# Patent

## Title
Method and apparatus for efficient aggregate computation over data streams

## Abstract

Improved techniques are disclosed for processing data stream queries wherein a data stream is obtained, a set of aggregate queries to be executed on the data stream is obtained, and a query plan for executing the set of aggregate queries on the data stream is generated. In a first method, the generated query plan includes generating at least one intermediate aggregate query, wherein the intermediate aggregate query combines a subset of aggregate queries from the set of aggregate queries so as to pre-aggregate data from the data stream prior to execution of the subset of aggregate queries such that the generated query plan is optimized for computational expense based on a given cost model. In a second method, the generated query plan includes identifying similar filters in two or more aggregate queries of the set of aggregate queries and combining the similar filters into a single filter such that the single filter is usable to pre-filter data input to the two or more aggregate queries.

## Background
FIELD OF THE INVENTION
The present invention relates generally to data processing systems and, more particularly, to improved techniques for processing data stream queries in such data processing systems.
BACKGROUND OF THE INVENTION
Examples of data streaming applications include applications that process data such as network traffic records, stock quotes, Web clicks, sensor data, and call records. One type of network traffic record is known as a NETFLOW record, which is a record generated in accordance with NETFLOW protocol available from Cisco Systems, Inc. (San Jose, Calif.). NETFLOW and CISCO are trademarks of Cisco Systems, Inc.
Such data streams can generate hundreds of gigabytes of information each day. Processing of such vast amounts of data can obviously place a heavy load on the data processing system that performs such processing. The situation is further exacerbated since analyzing huge volumes of data can require a large number of aggregate queries to be processed. As is known, an aggregate query is a query that performs an aggregate computation (e.g., summation, average, max, min, etc.) on a given data set (e.g., a data stream). These queries may be generated by system administrators seeking to obtain information about the system.
Thus, for real-world deployment, scalability is a key requirement for these types of collection systems. Naïve query answering systems that process the queries separately for each incoming record can not keep up with the high stream rates.
Accordingly, what is required for scalability is an improved technique for processing data stream queries.
SUMMARY OF THE INVENTION
Principles of the invention provide an improved technique for processing data stream queries.
For example, in one aspect of the invention, a method includes the following steps. A data stream is obtained. A set of aggregate queries to be executed on the data stream is obtained. A query plan for executing the set of aggregate queries on the data stream is generated. The generated query plan includes generating at least one intermediate aggregate query, wherein the intermediate aggregate query combines a subset of aggregate queries from the set of aggregate queries so as to pre-aggregate data from the data stream prior to execution of the subset of aggregate queries such that the generated query plan is optimized for computational expense based on a given cost model. By pre-aggregating the data, the intermediate aggregate query preferably reduces the number of computations that would otherwise be required to generate results of the subset of aggregate queries.
The generated query plan for executing the set of aggregate queries for the data stream may be substantially entirely executed using a main memory of a machine hosting the generated query plan.
The generated query plan may include a tree structure. The query plan generating step may further include determining an optimal query plan with a lowest computation cost by determining a minimum-cost aggregate tree. The minimum-cost aggregate tree may be determined using a heuristic which performs one or more locally-optimal modifications to the aggregate tree such that a maximum cost reduction is realized. The minimum-cost aggregate tree may be determined using a heuristic which adds one or more random aggregate queries to the aggregate tree to form an expanded aggregate graph, and uses a directed steiner tree heuristic to find the minimum-cost aggregate subtree of the expanded aggregate graph.
The generated query plan may further include generating other intermediate aggregate queries, wherein a first one of the other intermediate aggregate queries combines second and third ones of the other intermediate aggregate queries.
The data stream may include records received from a data network, wherein each of the data records includes attributes that describe flow statistics in the data network.
In another aspect of the invention, a method includes the following steps. A data stream is obtained. A set of aggregate queries to be executed on the data stream is obtained. A query plan for executing the set of aggregate queries on the data stream is generated. The generated query plan includes identifying similar filters in two or more aggregate queries of the set of aggregate queries and combining the similar filters into a single filter such that the single filter is usable to pre-filter data input to the two or more aggregate queries.
The generated query plan may further include generating other filters, wherein a first one of the other generated filters pre-filters data prior to the data entering a second one of the other generated filters, and the second one of the other generated filters pre-filters data prior to the data entering one or more of the set of aggregate queries.
In yet another aspect of the invention, apparatus includes a memory, and a processor coupled to the memory and operative to: obtain a data stream; obtain a set of aggregate queries to be executed on the data stream; and generate a query plan for executing the set of aggregate queries on the data stream, wherein the generated query plan comprises at least one of: (i) generating at least one intermediate aggregate query, wherein the intermediate aggregate query combines a subset of aggregate queries from the set of aggregate queries so as to pre-aggregate data from the data stream prior to execution of the subset of aggregate queries such that the generated query plan is optimized for computational expense based on a given cost model; and (ii) identifying similar filters in two or more aggregate queries of the set of aggregate queries and combining the similar filters into a single filter such that the single filter is usable to pre-filter data input to the two or more aggregate queries.
These and other objects, features and advantages of the present invention will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates a query processing system according to an embodiment of the invention.
FIG. 2 illustrates a processing system architecture for implementing a query processing system according to an embodiment of the invention.
FIGS. 3(a) through 3(d) illustrate query plans (for queries without filters) generated according to illustrative embodiments of the invention.
FIG. 4 illustrates a greedy heuristic for computing an aggregate tree according to an embodiment of the invention.
FIG. 5 illustrates a randomized heuristic for computing an aggregate tree according to an embodiment of the invention.
FIG. 6(a) through 6(d) illustrate query plans (for queries with filters) generated according to illustrative embodiments of the invention.



DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
Principles of the invention implement the concept of a query execution plan. Given a set of aggregate queries (also referred to herein more simply as “aggregates”), each of which may or may not involve filters, principles of the invention provide techniques for generating a query execution plan. A query execution plan is basically a structure that describes in which order the queries are to be executed.
As will be explained in detail below, the query execution plan may contain certain one or more intermediate aggregates. These intermediate aggregates are fine-grained aggregates, which are then used to generate coarse-grained aggregates. Advantageously, the intermediate aggregates will generally be much smaller than the input data stream itself, and so computing multiple query results from an intermediate aggregate will cost much less than answering these queries directly from the data stream.
With respect to filters, principles of the invention provide techniques for coalescing similar filter conditions into a single filter, which is then used as a pre-filter to reduce the amount of data input to the queries.
Furthermore, it is demonstrated below that query plans incorporating the above two computation sharing optimizations have a tree structure. Principles of the invention also provide a detailed cost model for aggregate query computation that takes into account hash computation and filter evaluation costs. Thus, the problem of finding the optimal query plan with the lowest computation cost is reduced to that of finding the minimum-cost aggregate tree.
It is proven that the problem of finding a minimum-cost aggregate tree is NP-hard. In accordance with principles of the invention, two heuristics are provided, one greedy and one randomized, to find low-cost aggregate trees. In the greedy heuristic, small locally optimal modifications that deliver the maximum cost reduction in each local step are made to the aggregate tree. The randomized heuristic takes a more global approach. In each iteration, the randomized heuristic adds randomized intermediate aggregates to the tree and then uses a directed steiner tree heursitic (R. Wong, “A Dual Ascent Approach for Steiner Tree Problems on a Directed Graph,” In Mathematical Programming, 1984) to find the minimum cost steiner tree out of the expanded graph.
These and other principles of the invention will be illustrated below in conjunction with NETFLOW records associated with an exemplary NETFLOW collector (NFC) system (available from Cisco Systems, Inc. (San Jose Calif.)) as the exemplary type of data stream and the exemplary data processing system. It should be understood, however, that the invention is not limited to use with any particular type of data stream or data processing system. The disclosed techniques are suitable for use with a wide variety of other data processing systems which process various types of data streams, and in numerous alternative applications.
Cisco's NETFLOW Collector (NFC) (“Cisco CNS NETFLOW Collection Engine Installation and Configuration Guide, 3.0”) is representative of an emerging class of applications that require multiple OLAP (Online Analytical Processing) style aggregate queries to be processed over a continuous stream of data. NFC collects IP (Internet Protocol) flow records exported by network devices and allows users to run queries for estimating traffic demands between IP endpoints, computing the top hosts in terms of IP traffic, profiling applications, and detecting network attacks and intrusions. For this reason, it is extensively used by network administrators to manage real-world IP networks. However, besides IP networks, such multiple-query streaming applications can be found in other domains as well, for example, financial tickers, retail transactions, Web log records, sensor node readings, and call detail records in telecommunications.
Principles of the invention were at least in part motivated to improve the scalability of NFC-like applications so that they can process hundreds of queries. In the following, we describe NFC in further detail.
In an IP network, a flow is essentially a continuous unidirectional sequence of packets from a source device to a destination device. NETFLOW, first implemented in Cisco's routers, is the most widely used IP flow measurement solution today. A network device (e.g., router, switch) can be configured to export a single NETFLOW data record for every IP flow that passes through it. Each NETFLOW record has a number of attributes that describe the various flow statistics. Individual attributes can be classified into one of two categories:

    
    
        Group-by attributes: These include source/destination IP addresses for the flow, source/destination ports, ToS byte, protocol, input and output interfaces, etc.
        Measure attributes: These include the number of packets or bytes in the flow, begin/end timestamp, flow duration, etc.
    
    


NFC collects the NETFLOW records exported by devices in the network, and processes user-specified aggregate queries on the collected NETFLOW data. Each aggregate query consists of: (1) a subset of group-by attributes—records with matching values for attributes in the subset are aggregated together; (2) an aggregate operator (e.g., SUM, COUNT) on a measure attribute—the measure attribute values for aggregated records are combined using the specified aggregate operator; (3) a boolean filter condition on attributes; and (4) a time period over which aggregation is to be performed—after each successive time period, result tuples for the aggregate query (computed over NETFLOW records that arrived during the time period) are output.
Below, we give an example NETFLOW query that is a variant of the Cisco NFC predefined HostMatrix aggregation scheme (Cisco NFC has 22 predefined aggregation schemes):
group-by: {srcaddr, dstaddr}
aggregate-op: SUM(bytes)
filter: (srcaddr∈135.254.*.*dstaddr∈135.254.*.*)
period: 15 min
The above query returns the total traffic in bytes between every pair of IP addresses in subnet 135.254.*.* aggregated over 15 minute intervals (note that * is a wild-card that matches any integer between 0 and 255).
A production service provider network contains hundreds of routers which can easily generate massive amounts of NETFLOW data. In fact, it is known that even with a high degree of sampling and aggregation, an IP backbone network alone can generate 500 GB (gigabytes) of NETFLOW data per day (about ten billion fifty-byte records). The situation is further exacerbated since analyzing the huge volumes of NETFLOW data (for diverse network management tasks) requires a large number of aggregation queries to be processed. Thus, for real-world deployment, scalability is a key requirement for a NETFLOW data management system like NFC. The system must be able to process, in real time, hundreds of queries over high-speed NETFLOW data streams. Naive query answering strategies that process the queries separately for each incoming NETFLOW record can not keep up with the high NETFLOW stream rates. Thus, we have realized that what is required for scalability are techniques that improve processing efficiency by avoiding redundant work and sharing computation among the various queries.
In an illustrative embodiment, we disclose two computation sharing techniques for scalable online processing of hundreds of aggregate queries on rapid-rate data streams. A key idea underlying our techniques, in this embodiment, is to first identify similarities among the group-by attributes and filter conditions of queries, and then use these commonalties as building blocks to generate the final query answers.
In accordance with principles of the invention, we assume that the streaming environment has sufficient memory to process the input aggregate queries. This realization is made possible, for example, due to the fact that: (i) RAM (random access memory) prices have dropped considerably in the past few years, allowing machines to be equipped with several GBs of RAM; and (ii) in our experiments with a number of real-life data sets and aggregate queries, we found that query results can be easily accommodated in main memory. For instance, in the NETFLOW record traces stored at the Abilene observatory (“Abilene Observatory Data Collections”), the New York Internet2 backbone router exported a total of 1.7 million NETFLOW records in a 20 minute period (from 11:20 to 11:40 on May 8, 2006). For this data, the aggregate results for the 22 default CISCO NFC queries contain approximately 6 million result tuples that take up only 75 MB (megabytes) of memory.
Therefore, based on current technology trends, we have realized that it is practical to process hundreds of stream queries in main memory. Advantageously, as will be illustrated below, this realization leads to query processing approaches that focus on optimizing CPU (central processing unit) cycles as opposed to main memory usage.
For the sake of convenience, the remainder of the detailed description is organized as follows. In Section 1, we describe a system architecture for implementing query processing techniques of the invention. We describe the system model and cost model for processing queries in Section 2. In Section 3, we present our two heuristics for generating tree-structured query plans for aggregate queries without filters. We extend our heuristics to handle filters in Section 4.
1. Illustrative System Architecture
FIG. 1 shows a block diagram of a data processing system in which techniques of the invention may be implemented. As shown, query processing system 102 includes module 104 for performing aggregation of data (in this embodiment, NETFLOW data) in hash tables, and module 105 for query plan generation.
In general, query plan generation module 105 receives input aggregate queries, filters (if any), and the epoch period. These inputs are defined by the user (e.g., system administrator). While input 106 is referred to as XML (Extensible Markup Language) input in the figure, the query plan generation module of the invention is not limited to processing input of this type. From this input (referred to as 106 in the figure), module 105 generates query plan 107.
Then, with query plan 107 generated by module 105, module 104 inputs the NETFLOW records from the various routers and switches (this is more generally considered as streaming data from one or more sources) in the subject network (referred to as 108 in the figure) and generates aggregated output 110. Given generation of the query plan in accordance with the techniques of the invention, aggregated output 110 is generated by module 104.
It is within the data aggregation module and the query plan generation module that techniques of the invention, to be described in detail below in the following sections, are preferably implemented.
FIG. 2 shows a processing architecture 202 for implementing query processing system 102 of FIG. 1. One or more software programs for implementing query processing (i.e., query plan generation and aggregate output generation) as described herein may be stored in memory 206 and executed by processor 204. Memory 206 may therefore be considered a processor-readable storage medium. Processor 204 may include one or more integrated circuits, digital signal processors or other types of processing devices, and associated supporting circuitry, in any combination.
The system shown in FIG. 2 may also be considered as the host machine on which the query processing system of FIG. 1 resides, i.e., the computing system upon which a query execution plan is generated and implemented. As mentioned above, principles of the invention are advantageously able to carry out all query processing in the host machine's main memory (memory 206 may be considered as representing the main memory of the host machine). Thus, storage for hash tables is not a major constraint in the system of the invention, and the system can accommodate multiple result tuples in a single hash bucket.
2. Illustrative System and Cost Models
In this section, we first describe the aggregation queries supported by our illustrative query processing system, which may be generalized in a straightforward manner to support a broad range of applications including NETFLOW data management similar to CISCO's NFC. We then present a naive approach that processes each query independently on the input stream, and finally, we develop a cost model for estimating the CPU cycles consumed for producing query answers.
2.1 System Model
We consider a single stream consisting of an infinite sequence of tuples, each with group-by attributes a1, . . . , am (e.g., source/destination IP addresses, source/destination ports), and a measure attribute a0 (e.g., byte count). We are interested in answering a set of aggregate queries Θ={Q1, . . . , Qn} defined over the stream of tuples. A typical aggregate query Qi has three main components, listed below:

