Multi-dimensional data parallel query method for real-time deduction of power system

Abstract

This invention provides a multi-dimensional parallel query method for real-time power system simulation, comprising the following steps: designing a multi-dimensional hierarchical index data structure Hi4H and establishing a multi-dimensional hierarchical data index; using a hash-based multi-dimensional index and an interval-based query algorithm to group query conditions, establishing an interval index for the query conditions within each group, and using the interval index for fast matching. This invention, by designing a more efficient multi-dimensional index and query algorithm, supports real-time power flow simulation in power systems, improving the efficiency of power flow simulation and user experience; it solves the problems of high-dimensional power operation data, management difficulties, and performance challenges in handling high-dimensional combined queries.

Description

Technical Field

[0001] This invention relates to the field of power systems, and in particular to a method for parallel querying of multidimensional data for real-time simulation of power systems. Background Technology

[0002] Power flow simulation in power systems requires reading power operation data from a simulation database and triggering programs such as electromagnetic steady-state simulation, dynamic simulation, and electromagnetic transient simulation to perform real-time power system simulation. Because it needs to simulate stable states under various scenarios, it is necessary to modify the physical parameters of different power components to generate different simulation data, thereby observing the evolution of the power grid. However, modern power systems have become more complex, with the number of nodes exceeding 100,000 and numerous associated power component parameters, resulting in tens of millions of simulation data records generated in a single step. Searching for the physical parameters to be modified within such a vast amount of data becomes extremely difficult. Furthermore, to avoid impacting the efficiency of power flow simulation and user experience, a high degree of real-time performance is typically required.

[0003] While existing database engines can optimize query performance through multidimensional indexes (such as Join-Index, KD-Tree, R-Tree, GridFile, and Bitmap), they are primarily designed for low-dimensional data and struggle to handle the hundreds of dimensions of data encountered in real-time power system simulations. There are two main reasons for this. First, such indexes require multiple sorting of the indexed objects based on the index's dimensions. When the dimensionality is high, there is no clear sorting function to differentiate the indexed data, leading to the curse of dimensionality. That is, performance rapidly declines as the dimensionality increases, eventually resulting in index query times that are no better than full table scan times. Second, dynamic simulations require frequent generation of new data. Optimizing queries through indexes not only demands short query times but also requires index construction to be completed quickly. Therefore, more efficient multidimensional indexes and query algorithms are needed to support real-time power flow simulations in power systems.

[0004] In recent years, GPU-based parallel algorithms have made significant progress in solving high-dimensional data computations, especially for image, video, and natural language data. For example, patent application CN109948462A discloses a fast hyperspectral image classification method based on multi-GPU collaborative interactive data stream organization, including the following steps: reading hyperspectral image training and test data, binding these data into page-locked memory; calculating the maximum likelihood probability matrix using a fast calculation method based on multi-GPU stream synchronization sparse multinomial logistic regression, and extracting the spectral information of the hyperspectral image. Domestic and international scholars have gradually introduced this method into the fields of data querying and data analysis. For instance, patent application CN107577641A provides a GPU-parallel fast density inversion method for gravity gradient tensor data. Compared to single gravity data inversion, it offers higher resolution, and the proposed fast preprocessing method, compared to the traditional conjugate gradient method, provides density values ​​that more closely match the geometric location of the set model. Telco utilizes the GPU database provided by HEAVY.AI to process billions of rows of telecommunications data in real time, enabling operators to monitor network performance and maintain a high level of network reliability. Simulmedia uses GPU databases to analyze billions of weekly TV viewing records to help make advertising decisions. Mobile location and positioning provider Skyhook uses GPU databases to help run up to 10 billion transactions per day, leveraging Wi-Fi, cellular, and other sensor data to refine user and device locations. Harvard University's Center for Spacetime Innovation uses GPU databases to enhance spatiotemporal computing, addressing the spatiotemporal connections of global issues, from disaster response to energy sustainability. With the rapid development of GPU technology, utilizing its tens of thousands of cores for parallel computing power can undoubtedly solve the past dilemmas faced by high-dimensional big data. This invention is based on the above situation and proposes a complete solution for multidimensional data querying in real-time power system simulation. To date, there have been no reports of directly applying this technology to real-time power system simulation. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a multi-dimensional parallel query method for real-time simulation of power systems, which solves the problems of high dimensionality and management difficulties in power operation data, as well as the difficulty in guaranteeing performance when facing high-dimensional combined queries.

