Summary. 4. Indexes. 4.0 Indexes. 4.1 Tree Based Indexes. 4.0 Indexes. 19Nov10. Last week: This week:


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1 Summary Data Warehousing & Data Mining WolfTilo Balke Silviu Homoceanu Institut für Informationssysteme Technische Universität Braunschweig Last week: Logical Model: Cubes, Dimensions, Hierarchies, Classification Levels Physical Level Relational: Star, Snowflakeschema Multidimensional (array based storage): linearization, problems e.g., order of dimensions, dense and sparse cubes This week: Indexes Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 2 4. Indexes 4. Indexes 4. Tree based indexes 4.2 Bitmap indexes 4.0 Indexes Why index? Consider a 00 GB table; at 00 MB/s read speed we need 7 minutes for a full table scan Consider an OLAP query: the number of Bosch S500 washing machines sold in Germany last month? Applying restrictions (product, location) the selectivity would be strongly reduced If we have 30 location, 0000 products and 24 months in the DW, the selectivity is /30 * / 0000 * /24 = 0, So we read 00 GB for,4kb of data not very smart Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 3 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Indexes 4. Tree Based Indexes Reduce the size of read pages to a minimum with indexes Product (Article) Product (Article) Product (Article) Product (Article) In the beginning there were BTrees Data structures for storing sorted data with amortized run times for insertion and deletion Basic structure of a node Key Value Data Pointer Tree Node Time (Days) Full table scan Time (Days) Cluster primary index Time (Days) More secondary Indexes, bitmap indexes Time (Days) Optimal multidimensional index Scanned data Selected data Node Pointers Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 5 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig
2 4. Tree Structures 4. Tree Structures Search in database systems Btree structures allow exact search with logarithmic costs Search in DWs The data is multidimensional, Btrees however, support only onedimensional search Are there any possibilities to extend tree functionality for multidimensional data? Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 7 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 8 4. Tree Structures 4. Tree Structures The basic idea of multidimensional trees Describe the sets of points through geometric regions, which comprise the points (clusters) The clusters are considered for the actual search and not the individual points Clusters can contain each other, resulting in a hierarchical structure Differentiating criterias for tree structures: Cluster construction: Completely fragmenting the space or Grouping data locally Cluster overlap: Overlapping or Disjoint Balance: Balanced or Unbalanced Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 9 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 0 4. Tree Structures 4. RTrees Object storage: Objects in leaves and nodes, or Objects only in the leaves Geometry: Hyperspheres, Hypercube,... The Rtree (Guttman, 984) is the prototype of a multidimensional extension of the classical Btrees Frequently used for lowdimensional applications (used to about 0 dimensions), such as geographic information systems More scalable versions: R + Trees, R*Trees and X Trees (each up to 20 dimensions for uniform distributed data) Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 2 2
3 4. RTree Structure 4. RTree Example Dynamic Index Structure (insert, update and delete are possible) Data structure Data pages are leaf nodes and store clustered point data and data objects Directory pages are the internal nodes and store directory entries Multidimensional data are structured with the help of Minimum Bounding Rectangles (MBRs) R R5 R4 R X Q X p X O R3 R R0 R R2 R3 R4 R5 R R7 R0 R Q P O Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 3 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 4 4. RTree Characteristics Local grouping for clustering Overlapping clusters (the more the clusters overlap the more inefficient is the index) Height balanced tree structure (therefore all the children of a node in the tree have about the same number of successors) Objects are stored, only in the leaves Internal nodes are used for navigation MBRs are used as a geometry 4. RTree Properties The has at least two children Each internal node has between m and M children M and m M / 2 are predefined parameters For each entry (I, childpointer) in an internal node, I is the smallest rectangle that contains the rectangles of the child nodes Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 5 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 4. RTree Properties For each index entry (I, tupleid) in a leaf, I is the smallest bounding rectangle that contains the data object (with the ID tupleid) All the leaves in the tree are on the same level All leaves have between m and M index records 4. Operations of RTrees The essential operations for the use and management of an Rtree are Search Insert Updates Delete Splitting Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 7 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 8 3
