# Patent

## Title
Sampling for database systems

## Abstract

A database server supports weighted and unweighted sampling of records or tuples in accordance with desired sampling semantics such as with replacement (WR), without replacement (WoR), or independent coin flips (CF) semantics, for example. The database server may perform such sampling sequentially not only to sample non-materialized records, such as those produced as a stream by a pipeline in a query tree for example, but also to sample records, whether materialized or not, in a single pass. The database server also supports sampling over a join of two relations of records or tuples without requiring the computation of the full join and without requiring the materialization of both relations and/or indexes on the join attribute values of both relations.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
The present application is a continuation of U.S. application Ser. No. 09/268,590, filed Mar. 15, 1999, now U.S. Pat. No. 6,532,458.


TECHNICAL FIELD
The present invention relates generally to the field of database systems. More particularly, the present invention relates to the field of sampling records in a database system.
BACKGROUND OF THE INVENTION
Computer database systems manage the storage and retrieval of data in a database. A database comprises a set of tables of data along with information about relations between the tables. Tables represent relations over the data. Each table comprises a set of records or tuples of data stored in one or more data fields. The records of a table are also referred to as rows, and the data fields of records in a table are also referred to as columns. A database server processes data manipulation statements or queries, for example, to retrieve, insert, delete, and update data in a database. Queries are defined by a query language supported by the database system.
For large databases such as data warehouses, for example, typical tools such as On Line Analytical Processing (OLAP) and data mining serve as middleware or application servers that communicate data retrieval requests to a backend database system through a query. Although the cost of executing ad-hoc queries against the backend can be expensive, many data mining applications and statistical analysis techniques can use a sample of the data requested through the query. Similarly, OLAP servers that answer queries involving aggregation (e.g., “find total sales for all products in the NorthWest region between Jan. 1, 1998 and Jan. 15, 1998”) benefit from the ability to present to the user an approximate answer computed from a sample of the result of the query posed to the database.
Sampling is preferably supported not only on existing stored or base relations but also on relations produced as a result of an arbitrary query. Sampling may be supported in relational databases as a primitive operation SAMPLE(R,f), for example, to produce a sample S of r tuples that is an f-fraction of a relation R. Fully evaluating a query Q to compute relation R only to discard most of relation R when applying SAMPLE(R,f), however, is inefficient. Preferably, query Q may be partially evaluated so as to produce only sample S of relation R.
For a given query tree T for computing a relation R that is the result of a query Q where SAMPLE(R,f) is the root or last operation of query tree T, pushing the sample operation down tree T toward its leaves would help minimize the cost of evaluating query Q as only a small fraction of stored and/or intermediate relations would be considered in evaluating query Q. The ability to commute the sample operation in this manner, however, depends on the relational operations used in query tree T. The standard relational operation of selection can be freely interchanged with sampling. With join operations, however, sampling may not be so easily commuted.
FIG. 1 illustrates a query tree 100 for obtaining a sample of a join of operand relations R1 and R2. Query tree 100 is executed in accordance with a flow diagram 200 of FIG. 2. For step 202 of FIG. 2, a relation J is computed by joining R1 and R2, or J=R1R2. For step 204, r tuples are randomly sampled from relation J to produce a sample relation S. Commuting the sample operation in query tree 100 to operand relations R1 and R2, as illustrated by a query tree 300 in FIG. 3, would minimize the cost of obtaining a join sample because only samples of operand relations R1 and R2 would need to be joined. A join of samples of operand relations R1 and R2, however, will not likely give a random sample of the join of operand relations R1 and R2.
As one example:

R1(A,B)={(a1,b0), (a2,b1), (a2,b2), (a2,b3), . . . , (a2,bn)}

and

R2(A,C)={(a2,c0), (a1,c1), (a1,c2), (a1,c3), . . . , (a1,cn)}.

That is, relation R1 is defined over attributes A and B. Among the n+1 tuples of relation R1, one tuple has an A-value a1 and n tuples have an A-value a2, but all n+1 tuples of relation R1 have distinct B-values. Similarly, relation R2 is defined over attributes A and C. Among the n+1 tuples of relation R2, n tuples have an A-value a1 and one tuple has an A-value a2, but all n+1 tuples of relation R2 have distinct C-values.

Computing the equi-join of relations R1 and R2 over attribute A produces the following relation:

J=R1R2={(a1,b0,c1), (a1,b0,c2), (a1,b0,c3), . . . (a2,b3,c0), . . . , (a2,bn,c0)}.

That is, relation J has n tuples with A-value a1 and n tuples with A-value a2.