[0006] A parallel query method for 100-dimensional data for real-time simulation of power systems includes:

[0007] (1) Organize the simulation data and construct a star-shaped multidimensional data model, in which the fact table is the bus, and other attribute tables are equipment and physical quantities. Sort the numbers of the bus in the fact table and store them in a one-dimensional array.

[0008] (2) Establish a multi-dimensional hierarchical data index;

[0009] Hierarchical indexes are built for the fields in the attribute table. For numerical data in trend analysis, Hi4H builds subtrees based on the set hierarchical height k and the maximum number of child nodes m of the intermediate nodes. For categorical data, numerical mapping of categories is first performed based on the number of categories, and then subtrees are built according to the numerical data.

[0010] (3) A hash-based multidimensional index is used to convert the physical parameters of power components into hash values ​​and establish a hash table index; an interval-based query algorithm is used to group the query conditions, establish an interval index for the query conditions in each group, and use the interval index for fast matching.

[0011] In this application, the most typical characteristics of power flow simulation data are continuous numerical data and categorical data. Moreover, the data is centered on the bus, from which information about other related power components can be obtained. In view of the above data characteristics, this invention designs a hierarchical index structure (hereinafter referred to as Hi4H). The biggest difference from other hierarchical indexes is that Hi4H is an index of the continuous memory index of the target object, rather than a direct index of the target object. For power flow simulation data, Hi4H indexes the one-dimensional data index of the storage bus. The advantage of indexing the index instead of the bus itself is that: (1) the index of the one-dimensional array is an integer and continuous, and (2) the intersection of indexes on different dimensions is more direct. Therefore, Hi4H combines the advantages of hierarchical index and bitmap index, which is not only conducive to querying numerical data, but also allows for quick access to related queries for multiple indexes. In addition, Hi4H adopts a unified organization method for the two different types of data in power flow simulation, making maintenance more convenient.

[0012] Preferably, the most typical characteristics of power flow projection data are continuous numerical data and categorical data. The data is centered on the bus, and information about other related power components can be obtained from the bus.

[0013] Furthermore, the multidimensional hierarchical data index is an index of the continuous memory sequence number of the target object. For power flow simulation data, the multidimensional hierarchical data index is a one-dimensional data sequence number of the storage bus.

[0014] In specific step (2), before constructing the hierarchical index structure, it is first necessary to sort the instances in the fact table (e.g., according to the primary key size), and then sort each instance according to its sorted sequence number (i.e., ... Figure 1 The sno) in the middle is redefined. The redefined serial number has two basic characteristics: (1) the serial number is an integer; (2) the serial number is a consecutive number.

[0015] Next, a hierarchical index needs to be built for the fields in the attribute table. For numerical data in the trend analysis, Hi4H builds subtrees based on the set hierarchical height k and the maximum number of child nodes m of intermediate nodes. For categorical data, it is necessary to first map the categories numerically based on the number of categories, and then build subtrees according to the numerical data.

[0016] Preferably, before constructing the hierarchical index structure, the instances in the fact table need to be sorted first, and each instance needs to be reassigned according to the sorted sequence number. The redefined sequence number has two basic characteristics: the sequence number is an integer and the sequence number is a consecutive value.

[0017] Preferably, the steps for establishing a multidimensional hierarchical data index include:

[0018] Step 1: Sort the n-dimensional data according to the bus bar number;

[0019] The unique number of the busbar is stored in an array, with the numerical number representing the sequence number;

[0020] Step 2: Organize the data D in each dimension i This will create a hierarchical tree structure with a height of k-1 and a degree not exceeding m.

[0021] The construction of a single hierarchical tree follows a bottom-up principle. First, the sorted parent trees are grouped, and each group of parent trees will constitute the leaf nodes of the hierarchical tree. The number of parent trees in each group can be obtained by the following formula:

[0022]

[0023] Where N is D i The number of buses; then the grouped buses are attached to the leaf nodes;

[0024] Step 3: Process the leaf nodes, updating the maximum and minimum values ​​of their indices;

[0025] Step 4: Process the non-leaf nodes up to the top level. Organize the data for each dimension in a similar way and finally link them to the global root node to form the final Hi4H hierarchical tree structure.

