# Patent

## Title
Method for estimating the number of distinct values in a partitioned dataset

## Abstract

The task of estimating the number of distinct values (DVs) in a large dataset arises in a wide variety of settings in computer science and elsewhere. The present invention provides synopses for DV estimation in the setting of a partitioned dataset, as well as corresponding DV estimators that exploit these synopses. Whenever an output compound data partition is created via a multiset operation on a pair of (possibly compound) input partitions, the synopsis for the output partition can be obtained by combining the synopses of the input partitions. If the input partitions are compound partitions, it is not necessary to access the synopses for all the base partitions that were used to construct the input partitions. Superior (in certain cases near-optimal) accuracy in DV estimates is maintained, especially when the synopsis size is small. The synopses can be created in parallel, and can also handle deletions of individual partition elements.

## Background
BACKGROUND OF THE INVENTION
1. Field of Invention
The present invention relates generally to the field of distinct-value estimation. More specifically, the present invention is related to a method for estimating the number of distinct values in a partitioned dataset.
2. Discussion of Related Art
The task of determining the number of distinct values (DVs) in a large dataset arises in a wide variety of settings in computer science, including data integration (for example, see papers to Brown et al. entitled, “Toward automated large-scale information integration and discovery,” and Dasu et al. entitled, “Mining database structure; or, how to build a data quality browser,”) query optimization (for example, see papers to Ioannidis et al, entitled, “The history of histograms,” and the paper to Selinger et al. entitled, “Access path selection in a relational database management system,”) network monitoring (for example, see paper to Estan et al. entitled, “Bitmap algorithms for counting active flows on high speed links,”) and OLAP (for example, see papers to Padmanabhan et al. entitled, “Multi-dimensional clustering: a new data layout scheme in DB2,” and the paper to Shukla et al. entitled, “Storage estimation for multidimensional aggregates in the presence of hierarchies”). The number of distinct values can be computed exactly by sorting the dataset and then executing a straightforward scan-and-count pass over the data; alternatively, a hash table can be constructed and used to compute the number of distinct values. Neither of these approaches scales to the massive datasets often encountered in practice, due to heavy time and memory requirements. A great deal of research over the past twenty five years has therefore focused on approximate methods that scale to very large datasets. These methods work either by drawing a random sample of the data items and using the observed frequencies of the values in the sample as a basis for estimation (see, for example, the paper to Charikar et al. entitled, “Towards estimation error guarantees for distinct values,” the paper to Haas et al. entitled, “An estimator of the number of species from quadrat sampling,” and the paper to Haas et al. entitled, “Estimating the number of classes in a finite population”) or by taking a single pass through the data and using hashing techniques to compute an estimate using a bounded amount of memory (see, for example, the paper to Alon et al. entitled, “The space complexity of approximating the frequency moments,” the paper to Astrahan et al. entitled, “Approximating the number of unique values of an attribute without sorting,” the paper to Bar-Yossef et al. entitled, “Counting distinct elements in a data stream,” the paper to Durand et al. entitled, “Loglog counting of large cardinalities,” the paper to Estan et al. entitled, “Bitmap algorithms for counting active flows on high speed links,” the paper to Flajolet et al. entitled, “Probabilistic counting algorithms for data base applications,” the paper to Gibbons et al. entitled, “Distinct sampling for highly-accurate answers to distinct values queries and event reports,” the paper to Giroire entitled, “Order statistics and estimating cardinalities of massive data sets,” and the paper to Whang et al. entitled, “A linear-time probabilistic counting algorithm for database applications”).
Almost all of this work has focused on producing a single synopsis of the entire dataset and then using the synopsis to obtain a DV estimate; methods for combining and exploiting synopses in the presence of set operations on partitioned datasets are virtually nonexistent. The present invention provides DV estimation methods in the context of a partitioned dataset, such as the “synopsis warehouse” environment described in the paper to Brown et al. entitled, “Techniques for warehousing of sample data.” In a synopsis warehouse, incoming data is split into partitions, i.e., multisets of values, and a synopsis is created for each partition; the synopses are used to quickly estimate various partition properties. As partitions are rolled in and out of a full-scale warehouse, the corresponding synopses are rolled in and out of the synopsis warehouse. The architecture requires that synopses can be created in parallel, ensuring scalability, and that synopses can be combined to create a synopsis corresponding to the multiset union, intersection, or difference of the corresponding partitions, providing flexibility. The term “partition” is used here in a very general sense. Data may be partitioned—e.g., by time-stamp, by data value, and so forth—for purposes of parallel processing and dealing with fluctuating data-arrival rates. Data may also, however, be partitioned by its source—e.g., SAP customer addresses versus PeopleSoft customer addresses. In the latter scenario, comparison of data characteristics in different partitions may be of interest for purposes of metadata discovery and automated data integration (see, for example, the paper to Brown et al. entitled, “Toward automated large-scale information integration and discovery”). For example, DV estimates can be used to detect keys and duplicates in a partition, can help discover subset-inclusion and functional-dependency relationships, and can be used to approximate the Jaccard distance or other similarity metrics between the domains of two partitions (see, for example, the paper to Brown et al. entitled, “Toward automated large-scale information integration and discovery” and the paper to Dasu et al. entitled, “Mining database structure; or, how to build a data quality browser”).
Now, previously-proposed synopses, DV estimators, and methods for handling compound partitions are discussed.
Synopses for Dv Estimation
In general, the literature on DV estimation does not discuss synopses explicitly, and hence does not discuss issues related to combining synopses in the presence of multiset operations on the corresponding partitions. One can, however, directly infer potential candidate synopses from the various algorithm descriptions.
Bit-Vector Synopses
The oldest class of synopses comprises various types of bit vectors. The “linear counting” technique (see, for example, the paper to Astrahan et al. entitled, “Approximating the number of unique values of an attribute without sorting,” the paper to Estan et al. entitled, “Bitmap algorithms for counting active flows on high speed links,” and the paper to Whang et al. entitled, “A linear-time probabilistic counting algorithm for database applications”) uses a bit vector V of length M=O(D), together with a hash function h from  to {1, 2, . . . , M}, where  denotes the domain of the dataset of interest and D=|| is the number of distinct values in the dataset. The function h is applied to each element v in A, and the h(v)th bit of V is set to 1. After the dataset has been scanned, the estimate of D is the total number of 1-bits in V, multiplied by a correction factor. The correction factor compensates for undercounting due to “hash collisions” in which h(v)=h(v′) for v≠v′; see, for example, the paper to Astrahan et al. entitled, “Approximating the number of unique values of an attribute without sorting.” The O(D) storage requirement for linear counting is often prohibitive in applications where D can be very large, especially if multiple DV estimators must be maintained.
The “logarithmic counting” method of Astrahan et al. (in the paper entitled, “Approximating the number of unique values of an attribute without sorting”) and Flajolet et al. (in the paper entitled, “Probabilistic counting algorithms for data base applications”) uses a bit vector of length L=O(log D). The idea is to hash each of the distinct values in the dataset to the set {0,1}L of binary strings of length L, and keep track of r, the position (counting from the left, starting at 0) of the leftmost 0 bit over all of the hashed values. The estimate is roughly of the form 2r (multiplied by a certain factor that corrects for “bias” and hash collisions). This tracking of r is achieved by taking each hashed value, transforming the value by zeroing out all but the leftmost 1, and computing the bitwise-OR of the transformed values. The value of r is then given by the leftmost 0 bit in the resulting bit vector. In the complete algorithm, several independent values of r are, in effect, averaged together (using a technique called “stochastic averaging”) and then exponentiated. Alon et al. in the paper entitled, “The space complexity of approximating the frequency moments” analyze a variant of the logarithmic counting algorithm under an assumption of pairwise-independent hashing. Recent work by Durand and Flajolet in the paper entitled, “Loglog counting of large cardinalities” improves on the storage requirement of the logarithmic counting algorithm by tracking and maintaining r, the position of the leftmost 0, directly. The number of bits needed to encode r is O(log logD), and hence the technique is called LogLog counting.