About one half of the tuples in a random sample S of relation J, or S⊂J, would likely have an A-value of a1 while the remaining tuples would have an A-value of a2. A random sample S1 of relation R1, or S1⊂R1, however, would not likely comprise tuple (a1,b0), random sample S2 of relation R2, or S2⊂R2, would not likely comprise tuple (a2,c0). The join of samples S1 and S2 would then likely comprise no tuples and therefore would not likely give random sample S of relation J.
One prior sampling strategy for obtaining a sample S of a join of two relations R1 and R2 with respect to a join attribute A is illustrated as a flow diagram 400 in FIG. 4.
For notational purposes, relations R1 and R2 have sizes n1 and n2, respectively. The domain of join attribute A is denoted by D. For each value v of domain D, or vεD, m1 (v) and m2(v) denote the number of distinct tuples in relations R1 and R2, respectively, that contain value v in attribute A. Then, ΣvεD m1(v)=n1 and ΣvεD m2(v)=n2. A relation J results from the computation of the join of relations R1 and R2, or J=R1R2, and n is the size of relation J, or n=|J|=|R1R2|. Then, n=ΣvεD m1(v)m2(v). For each tuple t of relation R1, the set of tuples in relation R2 that join with tuple t is denoted as Jt(R2)={t′ε R2|t′.A=t.A}; tR2 denotes the set of tuples in R1R2 obtained by joining tuple t with the tuples in Jt(R2); and |tR2|=|Jt(R2) |=m2(t.A). Similarly for each tuple t of relation R2, Jt(R1)={t′ε R1|t′.A=t.A}; R1t denotes the set of tuples in R1R1R2 obtained by joining tuples in Jt(R1) with tuple t; and |R1t|=|Jt(R1)|=m1(t.A).
For step 402 of FIG. 4, a variable r is initialized to the size of a sample relation S to be obtained from the join of relations R1 and R2. For step 404, a variable M is initialized to the upper bound on the number of join attribute values v in relation R2 for all values v of domain D on attribute A. That is, M is the maximum number of any one join attribute value in relation R2. A tuple t1 is randomly sampled from relation R1 for step 406. A tuple t2 is then randomly sampled for step 408 from among all tuples of relation R2 having a join attribute value t2.A that matches the join attribute value t1.A of tuple t1. For step 410, a tuple T is computed as T=t1t2 and output for sample relation S with a probability based on the number of tuples in relation R2 having a join attribute value that matches that of tuple t1 divided by M, or m2(t2.A)/M. If not output, the sample tuple t1 is rejected for step 410. If r tuples have not yet been output for sample relation S as determined for step 412, steps 406 through steps 412 are then repeated until r tuples have been output to form sample relation S as determined for step 412. Flow diagram 400 then ends for step 414.
The sampling technique of FIG. 4 in practice, however, requires indexes for random access to relations R1 and R2, noting relation R1 must be materialized for proper sampling because the rejection of tuples for step 410 requires that the number of samples from relation R1 be a random variable having a distribution dependent upon the distribution of join attribute values in relation R2. This strategy therefore has limited applicability in commuting sampling with joins involving intermediate relations that are produced as a result of an arbitrary query in a query tree and that are not materialized and indexed.
The ability to sample tuples produced as a stream, that is to perform sequential sampling, is significant not only because intermediate relations produced by a pipeline, such as in a query tree for example, may be sampled without materialization but also because a relation, whether materialized or not, may be sampled in a single pass. How and whether sequential sampling may be performed, however, may depend on the chosen semantics for the sampling.
The tuples of a relation may be sampled, for example, using with replacement (WR), without replacement (WoR), or independent coin flips (CF) semantics.
For WR sampling of an f-fraction of the n tuples in a relation R, each sampled tuple is chosen uniformly and independently from among all tuples in relation R, noting any one tuple could be sampled multiple times. The sample is a bag or multiset of f*n tuples from relation R.
For WoR sampling an f-fraction of the n tuples in a relation R, f*n distinct tuples are sampled from relation R, noting each successive sampled tuple is chosen uniformly from the set of tuples not yet sampled. The sample is a set of f*n distinct tuples from relation R.
For CF sampling an f-fraction of the n tuples in a relation R, each tuple in relation R is chosen for the sample with probability f, independent of other tuples. Sampling in this manner is analogous to flipping a coin with bias f for each tuple in turn. The sample is a set of X distinct tuples from relation R, where X is a random variable with the binomial distribution B(n,f) and has expectation f*n. The binomial distribution B(n,f) is, in effect, the distribution of a random value generated by counting the total number of heads when flipping n independent coins, each of which has a probability f of being heads. Sampling using independent coin flip semantics is also called binomial sampling.
The sampling of a relation may also be weighted or unweighted. For unweighted sampling, each element is sampled uniformly at random. For weighted sampling, each element is sampled with a probability proportional to its weight for some pre-specified set of weights.
One prior sequential sampling technique uses CF semantics by sampling each passing tuple of a relation R with probability f for heads and adds the tuple to a sample S if the probability is satisfied. Another prior sequential sampling technique uses WoR semantics by initializing a list or reservoir of r tuples with the first r tuples of relation R and repeatedly removing random tuples from the list while adding tuples from relation R to the end of the list to produce a sample S. Each of these techniques do not require the size of relation R in advance and may therefore be used for sampling relations that are not materialized. Each of these techniques also preserve sortedness by producing a sample of tuples in the same relative order as in relation R. The reservoir sampling technique, however, does not produce a sequential output of tuples as no tuples are output until the technique has terminated. In the case of scanning a materialized relation on a disk, however, the reservoir sampling technique may be efficient by reading only those tuples to be entered in the reservoir by generating random intervals of tuples to be skipped.
SUMMARY OF THE INVENTION
A sample operator for obtaining a sample of a plurality of records in a database system has the plurality of records and sampling semantics as parameters. The sampling semantics may be with replacement, without replacement, or coin flip sampling semantics. The sample operator may also have a size of the sample as a parameter and/or a weight function as a parameter to specify a sampling weight for each record.
Another sample operator for obtaining a sample of a plurality of records in a database system has the plurality of records as a parameter and a weight function as a parameter to specify a sampling weight for each record. The sample operator may also have a size of the sample as a parameter.
A method obtains a sample from a plurality of records in a database system. The method may be implemented by computer-executable instructions of a computer readable medium.
For the method, the plurality of records and sampling semantics are identified from parameters of a sample operator, and a sample is obtained from the identified plurality of records using the identified sampling semantics. The identified sampling semantics may be with replacement, without replacement, or coin flip sampling semantics. A size of the sample to be obtained may be identified from a parameter of the sample operator, and the sample may be obtained from the identified plurality of records based on the identified sample size. A weight function may be identified from a parameter of the sample operator to specify a weight for each record, and the sample may be obtained from the identified plurality of records based on the specified weight of each record.
The sample may be obtained by obtaining one record from the plurality of records, selectively outputting the one record one or more times based on a probability, and repeating these steps for one or more other records of the plurality of records to obtain the sample. The sample may also be obtained by obtaining one record from the plurality of records, selectively resetting one or more records of a reservoir to be the one record based on a probability, and repeating these steps for other records of the plurality of records such that the records of the reservoir form the sample.
Another method obtains a sample from a plurality of records in a database system. The method may be implemented by computer-executable instructions of a computer readable medium.
For the method, the plurality of records and a weight function are identified from parameters of a sample operator, wherein the weight function specifies a weight for each record, and a sample is obtained from the identified plurality of records based on the specified weight of each record. A size of the sample to be obtained may be identified from a parameter of the sample operator, and the sample may be obtained from the identified plurality of records based on the identified sample size.
The sample may be obtained by obtaining one record from the plurality of records and the weight specified for the one record, selectively outputting the one record one or more times based on the weight specified for the one record, and repeating these steps for one or more other records of the plurality of records to obtain the sample. The sample may also be obtained by obtaining one record from the plurality of records and the weight specified for the one record, selectively resetting one or more records of a reservoir to be the one record based on the weight specified for the one record, and repeating these steps for other records of the plurality of records such that the records of the reservoir form the sample.
Another method performs a sequential sampling of records in one pass in a database system. The method may be implemented by computer-executable instructions of a computer readable medium. The database system may perform the method with suitable means.
For the method, one record from a plurality of records is obtained and selectively output one or more times based on a probability. The plurality of records may be a relation produced as a stream of records as a result of a query or may be materialized as a base relation in a database of the database system.
The one record may be selectively output by determining a random number based on the probability such that the random number is greater than or equal to zero and outputting the one record the determined random number of times. The random number may be determined from a binomial distribution based on the probability. The random number may be determined based on a probability based on a number of record(s) of the plurality of records to be evaluated for output, based on a probability based on a weight of the one record divided by a sum of weight(s) of record(s) of the plurality of records to be evaluated for output, or based on a probability based on a fraction of the plurality of records. The random number may be determined such that the random number is less than or equal to a number of record(s) remaining to be output for the sample or such that the random number is less than or equal to a weight of the one record.
The one record may be selectively output one or more times based on a weight specified for the one record. The one record may be selectively output based on a probability a number of time(s) equal in number to the weight of the one record, and that probability may be based on a number of record(s) remaining to be output for the sample divided by a number of possible record(s) that may be output.
These steps are repeated for one or more other records of the plurality of records to form a sample of the plurality of records, wherein at least one obtained record may be output more than one time. The plurality of records may form a relation, and the sample may be joined with records of another relation.
Another method performs a sequential sampling of records in one pass in a database system. The method may be implemented by computer-executable instructions of a computer readable medium. The database system may perform the method with suitable means.
For the method, one record from a plurality of records is obtained, and one or more records of a reservoir are selectively reset to be the one record based on a probability.
Each record of the reservoir may be selectively reset to be the one record based on a probability. One or more records of the reservoir may be selectively reset to be the one record with a probability based on a number of record(s) that have been obtained. One or more records of the reservoir may be selectively reset to be the one record based on a weight of the one record. One or more records of the reservoir may be selectively reset to be the one record with a probability based on a weight of the one record divided by a sum of weight(s) of record(s) that have been obtained.
A random record of the reservoir may be selectively reset to be the one record based on a probability a number of time(s) equal in number to the weight of the one record. A random record of the reservoir may be selectively reset to be the one record with a probability based on a number of records in the reservoir divided by a sum of record(s) evaluated for reset in the reservoir.
These steps are repeated for other records of the plurality of records such that the records of the reservoir form a sample of the plurality of records, wherein at least one obtained record may be used to reset more than one record of the reservoir. The plurality of records may form a relation, and the sample may be joined with records of another relation.



BRIEF DESCRIPTION OF THE DRAWINGS
The present invention is illustrated by way of example and not limitation in the figures of the accompanying drawings, in which like references indicate similar elements and in which:
FIG. 1 illustrates a join query tree with a sample root operation;
FIG. 2 illustrates one prior art flow diagram for obtaining a sample over a join operation;
FIG. 3 illustrates a join query tree with the sample operation commuted to the operand relations of the join operation;
FIG. 4 illustrates another prior art flow diagram for obtaining a sample over a join operation;
FIG. 5 illustrates an exemplary operating environment for sampling of records and sampling over join operations;
FIG. 6 illustrates an exemplary database system for sampling of records and sampling over join operations;
FIG. 7 illustrates a flow diagram for one example of an unweighted sequential with replacement (WR) sampling technique;
FIG. 8 illustrates a flow diagram for another example of an unweighted sequential WR sampling technique;
FIG. 9 illustrates a flow diagram for one example of a weighted sequential WR sampling technique;
FIG. 10 illustrates a flow diagram for another example of a weighted sequential WR sampling technique;
FIG. 11 illustrates a flow diagram for one example of a weighted sequential without replacement (WoR) sampling technique;
FIG. 12 illustrates a flow diagram for another example of a weighted sequential WoR sampling technique;
FIG. 13 illustrates a flow diagram for one example of a weighted sequential independent coin flips (CF) sampling technique;
FIG. 14 illustrates a flow diagram for one example of a stream sampling technique over join operations;
FIG. 15 illustrates a flow diagram for one example of a group sampling technique over join operations;
FIG. 16 illustrates a flow diagram for one example of a count sampling technique over join operations;
FIG. 17 illustrates a flow diagram for one example of a frequency partition sampling technique over join operations; and
FIG. 18 illustrates a block diagram for one example of the frequency partition sequential sampling technique of FIG. 17.