[0026] Furthermore, if the index dimension is not numerical data but categorical data, then the categorical data needs to be properly encoded beforehand to convert it into numerical data.

[0027] Preferably, when establishing a multidimensional data index, different power components can use different hash functions to distribute the hash values ​​across different hash tables.

[0028] When establishing a multidimensional data index, a hash-based multidimensional index is used. The physical parameters of power components are converted into hash values, and a hash table index is created. Different power components can use different hash functions, distributing the hash values ​​across different hash tables. This index can quickly locate bus data that meets the specified criteria and provides fast data retrieval.

[0029] A range-based query algorithm is employed to group query conditions and create a range index for each group, using the range index for fast matching. This algorithm effectively solves high-dimensional combined query problems and improves query efficiency.

[0030] Preferably, during the power flow simulation, it is necessary to continuously interact and query the components associated with the bus and change the component parameters. According to the multi-dimensional hierarchical data index, the query steps include: traversing the hierarchical tree to obtain the set of bus numbers; and finding the intersection of the sets to obtain the bus that meets the requirements.

[0031] A further preferred method for multidimensional data query based on hierarchical trees includes: traversing the hierarchical tree for each dimension to obtain a set of candidate bus lines for that dimension, and finding the intersection of the candidate bus line sets for each dimension.

[0032] Further optimization involves querying the indexes for k dimensions, resulting in a corresponding index object for each dimension, i.e., a set of bus sequence numbers, which is then processed using a bitmap-based set intersection algorithm.

[0033] This application addresses the problems of high-dimensional power operation data, management difficulties, and performance challenges in handling high-dimensional combined queries.

[0034] This invention supports real-time power flow simulation in power systems by designing more efficient multidimensional indexing and query algorithms, thereby improving the efficiency of power flow simulation and user experience. Attached Figure Description

[0035] Figure 1 The star-shaped model for power flow projection data in this application;

[0036] Figure 2 This is a diagram of the Hi4H data structure. Detailed Implementation

[0037] Numerous specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the invention is not limited to the specific embodiments disclosed below.

[0038] This embodiment provides a multi-dimensional parallel query method for real-time simulation of power systems. Specifically, the query method can be decomposed into three stages, totaling ten steps.

[0039] (1) Data preparation and Hi4H structure design:

[0040] Step 1: Organize the power system simulation data according to the star model;

[0041] Since the simulation data is centered on the busbar, maintaining data on various equipment and physical quantities, this invention first organizes the simulation data and constructs a star-shaped multidimensional data model (e.g., Figure 1 (As shown). Among them, the fact table is the bus, and the other attribute tables are equipment and physical quantities.

[0042] Step 2: Sort the numbers of the parent lines in the fact table and store them in a one-dimensional array;

[0043] A one-dimensional array records the numbers of the parent lines in the fact table in sequence. This design has two advantages: 1) it can better utilize set theory to calculate the intersection of query results on different dimensions, and 2) it can better design GPU-based parallel algorithms to implement set intersection.

[0044] (2) Hi4H index construction algorithm:

[0045] Building the Hi4H index (as shown in Algorithm 1) involves the following four steps:

[0046] Step 1: Sort the n-dimensional data according to the bus bar number;

[0047] Store the unique number of the bus in an array, and use the numerical number to refer to the sequence number (lines 1-3 in Algorithm 1).

[0048] Step 2: Organize the data D in each dimension i This will create a hierarchical tree structure with a height of k-1 and a degree no greater than m.

[0049] The construction of a single hierarchical tree follows a bottom-up principle. First, the sorted parent trees are grouped, and each group of parent trees will constitute the leaf nodes of the hierarchical tree (lines 4-6 in Algorithm 1). The number of parent trees in each group can be obtained by the following formula:

[0050]

[0051] Where N is D i The number of buses. Then, the grouped buses are attached to the leaf nodes.

[0052] Step 3: Process leaf nodes;

[0053] Update the maximum and minimum values ​​of its index (line 7 in Algorithm 1).