The main drawback of the above bit-vector data structures, when used as synopses in the setting of a partitioned dataset, is that union is the only supported set operation. One must, e.g., resort to the inclusion/exclusion formula to handle intersections of partitions. As the number of set operations increases, this approach becomes extremely cumbersome, expensive, and inaccurate.
Several authors (for example, see the paper to Ganguly et al. entitled, “Tracking set-expression cardinalities over continuous update streams” and the paper to Shukla et al. entitled, “Storage estimation for multidimensional aggregates in the presence of hierarchies”) have proposed replacing each bit in the logarithmic-counting bit vector by an exact or approximate counter, in order to permit DV estimation in the presence of both insertions and deletions to a dataset. This modification does not ameliorate the inclusion/exclusion problem, however.
Random Samples
Another synopsis possibility is to use a random sample of the data items in the partition (see, for example, the paper to Charikar et al. entitled, “Towards estimation error guarantees for distinct values,” the paper to Haas et al. entitled, “An estimator of the number of species from quadrat sampling,” and the paper to Haas et al. entitled, “Estimating the number of classes in a finite population”). The key drawback is that DV estimates computed from such a synopsis can be very inaccurate, especially when the data is skewed or when there are many distinct values, each having a low frequency (but not all unique); see the paper to Charikar et al. entitled, “Towards estimation error guarantees for distinct values” for a negative result on the performance of sample-based estimators. Moreover, combining synopses to handle unions of partitions can be expensive (see, for example, the paper to Brown et al. entitled, “Techniques for warehousing of sample data”), and it appears that the inclusion/exclusion formula is needed to handle intersections.
Sample-Counting Synopsis
Another type of synopsis arises from the “sample counting” DV-estimation method, also called “adaptive sampling,” credited to Wegman (see the paper to Astrahan et al. entitled, “Approximating the number of unique values of an attribute without sorting” and the paper to Flajolet et al. entitled, “Adaptive sampling”). Here the synopsis for the dataset of interest comprises a subset of {h(v): vε}, where h is a hash function as before. In more detail, the synopsis comprises a fixed-size buffer that holds binary strings of length L=log(M), together with a “reference” binary string s, also of length L. The idea is to hash the distinct values in the dataset, as in logarithmic counting, and insert the hashed values into a buffer that can hold up to k>0 hashed values; the buffer tracks only the distinct hash values inserted into it. When the buffer fills up, it is purged by removing all hashed values whose leftmost bit is not equal to the leftmost bit of s; this operation removes roughly half of the hashed values in the buffer. From this point on, a hashed value is inserted into the buffer if and only if the first bit matches the first bit of s. The next time the buffer fills up, a purge step (with subsequent filtering) is performed by requiring that the two leftmost bits of each hashed value in the buffer match the two leftmost bits of the reference string. This process continues until all the values in the dataset have been hashed. The final DV estimate is roughly equal to K2r, where r is the total number of purges that have occurred and K is the final number of values in the buffer.
The algorithms in the paper to Bar-Yossef et al. entitled, “Counting distinct elements in a data stream,” the paper to Gibbons et al. entitled, “Distinct sampling for highly-accurate answers to distinct values queries and event reports,” and the paper to Gibbons et al. entitled, “Estimating simple functions on the union of data streams,” embody the same idea, essentially with a “reference string” equal to 00 . . . 0. Indeed, the number of purges in the sample-counting algorithm corresponds to the “die level” used in the above-described paper to Gibbons et al. One difference in these algorithms is that the actual data values, and not the hashed values, are stored: the level at which a data value is stored encodes the number of leading 0's in its hashed representation. In the paper to Gibbons et al. entitled, “Distinct sampling for highly-accurate answers to distinct values queries and event reports,” the stored values are augmented with additional information. Specifically, for each distinct value in the buffer, the algorithm maintains the number of instances of the value in the dataset (here a relational table) and also maintains a reservoir sample (see, for example, the paper to Vitter et al. entitled, “Random Sampling with a Reservoir”) of the rows in the table that contain the value. This additional information can be exploited to obtain approximate answers, with probabilistic error guarantees, to a variety of SELECT DISTINCT queries over a partition. Such queries include, as a special case, the SELECT COUNT(DISTINCT) query that corresponds to the desired DV estimate. In the paper to Bar-Yossef et al. entitled, “Counting distinct elements in a data stream,” the basic sample-counting algorithm is enhanced by compressing the stored values.
For sample-counting algorithms with reference string equal to 00 . . . 0, the synopsis holds the K smallest hashed values encountered, where K lies roughly between k/2 and k. The variability in K leads to inefficient storage and unstable DV estimates relative to the present invention.
The Bellman Synopsis
In the context of the Bellman system, the authors in the paper to Dasu et al. entitled, “Mining database structure; or, how to build a data quality browser” propose a synopsis related to DV estimation. This synopsis for a partition A comprises k entries and uses independent hash functions h1, h2, . . . , hk; the ith synopsis entry is given by the ith minHash value mi=minvε(A) hi(v), where (A) is the value domain of A. The synopsis for a partition is not actually used to directly compute the number of DVs in the partition, but rather to compute the Jaccard distance between partitions. When constructing the synopsis, each scanned data item in the partition incurs a cost of O(k), since the item must be hashed k times for comparison to the k current minHash values.
DV Estimators
Prior-art DV estimators have been provided in the context of a single (unpartitioned) dataset, so we discuss prior DV estimators in this setting. The present invention provides an estimator that is superior in this setting, and that also extends to the setting of set operations on multiple partitions.
The motivation behind virtually all DV estimators can be viewed as follows. If D points are placed randomly and uniformly on the unit interval (where D is assumed to be large), then, by symmetry, the expected distance between any two neighboring points is 1/(D+1)≈1/D, so that the expected value of U(k), the kth smallest point, is E[U(k)] k/D. Thus D≈k/E[U(k)]. The simplest estimator of E[U(k)] is simply U(k) itself, and yields the basic estimator:

{circumflex over (D)}kBE=k/U(k) 

The simplest connection between the above idea and the DV estimation problem rests on the observation that a hash function often “looks like” a uniform random number generator. In particular, let v1, v2, . . . , vD be an enumeration of the distinct values in the dataset and let h be a hash function as before. For many hash functions, the sequence h(v1), h(v2), . . . , h(vD) will look like the realization of a sequence of independent and identically distributed (i.i.d.) samples from the discrete uniform distribution on {0, 1, . . . , M}. Provided that M is sufficiently greater than D, the sequence U1=h(v1)/M, U2=h(v2)/M, . . . , UD=h(vD)/M will approximate the realization of a sequence of i.i.d. samples from the continuous uniform distribution on [0,1]. This assertion requires that M be much larger than D to avoid collisions, i.e., to ensure that, with high probability, h(vi) ≠h(vj) for all i≠j. A “birthday problem” argument shows that collisions will be avoided when M=O(D2). It is assumed henceforth that, for all practical purposes, any hash function that is discussed avoids collisions. The term “looks like” is used in an empirical sense, which suffices for applications. Thus, in practice, the estimator {circumflex over (D)}kBE can be applied with U(k) taken as the kth smallest hash value, multiplied by a normalizing factor of 1/M. The estimator {circumflex over (D)}kBE is biased upwards for each possible value of D. The present invention provides an unbiased estimator that also has a lower mean-squared error (MSE) than {circumflex over (D)}kBE in the setting of a single dataset.