DETAILED DESCRIPTION OF THE INVENTION
A database server supports weighted and unweighted sampling of records or tuples in accordance with desired sampling semantics such as with replacement (WR), without replacement (WoR), or independent coin flips (CF) semantics, for example. The database server may perform such sampling sequentially not only to sample non-materialized records, such as those produced as a stream by a pipeline in a query tree for example, but also to sample records, whether materialized or not, in a single pass. The database server also supports sampling over a join of two relations of records or tuples without requiring the computation of the full join and without requiring the materialization of both relations and/or indexes on the join attribute values of both relations.
Exemplary Operating Environment
FIG. 5 and the following discussion are intended to provide a brief, general description of a suitable computing environment in which the invention may be implemented. Although not required, the invention will be described in the general context of computer-executable instructions, such as program modules, being executed by a personal computer. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the invention may be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, and the like. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.
With reference to FIG. 5, an exemplary system for implementing the invention includes a general purpose computing device in the form of a conventional personal computer 520, including a processing unit 521, a system memory 522, and a system bus 523 that couples various system components including system memory 522 to processing unit 521. System bus 523 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. System memory 522 includes read only memory (ROM) 524 and random access memory (RAM) 525. A basic input/output system (BIOS) 526, containing the basic routines that help to transfer information between elements within personal computer 520, such as during start-up, is stored in ROM 524. Personal computer 520 further includes a hard disk drive 527 for reading from and writing to a hard disk, a magnetic disk drive 528 for reading from or writing to a removable magnetic disk 529, and an optical disk drive 530 for reading from or writing to a removable optical disk 531 such as a CD ROM or other optical media. Hard disk drive 527, magnetic disk drive 528, and optical disk drive 530 are connected to system bus 523 by a hard disk drive interface 532, a magnetic disk drive interface 533, and an optical drive interface 534, respectively. The drives and their associated computer-readable media provide nonvolatile storage of computer-readable instructions, data structures, program modules and other data for personal computer 520. Although the exemplary environment described herein employs a hard disk, a removable magnetic disk 529 and a removable optical disk 531, it should be appreciated by those skilled in the art that other types of computer-readable media which can store data that is accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks, Bernoulli cartridges, random access memories (RAMs), read only memories (ROMs), and the like, may also be used in the exemplary operating environment.
A number of program modules may be stored on the hard disk, magnetic disk 529, optical disk 531, ROM 524 or RAM 525, including an operating system 535, one or more application programs 536, other program modules 537, and program data 538. A user may enter commands and information into personal computer 520 through input devices such as a keyboard 540 and pointing device 542. Other input devices may include a microphone, joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to processing unit 521 through a serial port interface 546 that is coupled to system bus 523, but may be connected by other interfaces, such as a parallel port, game port or a universal serial bus (USB). A monitor 547 or other type of display device is also connected to system bus 523 via an interface, such as a video adapter 548. In addition to the monitor, personal computers typically include other peripheral output devices, such as speakers and printers.
Personal computer 520 may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 549. Remote computer 549 may be another personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to personal computer 520, although only a memory storage device 550 has been illustrated in FIG. 5. The logical connections depicted in FIG. 5 include a local area network (LAN) 551 and a wide area network (WAN) 552. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet.
When used in a LAN networking environment, personal computer 520 is connected to local network 551 through a network interface or adapter 553. When used in a WAN networking environment, personal computer 520 typically includes a modem 554 or other means for establishing communications over wide area network 552, such as the Internet. Modem 554, which may be internal or external, is connected to system bus 523 via serial port interface 546. In a networked environment, program modules depicted relative to personal computer 520, or portions thereof, may be stored in remote memory storage device 550. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used.
Database System
FIG. 6 illustrates one example of a computer database system 600 for performing sampling of records and sampling over join operations. Database system 600 comprises a database 610, a database server 620, and a client tool 630. Database system 600 manages the storage and retrieval of data in database 610 in accordance with data manipulation statements or queries presented to database server 620 by client tool 630, for example. Client tool 630 may be, for example, a database application, a user interface tool, On Line Analytical Processing (OLAP) or data mining middleware, or an OLAP or data mining application server.
Database 610 comprises a set of tables of data along with information about relations between the tables. Tables represent relations over the data. Each table comprises a set of records or tuples of data stored in one or more data fields. The records of a table are also referred to as rows, and the data fields of records in a table are also referred to as columns.
Database server 620 processes queries, for example, to retrieve, insert, delete, and/or update data in database 610. Database system 600 may support any suitable query language, such as Structured Query Language (SQL) for example, to define the queries that may be processed by database server 620. Suitable SQL queries include, for example, Select, Insert, Delete, and Update statements. Database server 620 for one embodiment comprises the Microsoft® SQL Server.
To enhance performance in processing queries, database server 620 uses indexes to help access data in database 610 more efficiently. Database server 620 also generates efficient execution plans for queries with respect to a set of indexes. In generating execution plans, database server 620 may use statistics such as histograms on the column(s) of the tables referenced in a query. Database server 620 may store such indexes and statistics in a catalog table 622, for example.
Because the cost of executing ad-hoc queries against database 610 can be expensive, for example if database 610 is a large database such as a data warehouse, database server 620 may sample not only existing base relations stored in database 610 but also relations produced as a result of an arbitrary query. Database server 620 may, for example, sample intermediate relations produced as a result of a query in a query tree.
Database server 620 supports weighted and unweighted sampling of records or tuples in accordance with desired sampling semantics such as with replacement (WR), without replacement (WoR), or independent coin flips (CF) semantics, for example. Database server 620 may perform such sampling sequentially not only to sample non-materialized records, such as those produced as a stream by a pipeline in a query tree for example, but also to sample records, whether materialized or not, in a single pass. Database server 620 also supports sampling over a join of two relations R1 and R2 without requiring the computation of the full join J=R1R2 and without requiring the materialization of both relations R1 and R2 and/or indexes on the join attribute values of both relations R1 and R2.
Database server 620 and client tool 630 are implemented for one example as program modules or computer-executable instructions and may be stored on any suitable computer-readable medium for execution in a suitable operating environment, such as the computing environment of FIG. 5 for example. The data of database 610 and catalog table 622 may be implemented in the form of one or more suitable data structures and may also be stored on any suitable computer-readable medium.
Sample Operator
Database server 620 supports a sample operator that may be inserted into a query execution plan or tree to obtain a weighted or unweighted sample S from a relation R using designated sampling semantics.
The sample operator may have as parameters a relation R from which a sample S is to be obtained, the size of sample S, the sampling semantics to be used in obtaining sample S, and a weight function. One exemplary sample operator is SAMPLE(relation R, sample size, semantics, weight function). The sample size, sampling semantics, and/or weight function parameters may optionally be set or predetermined by the sampling technique performed by database server 620 in performing the sample operator.
The sample size parameter may be expressed as a number r of tuples to be obtained from relation R for sample S or as a fraction f of the n tuples of relation R to be obtained for sample S. Whether the number r or fraction f is to be designated may depend on the sampling technique performed by database server 620 in performing the sample operator. Where the number n of tuples of relation R is known or may be determined, however, database server 620 may determine the fraction f for a designated number r and may also determine the number r for a designated fraction f.
The sampling semantics parameter may identify any suitable sampling semantics such as, for example, with replacement (WR), without replacement (WoR), or independent coin flips (CF) semantics.
The weight function parameter may designate unweighted sampling, for example by specifying the same weight, such as one for example, for each tuple t of relation R, or may designate weighted sampling by specifying how each tuple t of relation R is to be weighted.
Database server 620 may perform any suitable sampling technique in performing the sample operator. For sampling of materialized relations, database server 620 may perform any suitable non-sequential sampling technique, including any suitable known unweighted nonsequential sampling technique. Database server 620 may also perform any suitable sequential sampling technique, including any suitable known unweighted sequential WoR or CF sampling technique, not only to sample non-materialized relations, such as those produced by a pipeline in a query tree for example, but also to sample any relation, whether materialized or not, in a single pass.
Unweighted Sequential Wr Sampling
To obtain an unweighted with replacement (WR) sample S of an f-fraction or r tuples from a relation R of n tuples, for example in performing a sample operator such as SAMPLE(R, r, WR, w(t)) where w(t) returns a constant for each tuple t of relation R for unweighted sampling, database server 620 uniformly and independently chooses each tuple of sample S at random from relation R.
Database server 620 may perform unweighted sequential WR sampling in one pass by obtaining a tuple t from relation R, selectively outputting one or more copies of tuple t based on a probability, and repeating this technique for other tuples of relation R to form sample S. As one example, database server 620 may perform unweighted sequential WR sampling in accordance with a flow diagram 700 of FIG. 7.
For step 702 of FIG. 7, a variable r is initialized to the size of sample relation S to be obtained from relation R. A variable n is initialized to the number of tuples in relation R for step 704. For step 706, the variable ρ is initialized to r. The variable ρ maintains a count of the number of tuples remaining to be output for sample S. For step 708, a variable i is initialized to zero. The variable i maintains a count of the number of tuples that have been obtained from relation R for evaluation.
Each tuple of relation R is evaluated for steps 710, 712, 714, 716, 718, and 720 to determine whether any and, if so, how many copies of that tuple are to be output for sample S.
For step 710, a next tuple t in relation R is obtained for evaluation, and the variable i is incremented by one for step 712. For step 714, a random variable X distributed as a binomial distribution is determined. An exemplary binomial distribution has as parameters the number ρ of tuples yet to be obtained for sample S and a probability based on the number of tuples from relation R that have yet to be evaluated, such as B(ρ,1/(n−i+1)) for example. Noting the random variable X is an integer greater than or equal to zero and less than or equal to ρ, X copies of tuple t are output for sample S for step 716. Although illustrated as being distributed in accordance with binomial distribution B(ρ,1/(n−i+1)) for random sampling, the variable X may be determined in accordance with any suitable probability, such as for biased sampling for example. The number ρ of tuples yet to be obtained for sample S is accordingly reduced by X for step 718 and compared to zero for step 720.
If ρ is greater than zero for step 720, steps 710-720 are repeated until all tuples of relation R have been evaluated as determined for step 710 or until the number ρ of tuples yet to be obtained for sample S has been reduced to zero as determined for step 720. Flow diagram 700 then ends for step 722.
Sampling tuples in this manner advantageously produces the r tuples of sample S in the same relative order as in relation R and uses minimal auxiliary memory as only one tuple of relation R is stored at a time.
Although flow diagram 700 of FIG. 7 requires the size n of relation R, relation R may be materialized in which case the size of the relation may be already known or determined. Relation R may have also been produced as a sample of another relation in which case the size of relation R may have been specified for relation R or determined in producing relation R. The size n of relation R may optionally be designated as a parameter for a sample operator.
The sampling technique of FIG. 7 may be used for block-level sampling of a relation materialized on a disk, for example, and may be used to skip over a random set of tuples, that is those tuples for which the random variable X is zero.
Database server 620 may also perform unweighted sequential WR sampling in one pass by obtaining a tuple t from relation R, selectively resetting one or more tuples of a reservoir to be tuple t based on a probability, and repeating this technique for other tuples of relation R to form sample S. As one example, database server 620 may perform unweighted sequential WR sampling in accordance with a flow diagram 800 of FIG. 8.
For step 802 of FIG. 8, a variable r is initialized to the size of sample relation S to be obtained from relation R. For step 804, a variable i is initialized to zero. The variable i maintains a count of the number of tuples that have been obtained from relation R for evaluation. For step 806, a reservoir array A[1 . . . r] of r dummy tuples is initialized. The r tuples of the reservoir array are reset with tuples from relation R to obtain sample S.
Each tuple of relation R is evaluated for steps 808, 810, 812, 814, 816, and 818 to determine whether any one or more of the tuples of the reservoir array are to be reset to that tuple from relation R.
For step 808, a next tuple t in relation R is obtained for evaluation, and the variable i is incremented by one for step 810. For step 812, a variable j is initialized to one. The variable j is used to index each tuple in the reservoir array. For step 814, the jth tuple of the reservoir array is reset to be tuple t with a probability based on the number of tuples that have been obtained from relation R for evaluation, such as 1/i for example. The variable j is incremented by one for step 816 and compared to r for step 818. If the variable j is less than or equal to r for step 818, steps 814-818 are repeated so the other tuples of the reservoir array may be possibly reset to be tuple t. When the variable j is greater than r, steps 808-818 are repeated until all tuples of relation R have been evaluated as determined for step 808. The r tuples of the reservoir array are then output as sample S for step 820.