[0054] Step 4: Process non-leaf nodes;

[0055] Non-leaf nodes are processed in a similar manner up to the top level (line 8 in Algorithm 1). Data for each dimension is organized in a similar way and eventually linked to the global root node to form the final Hi4H hierarchical tree structure.

[0056] It is important to note that if the index dimension is categorical rather than numerical data, the categorical data needs to be properly encoded beforehand to convert it into numerical data. Common encoding methods include One-hot encoding, Ordinal encoding, and Sum encoding.

[0057] like Figure 2 As shown, Hi4H is a typical hierarchical structure, with index subtrees connected under the root node in different dimensions. Each subtree manages numerical or categorical data in a different dimension. Figure 1 The example demonstrates indexing data across three different dimensions (i.e., index_di, index_dj, index_dk). As you can see, the indexes for each dimension are maintained independently and then connected through the root node. Each dimension index is the parent index of its respective table. This index structure offers strong scalability; even with data spanning hundreds of dimensions, it first builds an index for each dimension and then connects it to the top-level node for unified management. This structure conforms to traditional tree indexes, making it easier to understand and implement.

[0058] (3) Optimize the query algorithm:

[0059] During power flow simulation, it is necessary to continuously query the components associated with the busbars to modify their parameters. For example, querying transformers connected to PQ type buses with rated voltages between 220kV and 330kV in Central China requires transformers with no-load current of 40-60%, impedance voltage of 60-70%, frequency of 50Hz, and insulation level of 35kV. Such numerous parameter constraints necessitate querying data across multiple dimensions. According to the hierarchical index proposed in this invention, achieving the above query requires two steps: traversing the hierarchical tree to obtain the set of busbar indices; and finding the intersection of these sets to obtain the buses that meet the requirements.

[0060] Step 1: Multidimensional data query based on hierarchical tree;

[0061] Multidimensional data querying can be achieved using a complete Hi4H hierarchical tree. Data querying involves two processes: (1) traversing the hierarchical tree for each dimension to obtain a set of candidate parent trees for that dimension; and (2) finding the intersection of the candidate parent tree sets for each dimension. The first process will be introduced here, and the second process will be introduced in the next section. Since the hierarchical tree is a labeled multi-way tree structure, various query methods can be implemented in practical applications, such as depth-first traversal and breadth-first traversal. The multidimensional data querying method will be introduced using the depth-first traversal hierarchical tree algorithm (see Algorithm 2). Other query methods can adopt similar algorithms.

[0062] During depth-first traversal, the subtree H of the specified dimension is first obtained from the root node. t Then, it checks if the queried value d is within the range of the subtree index. If not, the query ends (lines 1-3 of Algorithm 2). If the subtree height is 2, it means there are only leaf nodes, so only traversal is needed to find the parent node number corresponding to d (line 4 of Algorithm 2). If the height is greater than 2, it traverses the child nodes of the subtree and recursively calls the depth-first traversal algorithm (line 5 of Algorithm 2).

[0063] Step 2 (1): Intersection of the indexed object sets:

[0064] After indexing k dimensions, each dimension will yield a corresponding index object, i.e., a set of bus index numbers. Since power flow analysis requires integrating query conditions across multiple dimensions, it is necessary to finally intersect the aforementioned k bus index sets to obtain the final set of buses that meet the criteria.

[0065] Generally, the time complexity of intersection operations between two sets is O(l*s), where l and s are the lengths of the two sets, respectively. While hash tables can reduce this time complexity, the large number of hash tables and high query dimensions make their creation too costly. Using bitmap-represented set intersection algorithms can reduce the time complexity of intersection operations for k sets to O(k), where k is the query dimension.

[0066] It's important to note that if the indexed object is represented by a bus number instead of an ordinal number, converting the bus number to a bitmap will be extremely costly because the bus number cannot be guaranteed to be a series of consecutive integers. As mentioned earlier, in Hi4H, indexed objects using bus numbers have the characteristic of being consecutive integers. Constructing a bitmap structure from the ordinal number is also very straightforward, and intersection operations with O(k) complexity can be achieved using the bitmap.

[0067] As shown in Algorithm 3, step 1 (1.1-1.2) converts the bitmap from the bus index to the bitmap, where the bitmap is represented by a binary array (e.g., bitarray). Considering that the sizes of the intersecting sets are not uniform, the maximum length n is used to uniformly represent the bitmap. Step 2 sets the initial value to be returned, and step 3 performs the intersection operation on the bitmap. The overall algorithm complexity is O(n*k).