4 4. Searching in RTrees 4. Example The tree is searched recursively from the to the leaves One path is selected If the requested record has not been found in that subtree, the next path is traversed The path selection is arbitrary R R5 R4 R S R3 R R0 X R R2 R7 R3 X Check all the objects in node Check only 7 nodes instead of 2 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 9 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Searching in RTrees 4. Search Algorithm No guarantee for good performance In the worst case, all paths must traversed (due to overlaps of the MBRs) Search algorithms try to exclude as many irrelevant regions as possible ( pruning ) All the index entries which intersect with the search rectangle S are traversed The search in internal nodes Check each object for intersection with S For all intersecting entries continue the search in their children The search in leaf nodes Check all the entries to determine whether they intersect S Take all the correct objects in the result set Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 2 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Insert 4. Insert Procedure The best leaf page is chosen (ChooseLeaf) considering the spatial criteria Beast leaf: the leaf that needs the smallest volume growth to include the new object The object will be inserted there if there is enough room (number of objects in the node < M) If there is no more place left in the node, it is considered a case for overflow and the node is divided (SplitNode) Goal of the split is to result in minimal overlap and as small dead space as possible Interval of the parent node must be adapted to the new object (AdjustTree) If the is reached by division, then create a new whose children are the two split nodes of the old Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 23 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 24 4
5 4. RTree Insert Example 4. Heuristics R R5 R4 R R2 R3 R R0 x P x P x P Inserting P either in R7 or In R7, it needs more space, but does not overlap An object is always inserted in the nodes, to which it produces the smallest increase in volume If it falls in the interior of a MBR no enlargement is need If there are several possible nodes, then select the one with the smallest volume Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 25 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 2 4. Insert with Overflow 4. SplitNode R R5 R4 R R3 R R0 X P R7b X P R R2 R3 If an object is inserted in a full node, then the M+ objects will be divided among two new nodes The goal in splitting is that it should rarely be needed to traverse both resulting nodes on subsequent searches Therefore use small MBRs. This leads to minimal overlapping with other MBRs R4 R5 R R7 R7b R0 R Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 27 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Split Example Calculate the minimum total area of two rectangles, and minimize the dead space Bad split Better Split 4. Overflow Problem Deciding on how exactly to perform the splits is not trivial All objects of the old MBR can be divided in different ways on two new MBRs The volume of both resulting MBRs should remain as small as possible The naive approach of checking checks all splits and calculate the resulting volumes is not possible Two approaches With quadratic cost With linear cost Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 29 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 30 5
6 4. Overflow Problem 4. Overflow Problem Procedure with quadratic cost Compute for each 2 objects the necessary MBR and choose the pair with the largest MBR Since these two objects should not occur in an MBR, they will be used as starting points for two new MBRs Compute for all other objects, the difference of the necessary volume increase with respect to both MBRs Insert the object with the smallest difference in the corresponding MBR and compute the MBR again Repeat this procedure for all unallocated objects Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 3 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Overflow Problem Procedure with linear cost In each dimension: Find the rectangle with the highest minimum coordinates, and the rectangle with the smallest maximum coordinates Determine the distance between these two coordinates, and normalize it on the size of all the rectangles in this