    
    
        Aggregation. This includes: (1) the subset of group-by attributes on which aggregation is performed—a result tuple is output for each distinct combination of these group-by attribute values; and (2) the aggregation operator that is applied to the measure attribute values of aggregated tuples—this is one of the typical SQL (Structured Query Language) aggregates like MIN, MAX, AVERAGE, SUM, or COUNT.
        Filter. This is essentially a boolean expression (containing boolean operators  and ) over attribute range conditions. Only tuples whose attribute values satisfy the range conditions specified in the filter expression are considered for aggregation. For instance, the filter (srcaddr∈135.254.*.*dstaddr∈135.254.*.*) in the above example NETFLOW query only aggregates NETFLOW records between IP addresses in subnet 135.254.*.*.
        Period. This is the time interval (referred to in FIG. 1 as the epoch period) over which aggregation is performed—after each time period, result tuples for each unique combination of group-by attribute values and the associated aggregated measure attribute value are output.
    
    


In this embodiment, we will assume the following: (1) the measure attribute and aggregation operator are the same for all aggregates; and (2) all aggregate queries in θ have the same time period T; thus, result tuples for all aggregates are output at the same time. Our proposed aggregate and filter sharing techniques can, however, be easily extended to handle scenarios when these assumptions do not hold. For example, a straightforward way would be to partition the input query set into subsets of queries, each with identical measure attributes, aggregate operators, and time periods, and then apply our query processing techniques to each subset. Principles of the invention can be extended to other scenarios.
Thus, going back to the assumptions for this embodiment, aggregate queries in Θ differ only in their grouping attributes and filters. Consequently, if Ai, and Fi denote the group-by attributes and filter expression, respectively, for query Qi, then we can completely characterize each query Qi, by the pair (Ai, Fi). In the remainder of the detailed description, we will use A to denote the collection of grouping attributes Ai for the queries, and Φ for the set of filters Fi. We will also use N to denote the number of stream tuples that arrive in time period T. And finally, in view of the abundance of RAM on modern machines, we will assume that there is adequate main memory for processing queries.
2.2 Naive Query Evaluation Strategy
A naive strategy is to simply process each aggregation query independently for each incoming stream tuple. For each query Qi, we maintain a separate hash table on the group-by attributes Ai. The steps involved in processing query Qi for a tuple are: (1) check if the tuple satisfies the filter condition Fi—if not, then simply stop processing the tuple; and (2) hash on the group-by attributes to locate the hash bucket for the tuple, and then update the aggregate statistic for the group-by attribute values. Note that, in the second step, the first time a tuple with a specific combination of grouping attribute values is encountered, a new entry for that group is created (and initialized) in the bucket. If an entry for the group already exists in the bucket, then only the aggregate statistic for the group is updated.
Every time period T, the result tuples for all the aggregates are output by scanning the non-empty buckets in the hash table for each aggregate query, and writing to an output file the group-by attribute values and the aggregate value in every bucket entry. Once all the result tuples are written, all the hash tables are re-initialized by setting their buckets to be empty.
2.3 Query Evaluation Cost Model
Next, let us examine the CPU cost for answering a query Qi using the above naive strategy. First, we introduce some notation. Let σFi denote the selectivity of the filter condition Fi; thus, a fraction σFi of stream tuples satisfy Fi. Further, let sz(Ai, Fi) be the size of the result after tuples filtered through Fi are aggregated on attributes in Ai. Both σFi and sz(Ai, Fi) can be estimated by maintaining random samples of past stream tuples and applying known sampling-based techniques, for example, as disclosed in Moses Charikar et al., “Towards Estimation Error Guarantees for Distinct Values,” In PODS, 2000. Consider a random sample of size r of our stream data set with N tuples. Let f1 and f2 denote the number of values that occur exactly 1 time and 2 or more times, respectively, in the sample. Then the GEE estimator for the number of distinct values is




  
    
      
        N
        r
      
    
    ⁢
    
      f
      1
    
  
  +
  
    
      f
      2
    
    .
  




In this embodiment, we use the same random stream sample to estimate the size of all intermediate aggregates considered in our heuristics. Note that in the presence of filters, we require the values that contribute to the counts f1 and f2 to satisfy the filter.