Although the r tuples of sample S are not made available until each tuple of relation R has been evaluated, sampling tuples in this manner does not require the size n of relation R and may therefore be used to sample tuples from intermediate relations, for example, of unknown size. Sampling tuples in this manner also uses minimal auxiliary memory or storage as only r tuples of relation R are stored at a time.
The sampling technique of FIG. 8 may also be used for block-level sampling of a relation materialized on a disk, for example, and may be used to skip over a random set of tuples.
Weighted Sequential WR Sampling
To obtain a weighted WR sample S of an f-fraction or r tuples from a relation R of n tuples with each tuple t of relation R having a specified weight w(t), for example in performing a sample operator such as SAMPLE(R, r, WR, w(t)), database server 620 independently chooses each tuple t of sample S at random from relation R with a probability based on weight w(t) of tuple t.
Database server 620 may perform weighted sequential WR sampling in one pass by obtaining a tuple t from relation R and its specified weight w(t), selectively outputting one or more copies of tuple t based on its weight w(t) and based on a probability, and repeating this technique for other tuples of relation R to form sample S. As one example, database server 620 may perform weighted sequential WR sampling in accordance with a flow diagram 900 of FIG. 9.
For step 902 of FIG. 9, a variable r is initialized to the size of sample relation S to be obtained from relation R. For step 904, a variable D is initialized to zero. The variable D maintains a sum of the weights of the tuples that have been obtained from relation R for evaluation. For step 906, the variable ρ is initialized to r. The variable ρ maintains a count of the number of tuples remaining to be output for sample S. For step 908, a variable W is initialized to the sum of the weights of all n tuples of relation R.
Each tuple of relation R is evaluated for steps 910, 912, 914, 916, 918, and 920 to determine whether any and, if so, how many copies of that tuple are to be output for sample S.
For step 910, a next tuple t in relation R is obtained with its weight w(t) for evaluation, and the variable D is increased by the weight w(t) for step 912. For step 914, a random variable X distributed as a binomial distribution is determined. An exemplary binomial distribution has as parameters the number p of tuples yet to be obtained for sample S and a probability based on weight w(t) of tuple t and the sum of the weight of the tuples that have yet to be evaluated from relation R, such as B(ρ,w(t)/(W−D+w(t))) for example. Noting the random variable X is an integer greater than or equal to zero and less than or equal to ρ, X copies of tuple t are output for sample S for step 916. Although illustrated as being distributed in accordance with the binomial distribution B(ρ,w(t)/(W)−D+w(t))) for random sampling, the variable X may be determined in accordance with any suitable probability, such as for biased sampling for example. The number ρ of tuples yet to be obtained for sample S is accordingly reduced by X for step 918 and compared to zero for step 920.
If ρ is greater than zero for step 920, steps 910-920 are repeated until all tuples of relation R have been evaluated as determined for step 910 or until the number ρ of tuples yet to be obtained for sample S has been reduced to zero as determined for step 920. Flow diagram 900 then ends for step 922.
The sampling technique of FIG. 9 extends the unweighted sampling technique of FIG. 7 to weighted sampling. Indeed, the sampling technique of FIG. 9 may be used to implement the sampling technique of FIG. 7 by specifying the same weight for each tuple t of relation R, such as w(t)=1 for example. The sampling technique of FIG. 9 generally shares the features of the sampling technique of FIG. 7, although the sampling technique of FIG. 9 requires the total weight W of the tuples of relation R. As one example, the total weight W may be determined in determining the weight w(t) for each tuple t of relation R. The total weight W of relation R may optionally be designated as a parameter for a sample operator.
Database server 620 may also perform weighted sequential WR sampling in one pass by obtaining a tuple t from relation R and its specified weight w(t), selectively resetting one or more tuples of a reservoir to be tuple t based on its weight w(t) and based on a probability, and repeating this technique for other tuples of relation R to form sample S. As one example, database server 620 may perform weighted sequential WR sampling in accordance with a flow diagram 1000 of FIG. 10.
For step 1002 of FIG. 10, a variable r is initialized to the size of sample relation S to be obtained from relation R. For step 1004, a variable D is initialized to zero. The variable D maintains a sum of the weights of the tuples that have been obtained from relation R for evaluation. For step 1006, a reservoir array A[1 . . . r] of r dummy tuples is initialized. The r tuples of the reservoir array are reset with tuples from relation R to obtain sample S.
Each tuple of relation R is evaluated for steps 1008, 1010, 1012, 1014, 1016, and 1018 to determine whether any one or more of the tuples of the reservoir array are to be reset to that tuple from relation R.
For step 1008, a next tuple t in relation R is obtained with its weight w(t) for evaluation, and the variable D is increased by w(t) for step 1010. For step 1012, a variable j is initialized to one. The variable j is used to index each tuple in the reservoir array. For step 1014, the jth tuple of the reservoir array is reset to be tuple t with a probability based on weight w(t) of tuple t and the sum of the weight of the tuples that have been obtained from relation R for evaluation, such as w(t)/D for example. The variable j is incremented by one for step 1016 and compared to r for step 1018. If the variable j is less than or equal to r for step 1018, steps 1014-1018 are repeated so the other tuples of the reservoir array may be possibly reset to be tuple t. When the variable j is greater than r, steps 1008-1018 are repeated until all tuples of relation R have been evaluated as determined for step 1008. The r tuples of the reservoir array are then output as sample S for step 1020.
The sampling technique of FIG. 10 extends the unweighted sampling technique of FIG. 8 to weighted sampling. Indeed, the sampling technique of FIG. 10 may be used to implement the sampling technique of FIG. 8 by specifying the same weight for each tuple t of relation R. such as w(t)=1 for example. The sampling technique of FIG. 10 generally shares the features of the sampling technique of FIG. 8.
Weighted Sequential WoR Sampling
A weighted without replacement (WoR) sample S of an f-fraction or r tuples from a relation R of n tuples with each tuple t of relation R having a specified non-negative integer weight w(t) is assumed to be the same as an unweighted WoR sample from a modification of relation R to a relation Rw containing w(t) copies of each tuple t of relation R. Database server 620 may perform weighted WoR sampling, for example, in performing a sample operator such as SAMPLE(R, r, WoR, w(t)).
Database server 620 may perform weighted sequential WoR sampling in one pass by obtaining a tuple t from relation R and its specified weight w(t), selectively outputting one or more copies of tuple t based on its weight w(t) and based on a probability, and repeating this technique for other tuples of relation R to form sample S. As one example, database server 620 may perform weighted sequential WoR sampling in accordance with a flow diagram 1100 of FIG. 11.
For step 1102 of FIG. 11, a variable r is initialized to the size of sample relation S to be obtained from relation R. For step 1104, a variable D is initialized to zero. The variable D maintains a sum of the weights of the tuples that have been obtained from relation R for evaluation. For step 1106, the variable ρ is initialized to r. The variable ρ maintains a count of the number of tuples remaining to be output for sample S. For step 1108, a variable W is initialized to the sum of the weights of all n tuples of relation R.
Each tuple of relation R is evaluated for steps 1110, 1112, 1114, 1116, 1118, 1120, 1122, 1124, 1126, and 1128 to determine whether any and, if so, how many copies of that tuple are to be output for sample S.
For step 1110, a next tuple t in relation R is obtained with its weight w(t) for evaluation. A variable i is initialized to one for step 1112. The variable i maintains an iteration count for evaluating tuple t w(t) times for output for sample S. The variable D is incremented for step 1114. For step 1116, a binary random variable X is set to one with a probability based on the number ρ of tuples yet to be obtained for sample S and the number of possible tuples that may be output for sample S, such as ρ/(W−D+1) for example. The variable X is compared to one for step 1118. If the variable X is one, a copy of tuple t is output for sample S for step 1120. The number ρ of tuples yet to be obtained for sample S is accordingly reduced by X or one for step 1122 and compared to zero for step 1124. If ρ is greater than zero for step 1124, the variable i is then incremented for step 1126. If the variable X is zero for step 1118, the variable i is incremented for step 1126 without output of tuple t for this ith iteration of steps 1114-1128.
The variable i is compared to weight w(t) for step 1128. If the variable i is not greater than weight w(t), steps 1114-1128 are repeated until the variable i is greater than zero as determined for step 1128 or until the number ρ of tuples yet to be obtained for sample S has been reduced to zero as determined for step 1124. If the variable i is greater than weight w(t), steps 1110-1128 are repeated until all tuples of relation R have been evaluated as determined for step 1110 or until the number ρ of tuples yet to be obtained for sample S has been reduced to zero as determined for step 1124. Flow diagram 1100 then ends for step 1130.
Rather than iteratively evaluating each tuple t w(t) times for steps 1114-1128, a suitable random generation technique that outputs X number of copies of tuple t, where 0≦X≦w(t), may be used for other examples.
The sampling technique of FIG. 11 generally shares the features of the sampling technique of FIG. 9.
Database server 620 may also perform weighted sequential WoR sampling in one pass by obtaining a tuple t from relation R and its specified weight w(t), selectively resetting one or more tuples of a reservoir to be tuple t based on its weight w(t) and based on a probability, and repeating this technique for other tuples of relation R to form sample S. As one example, database server 620 may perform weighted sequential WoR sampling in accordance with a flow diagram 1200 of FIG. 12.
For step 1202 of FIG. 12, a variable r is initialized to the size of sample relation S to be obtained from relation R. For step 1204, a variable D is initialized to zero. The variable D maintains a sum of the weights of the tuples that have been obtained from relation R for evaluation. For step 1206, a reservoir array A[1 . . . r] of r dummy tuples is initialized. The r tuples of the reservoir array are reset with tuples from relation R to obtain sample S.
Each tuple of relation R is evaluated for steps 1208, 1210, 1212, 1214, 1216, 1218, 1220, and 1222 to determine whether any one or more of the tuples of the reservoir array are to be reset to that tuple from relation R.
For step 1208, a next tuple t in relation R is obtained with its weight w(t) for evaluation. A variable j is initialized to one for step 1210. The variable j maintains an iteration count for evaluating tuple t w(t) times for output for sample S. The variable D is incremented for step 1212. For step 1214, the variable D is compared to r. If the variable D is less than or equal to r, the tuple A[D] of the reservoir array is reset to be tuple t for step 1216. Otherwise, for step 1218, a random tuple in the reservoir array is reset to be tuple t with a probability based on the sample size r and the total number of times a tuple from relation R has been evaluated for output for sample S, such as r/D for example. Following step 1216 or 1218, the variable j is incremented for step 1220 and compared to w(t) for step 1222.
If the variable j is less than or equal to w(t) for step 1222, steps 1212-1222 are repeated so the other tuples of the reservoir array may be possibly reset to be tuple t. When the variable j is greater than w(t), steps 1208-1222 are repeated until all tuples of relation R have been evaluated as determined for step 1208. The r tuples of the reservoir array are then output as sample S for step 1224.
Rather than iteratively evaluating each tuple t w(t) times for steps 1212-1222, a suitable random generation technique that outputs X number of copies of tuple t, where 0≦X≦w(t), may be used for other examples.
The sampling technique of FIG. 12 generally shares the features of the sampling technique of FIG. 10.
Weighted Sequential CF Sampling
A weighted independent coin flips (CF) sample S of an f-fraction or r tuples from a relation R of n tuples with each tuple t of relation R having a specified non-negative integer weight w(t) is assumed to be the same as an unweighted CF sample from a modification of relation R to a relation Rw containing w(t) copies of each tuple t of relation R. Database server 620 may perform weighted CF sampling, for example, in performing a sample operator such as SAMPLE(R, r, CF, w(t)).
Database server 620 may perform weighted sequential CF sampling in one pass by obtaining a tuple t from relation R and its specified weight w(t), selectively outputting one or more copies of tuple t based on its weight w(t) and based on a probability, and repeating this technique for other tuples of relation R to form sample S. As one example, database server 620 may perform weighted sequential CF sampling in accordance with a flow diagram 1300 of FIG. 13.
For step 1302 of FIG. 13, a variable f is set to the fraction of tuples to be sampled from relation R. For step 1304, a next tuple t in relation R is obtained with its weight w(t) for evaluation. For step 1306, a random variable X distributed as a binomial distribution is determined. An exemplary binomial distribution has as parameters the weight w(t) and a probability based on fraction f, such as B(w(t),f) for example. Noting the random variable X is an integer greater than or equal to zero and less than or equal to w(t), X copies of tuple t are output for sample S for step 1308. Although illustrated as being distributed in accordance with the binomial distribution B(w(t),f) for random sampling, the random variable X may be determined in accordance with any suitable probability, such as for biased sampling for example. Steps 1304-1308 are repeated until all tuples of relation R have been evaluated as determined for step 1304. Flow diagram 1300 then ends for step 1310.
Sampling tuples in this manner advantageously sequentially produces the r tuples of sample S in the same relative order as in relation R and uses minimal auxiliary memory as only one tuple of relation R is stored at a time. Sampling tuples in this manner also does not require the size n or total weight W of the relation R and may therefore be used to sample tuples from intermediate relations, for example, of unknown size.
Sampling Conversions
Database server 620 may also perform sampling for one type of semantics using a sampling technique of different semantics.
As one example, database server 620 can convert a WR sampling technique to a WoR sampling technique by checking each newly sampled tuple and rejecting that tuple if it has already been generated.
As another example, database server 620 can convert a CF sampling technique to a WoR sampling technique by sampling a slightly larger fraction f′ to ensure that at least f-fraction tuples are obtained and rejecting an appropriate number of tuples such that f-fraction of tuples are obtained. The f-fraction of tuples may also be obtained from the f′-fraction of tuples by obtaining a WoR sample of f-fraction tuples from the CF sample of f′-fraction tuples.
As yet another example, database server 620 can convert a WoR sampling technique to a WR sampling technique by sampling with replacement from the WoR sample.
Sampling Over Joins
Database server 620 supports sampling over a join of two relations R1 and R2 without requiring the computation of the full join J=R1R2 and without requiring the materialization of both relations R1 and R2 and/or indexes on the join attribute values of both relations R1 and R2.
Referring to the previous example where:

R1(A,B)={(a1,b0), (a2,b1), (a2,b2), (a2,b3), . . . , (a2,bn)}

and

R2(A,C)={(a2,c0), (a1,c1), (a1,c2), (a1,c3), . . . , (a1,cn)},

the projection, after removal of duplicates, of the relation J=R1R2 onto attributes A and B, for example, does not yield a uniform random sample of relation R1 but rather gives a biased or weighted sample of relation R1 where each tuple of relation R1 is sampled with a probability dependent on the number of tuples in relation R2 joining with relation R1. Specifically, the tuple (a1,b0) of relation R1 is sampled with probability 1⁄2 while the remaining tuples are each sampled with probability 1⁄2n. Similarly, the tuple (a2,c0) of relation R2 is sampled with probability 1⁄2while the remaining tuples are each sampled with probability 1⁄2n. The skewness of the resulting distribution stems from the removal of duplicates. Also, the relatively high skew in relations R1 and R2 prevents samples of relations R1 and R2 from reflecting frequently appearing attribute values in the join output adequately.

Although a join of random samples of operand relations R1 and R2 will not likely give a random sample of the join of operand relations R1 and R2, a sample of the join of relations R1 and R2 may be obtained from non-uniform samples of relations R1 and R2.
Considering a tuple t=(a1,b0)εR1 and its influence on relation J=R1R2, the number of tuples in relation R1 containing the value a1 in attribute A is one, that is m1(a1)=1, yet the number Jt(R2) of tuples in relation R2 joining with tuple t is m2(a1). Although a random sample of relation R1 will unlikely have the one tuple with the A-value a1, the fraction of tuples in relation J having A-value a1 is m2(a1)/|J| which could be relatively large. The structure of relation J is more likely reflected in a sample S1 of relation R1 where a tuple t from relation R1 is sampled with a probability based on the number m2(t.A) of matching join attribute values t.A in relation R2.
Database server 620 may therefore obtain a sample S of the join of relations R1 and R2 by sampling tuples from relation R1 based on the number of matching join attribute values in relation R2 and joining tuples of the resulting sample S1 with those of relation R2. Database server 620 may use frequency statistics on the join attribute values of relation R2 in sampling tuples of relation R1. Because relation R2 may or may not be materialized and/or indexed, how database server 620 obtains a sample of the join of relations R1 and R2 depends on the availability of information for relations R1 and/or R2 such as in the following cases:

    
    
        Case A: no information is available for either relation R1 or R2;
        Case B: no information is available for relation R1 but indexes and/or statistics are available for relation R2; or
        Case C: indexes and/or statistics are available for relations R1 and R2.
    
    