The estimator {circumflex over (D)}kBE was proposed in the paper to Bar-Yossef et al. entitled, “Counting distinct elements in a data stream,” along with conservative error bounds based on Chebyshev's inequality. Both the logarithmic and sample-counting estimators can be viewed as approximations to the basic estimator. For logarithmic counting, specifically the Flajolet-Martin algorithm, consider the binary decimal representation of the normalized hash values h(v)/M, where M=2L. For example, a hash value h(v)=00100110, after normalization, will have the binary decimal representation 0.00100110. It can be seen that the smallest normalized hash value is approximately equal to 2−r, so that, modulo the correction factor, the Flajolet-Martin estimator (without stochastic averaging) is 1⁄2−r, which roughly corresponds to {circumflex over (D)}1BE. Because this latter estimate is unstable, in that E[{circumflex over (D)}1BE]=∞, the final Flajolet-Martin estimator uses stochastic averaging to average independent values of r and hence compute an estimator Ê of E[log2 {circumflex over (D)}1BE], leading to a final estimate of {circumflex over (D)}=C2Ê, where the constant c approximately unbiases the estimator. (the new estimators are exactly unbiased.) For sample counting, suppose, without loss of generality, that the reference string is 00 . . . 0 and, as before, consider the normalized binary decimal representation of the hashed values. Thus the first purge leaves behind normalized values of the form 0.0 . . . , the second purge leaves behind values of the form 0.00 . . . , and so forth, the last (rth) purge leaving behind only normalized hashed values with r leading 0's. Thus the number 2−r (which has r−1 leading 0's) is roughly equal to the largest of the K normalized hashed values in the size-k buffer, so that the estimate K/2−r is roughly equal to {circumflex over (D)}kBE.
The paper to Giroire et al. entitled, “Order statistics and estimating cardinalities of massive data sets” studies a variant of {circumflex over (D)}kBE in which the hashed values are divided into m>1 subsets, leading to m i.i.d. copies of the basic estimator. These copies are obtained by dividing the unit interval into m equal segments; the jth copy of the basic estimator is based on all of the hashed values that lie in the jth segment, after shifting and scaling the segment (and the points therein) into a copy of the unit interval. (Note that for a fixed synopsis size k, each copy of the basic estimator is based on approximately k/m observations.) Each copy of the basic estimator is then subjected to a nonlinear transformation g, and multiplied by a correction factor c. The function g is chosen to “stabilize” the estimator, and the constant c is chosen to ensure that the estimator is asymptotically unbiased as k becomes large. Finally, the i.i.d. copies of the transformed estimators are averaged together. The motivation behind the transformation g is to avoid the instability problem, discussed previously, that arises when k=1. Later, the present invention's proposed estimator is shown to be unbiased for any values of D and k>1, while being less cumbersome to compute. Moreover, when D>>k>>0, the estimator provided by the current invention has approximately the minimum possible MSE, and hence is at least as statistically efficient as any estimator in the paper to Giroire et al. entitled, “Order statistics and estimating cardinalities of massive data sets.”
DV Estimators for Compound Partitions
As mentioned above, previous work has mostly focused on DV estimation for a single dataset. To allow for more scalable and flexible data processing, it is convenient to decompose a dataset into a collection of disjoint “base” partitions. A compound partition in a partitioned dataset is formed from two or more of the base partitions via one or more multiset union, multiset intersection, and multiset difference operations. To our knowledge, the only prior discussion of how to construct DV-related estimates for compound partitions is found in the paper to Dasu et al. entitled, “Mining database structure; or, how to build a data quality browser.” DV estimation for the intersection of partitions A and B is not computed directly. Instead, the Jaccard distance ρ=DA∩B/DA∪B (called the “resemblance” in the paper to Dasu et al. entitled, “Mining database structure; or, how to build a data quality browser”) is estimated first and then, using the estimator {circumflex over (ρ)}, the number of values in the intersection is estimated as




  
    
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Here and elsewhere, DX denotes the number of distinct values in partition X.

The quantities DA and DB are computed exactly, by means of GROUP BY queries; the present invention provides estimators that avoid the need to compute or estimate these quantities. There is no discussion in the paper to Dasu et al. entitled, “Mining database structure; or, how to build a data quality browser” of how to handle any set operations other than the intersection of two partitions. If one uses the principle of inclusion/exclusion to handle other set operations, the resulting estimation procedure will not scale well as the number of operations increases. The present invention's methods handle arbitrarily complex combinations of operations on partitions (multiset unions, intersections, and differences) in an efficient manner.
Whatever the precise merits, features, and advantages of the prior art, none of them achieves or fulfills the purposes of the present invention.
SUMMARY OF THE INVENTION
The present invention provides synopses for DV estimation in the setting of a partitioned dataset, as well as corresponding DV estimators that exploit these synopses. Whenever an output compound data partition is created via a multiset operation on a pair of (possibly compound) input partitions, the synopsis for the output partition can be obtained by combining the synopses of the input partitions. If the input partitions are compound partitions, it is not necessary to access the synopses for all of the base partitions that were used to construct the input partitions. Superior (in certain cases near-optimal) accuracy in the DV estimates is maintained, especially when the size of the synopsis is small.
Specifically, the present invention teaches the use of an “AKMV synopsis,” extending an idea in the paper to Bar-Yossef et al. entitled, “Counting distinct elements in a data stream” in order to handle multiset operations gracefully. The present invention's extensions involve adding counters to the basic synopsis (see for example, the paper to Ganguly et al. entitled, “Tracking set-expression cardinalities over continuous update streams,” the paper to Gibbons et al. entitled, “Distinct sampling for highly-accurate answers to distinct values queries and event reports,” and the paper to Shukla et al. entitled, “Storage estimation for multidimensional aggregates in the presence of hierarchies”). The present invention also teaches methods for combining AKMV synopses, such that the collection of these synopses is “closed” under multiset operations on the parent partitions. The AKMV synopsis can also handle deletions of individual elements from the dataset.
The present invention's new DV estimators generalize and improve the basic DV estimator proposed in the paper to Bar-Yossef et al. entitled, “Counting distinct elements in a data stream.” Specifically, the new DV estimator is unbiased and can be used with both base and compound partitions. When applied to a base partition, the new estimator has lower MSE than the basic DV estimator and, if the number of DVs is large, has essentially the minimum possible MSE of any DV estimator. The invention provides measures of estimation error to accompany the provided DV estimates, as well as a method to determine an appropriate size for the AKMV synopses prior to seeing the dataset. In an alternative embodiment, the current invention provides a maximum-likelihood DV estimator.
The present invention, in one embodiment, teaches a computer-based method to estimate distinct values in a partitioned dataset, wherein the computer-based method comprises the steps of: (a) creating a synopsis of each partition in a partitioned dataset; (b) combining created synopsis to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference; and (c) outputting said constructed DV estimations in (b).
The present invention, in one embodiment, teaches a computer-based method to estimate distinct values in a partitioned dataset, wherein the computer-based method comprises the steps of: (a) selecting a hash function h with domain , which is a value domain of the partitioned dataset, and range {0, 1, . . . , M}, wherein M=O(||2) and || is the number of distinct values in , said hash function h to be used for all synopses in said partitioned dataset; (b) in case of a partition being a base partition A with domain (A): (i) hashing each value in (A) using hash function h; (ii) recording k smallest values h(v1), h(v2), . . . , h(vk) among said hashed values, as well as recording the multiplicity of each of the values v1, v2, . . . , vk in A, and (iii) creating the synopsis as an AKMV synopsis LA+=(LA,cA) based on said recorded k smallest values and said recorded multiplicities; or in case of a partition being a compound partition G=E op F, where op is one of ∩m, ∪m, and \m, and E and F have respective AKMV synopses LE+=(LE, cE) and LF+=(LF, cF) of sizes kE and kF, creating the synopsis as an AKMV synopsis LG+=(LE⊕LF, cG), where, for each wεLE⊕LF,