We will use CH (Ai) to denote the cost of hashing a tuple on its group-by attributes Ai. Similarly, CF(Fi) will denote the cost of checking the filter condition Fi for the tuple. We use the UNIX ELF hash function (e.g., Andrew Binstock, “Hashing rehashed,” Dr. Dobbs, April 1996) in our hash table implementation; the function first computes a hash value by performing bit manipulation operations on successive bytes of the input value to be hashed. UNIX is a trademark of the Open Group of San Francisco, Calif. It then applies a mod function to compute the hash bucket from the hash value. Our filter evaluation operation considers a conjunction of attribute range conditions, and checks the range condition (by performing two comparisons) for each attribute in the filter. We measured the running times (in nanoseconds or ns) for hashing and filtering on a PC with a 3 GHz Intel INTEL PENTIUM 4 processor running REDHAT ENTERPRISE LINUX 3.0. INTEL PENTIUM is a trademark of Intel Corporation of Santa Clara, Calif. REDHAT ENTERPRISE LINUX is a trademark of Redhat, Inc. of Raleigh, N.C. Both hashing and filtering costs increase linearly with the number of attributes. Hashing incurs about 50 ns for each additional attribute in Ai, while filtering requires about 5 ns per attribute range condition in Fi. Thus, it follows that hashing is about 10 times more expensive than filtering, for the same number of attributes. In our hash computation experiments, we found the overhead of the final mod function step to be negligible at only about 15 ns. Additionally, when inserting tuples into a hash table, we found that hashing is the dominant cost, and other actions like finding the appropriate bucket entry and updating it consume only a small fraction of the CPU cycles.
Now, the computation cost for query Qi on each stream tuple includes the cost of applying the filter Fi to the tuple, and then inserting the tuple into the hash table on attributes Ai if it satisfies Fi. Thus, since there are N stream tuples in time period T, we get that the CPU cost for processing Qi over time interval T is N·CF(Fi)+N·σFi·CH(Ai). At the end of time T, the sz(Ai, Fi) result tuples for Qi are output. In general, sz(Ai, Fi) will be small compared to N, and so we expect output costs to be negligible compared to the computation costs. Also, every query processing scheme will incur identical output costs. So in the remainder of the detailed description, we ignore the cost of writing the result tuples to an output file, and focus primarily on the result computation cost which comprises the CPU cycles for hashing and filtering the incoming stream tuples.
Processing each query in Θ independently (as is done by the naive strategy) may lead to redundant computation. In the following sections, we show that by sharing aggregate computation among the queries in Θ in accordance with principles of the invention, it is possible to achieve a significant reduction in computation overhead and boost overall system throughput.
3. Processing Aggregate Queries Without Filters
We begin by considering queries without filters. Thus, each query Qi∈Θ is simply the group-by attributes Ai on which tuples are aggregated, and query processing costs are completely dominated by the hash function computation costs.
For the multiple-query scenario, the naive approach of maintaining separate hash tables for each aggregation query has the drawback that for each streaming tuple, the hash function value is computed n times, once for each input aggregate Ai. In this section, we show how we can reduce the hash function computation overhead by sharing hash tables across aggregates.
3.1 Execution Model and Problem Formulation
To reduce the number of hash operations, our technique instantiates a few intermediate aggregates B1, . . . , Bq each of whose size is much smaller than N, and then uses them to compute the various Ais. The reason for the small Bj sizes is that there will typically be many duplicate tuples in the stream when we restrict ourselves to only the grouping attributes in Bj—these will all be aggregated into a single result tuple. Now, it is easy to see that each intermediate aggregate Bj can be used to compute any aggregate Ai∈A that it covers (that is, Ai⊂Bj). This is because all the group-by attribute values for Ai are present in the result tuples for Bj. Thus, by making a single pass over the result tuples for Bj and inserting them into the hash table for Ai, aggregate Ai can be computed. In this manner, the result tuples for these intermediate aggregates Bj can be used as input (instead of stream tuples) to compute the aggregates in A covered by them. Since the intermediate aggregates Bj are much smaller than the tuple stream, it follows that the number of hash computations is significantly reduced.
In general, our technique instantiates an intermediate aggregate if it is beneficial to the overall query processing plan. For an intermediate aggregate to be beneficial, it preferably has the following property. Assume that: N=input size; S=output size; X=sum of the number of group-by attributes in the queries composing the intermediate aggregate; and Y=number of group-by attributes in the intermediate aggregate. If S<(N*(X−Y)/X), then the intermediate aggregate is beneficial. For example, assume there are 1,000,000 records in the stream, and there are two children composing the intermediate aggregate with group-by attributes: A, B and B, C. N=1,000,000 and X=4. The intermediate aggregate would have group-by attributes: A, B, C. Thus, Y=3. Therefore, for the intermediate to be beneficial, S must be less than N*(X−Y)/X, i.e., 1,000,000*(4−3)/4=250,000. Therefore, if the output size is less than 250,000, then S is beneficial. So in this example, S must be 1⁄4th the size of N. However, in practice, it is common to see this ratio exaggerated such that S is many orders or magnitude smaller than N. If the input stream is 1,000,000 records, it is possible for the output size of an intermediate aggregate to be 100 records or less, depending on the data set and the query in question.
More formally, suppose sz(Bj) denotes the size of aggregate Bj, that is, sz(Bj) is the number of distinct value combinations observed for group-by attributes Bj in the tuple stream over period T. Then the cost of computing aggregate Ai directly from the stream is N·CH(Ai). On the other hand, the cost of further aggregating the result tuples for an intermediate Bj to compute an aggregate Ai that it covers is sz(Bj)·CH(Ai). Thus, by ensuring that sz(Bj)=N, we can realize substantial cost savings. There is, of course, the additional cost of computing each Bj from the input stream, which is N·CH(Bj). However, if we select the Bjs carefully, then this cost can be amortized across the multiple aggregates Ai that are covered by (and thus computed from) each Bj.
Next we address the question of what is the best set of intermediate aggregates Bj to instantiate? Our discussion above points to Bjs that are small and cover many input aggregates Ai as good candidates for instantiation. We illustrate the trade-offs between the different alternatives in the following example.
Example 1
Consider a stream with attributes a, b, c and d. Also let the aggregates Ai∈A be defined as follows: A1={a, b}, A2={a, c}, and A3={c, d}. Below, we look at 3 strategies for computing the aggregates Ai (we assume that the hashing cost CH(Ai) is proportional to the number of attributes in Ai).
Strategy 1. This is the naive strategy in which each aggregate Ai is computed directly from the stream (see FIG. 3(a)). Thus, the total cost of computing the aggregates is ΣiN·CH(Ai).
Strategy 2. This is the other extreme in which we instantiate a single intermediate aggregate that covers all the aggregates Ai (see FIG. 3(b)). Let B1={a, b, c, d} denote this aggregate. Each time period T, the result tuples in B1 are scanned and inserted into the hash tables for each Ai to compute the final result tuples. The cost of processing the aggregates is thus the sum of the following two costs: (1) N·CH(B1), the cost of instantiating B1 from the stream; and (2) Σisz(B1)·CH(Ai), the cost of generating the aggregates Ai from B1. Thus, the total cost is N·CH(B1)+Σisz(B1)·CH(Ai).
Strategy 3. A possible middle ground between the above two extremes is to maintain a single intermediate aggregate B2={a, b, c} and the aggregate A3={c, d} directly on the input stream (see FIG. 3(c)). Then, each time period T, B2 is used to generate the result tuples for A1 and A2 (by inserting B2's result tuples into the hash tables for A1 and A2). Thus, the cost of processing the aggregates is the sum of the following two costs: (1) N·CH(B2)+N·CH(A3), the costs of instantiating B2 and A3 from the stream; and (2) sz(B2)·CH (A1)+sz(B2)·CH(A2), the cost of generating the aggregates A1 and A2 from B2. Thus, the total cost is N·CH(B2)+N·CH(A3)+sz(B2)·CH(A1)+sz(B2)·CH(A2).
Now, suppose that N>>sz(B2). Further, suppose that sz(B1)≈N. This is entirely possible because B1 contains result tuples for every possible combination of attribute values, and the number of such value combinations could be high. In such a scenario, both strategies 1 and 2 have high computation costs because of the large N and sz(B1) values. In contrast, since sz(B2) is small relative to N and sz(B1), it is easy to verify that Strategy 3 results in the lowest cost among the 3 strategies. In fact, if for B3={a , c, d}, it is the case that sz(B3)>sz(B2), then Strategy 3 can be shown to be the best possible strategy for answering the aggregate queries.
Note that it is not necessary to compute every intermediate aggregate Bj directly from the stream. Rather, it may be possible to reduce hash computation costs by computing an intermediate Bj from another intermediate aggregate, and then using Bj to compute multiple aggregates Ai. For instance, in Example 1, if N>>sz(B1) and sz(B1)>>sz(B2), then the following strategy (depicted in FIG. 3(d)) would be better than Strategy 3: compute only B1 from the stream, then compute B2 and A3 from B1, and finally compute A1 and A2 from B2.
Also, observe that each of the query plans considered above (and shown in FIGS. 3(a) through (d)) is essentially a tree with the root node corresponding to the stream, and other nodes corresponding to (intermediate and input) aggregates. Further, a directed edge in the tree indicates that the destination aggregate is computed from the source aggregate. We formalize this using the notion of aggregate trees below.
Aggregate Trees. An aggregate tree is a directed tree with: (1) a special root node corresponding to the input stream; and (2) other nodes corresponding to aggregates. The aggregate for vertex vi is denoted by A(vi). At the root node, since the input stream is not aggregated, we use the special symbol T for A(root). T covers every other aggregate A(vi) but not vice versa, that is, A(vi)⊂T for all A(vi)—this is because any aggregate can be generated from the input stream. Further, since the root includes all the stream tuples, sz(T)=N.
A directed edge v1, v2 from vertex v1 to vertex v2 can be present in the tree only if the aggregate for v1 covers the aggregate for v2 (that is, A(v2)⊂A(v1)). Note that there are no incoming edges into the root node. However, there are no restrictions on outgoing edges from the root, that is, there can be edges from the root to any other node in the tree. Further, all nodes in the aggregate tree are reachable from the root. Each edge v1, v2 in the tree has an associated cost given by sz(A(v1))·CH(A(v2)). Note that the cost of any edge v1, v2 originating at the root is N·CH(A(v2)). The cost of a tree is simply the sum of the costs of all its edges.
Intuitively, an aggregate tree corresponds to a query plan capable of generating answers for every aggregate contained in the tree. The directed edge v1, v2 implies that node v2 's aggregate is generated from that of node v1 's. This is possible because A(v2)⊂A(v1) for a non-root v1, and any aggregate can be generated from the input stream associated with the root node. The plan for a tree generates aggregates in two phases:

    
    
        Real-time streaming phase. Only the child aggregates of the root node are maintained as tuples are streaming in. Each streaming tuple is inserted into the hash tables of each of the root's children.
        Periodic results output phase. At time intervals of period T, the root's children are used to generate the remaining aggregates in the tree. Starting with each child, aggregates are generated by performing a depth first traversal of the tree. Every time a directed edge v1, v2 is traversed, the aggregate for v2 A(v2) is produced from the result tuples for A(v1).
    
    


Observe that the cost of the edge v1, v2 is the hash computation cost of producing the aggregate A(v2) from aggregate A(v1)—this is the cost of scanning the sz(A(v1)) result tuples for aggregate A(v1) (or N stream tuples if v1 is root) and inserting them into the hash table for aggregate A(v2). Thus, the cost of an aggregate tree reflects the total computation cost of producing all the aggregates in the tree.
Thus, our problem of finding a good query plan (with low hash computation costs) to process the aggregate queries in A reduces to the following:
Given an aggregate set A, compute the minimum-cost aggregate tree T that contains all the aggregates in A.
Our aggregate tree concept allows us to effectively capture, within a single unified framework, the computation costs incurred during the real-time streaming and periodic results output phases. In contrast, existing schemes such as that disclosed by Rui Zhang et al. (“Multiple Aggregations over Data Streams,” In SIGMOD, 2005) focus exclusively on optimizing the real-time streaming phase cost, which is the dominant cost when the available space is low and collision rates are high. However, this can lead to poor query plans for environments that are not necessarily memory-constrained—this is because in such environments, the periodic results output phase cost becomes significant due to low collision rates, and this is not considered by Rui Zhang et al. Note that as shown above in Example 1, the minimum-cost aggregation tree for A may contain intermediate aggregates not in A.
We have proven that the following decision version of our aggregate tree computation problem is NP-hard: Given an aggregate set A and a constant τ, is there an aggregate tree T with cost at most τ that also contains all the aggregates in A?
3.2 Heuristics for Computing Aggregate Trees
In this section, we present two heuristics for computing an appropriate aggregate tree. The first is a greedy heuristic that applies a series of local modifications to the tree, at each step, selecting the modification that leads to the biggest cost reduction. The second is a randomized heuristic that adopts a more global approach; it relies on the observation that the aggregate tree computation problem has strong similarities to computing a directed steiner tree over the global aggregate space. So, directed steiner approximation algorithms such as the one proposed in M. Charikar et al., “Approximation Algorithms for Directed Steiner Problems,” In SODA, 1998 or heuristics like the one in R. Wong, “A Dual Ascent Approach for Steiner Tree Problems on a Directed Graph,” In Mathematical Programming, 1984 can be used to compute an appropriate aggregate tree.
3.2.1 Greedy Heuristic
Algorithm 1 shown in FIG. 4 contains the pseudocode for our greedy heuristic. The greedy heuristic considers the following two types of local tree modifications in each iteration: (1) addition of a new aggregate C obtained as a result of merging sibling aggregates A, B (steps 4-9); and (2) deletion of an aggregate A (steps 10-14). In each iteration, the local modification that results in the biggest cost decrease is applied to the tree. The heuristic terminates when the cost improvement due to the best local modification falls below a (small) constant threshold ∈.
Now, lets look at the rationale behind our two local modifications. For a pair of aggregates A, B whose union C is much smaller than their current parent P, our first modification enables cost savings of sz(P)−2·sz(C)≈sz(P) to be realized by adding the new aggregate C to the tree. This is because generating C from P requires sz(P) hash computations, and then generating A, B from C incurs an additional 2·sz(C) hash operations, while generating A, B directly from P requires 2·sz(P) operations. The second modification considers the opposite situation when the size of an aggregate A is close to the size of its parent P in the tree—in this case, the extra cost of generating A from P does not offset the cost reduction when A's children are generated from A instead of P. Thus, it is more beneficial in this case to delete A from the tree and compute A's children directly from P.
Note that, in the worst-case, we may need to consider a quadratic (in n, the number of input aggregates) number of local modifications in a single iteration. Since the cost benefit of each local modification can be computed in constant time, each iteration has a worst case time complexity that is quadratic in the size of the input.
3.2.2 Randomized Heuristic
As is evident, the greedy heuristic considers local modifications like merging a pair of siblings. In contrast, the randomized heuristic that we propose in this section takes a more global perspective—in each merge step, it coalesces multiple randomly chosen aggregates from A to generate new intermediate aggregates.
Before discussing our randomized heuristic, we make an important observation that relates our aggregate tree computation problem to the problem of computing a directed steiner tree. Consider the graph containing a node for every possible aggregate (that is, every possible subset of group-by attributes), and also T for the input stream. In the aggregate graph, there is a directed edge from aggregate A to aggregate B if A covers B, and the cost of the edge is sz(A)·CH(B). Now, it is easy to see that computing the optimal aggregate tree T is nothing but computing a directed steiner tree (in the graph) that connects the root T to the set of aggregates A.
Although computing a directed steiner tree is an NP-hard problem, there exist approximation algorithms (e.g., M. Charikar et al., “Approximation Algorithms for Directed Steiner Problems,” In SODA, 1998) and heuristics (e.g., R. Wong, “A Dual Ascent Approach for Steiner Tree Problems on a Directed Graph,” In Mathematical Programming, 1984) in the literature for computing such a tree. Thus, we could theoretically use a directed steiner approximation algorithm to find a good aggregate tree in the full aggregate graph. However, the problem with this is that the full graph contains 2m nodes (a node for every subset of group-by attributes). This is exponential in the number of attributes, and so any approach that is based on creating the full graph will only work for a small number of attributes.
As illustrated in FIG. 5. our randomized heuristic (Algorithm 2) circumvents this exponential problem by employing randomization in successive iterations to construct a sequence of partial (instead of full) aggregate graphs. At the end of each iteration, variables Tbest and S keep track of the current best aggregate tree and the aggregates contained in it, respectively. In each iteration, we pick a set R of c2 random intermediate aggregates (steps 4-8), and construct a partial aggregate graph G on S∪R. G contains edges from an aggregate to every other aggregate that it covers. We then invoke the dual-ascent directed steiner heuristic of R. Wong (“A Dual Ascent Approach for Steiner Tree Problems on a Directed Graph,” In Mathematical Programming, 1984) to compute a minimum-cost tree connecting root T to aggregates in A in graph G. The user-defined parameters c1 and c2 determine the number of iterations and the number of random aggregates selected in each iteration, respectively—in our experiments, we were able to obtain satisfactory trees with settings c1=50 and c2=n, the number of input aggregates.
Advantageously, since the running time of each iteration of Algorithm 2 is dominated by steiner tree computation, our randomized heuristic scales well with the number of queries.
4. Processing Aggregate Queries With Filters