Any sampling technique for a case with less available information will also apply for a case with more available information. When no information is available for a relation, database server 620 may assume the relation is not materialized but rather is produced as a stream by a pipeline process.
For Case A, database 620 has no frequency statistics or indexes for relation R2 to help guide any sampling of relation R1. Database server 620 may therefore use the sampling technique of FIG. 2 to compute the full join J=R1R2 and obtain a sample relation S of r tuples from relation J by rejection sampling each tuple of relation J with probability 1/|J|.
Although the prior art sampling technique of FIG. 4 avoids having to compute the full join J=R1R2, in practice it requires the materialization of relation R1 and an index for random access for relation R1 for step 406, full statistics on the join attribute values of relation R2 for step 410, and the materialization of relation R2 and an index for random access of relation R2 for step 408. The sampling technique of FIG. 4 therefore applies only to Case C of the above three cases.
For Case B, however, database server 620 may avoid having to compute the full join J=R1R2 as well as avoid the requirement for the materialization of both relations R1 and R2 and indexes on the join attribute values of both relations R1 and R2 because database server 620 may obtain a sample relation S of r tuples of the join of relations R1 and R2 using frequency statistics and/or an index on the join attribute values of relation R2 in sampling tuples of relation R1.
Join Sample Operator
Database server 620 supports a join sample operator that may be inserted into a query execution plan or tree to obtain a sample S from a join of two relations R1 and R2 using designated sampling semantics.
The join sample operator may have as parameters relations R1 and R2, the size of sample S, and the sampling semantics to be used in obtaining sample S. One exemplary join sample operator is JOINSAMPLE(relation R1, relation R2, sample size, semantics). The sample size and sampling semantics parameters may optionally be set or predetermined by the join sampling technique performed by database server 620 in performing the join sample operator.
The sample size parameter may be expressed as a number r of tuples to be obtained from the resulting relation J=R1R2 for sample S or as a fraction f of the n tuples of relation J to be obtained for sample S. Whether the number r or fraction f is to be designated may depend on the join sampling technique performed by database server 620 in performing the join sample operator. Where the number n of tuples of relation J is known or may be determined, however, database server 620 may determine the fraction f for a designated number r and may also determine the number r for a designated fraction f.
The sampling semantics parameter may identify any suitable sampling semantics such as, for example, with replacement (WR), without replacement (WoR), or independent coin flips (CF) semantics.
Database server 620 may perform any suitable join sampling technique in performing the join sample operator. Database server 620 may perform the join sampling technique of FIG. 2, for example, using for step 204 the sampling semantics as designated by a join sample operator or as predetermined by database server 620. Database server 620 may avoid materializing relation J for step 204 by performing a sequential sample of relation J using any suitable sequential sampling technique, such as the sampling technique of FIG. 7 or 8 for WR semantics for example. Database server 620 may also perform the join sampling technique of FIG. 4, for example, using for step 406 the sampling semantics as designated by a join sample operator or as predetermined by database server 620. To avoid having to compute the full join J=R1R2 as for the join sampling technique of FIG. 2 and to avoid the requirement for the materialization of and indexes on both relations R1 and R2 as for the join sampling technique of FIG. 4, database server 620 may perform the join sampling techniques of FIG. 14, 15, 16, or 17.
Stream Sampling
Database server 620 for one example may perform a stream sampling technique over joins in accordance with a flow diagram 1400 of FIG. 14.
For step 1402 of FIG. 14, a variable r is initialized to the size of sample relation S to be obtained from the join of relations R1 and R2. For step 1404, a weighted sample relation S1 of r tuples s1, . . . , sr is obtained from relation R1 based on the number m2(t.A) of matching join attribute values t.A in relation R2 for each tuple t of relation R1. Any suitable weighted sampling technique using the sampling semantics as designated by a join sample operator or as predetermined by database server 620 may be used to obtain sample relation S1.
For WR semantics, the weighted sampling technique of FIG. 9 or 10, for example, may be used with each tuple t of relation R1 having a specified weight proportional to the number of tuples in relation R2 having a join attribute value that matches that of tuple t. That is, the weight w(t) for each tuple tεR1 is set to m2(t.A). For WoR semantics, the weighted sampling technique of FIG. 11 or 12, for example, may be used with this same weight specification.
For CF semantics, the weighted sampling technique of FIG. 13, for example, may be used with this same weight specification, noting that the sampling fraction f for the sampling technique of FIG. 13 may be determined from the sample size r of step 1402 and that the sample size r may be redefined for step 1404 as the sampling technique of FIG. 13 may not return a sample relation S1 of the original sample size r. Alternatively, the sampling fraction f may be initialized for step 1402, and the sample size r may be determined based on the number of tuples the sampling technique of FIG. 13 returns for sample relation S1.
For step 1406, a variable i is initialized to one. The variable i is used to denote the ith tuple S1 of sample relation S1. For step 1408, a tuple t2 is randomly sampled from among all tuples of relation R2 having a matching join attribute value with tuple s1 of sample relation S1, that is t2.A=si.A. For step 1410, a tuple T is computed as the join T=sit2 and output for sample relation S. The variable i is incremented for step 1412 and compared to r for step 1414. If i is less than or equal to r, steps 1408-1414 are repeated for the remaining tuples si of sample relation S1. If i is greater than r, indicating all tuples of sample relation S1 have been joined with a tuple from relation R2, flow diagram 1400 ends for step 1416.
In practice, the stream sampling technique of FIG. 14 may be used where no information is available for relation R1 and may assume for step 1404 the availability of frequency statistics on the join attribute values for relation R2 and for step 1408 an index for random access of relation R2. Whether relation R1 is materialized or produced as a stream from a pipeline process, the sampling technique of FIG. 14 may sequentially sample tuples from relation R1 for step 1404 in one pass. As each tuple si of sample relation S1 is sequentially output, then, steps 1408 and 1410 may be performed to produce and output each tuple T for sample relation S sequentially.
Group Sampling
Database server 620 for another example may perform a group sampling technique over joins in accordance with a flow diagram 1500 of FIG. 15.
For step 1502 of FIG. 15, a variable r is initialized to the size of sample relation S to be obtained from the join of relations R1 and R2. For step 1504, a weighted sample relation S1 of r tuples s1, . . . ,sr is obtained from relation R1 based on the number m2(t.A) of matching join attribute values t.A in relation R2 for each tuple t of relation R1. Any suitable weighted sampling technique using the sampling semantics as designated by a join sample operator or as predetermined by database server 620 may be used to obtain sample relation S1.
For WR semantics, the weighted sampling technique of FIG. 9 or 10, for example, may be used with each tuple t of relation R1 having a specified weight proportional to the number of tuples in relation R2 having a join attribute value that matches that of tuple t. That is, the weight w(t) for each tuple tεR1 is set to m2(t.A). For WoR semantics, the weighted sampling technique of FIG. 11 or 12, for example, may be used with this same weight specification.
For CF semantics, the weighted sampling technique of FIG. 13, for example, may be used with this same weight specification, noting that the sampling fraction f for the sampling technique of FIG. 13 may be determined from the sample size r of step 1502 and that the sample size r may be redefined for step 1504 as the sampling technique of FIG. 13 may not return a sample relation S1 of the original sample size r. Alternatively, the sampling fraction f may be initialized for step 1502, and the sample size r may be determined based on the number of tuples the sampling technique of FIG. 13 returns for sample relation S1.
For step 1506, a relation J is computed as the join J=S1R2. The relation J has tuples grouped by the tuples s1, . . . , sr of sample relation S1. For step 1508, one tuple is randomly sampled from each group of tuples of relation J corresponding to each tuple s1, . . . , sr to produce sample relation S. Any suitable sampling technique, such as the sampling technique of FIG. 7 or 8 for example, may be used to sample a tuple from the group corresponding to each tuple si. For the sampling technique of FIG. 7, the size of each group is m2(si.A).
The group sampling technique of FIG. 15 may be used where no information is available for relation R1 and where frequency statistics on the join attribute values yet no index is available for relation R2. Whether relation R1 is materialized or produced as a stream from a pipeline process, the sampling technique of FIG. 15 may sequentially sample tuples from relation R1 for step 1504 in one pass.
Count Sampling
Database server 620 for another example may perform a count sampling technique over joins in accordance with a flow diagram 1600 of FIG. 16.
For step 1602 of FIG. 16, a variable r is initialized to the size of sample relation S to be obtained from the join of relations R1 and R2. For step 1604, a weighted sample relation S1 of r tuples s1, . . . , sr is obtained from relation R1 based on the number m2(t.A) of matching join attribute values t.A in relation R2 for each tuple t of relation R1. Any suitable weighted sampling technique using the sampling semantics as designated by a join sample operator or as predetermined by database server 620 may be used to obtain sample relation S1.
For WR semantics, the weighted sampling technique of FIG. 9 or 10, for example, may be used with each tuple t of relation R1 having a specified weight proportional to the number of tuples in relation R2 having a join attribute value that matches that of tuple t. That is, the weight w(t) for each tuple tεR1 is set to m2(t.A). For WoR semantics, the weighted sampling technique of FIG. 11 or 12, for example, may be used with this same weight specification.
For CF semantics, the weighted sampling technique of FIG. 13, for example, may be used with this same weight specification, noting that the sampling fraction f for the sampling technique of FIG. 13 may be determined from the sample size r of step 1602 and that the sample size r may be redefined for step 1604 as the sampling technique of FIG. 13 may not return a sample relation S1 of the original sample size r. Alternatively, the sampling fraction f may be initialized for step 1602, and the sample size r may be determined based on the number of tuples the sampling technique of FIG. 13 returns for sample relation S1.
For step 1606, sample relation S1 is materialized and the number u1(v) of join attribute values v in sample relation S1 is determined for all values v of domain D of join attribute A. For step 1608, r tuples are sampled from relation R2 to produce a sample relation S2 such that the number u2(v) of tuples in relation S2 with join attribute value v is the same as that in relation S1, that is u2(v)=u1(v). Any suitable sampling technique may be used for step 1608. As one example, an unweighted sampling technique using the sampling semantics as designated by a join sample operator or as predetermined by database server 620 may be used to obtain sample relation S2. For WR semantics, step 1608 for one example may be performed by scanning relation R2 and separately performing, for each join attribute value v, the sampling technique of FIG. 7 on the tuple(s) of relation R2 having the join attribute value v, noting the size of each relation being sampled is m2(v) and the size of each sample to be obtained is u1(v) for the sampling technique of FIG. 7. For each join attribute value v, then, a sample of u1(v) tuples are obtained with replacement from the m2(v) tuples of relation R2.
For each tuple of sample relation S2, for step 1610, a tuple having the same join attribute value from sample relation S1 is randomly sampled without replacement and joined with the tuple from sample relation S2 to produce a tuple for sample relation S. Sample relation S2 may optionally be materialized and randomly matched to sample relation S1 for step 1610.
The count sampling technique of FIG. 16 may be used where no information is available for relation R1 and where frequency statistics on the join attribute values yet no index is available for relation R2. Whether relation R1 is materialized or produced as a stream from a pipeline process, the sampling technique of FIG. 16 may sequentially sample tuples from relation R1 for step 1604 in one pass. The count sampling technique of FIG. 16 also does not require an index on relation R2 as relation R2 may be scanned only once for step 1608.