[0068] Step 2(2): Parallel optimization of the set intersection algorithm;

[0069] Although bitmap-based set intersection algorithms can greatly reduce time complexity, for large-scale data, the value of n can reach hundreds of millions, which remains a bottleneck for the entire method. This invention designs a GPU-based parallel optimization algorithm, which can further reduce the overall computation time for set intersection. The implementation process of the parallel optimization algorithm is described below. As shown in Algorithm 4, the first step is to convert the bus index into a bitmap form. The second step is to copy the bitmap from CPU memory to GPU memory. The third step is to perform vertical bit operations and save them to the first bitmap array. Since the third step is performed in the GPU, its algorithm complexity can be reduced to O(k). Therefore, the intersection complexity can be reduced from O(n*k) in step two (1) to O(k).

[0070] Algorithm 1 (Index Construction): Assuming the index object has dimension o, and given numerical data D(o,t) on dimension t, construct a hierarchical tree with the largest child node m and a height k.

[0071] Input: ordered numerical data D(o,t), maximum number of child nodes m, depth k;

[0072] Output: Hi4H hierarchical subtree;

[0073] 1. N = ||D||;

[0074] 2. buckets = [] * N / / Store leaf nodes;

[0075] 3. H = [[] * (k-1)] / / Store intermediate nodes;

[0076] 4.b= / / bin size of buckets;

[0077] 5.s = / / number of buckets;

[0078] 6. For i∈[0,N-1];

[0079] 6.1.p= / / Record the position of i in buckets;

[0080] 6.2. Add D[i].o to buckets[p];

[0081] 6.3. Update the maximum and minimum values ​​in buckets[p];

[0082] 7. For I∈[0,s-1] / / Process leaf nodes, i.e., the (k-1)th level

[0083] 7.1.H[k-1][i]={'min':buckets[i].min,'max':buckets[i].max,'child':

[0084] buckets[i]};

[0085] 8. For i∈[k-2,1] / / Process other non-leaf nodes (1:k-2];

[0086] 8.1 size = ||H[j+1]||;

[0087] 8.2. For j∈[0,size-1];

[0088] 8.2.1p=

[0089] 8.2.2H[j][p]={'min':H[j+1][i].min,'max':H[j+1][i+m-1].max,'child':H[j+1][i:i+m]};

[0090] 9. return H[0][0].

[0091] Algorithm 2 (Depth-first search): Given data D, data d to be queried, and corresponding dimension t, query the set of generatrices containing data d.

[0092] Input: 100-dimensional data D, dimension t, hierarchical tree H (Hi4H), k-tree height, query data d;

[0093] Output: Busbar number;

[0094] 1. Obtain the subtree H containing dimension t from H. t ;

[0095] 2. Obtain the minimum and maximum values ​​of the index from the root node of the subtree;

[0096] 3. If d<min or d> max, return -1;

[0097] 4. If subtree H t The height k is 2;

[0098] 4.1 From H t Traverse and search through the leaf nodes;

[0099] 4.2 If d==D[t,h], return h;

[0100] 5. Otherwise;

[0101] 5.1 Traversing H t Child nodes;

[0102] 5.2 Recursively call this algorithm, and let k = k-1, H t =H t .child i ;

[0103] 5.3 Return the result of the recursive call.

[0104] Algorithm 3 (Set Intersection): Given several sets A, find the common intersection of sets A in O(n*k) time.

[0105] Input: An array A of sets to be intersected, and the maximum length n of the sets;

[0106] Output: The intersection set;

[0107] 1. For a in A;

[0108] 1.1 Convert 'a' into a binary array of length 'n';

[0109] 1.2 Store 'a' in array B;

[0110] 2.r = B[0];

[0111] 3. For b in B[1:];

[0112] 3.1r&=b;

[0113] 4. return r.

[0114] Algorithm 4 (Parallel Set Intersection): Given several sets A, find the common intersection of sets A in O(k) time.