We now turn our attention to aggregate queries with filters. So, each query Qi now consists of a set Ai of grouping attributes and a filter Fi. In the following subsections, we will show how the aggregate tree concept and our heuristics for computing good trees can be extended to handle these richer query types.
4.1 Execution Model and Problem Formulation
In the presence of filters, principles of the invention can reduce computational overhead by sharing filter evaluation among the various queries. For instance, we can coalesce a group of similar query filters, and then with a single application of the coalesced filter, discard a significant fraction of stream tuples that are not relevant to the queries. Further, depending on the selectivity of filters, the location and order in which filters and hashing operations are executed in the aggregate tree can make a substantial difference to the overall computation costs. We illustrate these ideas in the following example.
Example 2
Consider a stream with attributes a, b, c, and d each with domain {0, . . . , 1000}. For purposes of illustration, we assume that attribute values are uniformly distributed and independent. Let there be three queries: (1) Q1 with group-by attributes {a, b} and filter 0≦a≦95; (2) Q2 with group-by attributes {a, c} and filter 50≦a≦100; and (3) Q3 with group-by attributes {a, d} and filter 200≦a≦300. Now there are multiple query evaluation strategies possible here, which we consider below.
Strategy 1. The naive strategy is to process each query separately (see FIG. 6(a))—thus for each stream tuple, query pair, we first check to see if the tuple satisfies the query filter, and if so, we insert the tuple into the hash table for the query.
Strategy 2. Now a more efficient strategy can be devised based on the observation that the filters F1 and F2 have a fair amount of overlap and so can be merged to create a new filter, G1=0≦a≦100. Note that G1 is equivalent to F1F2. The idea then would be to evaluate the filter G1 for every stream tuple, and only if the tuple satisfies G1 would we check the filters F1 and F2 for the queries Q1 and Q2, respectively. Of course, if the tuple does not satisfy G1, then it cannot possibly satisfy F1 or F2, and thus, the tuple can be safely discarded. Thus, with Strategy 2 (depicted in FIG. 6(b)), we perform only one filter check for tuples that do not satisfy G1, and three filter evaluations for tuples that satisfy G1. It follows that over N tuples, the filter sharing strategy results in (1+2σG1)·N filter operations, where σG1 denotes the selectivity of filter G1. In contrast, the naive strategy requires 2N filter checks for processing queries Q1 and Q2. Now, since attribute values are uniformly distributed, σG1=0.1. Thus, since 2σG1=0.2<1, the filter sharing strategy has lower filter evaluation costs compared to the naive strategy.
Strategy 3. Next observe that filter F1 has significant overlap with filter G1. Consequently, when F1 is applied immediately after G1 on stream tuples (as in FIG. 6(b)), the number of additional tuples filtered out by F1 is (σG1−σF1)·N. This translates to filtering out 0.005 fraction of the N stream tuples that do not need to be inserted into the hash table for Q1, thus leading to computational savings of 0.005·N·CH(A1). However, there is the additional cost of applying the filter F1 on tuples filtered through G1 which is given by σG1·N·CF(F1).
Now suppose that the aggregated result size sz(A1, G1)=σG1·N. Then, Strategy 3 (depicted in FIG. 6(c)) avoids the filtering cost of σG1·N·CF (F1) by applying filter F1 while the result tuples for Q1 are being output from the hash table instead of applying it before stream tuples are inserted into the hash table. Note that since sz(A1, G1)=σG1·N, the cost of applying filter F1 on the aggregated result is negligible. However, postponing application of the filter F1 will result in (σG1−σF1)·N additional tuples (see above) being inserted into the hash table, leading to an additional cost of 0.005·N·CH(Al). Thus, depending on which of the two quantities 0.005·N·CH(A1) or 0.1·N·CF (F1) is greater, we should apply F1 either before inserting tuples into the hash table or while they are being output from the hash table. In our case, since CH(A1)≈10CF(F1), it is more cost-effective to apply F1 at the end when result tuples are being output.
Observe that the same argument does not hold for F2 which filters (σG1−σF2)·N tuples thus saving 0.05·N·CH(A2) in hashing costs. Since checking F2 on the filtered stream from G1 costs only 0.1·N·CH(F2), the cost savings from hashing fewer tuples far outweigh the additional cost of evaluating F2—thus, in Strategy 3, we apply F2 before tuples are inserted into the hash table for Q2.
Strategy 4. Now if sz(B1)=N for aggregate B1={a, b, c}, then in addition to applying the filter G1 on the tuple stream, Strategy 4 (shown in FIG. 6(d)) further reduces computation costs by aggregating the stream on attributes B1 prior to feeding the tuples into the hash tables for queries Q1 and Q2. Furthermore, even though G1 and F3 do not overlap, it obtains further improvements in filter evaluation costs by introducing a new filter G2=0≦a≦300 obtained as a result of merging filters G1 and F3. This is because 2σG2=0.6<1.
For simplicity of exposition, we will initially only consider filters that are conjunctions () of attribute range conditions. Thus, each filter is a multi-dimensional box whose boundaries along a dimension coincide with the range specified for the attribute corresponding to the dimension. Only tuples belonging to the box (with attribute values in the ranges specified in the filter) are considered for aggregation. The union F=F1∪F2 of two filters F1 and F2 is a box that contains the boxes for F1 and F2. Essentially, in the union F, the range for each attribute a contains its ranges in F1 and F2. For example, if F1=(0≦a≦50≦b≦5) and F2=(5≦a≦105≦b≦10), then their union F=(0≦a≦100≦b≦10). In Section 4.3, we will discuss how our techniques can be extended to handle filters containing disjunctions () as well.
We will also assume that for each query Qi the filter attributes in Fi are a subset of the group-by attributes Ai. We expect that this will be the case for a majority of the queries. For the few queries Qi that do not satisfy this assumption, we can either: (1) process Qi separately; or (2) process a variant Qi′ of Qi jointly with other queries in θ if we find that this leads to lower query processing costs. Here, Qi′=(Ai′, Fi′) is derived from Qi and has the same filter as Qi (that is, Fi′=Fi), but its group-by attributes set Ai′ contains attributes in both Ai and Fi. Since Ai⊂Ai′, the answer for Qi can be derived from the result for Qi′ by performing a final additional aggregation step. Note that the cost for the additional aggregation step needs to be added to the processing cost for Qi′.
Aggregate Trees. In the presence of filters, each node of the aggregate tree is a (filter, grouping attributes) pair. Note that there is an implicit ordering of filter and aggregation operations in each node depending on the input tuples to the node. We discuss details below. The root node is special with a (filter, attributes) pair equal to (T, T), and corresponds to the input stream. Here, T is a special symbol that contains all other filters and grouping attributes, but not vice versa. Further, all tuples satisfy the filter condition T. Intuitively, nodes with group-by attributes equal to T perform no aggregation, and nodes with filters equal to T do no filter checks. In the aggregate tree, there can be an edge from a vertex v1 to a vertex v2 only if v1 covers v2, that is, the filter and group-by attributes of v1 contain the filter and group-by attributes, respectively, of v2. Note that since T contains every other filter and group-by attributes, the root can have edges to every other node in the tree.
Execution Plan for Aggregate Trees. Now, an aggregate tree essentially specifies an execution plan for answering the input aggregate queries. Let V denote the set of tree nodes where incoming stream tuples are first aggregated. More formally, V contains all tree nodes v such that: (1) the group-by attributes of v is not T (that is, v performs aggregation); and (2) the path from the root to v only has nodes with group-by attributes equal to T (that is, none of v's ancestors perform aggregation).
As before, the execution plan has two phases:

    
    
        Real-time streaming phase: We maintain a hash table for each intermediate node v in V on the grouping attributes for v. Each incoming stream tuple is inserted into the hash table for v if and only if it satisfies all the filters in the path from the root to v.
        Periodic results output phase: After time period T, the result tuples in the hash table for each intermediate node v in V are used to compute the result tuples for nodes in the aggregate subtree rooted at v. Essentially, the result tuples for v are used to compute the result tuples for v's children, and their result tuples, in turn, are used to compute the result tuples for their children, and so on. Let v1 be v2's parent in the subtree (v1 and v2 differ in their filters or their grouping attributes). Also, let (Gi, Bi) denote the (filter, group-by attributes) pair at node v1. Then, when computing v2's result tuples from v1's tuples, we need to consider the following three cases.
        Case 1: v2's filter is identical to v1's filter. Note that this covers the case that v2's filter is T. In this case, all the result tuples for v1 are aggregated on v2's group-by attributes by inserting them into a hash table on v2's attributes (without any filtering). The aggregated tuples in the hash table are the result tuples for v2, and the cost of computing these tuples is sz(B1, G1)·CH(B2).
        Case 2: v2's group-by attributes are identical to v1's attributes. Note that this covers the case that v2's grouping attributes are T. In this case, only v2's filter condition is applied to all the result tuples for v1 (without any aggregation), and those that satisfy the filter constitute the result tuples for v2. The cost of computing these tuples is sz(B1, G1)·CF (G2).
        Case 3: v1 and v2 have different filters and group-by attributes. In this case, we have two options: (1) first apply v2's filter to v1's result tuples, and then aggregate the ones that satisfy the filter on v2's group-by attributes; or (2) first aggregate v1's result tuples on v2's group-by attributes, and then filter out the aggregate tuples that do not satisfy v2's filter. Depending on which of the two options has a lower cost, we will order the filtering and aggregation operations in v2 differently. The costs of the two options are as follows:

Option(1)cost=sz(B1,G1)·CF(G2)+sz(B1,G2)·CH(B2)

Option(2)cost=sz(B1,G1)·CH(B2)+sz(B2,G1)·CF(G2)

    
    