Frequency Partition Sampling
As shown by the sampling techniques of FIGS. 14-16, having complete frequency statistics on relation R2 can be advantageous. Complete frequency statistics on relation R2, however, may not be available. Because the availability of a histogram having frequency statistics on high frequency values in relation R2 may be reasonably assumed and because a significant source of inefficiency in computing a full join of two relations is the presence of join attribute values having a relatively high frequency, as the size of the join becomes large because of the high frequency values, database server 620 may perform a frequency partition sampling technique over joins by computing a full join of the tuples having low frequency join attribute values in relations R1 and R2 and obtaining a sample from this join, as in the join sampling technique of FIG. 2, and by using frequency statistics on high frequency join attribute values in relation R2 to obtain a sample of the join of tuples having high frequency join attribute values in relations R1 and R2, as in the join sampling techniques of FIGS. 14, 15, and 16.
The join attribute values need not be of high frequency simultaneously in both relations R1 and R2. Rather, the join attribute values which have a high frequency in relation R2 may be used as a suitable approximation to the set of join attribute values which create a large number of tuples in the join.
Database server 620 for one example may perform a frequency partition sampling technique over joins in accordance with a flow diagram 1700 of FIG. 17.
For step 1702 of FIG. 17, a variable r is initialized to the size of sample relation S to be obtained from the join of relations R1 and R2. For step 1704, a partition parameter p is determined for domain D of join attribute A to separate high frequency join attribute values from low frequency join attribute values in relations R1 and R2. Partition parameter p may be determined in any suitable manner. As one example, partition parameter p may be determined based on a suitable threshold percentage of the number of tuples of relation R2, such as 2%, or p=0.02*n2, for example. The suitability of a particular partition parameter p may depend, for example, on the data distribution skew of the join attribute values in relation R2. A suitable partition parameter for a given pair of relations R1 and R2 may be determined experimentally, for example, by using different partition parameters for the join sampling technique of FIG. 17 and comparing the resulting samples with a sample produced from the join sampling technique of FIG. 2.
For step 1706, domains Dhi and Dlo are defined based on parameter p. The domain Dhi may be defined as the set of values in domain D having a frequency greater than or equal to partition parameter p in relation R2, and the domain Dlo may be defined as the remaining values in domain D, that is those values in domain D having a frequency less than parameter p in relation R2. Domains Dhi and Dlo are used to induce a corresponding partition of relation R1 into relations R1hi=R1|Dhi and R1lo=R1|Dlo and of relation R2 into relations R2hi=R2|dhi and R2lo=R2|Dlo, where R|D′denotes the selection of tuples from relation R with a join attribute value in domain D′.
For step 1708, a relation Jlo is computed as the join Jlo=R1loR2lo, and a sample relation J*hi is computed as a sample of the join Jhi=R1hiR2hi using the sampling semantics as designated by a join sample operator or as predetermined by database server 620. Sample relation J*hi may be computed, for example, in accordance with the stream sampling technique of FIG. 14, the group sampling technique of FIG. 15, or the count sampling technique of FIG. 16.
For step 1710, a suitable number, such as the sample size r for example, of tuples from relation Jlo are sampled to produce a sample relation J*lo. Sample relation J*lo may be obtained from relation Jlo using any suitable sampling technique such as the sampling technique of FIG. 7 or 8, for example. The computation of the join Jlo=R1loR2lo for step 1708 and the sampling of relation Jlo for step 1710 generally corresponds to the join sampling technique of FIG. 2.
A sample relation S of the join J=R1R2 is then determined for step 1712 based on sample relations J*lo and J*hi. Sample relation S may be determined in any suitable manner based on sample relations Jlo and J*hi. As one example, sample relation S may be determined by sampling a number of tuples from each sample relation J*lo and J*hi in proportion to the relative size of the join in each domain Dlo and Dhi, respectively. Obtaining sample relation S in this manner may be facilitated by sampling the same number r of tuples for each sample relation J*lo and J*hi.
Database server 620 for one example may perform the frequency partition sampling technique of FIG. 17 in accordance with the block diagram of FIG. 18.
As illustrated in FIG. 18, relation R1 is received as a stream and, for steps 1704 and 1706 of FIG. 17, is partitioned into relations R1hi and R1lo by evaluating each tuple of relation R1 as it is received and directing the tuple either through a weighted sampling tool 1802 if the tuple has a join attribute value that exceeds parameter p or past weighted sampling tool 1802 if the tuple has a join attribute value that does not exceed parameter p. As the tuples of relation R1hi stream by, frequency statistics on the join attribute values of relation R1hi may be collected and combined with the frequency statistics on the join attribute values of relation R2hi to determine the net size nhi of the join Jhi=R1hiR2hi.
For step 1708, weighted sampling tool 1802 uses frequency statistics on the join attribute values of relation R2 as weights to obtain a sample relation S1 of r tuples from relation R1hi. Weighted sampling tool 1802 may perform any suitable sequential sampling technique such as that of FIG. 9 or 10, for example, for WR semantics. The tuples of sample relation S1 are merged back into the stream with the tuples of relation R1lo, and the resulting merged output stream is denoted as a relation R1*=S1∪R1lo. The tuples of relation R1* are directed through a join tool 1804 with the tuples of relation R2 to compute a relation J* as the join J*=R1*R2. As the tuples of relation J* are output, the number nlo of tuples in relation J* that contain Dlo values, that is the number nlo of tuples in the join Jlo=R1loR2lo, may be determined.
Relation J* is then partitioned with the tuples having high frequency join attribute values directed through a group sampling tool 1806 where one random tuple is sampled from each of the r groups corresponding to the r tuples of sample relation S1 to produce sample relation J*hi. Weighted sampling tool 1802, join tool 1804, and group sampling tool 1806 thus compute sample relation J*hi as a sample of the join Jhi=R1hiR2hi using WR sampling semantics in accordance with the group sampling technique of FIG. 15.
The tuples of relation J* having low frequency join attribute values, that is the tuples of relation Jlo, are directed through a sampling tool 1808 where r tuples from relation Jlo are sampled to produce sample relation J*lo for step 1710. Sampling tool 1808 may perform any suitable sampling technique, such as the sampling technique of FIG. 7 or 8 for example. Join tool 1804 and sampling tool 1808 thus compute sample relation J*lo as a sample of the join Jlo=R1loR2lo generally in accordance with the join sampling technique of FIG. 2.
For step 1712, a number of tuples to be sampled from each sample relation J*hi and J*lo is determined using a coin flip technique where r coins are flipped with heads probability proportional to nhi and tails probability proportional to nlo. The number of heads is rh, and the number of tails is rt, noting r=rh+rt. The tuples of sample relation J*hi are directed through a sampling tool 1810 where rh tuples of sample relation J*hi are sampled, and the tuples of sample relation J*lo are directed through a sampling tool 1812 where rt tuples of sample relation J*lo are sampled. Sampling tools 1810 and 1812 may each perform any suitable sampling technique, such as an unweighted sequential WoR sampling technique for example. The sample rh tuples of sample relation J*hi and the sample rt tuples of sample relation J*lo are then combined to form sample relation S of r tuples.
The frequency partition sampling technique example of FIG. 18 requires neither an index nor complete frequency statistics on the join attribute values of relation R2. Rather, a histogram having frequency statistics for the high frequency values in relation R2 is used. If an index on relation R2hi is available in addition to the frequency statistics on relation R2hi, then sample relation J*hi may be computed for step 1708 using the stream sampling technique of FIG. 14. Alternatively, the use of an index on relation R2 may be replaced with a scan of relation R2hi by computing sample relation J*hi for step 1708 using the count sampling technique of FIG. 16.
Database server 620 for another example may perform the frequency partition sampling technique of FIG. 17 using CF semantics by computing sample relation J*hi for step 1708 using CF semantics in accordance with the group sampling technique of FIG. 15. The tuples of relation Jlo may be sampled for step 1710 using a suitable unweighted CF sampling technique to produce sample relation J*lo, and sample relations J*hi and J*lo may then be combined for step 1712 to produce sample relation S.
Join Sampling Implantation Examples
As examples, the sequential WR sampling techniques of FIGS. 7 and 9 may each be implemented with the Microsoft® SQL Server as an operator. Because of the object-oriented design of code in the Microsoft® SQL Server, adding each operator to a query execution tree may be done by creating a derived class of a base operator class and implementing standard methods, such as Open, Close, and GetRow for example.
The join sampling techniques of FIGS. 2, 4, 14, and 18 may be implemented with the Microsoft® SQL Server in the following exemplary manners.
For the join sampling technique of FIG. 2, an execution tree generated by the optimizer of the Microsoft® SQL Server may be modified by adding the operator for the sequential WR sampling technique of FIG. 7 as the root of the execution tree. The resulting query execution tree produces a random sample of the query.
For the join sampling technique of FIG. 4, a random sample of tuples of relation R1 may be obtained by creating a uniform random sample of the key values of relation R1, storing them in a temporary table T1, and joining table T1 with relation R1 on the key value. Also, each join method in the Microsoft® SQL Server, such as the Nested Loops, Hash Join, and Merge Join for example, may be modified so that each sampled tuple from relation R1 is joined with a random tuple from relation R2 among all tuples in relation R2 having a matching join attribute value with the sampled tuple from relation R1. For the Nested Loops join method, for example, a random number k between 1 and f(v) for the join attribute value v in a sampled tuple from relation R1 is determined. The first k−1 matches for the join are skipped before considering the kth match. This kth tuple may be accepted with probability m2(v)/M. The other sampled tuples from relation R1 are similarly evaluated.
For the stream sampling technique of FIG. 14, the operator for the sequential WR sampling technique of FIG. 9 may be implemented as a child of the join method, that is between scan operator on relation R1 and the join operator. The operator may determine the number of copies to be generated and produce that many copies for each tuple of relation R1 in the GetRow method. The statistics on the join attribute values of relation R2 may be read from a file and stored in a work table indexed on the join attribute values to facilitate access of the frequency of a given join attribute value. Also, the join operator may be modified so that for each tuple sampled from relation R1, exactly one tuple is output at random from among all the tuples that join with relation R2.
For the frequency partition sampling technique of FIG. 18, the operator for the stream sampling technique may be modified to generate the weighted random sample from only the tuples having high frequency join attribute values in relation R1. The tuples of relation R1 having low frequency join attribute values may pass through the operator. The statistics for high frequency join attribute values used by this operator may be read from a file and stored in a work table as for the stream sampling technique. Also, the join operator may be modified to perform the group sampling technique to produce the sample J*hi. The operator for the sequential WR sampling technique of FIG. 9 may be added over the join operator to produce the sample J*lo. An operator for materializing the samples J*hi and J*lo and for sampling them without replacement may be added as the root of the query tree to produce the sample of the join.
In the foregoing description, the invention has been described with reference to specific exemplary embodiments thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit or scope of the present invention as defined in the appended claims. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense.