dimension Determine the two starting points of the new MBRs as the two objects with the highest normalized distance 4. Example A B 4 5 C xdirection: select A and E, as d x = diff x /max x = 5 / 4 ydirection: select C and D, as d y = diff y /max y = 8 / 3 Since d x < d y, C and D are chosen for the split D E 8 3 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 33 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Overflow Problem Classify all remaining objects the MBR with the smallest volume growth The linear process is a simplification of the quadratic method It is usually sufficient providing similar quality of the split (minimal overlap of the resulting MBRs) 4. Delete Procedure Search the leaf node with the object to delete (FindLeaf) Delete the object (deleterecord) The tree is condensed (CondenseTree) if the resulting node has < m objects When condensing, a node is completely erased and the objects of the node which should have remained are reinserted If the remains with just one child, the child will become the new Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 35 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 3
7 4. Example An object from is deleted ( object remains in, but m = 2) Due to few objects is deleted, and R2 is reduced (condensetree) 4. Update If a record is updated, its surrounding rectangle can change The index entry must then be deleted updated and then reinserted R R2 R3 R4 R5 R R7 R0 R Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 37 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Block Access Cost The most efficient search in Rtrees is performed when the overlap and the dead space are minimal A D F C M N E E L G S H K J I B A B C D E F G H I J K L M N Avoiding overlapping is only possible if data points are known in advance 4. Improved Versions of RTrees Where are Rtrees inefficient? They allow overlapping between neighboring MBRs R + Trees (Sellis ua, 987) Overlapping of neighboring MBRs are prohibited This may lead to identical leafs occurring more than once in the tree Improve search efficiency, but similar scalability as Rtrees Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 39 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 40 A F D E C M N 4. R + Trees L P G S H K J I B D E F G A B C P I J K L M N G H Overlaps are not permitted (A and P) Data rectangles are divided and may be present (e.g., G) in several leafs 4. Performance The main advantage of R + trees is to improve the search performance Especially for point queries, this saves 50% of access time Drawback is the low occupancy of nodes resulting through many splits R + trees often degenerate with the increasing number of changes Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 4 Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 42 7
8 4. More Versions R* trees and Xtrees improve the performance of the R + trees (Kriegel and others, 990/99) Improved split algorithm in R*trees Extended nodes in Xtrees allow sequential search of larger objects Scalable up to 20 dimensions Combination of B * Tree and Zcurve = Universal BTree (UBtree) Zcurve is used to map multidimensional points to onedimensional values (Zvalues) Zvalues are used as keys in B * Tree Index part Data part Multimedia Databases WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 43 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 44 Concept of ZRegions To create a disjunctive partitioning of the multidimensional space This allows for very efficient processing of multidimensional range queries ZRegions The space covered by an interval on the ZCurve Defined by two ZAddresses a and b We call b the region address of [a : b] Each ZRegion maps exactly onto one page on secondary storage I.e., to one leaf page of the B * Tree E.g., of ZRegions [:9],[0, 8], [9, 28] Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 45 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 4 ZValue address representation Calculated through bit interleaving of the coordinates of the tuple Y = 5 = 0 Zvalue = 000 X = 4 = 00 Tuple= 5, x = 4, y = 5 y x Why ZValues? With ZValues we reduce the dimensionality of the data to one dimension ZValues are then used as keys in B * trees Using B * Trees results in high node filling degree (at least 50%) Logarithmical complexity at search, insert and delete Guaranteed maximum nodeaccesses to locate a key is log ( + 2 ) ZValue are very important for range queries! Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 47 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 48 8