Thus, the cost of computing v2's result tuples is the minimum of the costs of options (1) and (2) above. Intuitively, if sz(B1, G2)=sz(B1, G1), then Option (1) is preferable. If this is not the case and if sz(B2, G1)=sz(B1, G1), then Option (2) may prove to be better.
Problem Definition. We assign a cost to each tree edge v1, v2 equal to the CPU cost of materializing the result tuples for v2 using the tuples of v1 (as described in the 3 cases above). Thus, the aggregate tree cost (which is the sum of the edge costs) reflects the total CPU cost of processing all the input aggregate queries. Our objective then is to find the minimum-cost aggregate tree containing all the input aggregate queries in θ.
4.2 Heuristics for Computing Aggregate Trees
It can be proven that the more general problem of computing the optimal aggregate tree for queries containing filters is NP-hard. In the following subsections, we extend the greedy and randomized heuristics presented above in sections 3.2.1 and 3.2.2, respectively, to compute a satisfactory low-cost aggregate tree.
4.2.1 Greedy Heuristic
In each iteration, our modified greedy heuristic applies four types of local modifications to the tree, and selects the one that results in the largest cost reduction. Of the four modifications listed below, the first two are variants of previously proposed modifications for queries without filters (see Algorithm 1 in FIG. 4).
1. For every pair of sibling nodes v1, v2 (with parent p), create a new node v with p as parent, and make v1, v2 children of v. Set node v's filter and group-by attributes equal to the union of the filters and group-by attributes, respectively, of v1 and v2.
2. For every node v∉θ (with parent p), delete v from the tree, and make p the parent of v's children.
3. For every node v∉θ, modify v's group-by attributes to be equal to its parent's group-by attributes.
4. For every node v∉θ, modify v's filter to be equal to its parent's filter.
FIGS. 6(c) and 6(d) (in Example 2) depict aggregate trees containing nodes that apply filters but do not perform aggregation (the reverse situation is possible as well). Modifications 3 and 4 described above have the effect of suppressing aggregation and filtering, respectively, within node v, and thus allow such nodes to be included in the aggregate tree by our greedy heuristic. Note that a simple optimization to our greedy heuristic would be to consider pairs of local modifications in each iteration instead of only a single modification. This would allow, for example, modifications 1 and 3 above to be combined to obtain a variant of modification 1 in which the merged node v's aggregation component is suppressed.
4.2.2 Randomized Heuristic
Similar to Algorithm 2 (in FIG. 5) presented above in Section 3.2.2, in each iteration, our randomized heuristic randomly selects a set of aggregate nodes R, and then computes a directed steiner tree within the aggregate graph on S∪R. However, in order to ensure that R contains candidate nodes with suppressed aggregation or filtering components, its elements are generated by repeating the following steps a constant (c2) number of times:
1. Randomly select a subset of input query nodes from θ.
2. Let v denote the union of (filters and group-by attributes of) the nodes selected above. Add v to R.
3. For every other node u in S that covers v, we add the following two additional nodes x and y to R:

    
    
        Node x with v's filter, but u's group-by attributes.
        Node y with v's group-by attributes, but u's filter.

4.3 Handling Complex Filters

    
    


Our proposed techniques can be extended to handle complex filters containing disjunctions (in addition to conjunctions). We will assume that each filter F is in disjunctive normal form, that is, each filter has the form D1 . . . Dl where each Di is a conjunction of attribute range conditions. Thus, our filter F now is a union of multiple boxes instead of a single box. Consequently, we can model the cost CF(F) of evaluating filter F as ΣiCF(Di), and for estimating the size of aggregates with filters, we can use the sampling-based estimator described in the previous subsection.
Now, in our heuristics, we compute the filter F for a new node in the aggregate tree as the union F1∪ . . . ∪Fq of multiple filters. When each Fi is a single box, their union is simply the box that contains all the filter boxes. However, when each Fi is a set of boxes {D1i, . . . , Dlii}, the union computation for F=F1∪ . . . ∪Fq is somewhat more involved. We begin by initializing the union F to be the set of all the boxes Dji, that is, F={Dji:1≦i≦q, 1≦j≦li}. Now, if F is used to pre-filter tuples into the filters Fi, then the filtering cost per tuple is CF(F)+σF·ΣiCF(Fi)—here the first term is the cost of checking whether the tuple satisfies F and the second term is the cost of checking filters Fi if the tuple satisfies F. Clearly, the ideal value for the union F is one that minimizes the filtering cost CF(F)+σFΣiCF(Fi). So we repeat the following step until no further improvement in filtering cost is possible: Let D1, D2 be the pair of filter boxes in F whose merging results in an F with the smallest filtering cost; merge D1, D2 (by taking their union) into a single box.
As described above in detail, principles of the invention provide two techniques for sharing computation among multiple aggregate queries over a data stream: (1) instantiating certain intermediate aggregates; and (2) coalescing similar filters and using the coalesced filter to pre-filter stream tuples. We proposed two heuristics, one greedy and another randomized, for finding low-cost query plans incorporating the above optimizations. In our experiments with real-life NETFLOW data sets, we found that our randomized heuristic generated the best query plans with maximum sharing—this is because it adopts a more global approach, continuously interleaving optimization steps with random perturbations to the query plan. In fact, query plans output by our randomized heuristic boosted system throughput by over a factor of three compared to a naive approach that processes queries separately.
Although illustrative embodiments of the present invention have been described herein with reference to the accompanying drawings, it is to be understood that the invention is not limited to those precise embodiments, and that various other changes and modifications may be made by one skilled in the art without departing from the scope or spirit of the invention.

## Claims

1. A method, comprising:
obtaining a data stream;
obtaining a set of aggregate queries to be executed on the data stream; and
generating a query plan for executing the set of aggregate queries on the data stream, wherein the generated query plan comprises generating at least one intermediate aggregate query, wherein the intermediate aggregate query combines a subset of aggregate queries from the set of aggregate queries so as to pre-aggregate data from the data stream prior to execution of the subset of aggregate queries such that the generated query plan is optimized for computational expense based on a given cost model;
wherein the generated query plan comprises a tree structure, the query plan generating step further comprises determining an optimal query plan with a lowest computation cost by determining a minimum-cost aggregate tree, and the minimum-cost aggregate tree is determined using a heuristic which adds one or more random aggregate queries to the aggregate tree to form an expanded aggregate graph, and uses a directed Steiner tree heuristic to find the minimum-cost aggregate subtree of the expanded aggregate graph;
wherein the generation of the query plan is implemented by executing one or more software programs on a processor device.

2. An article of manufacture comprising a processor-readable non-transitory storage medium storing one or more software programs which when executed by a processor perform the steps of the method of claim 1.

3. Apparatus, comprising:
a memory; and
a processor coupled to the memory and operative to: obtain a data stream; obtain a set of aggregate queries to be executed on the data stream; and generate a query plan for executing the set of aggregate queries on the data stream, wherein the generated query plan comprises at least one of: (i) generating at least one intermediate aggregate query, wherein the intermediate aggregate query combines a subset of aggregate queries from the set of aggregate queries so as to pre-aggregate data from the data stream prior to execution of the subset of aggregate queries such that the generated query plan is optimized for computational expense based on a given cost model; and (ii) identifying similar filters in two or more aggregate queries of the set of aggregate queries and combining the similar filters into a single filter such that the single filter is usable to pre-filter data input to the two or more aggregate queries;
wherein the generated query plan comprises a tree structure, the query plan generating operation further comprises determining an optimal query plan with a lowest computation cost by determining a minimum-cost aggregate tree, and the minimum-cost aggregate tree is determined using a heuristic which adds one or more random aggregate queries to the aggregate tree to form an expanded aggregate graph, and uses a directed Steiner tree heuristic to find the minimum-cost aggregate subtree of the expanded aggregate graph.