  
    
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(c) combining synopsis created in (b) to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference; and (d) outputting said constructed DV estimations in (b).

In one embodiment, the following unbiased estimator is used to construct the DV estimations for a partition E having an AKMV synopsis LE+=(LE,cE) of size k:




  
    
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wherein KE is the number of non-zero elements in cE and U(k) is the largest value in LE, multiplied by a factor of 1/M.

In one embodiment, the estimator error of the DV estimator {circumflex over (D)}E is assessed as any of, or a combination of, the following: a number εp that bounds the expected absolute ratio error (ARE) from above with probability approximately 100 p %, wherein said number εp is computed as the solution to the equation p=Δ(k{circumflex over (D)}E/j, k, εp), with k the size of the AKMV synopsis LE+=(LE,cE) for E and j the observed number of nonzero elements in cE, or an approximate variance of estimator {circumflex over (D)}E, computed as




  
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In one embodiment, AKMV synopses are maintained in the presence of deletions by: updating the synopsis (L, c) of a partition after deletion of a value v from the partition, wherein h(v)=w, by updating counter c(w) according to c(w)=max(c(w)−1, 0), if w is present in L, and by periodically recomputing the synopsis from scratch when estimation errors become unacceptable.
In one embodiment, the size k of each of the synopses for the base partitions is chosen a priori by setting k=┌2+2/(πε2)┐, where ε is a desired upper bound on the expected ARE of a base-partition DV estimate.
The present invention also teaches an article of manufacture having computer readable program code implementing the steps of each of the computer-based methods described above to estimate distinct values in a partitioned dataset.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates one embodiment of the present invention's computer-based method to estimate distinct values in a partitioned dataset.
FIG. 2 another embodiment of the present invention's computer-based method to estimate distinct values in a partitioned dataset.



DESCRIPTION OF THE PREFERRED EMBODIMENTS
While this invention is illustrated and described in a preferred embodiment, the device may be produced in many different configurations, forms and materials. There is depicted in the drawings, and will herein be described in detail, a preferred embodiment of the invention, with the understanding that the present disclosure is to be considered as an exemplification of the principles of the invention and the associated functional specifications for its construction and is not intended to limit the invention to the embodiment illustrated. Those skilled in the art will envision many other possible variations within the scope of the present invention.
The AKMV Synopsis for Base Partitions
The AKMV synopsis for a base partition A is constructed as follows. First, each value in the domain (A) is hashed using a hash function h with range {0, 1, . . . , M}, and then the k smallest of the hashed values are recorded in a data structure LA. Those skilled in the art can envision appropriate data structures for this purpose; in the current description, LA is referred to as a “set.” This set is sometimes called a KMV synopsis (of size k). The AKMV synopsis LA+, =(LA, CA) is obtained by augmenting LA with a counter vector cA, with one entry for each element wεLA. Specifically, if w is an element of LA with w=h(v), then cA(w) equals the number of times the value v appears in the multiset A, i.e., the multiplicity of v in A. An AKMV synopsis is said to be of size k if its constituent KMV synopsis is of size k. An AKMV vector requires O(k log DA+k log QA) bits of storage, where DA is the number of distinct values in A and QA is the maximum multiplicity of a value in A. Note that, if A is a set, rather than a multiset, then it suffices to maintain a bit vector rather than a vector of counts, so that the size of the synopsis is O(k log DA),
An AKMV synopsis of size k can be computed from a single scan of the data partition, using Algorithm 1, shown below.










Algorithm 1 - (AKMV Computation for a Base Partition)

















1: h: hash function from domain of dataset to {0, 1, ... , M}



2: L: list of k smallest hashed partition values seen so far



3: maxVal(L): returns the largest value in L



4: c: list of counters corresponding to the values in L



5:



6: for each item v in the partition do



7:   w = h(v)



8:   if w∉L then



9:     if |L| < k then



10:     insert w into L and set new counter c(w) = 1



11:    else if w < maxVal(L) then



12:     insert w into L and set new counter c(w) = 1



13:     remove largest element w’ of L



14:     remove counter c(w’)



15:    end if



16:  else set c(w) = c(w) + 1



17:  end if



18: end for










One skilled in the art will envision appropriate data structures for implementing this algorithm, for example, a priority queue to facilitate identification of the largest element of L in lines 11 and 13, and a temporary hash table to facilitate the membership check in line 8 and the counter updates in lines 14 and 16. Assuming that the scan order of the items in a partition is independent of the hashed item values, as is virtually always the case, it can be shown that the expected cost to construct an AKMV synopsis of size k from a base partition A comprising NA data items having DA distinct values is O(NA+k (log k)(log DA)). This construction cost compares favorably to the O(kNA) cost for the Bellman synopsis. Moreover, when DA is small, the AKMV synopsis contains more information in a statistical sense than the Bellman synopsis, since the former synopsis essentially samples distinct values without replacement, whereas the latter synopsis samples distinct values with replacement. The cost for the AKMV synopsis is comparable to that of the sample-counting synopsis in the paper to Gibbons et al. entitled, “Estimating simple functions on the union of data streams.” Indeed, the sample-counting synopsis is very similar to KMV, except that the size is a random variable K whose value ranges between roughly k/2 and k. Thus the AKMV synopsis contains more statistical information for a given space allocation, and yields DV estimators that are statistically more stable.
Combining AKMV Synopses
A method for combining AKMV synopses is now described. Suppose partitions E and F—having respective AKMV synopses LE+=(LE, cE) and LF+=(LF, cF) of sizes kE and kF—are combined to create a compound partition G=E op F, where op is one of multiset-union, multiset-intersection, or multiset-difference, denoted by ∩m, ═m, and \m, respectively. It is assumed throughout that all synopses are created using the same hash function h from  to {1, 2, . . . , M}, where  denotes the domain of the values in the dataset and M=O(||2) as discussed previously. The AKMV synopsis for G is defined as LG+=(LE⊕LF, cG). Here, LE⊕LF is defined to be the set comprising the k smallest values in LE∩LF, where k=min(kE, kF). (Observe that the ⊕ operator is symmetric and associative.) Moreover, cG is defined by setting