[0115] Input: An array A of sets to be intersected, and the maximum length n of the sets;

[0116] Output: The intersection set;

[0117] 1. For a in A;

[0118] 1.1 Convert 'a' into a binary array of length 'n';

[0119] 1.2 Store 'a' in array B;

[0120] 2. Copy B to GPU memory;

[0121] 3. For i in B;

[0122] For j in B[i];

[0123] 3.1B[0][j]&=B[i][j];

[0124] 4. return B[0].

[0125] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for multi-dimensional data parallel query for real-time inference of power system, characterized in that, include: (1) Organize the simulation data and construct a star-shaped multidimensional data model, in which the fact table is the bus, and other attribute tables are equipment and physical quantities. Sort the numbers of the bus in the fact table and store them in a one-dimensional array. (2) Establish a multi-dimensional hierarchical data index; Hierarchical indexes are built for the fields in the attribute table. For numerical data in trend analysis, Hi4H builds subtrees based on the set hierarchical height k and the maximum number of child nodes m of the intermediate nodes. For categorical data, numerical mapping of categories is first performed based on the number of categories, and then subtrees are built according to the numerical data. The steps to build a multidimensional hierarchical data index include: Step 1: Sort the n-dimensional data according to the bus bar number; The unique number of the busbar is stored in an array, with the numerical number representing the sequence number; Step two: organize data D in each dimension i Make it constitute a hierarchical tree structure with height k-1 and degree no more than m; The construction of a single hierarchical tree follows a bottom-up principle. First, the sorted parent trees are grouped, and each group of parent trees will constitute the leaf nodes of the hierarchical tree. The number of parent trees in each group is obtained by the following formula: where N is D i number of buses; then mount the grouped buses to leaf nodes; Step 3: Process the leaf nodes, updating the maximum and minimum values ​​of their indices; Step 4: Process non-leaf nodes up to the top level. Organize the data for each dimension in the same way and finally link them to the global root node to form the final Hi4H hierarchical tree structure. (3) A hash-based multidimensional index is used to convert the physical parameters of power components into hash values ​​and establish a hash table index; an interval-based query algorithm is used to group the query conditions, establish an interval index for the query conditions in each group, and use the interval index for fast matching.

2. The method of claim 1, wherein, The most typical characteristics of power flow projection data are continuous numerical data and categorical data. The data is centered on the bus, and information about other related power components can be obtained from the bus.

3. The method of claim 2, wherein, The multidimensional hierarchical data index is an index of the continuous memory sequence number of the target object. For power flow simulation data, the multidimensional hierarchical data index is a one-dimensional data sequence number of the storage bus.

4. The method for parallel querying of 100-dimensional data for real-time simulation of power systems according to claim 1, characterized in that, Before constructing the hierarchical index structure, the instances in the fact table must first be sorted, and each instance must be reassigned according to the sorted sequence number. The redefined sequence number has two basic characteristics: the sequence number is an integer and the sequence number is a consecutive number.

5. The method for parallel querying of 100-dimensional data for real-time simulation of power systems according to claim 1, characterized in that, If the index dimension is categorical rather than numerical, then the categorical data needs to be properly encoded beforehand to convert it into numerical data.

6. The method for parallel querying of 100-dimensional data for real-time simulation of power systems according to claim 1, characterized in that, When building a multidimensional data index, different power components use different hash functions, distributing the hash values ​​across different hash tables.

7. The method for parallel querying of 100-dimensional data for real-time simulation of power systems according to claim 1, characterized in that, During the power flow simulation, it is necessary to continuously interact and query the components associated with the bus and change the component parameters. According to the multi-dimensional hierarchical data index, the query steps include: traversing the hierarchical tree to obtain the set of bus numbers; and finding the intersection of the sets to obtain the bus that meets the requirements.

8. The method for parallel querying of 100-dimensional data for real-time simulation of power systems according to claim 7, characterized in that, Multidimensional data query based on hierarchical trees includes: traversing the hierarchical tree for each dimension to obtain the set of candidate bus lines under that dimension, and finding the intersection of the candidate bus line sets of each dimension.

9. The method for parallel querying of 100-dimensional data for real-time simulation of power systems according to claim 7, characterized in that, After indexing k dimensions, each dimension will yield a corresponding index object, i.e., a set of bus sequence numbers, which will then be processed using a bitmap-based set intersection algorithm.

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