## Claims

1. A method that samples with replacement a plurality of records in a database system in a single sequential pass of the records, comprising:
identifying the plurality of records;
determining a sum of weights to be assigned to all records in the plurality of records;
when a record is selected to be included in the sample, tabulating a remaining weight that subtracts all weights associated with records included in the sample from the sum of weights;
for each of the plurality of records:
inputting the record to a sampling operator that determines a number of times the record is included in the sample by generating a random value from a binomial distribution using the remaining weight and a weight of the current record as parameters; and
selectively outputting and storing the record for inclusion in the sample the determined number of times, wherein the record may be included more than one time in the sample.

2. The method of claim 1 wherein the sampling operator determines the number of times the record should be included in the sample by generating a random value from a binomial distribution having a desired sample size and a number of records being sampled as parameters.

3. A method that samples with replacement a plurality of database records in a database system in a single sequential pass of the database records, the method comprising:
(a) initializing a sample reservoir with dummy records;
(b) associating a weight with each of the database records; and
(c) for each database record, selectively replacing one or more dummy records in the sample reservoir with the database record according to a given probability such that multiple copies of a database record may be included in the sample, wherein the dummy records are selectively replaced in the sample reservoir with the record by considering the weight associated with the record.

4. The method of claim 3 wherein the probability is the reciprocal of the number of database records that have been scanned.

5. One or more computer readable media having computer-executable instructions stored thereon that, when executed, perform the method of claim 1.

6. One or more computer readable media having computer-executable instructions stored thereon that, when executed, perform the method of claim 3.

7. An apparatus for sampling a plurality of records in a database system in a sequential sampling of the records, the method comprising:
means for identifying the plurality of records;
means for inputting each of the plurality of records to a sampling operator that determines a number of times the record will be included in the sample by generating a random value from a binomial distribution having a desired sample size and a number of records beings sampled as parameters; and
means for selectively outputting and storing the record for inclusion in the sample the determined number of times, wherein the record is output one or more times, which may result in multiple copies of a record being included in the sample.

8. The apparatus of claim 7 comprising means for associating a weight with each record in the plurality of records that selectively outputs the record by considering the weight associated with the record.

9. The apparatus of claim 8 comprising:
(a) means for determining a sum of weights to be assigned to all records in the plurality of records;
(b) means for tabulating a remaining weight that subtracts all weights associated with records already included in the sample from the sum of weights computed by the means for determining a sum of weights; and
(c) wherein the means for inputting determines the number of times the record should be included in the sample by generating a random value from a binomial distribution having the remaining weight of the weight of the current record as parameters.

10. An apparatus for sampling with replacement a plurality of database records in a database system in a sequential sampling of the database records, comprising:
(a) means for initializing a sample reservoir with dummy records;
(b) weighting means configured to associate a weight with each database record and determine a sum of weights of sampled records;
(c) examining means for examining each database record and selectively replacing one or more dummy records in the sample reservoir with a database record according to a probability that is based on the weight of the database record and the sum of weights, which can result in multiple copies of a record being included in the sample.

11. The apparatus of claim 10 wherein the probability is the reciprocal of the number of database records that have been sampled.