  
    
      c
      G
    
    ⁡
    
      (
      w
      )
    
  
  =
  
    {
    
      
        
          
            
              
                c
                E
              
              ⁡
              
                (
                w
                )
              
            
            +
            
              
                c
                F
              
              ⁡
              
                (
                w
                )
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              ⋃
              m
            
          
        
      
      
        
          
            min
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  
                    c
                    E
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
                ,
                
                  
                    c
                    F
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
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            max
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  
                    
                      c
                      E
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                  -
                  
                    
                      c
                      F
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                
                ,
                0
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              \
              m
            
          
        
      
    
  




for each wεLE⊕LF. In these definitions, take cX(w)=0 if w∉LX.

In the above method, E and F may themselves be compound partitions. It follows that, in general, if G is a compound partition created by applying the operations ∩m, ∪m, and \m to n≧2 base partitions A1, A2, . . . , An, the AKMV synopsis LG+=(LG, cG) satisfies LG=LA1⊕+LA2 ⊕ . . . ⊕LAn and, moreover, if wεLG with w=h(v), then cE(W) is precisely the multiplicity of v in E. Importantly, when combining E and F to create G, the AKMV synopsis LG+ can be created directly from LE+ and LF+; there is no need to access the “base” synopses LA1+, LA2+, . . . , LAn+.
In summary, the collection of AKMV synopses over compound partitions is closed under multiset operations. Therefore, synopses for base partitions can be computed using Algorithm 1 and, as compound partitions are built up from base partitions, the synopses for these compound partitions can be built up concomitantly.
One skilled in the art will perceive that, if the only multiset operations of interest are multiset-union and multiset-intersection, then it suffices to maintain KMV synopses, rather than AKMV synopses.
Handling Deletions
It is now shown how AKMV synopses can easily support deletions of individual items. Consider a partition A that receives a stream of transactions of the form +v or −v, corresponding to the insertion or deletion, respectively, of value v.
The invention maintains two AKMV synopses: a synopsis Li+ for the multiset Ai of inserted items and a synopsis Ld+ for the multiset Ad of deleted items. Computing the AKMV synopsis of the multiset difference Ai\m Ad yields the AKMV synopsis LA+ of the true multiset A. Because Ai and Ad are always growing with time, they can become significantly larger than the actual partition A, so that DV estimates will be of low quality. Therefore, whenever the number of deletions causes the error bounds to become unacceptable, a fresh scan of the data can be used to produce a new pair of synopses Li+ and Ld+ corresponding to Ai=A and Ad=ø.
This method does not actually require two synopses. The counters in a single AKMV synopsis L are maintained by incrementing the counter at each insertion and decrementing at each deletion, while ensuring that each counter is nonnegative. If synopsis entries are retained having counter values equal to 0, the synopsis L described above is produced.
New DV Estimator
Given a compound partition E and a corresponding AKMV synopsis LE+=(LE, cE) of size k, the preferred embodiment estimates the number of DVs in E as:




  
    
      D
      ^
    
    E
  
  =
  
    
      
        K
        E
      
      k
    
    ⁢
    
      (
      
        
          k
          -
          1
        
        
          U
          
            (
            k
            )
          
        
      
      )
    
  




where KE is the number of non-zero elements in cE and U(k) is the largest value in LE, multiplied by a normalizing factor of 1/M. This estimator is unbiased for DE, the true number of distinct values in E.

In the special case where E=A for a base partition A, it is seen that KA=k, and so the general estimator reduces to the simpler expression




  
    
      D
      ^
    
    A
  
  =
  
    
      k
      -
      1
    
    
      U
      
        (
        k
        )
      
    
  




where U(k) is the largest value in LA, multiplied by 1/M. When the number of distinct values in A is large, {circumflex over (D)}A has approximately the minimum possible MSE of any estimator of DA, and has smaller MSE than the basic estimator {circumflex over (D)}kBE for any number of distinct values.

Assessing Estimation Error

The error associated with {circumflex over (D)}E may be assessed, as follows. For n≧k≧1, set
Δ(n, k, ε)=Iu(n,k,ε)(k, n−k+1)−Ii(n,k,ε)(k, n−k+1)
where Ix(a,b) is the regularized incomplete beta function defined by



  
    
      
        
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          x
        
        ⁡
        
          (
          
            a
            ,
            b
          
          )
        
      
      =
      
        
          
            ∫
            0
            x
          
          ⁢
          
            
              
                
                  t
                  
                    a
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    1
                    -
                    t
                  
                  )
                
              
              
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                -
                1
              
            
            ⁢
            
              d
              t
            
          
        
        
          
            ∫
            0
            1
          
          ⁢
          
            
              
                
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                  (
                  
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            ⁢
            
              d
              t
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      and
      ⁢
      
        
      
      ⁢
      where
    
  




  
    
      u
      ⁡
      
        (
        
          D
          ,
          k
          ,
          ɛ
        
        )
      
    
    =
    
      
        
          
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            -
            1
          
          
            
              (
              
                1
                -
                ɛ
              
              )
            
            ⁢
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        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          l
          ⁡
          
            (
            
              D
              ,
              k
              ,
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            )
          
        
      
      =
      
        
          
            k
            -
            1
          
          
            
              (
              
                1
                +
                ɛ
              
              )
            
            ⁢
            D
          
        
        .
      
    
  



Take Δ(∞,k,ε)=0. Then, given an estimate {circumflex over (D)}E computed as above using an AKMV synopsis LE+ of size k, together with an observed value KE=j and a specified value 0<p<1, use standard numerical root-finding algorithms to compute the value εp that solves the equation

p=Δ(k{circumflex over (D)}E/j,k,εp).

Then the absolute ratio error (ARE) of the estimator, defined as





  
    
      A
      ⁢
      
        
      
      ⁢
      R
      ⁢
      
        
      
      ⁢
      
        E
        ⁡
        
          [
          
            
              D
              ^
            
            E
          
          ]
        
      
    
    =
    
      
        
        
          
            
              D
              ^
            
            E
          
          -
          
            D
            E
          
        
        
      
      
        D
        E
      
    
  
  ,




is less than or equal to εp with probability approximately 100 p %.

In an alternative embodiment, the variance of the estimator may be computed approximately as




  
    
      Var
      ⁡
      
        [
        
          
            D
            ^
          
          E
        
        ]
      
    
    ≈
    
      
        
          
            D
            ^
          
          E
        
        (
        
          
            
              
                (
                
                  k
                  -
                  1
                
                )
              
              2
            
            
              U
              
                (
                k
                )
              
            
          
          -
          
            k
            2
          
          +
          k
          +
          
            
              D
              ^
            
            E
          
        
        )
      
      
        k
        ⁡
        
          (
          
            k
            -
            2
          
          )
        
      
    
  
  ,




where U(k) is the largest value in LE, multiplied by 1/M.

