resulting sketches in SketchMart database  1760  or feed sketches to Cardinality Estimator  1740 . In an embodiment, Sketch Operator  1750  may perform multiple set operations in parallel and feed resulting sketches to other components of sketch system  1700 . 
         [0112]    Upon receipt of a sketch, Cardinality Estimator  1740 , a component of sketch system  1700 , processes sketch and its subset of values to estimate result. In an embodiment, Cardinality Estimator  1740  may perform multiple estimations in parallel to yield multiple results. 
         [0113]    Each of Value Transformer  1720 , Sketch Generator  1730 , Cardinality Estimator  1740 , Sketch Operator  1750  and SketchMart database  1760  are part of computer system and may be implemented in software, hardware, or a combination of software and hardware. For example, one or more of Value Transformer  1720 , Sketch Generator  1730 , Cardinality Estimator  1740 , Sketch Operator  1750  and SketchMart database  1760  may be implemented using stored program logic. 
         [0114]      FIG. 18  is a block diagram that depicts various infrastructure components through which a sketch system may be implemented, in an embodiment. The infrastructure components in aggregation are referred to as an analytical data warehouse (ADW)  1800 . ADW  1800  may be built using the Hadoop File System (HDFS)  1808  for distributed data storage and Hadoop-MR6, which is a MR (map reduced) driven processing system. Multiple Hadoop systems or grids  1808  may be used. Hadoop grid  1808  includes a Hive-7 system  1820 . Optionally, ADW  1800  further includes a Spark/Shark satellite cluster  1830 . 
         [0115]    In a related embodiment, SketchMart database  1760  may be implemented as part of HDFS  1808 . Value Transformer  1720 , Sketch Generator  1730 , Cardinality Estimator  1740 , and Sketch Operator  1750  may be implemented using one or more Hive-7 system  1820  clusters or Spark/Shark satellite clusters  1830 . 
       Hardware Overview 
       [0116]    According to one embodiment, the techniques described herein are implemented by one or more special-purpose computing devices. The special-purpose computing devices may be hard-wired to perform the techniques, or may include digital electronic devices such as one or more application-specific integrated circuits (ASICs) or field programmable gate arrays (FPGAs) that are persistently programmed to perform the techniques, or may include one or more general purpose hardware processors programmed to perform the techniques pursuant to program instructions in firmware, memory, other storage, or a combination. Such special-purpose computing devices may also combine custom hard-wired logic, ASICs, or FPGAs with custom programming to accomplish the techniques. The special-purpose computing devices may be desktop computer systems, portable computer systems, handheld devices, networking devices or any other device that incorporates hard-wired and/or program logic to implement the techniques. 
         [0117]    For example,  FIG. 19  is a block diagram that illustrates a computer system  1900  upon which an embodiment of the invention may be implemented. Computer system  1900  includes a bus  1902  or other communication mechanism for communicating information, and a hardware processor  1904  coupled with bus  1902  for processing information. Hardware processor  1904  may be, for example, a general purpose microprocessor. 
         [0118]    Computer system  1900  also includes a main memory  1906 , such as a random access memory (RAM) or other dynamic storage device, coupled to bus  1902  for storing information and instructions to be executed by processor  1904 . Main memory  1906  also may be used for storing temporary variables or other intermediate information during execution of instructions to be executed by processor  1904 . Such instructions, when stored in non-transitory storage media accessible to processor  1904 , render computer system  1900  into a special-purpose machine that is customized to perform the operations specified in the instructions. 
         [0119]    Computer system  1900  further includes a read only memory (ROM)  1908  or other static storage device coupled to bus  1902  for storing static information and instructions for processor  1904 . A storage device  1910 , such as a magnetic disk, optical disk, or solid-state drive is provided and coupled to bus  1902  for storing information and instructions. 
         [0120]    Computer system  1900  may be coupled via bus  1902  to a display  1912 , such as a cathode ray tube (CRT), for displaying information to a computer user. An input device  1914 , including alphanumeric and other keys, is coupled to bus  1902  for communicating information and command selections to processor  1904 . Another type of user input device is cursor control  1916 , such as a mouse, a trackball, or cursor direction keys for communicating direction information and command selections to processor  1904  and for controlling cursor movement on display  1912 . This input device typically has two degrees of freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), that allows the device to specify positions in a plane. 