Those skilled in the art can see how to modify the forgoing methods (both for computing and combining AKMV synopses and for estimating the number of DVs in a partition) to handle degenerate cases in which the size of a synopsis of a partition coincides with the number of DVs in the partition, so that the synopsis contains complete information about the number of DVs in the partition.
Alternative DV Estimators
The preferred embodiment {circumflex over (D)}E of the DV estimator for a compound partition E, based on an AKMV synopsis of size k, can be written as {circumflex over (D)}E={circumflex over (D)}E(1){circumflex over (D)}E(2), where {circumflex over (D)}E(1)=KE/k and {circumflex over (D)}E(2)=(k−1)/U(k). Denoting by A1, A2, . . . , An, the base partitions used to construct E and by V the union of sets A1, A2, . . . , An, it can be seen that the first term {circumflex over (D)}E(1) is an unbiased estimator of the ratio DE/DV, and the second term {circumflex over (D)}E(2) is an unbiased estimator of DV, and these two terms are statistically independent. In particular, if E=A1∪A2, then {circumflex over (D)}E(1) is an unbiased estimator of the Jaccard distance between A1 and A2.
In alternative embodiments, the term {circumflex over (D)}E(2) may be replaced by the basic estimator {circumflex over (D)}kBE for DV, or by the maximum likelihood estimator for DV. The latter estimator is computed as the solution D to the equation ln(1−U(k))−Ψ(D−k+1)+Ψ(D+1)=0, where U(k) is the largest value in the KMV synopsis for E, multiplied by 1/M, and Ψ is the digamma function. This equation can be solved numerically using standard root-finding algorithms. When the number of distinct values is large, the values of {circumflex over (D)}E(2), the basic estimator, and the maximum-likelihood estimator are almost identical.
Pre-Selecting a Synopsis Size
An appropriate value of k may be selected in advance of observing the dataset. Suppose that it is desired to limit the expected ARE of the DV estimate for each partition to a maximum value of ε. This goal can be achieved by choosing k=┌2+2/(πε2)┐, where ┌x┐ denotes the smallest integer greater than or equal to x.
FIG. 1 illustrates one embodiment of the present invention's computer-based method to estimate distinct values in a partitioned dataset, wherein the computer-based method comprises the steps of: (a) creating a synopsis of each partition in a partitioned dataset 102; (b) combining created synopsis to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference 104; and (c) outputting said constructed DV estimations in (b) 106.
When the partition is a base partition A with domain (A), the method creates a synopsis of a partition as per the following steps: selecting a hash function h with domain , which is a value domain of the partitioned dataset, and range {0, 1, . . . , M}, wherein M=O(||2) and || is the number of distinct values in , said hash function h to be used for all synopses in said partitioned dataset; hashing each value in (A) using hash function h; recording k smallest values h(v1), h(v2), . . . , h(vk) among said hashed values, as well as recording the multiplicity of each of the values v1, v2, . . . , vk in A, and creating said synopsis as an AKMV synopsis LA+=(LA,cA) based on said recorded k smallest values and said recorded multiplicities.
When a partition is a compound partition G=E op F (where op is one of ∪m, ∩m, and \m, and E and F have respective AKMV synopses LE+=(LE, cE) and LF+=(LF, cF) of sizes kE and kF), the method creates a synopses of a partition as per the following steps: selecting a hash function h with domain , which is a value domain of the partitioned dataset, and range {0, 1, . . . , M}, wherein M=O(||) and || is the number of distinct values in , said hash function h to be used for all synopses in said partitioned dataset; and creating said synopsis as an AKMV synopsis LG+=(LE⊕LF, cG), where, for each wε




  
    
      L
      E
    
    ⊕
    
      L
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  ,
  
    
      
        c
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    =
    
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FIG. 2 illustrates another embodiment of the present invention's computer-based method to estimate distinct values in a partitioned dataset, wherein the computer-based method comprises the steps of: (a) selecting a hash function h with domain , which is a value domain of the partitioned dataset, and range {0, 1, . . . , M}, wherein M=O(||) and || is the number of distinct values in , said hash function h to be used for all synopses in said partitioned dataset 202; (b) in case of a partition being a base partition A with domain (A) 204: (i) hashing each value in (A) using hash function h; (ii) recording k smallest values h(v1), h(v2), . . . , h(vk) among said hashed values, as well as recording the multiplicity of each of the values v1, v2, . . . , vk in A, and (iii) creating the synopsis as an AKMV synopsis LA+=(LA, cA) based on said recorded k smallest values and said recorded multiplicities; or in case of a partition being a compound partition G=E op F 206, where op is one of ∪m, ∩m, and \m, and E and F have respective AKMV synopses LE+=(LE, cE) and LF+=(LF, cF) of sizes kE and kF, creating the synopsis as an AKMV synopsis LG+=(LE⊕LF, cG), where, for each wεLE⊕LF,




  
    
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  =
  
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                )
              
            
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              ⁢
              
                
              
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            =
            
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                    E
                  
                  ⁡
                  
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            ⁡
            
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(c) combining synopsis created in (b) to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference 208; and (d) outputting said constructed DV estimations in (b) 210.

In one embodiment, the following unbiased estimator is used to construct the DV estimations for a partition E having an AKMV synopsis LE+=(LE, cE) of size k:




  
    
      D
      ^
    
    E
  
  =
  
    
      
        K
        E
      
      k
    
    ⁢
    
      (
      
        
          k
          -
          1
        
        
          U
          
            (
            k
            )
          
        
      
      )
    
  




wherein KE is the number of non-zero elements in cE and U(k) is the largest value in LE, multiplied by a factor of 1/M.

In one embodiment, the estimator error of the DV estimator {circumflex over (D)}E is assessed as any of, or a combination of, the following: a number εp that bounds the expected absolute ratio error (ARE) from above with probability approximately 100 p %, wherein said number εp is computed as the solution to the equation p=Δ(k{circumflex over (D)}E/j, k, εp), with k the size of the AKMV synopsis LE+=(LE,cE) for E and j the observed number of nonzero elements in cE, or an approximate variance of estimator {circumflex over (D)}E, computed as




  
    Var
    ⁡
    
      [
      
        
          D
          ^
        
        E
      
      ]
    
  
  ≈
  
    
      
        
          
            D
            ^
          
          E
        
        (
        
          
            
              
                (
                
                  k
                  -
                  1
                
                )
              
              2
            
            
              U
              
                (
                k
                )
              
            
          
          -
          
            k
            2
          
          +
          k
          +
          
            
              D
              ^
            
            E
          
        
        )
      
      
        k
        ⁡
        
          (
          
            k
            -
            2
          
          )
        
      
    
    .
  




In one embodiment, AKMV synopses are maintained in the presence of deletions by: updating the synopsis (L, c) of a partition after deletion of a value v from the partition, wherein h(v)=w, by updating counter c(w) according to c(w)=max(c(w)−1, 0), if w is present in L, and by periodically recomputing the synopsis from scratch when estimation errors become unacceptable.
In one embodiment, the size k of each of the synopses for the base partitions is chosen a priori by setting k=┐2+2/(πε2)┌, where ε is a desired upper bound on the expected ARE of a base-partition DV estimate.
Additionally, the present invention provides for an article of manufacture comprising computer readable program code contained within implementing one or more modules to implement a method for estimating the number of distinct values in a collection of data partitions. Furthermore, the present invention includes a computer program code-based product, which is a storage medium having program code stored therein which can be used to instruct a computer to perform any of the methods associated with the present invention. The computer storage medium includes any of, but is not limited to, the following: CD-ROM, DVD, magnetic tape, optical disc, hard drive, floppy disk, ferroelectric memory, flash memory, ferromagnetic memory, optical storage, charge coupled devices, magnetic or optical cards, smart cards, EEPROM, EPROM, RAM, ROM, DRAM, SRAM, SDRAM, or any other appropriate static or dynamic memory or data storage devices.
Implemented in computer program code based products are software modules for: (a) creating a synopsis of each partition in a partitioned dataset; (b) combining created synopsis to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference; and (c) outputting said constructed DV estimations in (b).
CONCLUSION
A system and method has been shown in the above embodiments for the effective implementation of a method for estimating the number of distinct values in a partitioned dataset. While various preferred embodiments have been shown and described, it will be understood that there is no intent to limit the invention by such disclosure, but rather, it is intended to cover all modifications and alternate constructions falling within the spirit and scope of the invention, as defined in the appended claims. For example, the present invention should not be limited by size, materials, or specific manufacturing techniques.

## Claims

1. A computer-based method to estimate distinct values in a partitioned dataset, said computer-based implemented in computer readable program code stored in computer memory, said computer-based method comprising the steps of:
a. creating a synopsis of each partition in a partitioned dataset;
b. combining created synopsis to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference; and
c. outputting said constructed DV estimations in (b); and
wherein said step of creating synopsis of a partition further comprises the steps of: selecting a hash function h with domain , which is a value domain of the partitioned dataset, and range {0, 1, . . . , M}, wherein M=O(||2) and ||is the number of distinct values in , said hash function h to be used for all synopses in said partitioned dataset;
when a partition is a base partition A with domain (A):
i. hashing each value in (A) using hash function h;
ii. recording k smallest values h(v1), h(v2), . . . , h(vk) among said hashed values, as well as recording the multiplicity of each of the values v1, v2, . . . , vk in A, and
iii. creating said synopsis as an AKMV synopsis LA+=(LA,cA) based on said recorded k smallest values and said recorded multiplicities; or when a partition is a compound partition G=E op F, where op is one of ∪m, ∩m, and \m, and E and F have respective AKMV synopses LE+=(LE, cE) and LF +=(LF, cF) of sizes kE and kF, creating said synopsis as an AKMV synopsis LG+=(LE ⊕ LF, cG), where, for each w ∈ LE ⊕ LF, cG 






  
    
      c
      G
    
    ⁡
    
      (
      w
      )
    
  
  =
  
    {
    
      
        
          
            
              
                c
                E
              
              ⁡
              
                (
                w
                )
              
            
            +
            
              
                c
                F
              
              ⁡
              
                (
                w
                )
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              ⋃
              m
            
          
        
      
      
        
          
            min
            ⁡
            
              (
              
                
                  
                    c
                    E
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
                ,
                
                  
                    c
                    F
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              ⋂
              m
            
          
        
      
      
        
          
            max
            ⁡
            
              (
              
                
                  
                    
                      c
                      E
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                  -
                  
                    
                      c
                      F
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                
                ,
                0
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              
                \
                m
              
              .

2. The computer-based method of claim 1, wherein the following unbiased estimator is used to construct said DV estimations for a partition E having an AKMV synopsis LE+=(LE,cE) of size k:




  
    
      D
      ^
    
    E
  
  =
  
    
      
        K
        E
      
      k
    
    ⁢
    
      (
      
        
          k
          -
          1
        
        
          U
          
            (
            k
            )
          
        
      
      )
    
  




wherein KE is the number of non-zero elements in cE and U(k) is the largest value in LE, multiplied by a factor of 1/M.

3. The computer-based method of claim 2, wherein estimator error of said DV estimator {circumflex over (D)}E is assessed as any of, or a combination of, the following:
(i) a number εp that bounds the expected absolute ratio error (ARE) from above with probability approximately 100 p %, wherein said number εp is computed as the solution to the equation p=Δ(k{circumflex over (D)}E/j,k,εp), with k the size of the AKMV synopsis LE+=(LE,cE) for E and j the observed number of nonzero elements in cE, or
(ii) an approximate variance of estimator {circumflex over (D)}E, computed as





  
    Var
    ⁡
    
      [
      
        
          D
          ^
        
        E
      
      ]
    
  
  ≈
  
    
      
        
          
            D
            ^
          
          E
        
        (
        
          
            
              
                (
                
                  k
                  -
                  1
                
                )
              
              2
            
            
              U
              
                (
                k
                )
              
            
          
          -
          
            k
            2
          
          +
          k
          +
          
            
              D
              ^
            
            E
          
        
        )
      
      
        k
        ⁡
        
          (
          
            k
            -
            2
          
          )
        
      
    
    .

4. The computer-based method of claim 1, wherein said AKMV synopses are maintained in the presence of deletions by:
(i) updating the synopsis (L, c) of a partition after deletion of a value v from the partition, wherein h(v) =w, by updating counter c(w) according to c(w)=max(c(w) −1, 0), if w is present in L, and
(ii) by periodically recomputing the synopsis from scratch when estimation errors become unacceptable.

5. The computer-based method of claim 1, wherein the size k of each of the synopses for the base partitions is chosen a priori by setting k=┌2+2/(πε2)┐, where ε is a desired upper bound on the expected ARE of a base-partition DV estimate.

6. An article of manufacture comprising computer usable medium implementing a computer-based to estimate distinct values in a partitioned dataset, said computer usable medium comprising:
a. computer readable program code creating a synopsis of each partition in a partitioned dataset;
b. computer readable program code combining created synopsis to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference; and
c. computer readable program code aiding in outputting said constructed DV estimations in (b)
wherein said step of creating synopsis of a partition further comprises the steps of: selecting a hash function h with domain , which is a value domain of the partitioned dataset, and range {0, 1, . . . , M}, wherein M=O(||2) and || is the number of distinct values in , said hash function h to be used for all synopses in said partitioned dataset;
when a partition is a base partition A with domain (A):
i. hashing each value in (A) using hash function h;
ii. recording k smallest values h(v1), h(v2), . . . , h(vk) among said hashed values, as well as recording the multiplicity of each of the values v1, v2, . . . , vk in A, and
iii. creating said synopsis as an AKMV synopsis LA+=(LA,cA) based on said recorded k smallest values and said recorded multiplicities; or
when a partition is a compound partition G=E op F, where op is one of ∪m,∩m, and \m, and E and F have respective AKMV synopses LE+=(LE,cE) and LF+=(LF, cF) of sizes kE and kF, creating said synopsis as an AKMV synopsis LG+=(LE⊕ LF, cG), where, for each w ∈ LE ⊕ LF,






  
    
      c
      G
    
    ⁡
    
      (
      w
      )
    
  
  =
  
    {
    
      
        
          
            
              
                c
                E
              
              ⁡
              
                (
                w
                )
              
            
            +
            
              
                c
                F
              
              ⁡
              
                (
                w
                )
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              ⋃
              m
            
          
        
      
      
        
          
            min
            ⁡
            
              (
              
                
                  
                    c
                    E
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
                ,
                
                  
                    c
                    F
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              ⋂
              m
            
          
        
      
      
        
          
            max
            ⁡
            
              (
              
                
                  
                    
                      c
                      E
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                  -
                  
                    
                      c
                      F
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                
                ,
                0
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              
                \
                m
              
              .