         [0121]    Computer system  1900  may implement the techniques described herein using customized hard-wired logic, one or more ASICs or FPGAs, firmware and/or program logic which in combination with the computer system causes or programs computer system  1900  to be a special-purpose machine. According to one embodiment, the techniques herein are performed by computer system  1900  in response to processor  1904  executing one or more sequences of one or more instructions contained in main memory  1906 . Such instructions may be read into main memory  1906  from another storage medium, such as storage device  1910 . Execution of the sequences of instructions contained in main memory  1906  causes processor  1904  to perform the process steps described herein. In alternative embodiments, hard-wired circuitry may be used in place of or in combination with software instructions. 
         [0122]    The term “storage media” as used herein refers to any non-transitory media that store data and/or instructions that cause a machine to operate in a specific fashion. Such storage media may comprise non-volatile media and/or volatile media. Non-volatile media includes, for example, optical disks, magnetic disks, or solid-state drives, such as storage device  1910 . Volatile media includes dynamic memory, such as main memory  1906 . Common forms of storage media include, for example, a floppy disk, a flexible disk, hard disk, solid-state drive, magnetic tape, or any other magnetic data storage medium, a CD-ROM, any other optical data storage medium, any physical medium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, NVRAM, any other memory chip or cartridge. 
         [0123]    Storage media is distinct from but may be used in conjunction with transmission media. Transmission media participates in transferring information between storage media. For example, transmission media includes coaxial cables, copper wire and fiber optics, including the wires that comprise bus  1902 . Transmission media can also take the form of acoustic or light waves, such as those generated during radio-wave and infra-red data communications. 
         [0124]    Various forms of media may be involved in carrying one or more sequences of one or more instructions to processor  1904  for execution. For example, the instructions may initially be carried on a magnetic disk or solid-state drive of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system  1900  can receive the data on the telephone line and use an infra-red transmitter to convert the data to an infra-red signal. An infra-red detector can receive the data carried in the infra-red signal and appropriate circuitry can place the data on bus  1902 . Bus  1902  carries the data to main memory  1906 , from which processor  1904  retrieves and executes the instructions. The instructions received by main memory  1906  may optionally be stored on storage device  1910  either before or after execution by processor  1904 . 
         [0125]    Computer system  1900  also includes a communication interface  1918  coupled to bus  1902 . Communication interface  1918  provides a two-way data communication coupling to a network link  1920  that is connected to a local network  1922 . For example, communication interface  1918  may be an integrated services digital network (ISDN) card, cable modem, satellite modem, or a modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface  1918  may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also be implemented. In any such implementation, communication interface  1918  sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of information. 
         [0126]    Network link  1920  typically provides data communication through one or more networks to other data devices. For example, network link  1920  may provide a connection through local network  1922  to a host computer  1924  or to data equipment operated by an Internet Service Provider (ISP)  1926 . ISP  1926  in turn provides data communication services through the world wide packet data communication network now commonly referred to as the “Internet”  1928 . Local network  1922  and Internet  1928  both use electrical, electromagnetic or optical signals that carry digital data streams. The signals through the various networks and the signals on network link  1920  and through communication interface  1918 , which carry the digital data to and from computer system  1900 , are example forms of transmission media. 
         [0127]    Computer system  1900  can send messages and receive data, including program code, through the network(s), network link  1920  and communication interface  1918 . In the Internet example, a server  1930  might transmit a requested code for an application program through Internet  1928 , ISP  1926 , local network  1922  and communication interface  1918 . 
         [0128]    The received code may be executed by processor  1904  as it is received, and/or stored in storage device  1910 , or other non-volatile storage for later execution. 
         [0129]    In the foregoing specification, embodiments of the invention have been described with reference to numerous specific details that may vary from implementation to implementation. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. The sole and exclusive indicator of the scope of the invention, and what is intended by the applicants to be the scope of the invention, is the literal and equivalent scope of the set of claims that issue from this application, in the specific form in which such claims issue, including any subsequent correction.