7. The article of manufacture of claim 6, wherein computer readable program code uses the following unbiased estimator to construct said DV estimations for a partition E having an AKMV synopsis LE+=(LE,cE) of size k:




  
    
      D
      ^
    
    E
  
  =
  
    
      
        K
        E
      
      k
    
    ⁢
    
      (
      
        
          k
          -
          1
        
        
          U
          
            (
            k
            )
          
        
      
      )
    
  




wherein KE is the number of non-zero elements in cE and U(k) is the largest value in LE, multiplied by a factor of 1/M.

8. The article of manufacture of claim 7, wherein computer readable program code assesses estimator error of said DV estimator {circumflex over (D)}E as any of, or a combination of, the following:
(i) a number εp that bounds the expected absolute ratio error (ARE) from above with probability approximately 100 p %, wherein said number εp is computed as the solution to the equation p=Δ(k{circumflex over (D)}E/j,k,εp), with k the size of the AKMV synopsis LE+=(LE,cE) for E and j the observed number of nonzero elements in cE, or
(ii) an approximate variance of estimator {circumflex over (D)}E, computed as





  
    Var
    ⁡
    
      [
      
        
          D
          ^
        
        E
      
      ]
    
  
  ≈
  
    
      
        
          
            D
            ^
          
          E
        
        (
        
          
            
              
                (
                
                  k
                  -
                  1
                
                )
              
              2
            
            
              U
              
                (
                k
                )
              
            
          
          -
          
            k
            2
          
          +
          k
          +
          
            
              D
              ^
            
            E
          
        
        )
      
      
        k
        ⁡
        
          (
          
            k
            -
            2
          
          )
        
      
    
    .

9. The article of manufacture of claim 6, wherein computer readable program code aids in maintaining said AKMV synopses in the presence of deletions by:
(iii) updating the synopsis (L, c) of a partition after deletion of a value v from the partition, wherein h(v)=w, by updating counter c(w) according to c(w)=max(c(w) −1, 0), if w is present in L, and
(iv) by periodically recomputing the synopsis from scratch when estimation errors become unacceptable.

10. The article of manufacture of claim 6, wherein computer readable program code choosing, a priori, the size k of each of the synopses for the base partitions by setting k=┌2+2/(πε2)┐, where ε is a desired upper bound on the expected ARE of a base-partition DV estimate.

11. A computer-based method to estimate distinct values in a partitioned dataset, said computer-based implemented in computer readable program code stored in computer memory, said computer-based method comprising the steps of:
a. selecting a hash function h with domain , which is a value domain of the partitioned dataset, and range {0, 1, . . . , M}, wherein M=O(||2) and || is the number of distinct values in , said hash function h to be used for all synopses in said partitioned dataset;
b. when a partition being a base partition A with domain (A) hashing each value in (A) using hash function h; recording k smallest values h(v1), h(v2), . . . , h(vk) among said hashed values, as well as recording the multiplicity of each of the values v1, v2, . . . , vk in A, and creating said synopsis as an AKMV synopsis LA+=(LA,cA) based on said recorded k smallest values and said recorded multiplicities;
c. when a partition being a compound partition G=E op F, where op is one of ∪m, ∩m, and \m, and E and F have respective AKMV synopses LE+=(LE, cE) and LF+=(LF, cF) of sizes kE and kF, creating said synopsis as an AKMV synopsis LG+=(LE ⊕ LF, cG), where, for each w ∈ LE ⊕ LF,





  
    
      c
      G
    
    ⁡
    
      (
      w
      )
    
  
  =
  
    {
    
      
        
          
            
              
                c
                E
              
              ⁡
              
                (
                w
                )
              
            
            +
            
              
                c
                F
              
              ⁡
              
                (
                w
                )
              
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              ⋃
              m
            
          
        
      
      
        
          
            min
            ⁡
            
              (
              
                
                  
                    c
                    E
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
                ,
                
                  
                    c
                    F
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              ⋂
              m
            
          
        
      
      
        
          
            max
            ⁡
            
              (
              
                
                  
                    
                      c
                      E
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                  -
                  
                    
                      c
                      F
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                
                ,
                0
              
              )
            
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              op
            
            =
            
              
                \
                m
              
              .
            
          
        
      
    
  



d. combining synopsis created in (b) and (c) to construct distinct value (DV) estimations of a base partition or of a compound partition that is based on at least one of the following multiset operations: multiset-union, multiset-intersection, and multiset-difference; and
e. outputting said constructed DV estimations in (d),
wherein the following unbiased estimator is used to construct said DV estimations for a partition E having an AKMV synopsis LE+=(LE,cE) of size k:





  
    
      D
      ^
    
    E
  
  =
  
    
      
        K
        E
      
      k
    
    ⁢
    
      (
      
        
          k
          -
          1
        
        
          U
          
            (
            k
            )
          
        
      
      )
    
  



wherein KE is the number of non-zero elements in cE and U(K) is the largest value in LE, multiplied by a factor of 1/M.

12. The computer-based method of claim 11, wherein estimator error of said DV estimator {circumflex over (D)}E is assessed as any of, or a combination of, the following:
(i) a number εp that bounds the expected absolute ratio error (ARE) from above with probability approximately 100 p %, wherein said number εp is computed as the solution to the equation p=Δ(k{circumflex over (D)}E/j,k,εp), with k the size of the AKMV synopsis LE+=(LE, cE) for E and j the observed number of nonzero elements in cE, or
(ii) an approximate variance of estimator {circumflex over (D)}E, computed as





  
    Var
    ⁡
    
      [
      
        
          D
          ^
        
        E
      
      ]
    
  
  ≈
  
    
      
        
          
            D
            ^
          
          E
        
        (
        
          
            
              
                (
                
                  k
                  -
                  1
                
                )
              
              2
            
            
              U
              
                (
                k
                )
              
            
          
          -
          
            k
            2
          
          +
          k
          +
          
            
              D
              ^
            
            E
          
        
        )
      
      
        k
        ⁡
        
          (
          
            k
            -
            2
          
          )
        
      
    
    .

13. The computer-based method of claim 11, wherein said AKMV synopses are maintained in the presence of deletions by:
(i) updating the synopsis (L, c) of a partition after deletion of a value v from the partition, wherein h(v)=w, by updating counter c(w) according to c(w)=max(c(w) −1, 0), if w is present in L, and
(ii) by periodically recomputing the synopsis from scratch when estimation errors become unacceptable.

14. The computer-based method of claim 11, wherein the size k of each of the synopses for the base partitions is chosen a priori by setting k=┌2+2/(πε2)┐, where ε is a desired upper bound on the expected ARE of a base-partition DV estimate.