9 Range queries (RQ) in UBTrees Each query can be specified by 2 coordinates q a (the upper left corner of the query rectangle) q b (the lower right corner of the query rectangle) RQalgorithm Starts with q a and calculates its ZRegion ZRegion of q a is [0:8] Range queries (RQ) in UBTrees 2. The corresponding page is loaded and filtered with the query predicate. Tupels 5 and fulfill the predicate 3. The next region (inside the query rectangle) on the Z curve is calculated. The next jump point on the Zcurve is Repeat steps 2 and 3 until the endaddress of the last filtered region is bigger than q b Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 49 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 50 The critical part of the algorithm is calculating the jump point on the Zcurve which is inside the query rectangle If this takes too long it eliminates the advantage obtained through optimized disk access How is the jump point optimally calculated? From 3 points: q a, q b and the current ZRegion By performing bit operations Bitmap Indexes Lets assume a relation Expenses with three attributes: Nr, Shop and Sum A bitmap index for attribute Shop looks like this Value Nr Shop Sum Saturn 50 2 Real 5 3 P&C 0 4 Real 45 5 Saturn 350 Real 80 Vector Real 000 Saturn 0000 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 5 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 52 A bitmap index for an attribute of relation is: A collection of bitvectors The number of bitvectors represents the number of distinct values of the attribute in the relation The length of each bitvector is called the cardinality of the relation The bitvector for value v has in position i, if the i th record has v in attribute A, and it has 0 otherwise Records are allocated permanent numbers There is a mapping between record numbers and record addresses Assume: There are n records in the table Attribute A has m distinct values in the table The size of a bitmap index on attribute A is m*n Significant number of 0 s if m is big, and of s if m is small Opportunity to compress Run Length Encoding (RLE) Gzip (LempelZiv, LZ) ByteAligned Bitmap Compression (BBC): variable byte length encoding (Oracle patent) Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 53 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 54 9
10 Nr Shop Handling modification Assume record numbers are 2 Real 5 not changed Deletion 4 Real Real Tombstone replaces deleted record ( doesn t become 5!) Corresponding bit is set to 0 Before Value Vector Real 000 Saturn 0000 Value Sum Saturn 50 3 P&C 0 5 Saturn 350 After Vector Real 000 Saturn Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 55 Inserted record is assigned the next record number A bit of value 0 or is appended to each bit vector If new record contains a new value of the attribute, add one bitvector Nr Shop Sum Saturn 50 2 Real 5 3 P&C 0 4 Real 45 5 Saturn 350 Real 80 7 REWE 23 E.g., insert new record with REWE as shop After Before Value Vector Value Vector 0 Real 0000 Real 000 Saturn Saturn 0000 REWE Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 5 Nr Shop Sum Update Change the bit corresponding to 2 REWE 5 the old value of the modified record to 0 Change the bit corresponding to 4 Real 45 Real the new value of the modified record to 80 If the new value is a new value of A, then insert a new bitvector: e.g., replace Shop for record 2 to REWE Before Value Vector Real 000 Saturn 0000 Value Saturn 50 3 P&C 0 5 Saturn 350 After Vector Real 0000 Saturn 0000 REWE Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 57 Select Basic AND, OR bit operations: E.g., select the sums we have spent in Saturn and P&C Saturn OR P&C = Result Nr Shop Bitmap indexes should be used when selectivity is high Sum Saturn 50 2 Real 5 3 P&C 0 4 Real 45 5 Saturn 350 Real 80 Value Vector Real 000 Saturn 0000 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Bitmap indexes Advantages Operations are efficient and easy to implement (directly supported by hardware) Disadvantages For each new value of an attribute a new bitmapvector is introduced If we bitmap index an attribute like birthday (only day) we have 35 vectors: 35/8 bits 4 Bytes for a record, just for that Solution to such problems is multicomponent bitmaps Not fit for range queries where many bitmap vectors have to be read Solution: rangeencoded bitmap indexes Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Bitmap indexes Multicomponent bitmap indexes Encoding using a different numeration system E.g., for the month attribute, between 0 and values can be encoded as x = 4 *y+z, where 0 y 2, and 0 z 3, called <3,4> basis encoding Month Dec Nov Oct Sep Aug Jul Jun Mai Apr Mar Feb Jan M A A 0 A 9 A 8 A 7 A A 5 A 4 A 3 A 2 A A = 4*+ X Y Z M A 2, A, A 0, A 3,0 A 2,0 A,0 A 0, Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 0 0
11 4.2 Multicomponent bitmap indexes Advantage of multicomponent bitmap indexes If we have 00 (0..99) different Days to index we can use a multicomponent bitmap index with basis of <0,0> The storage is reduced from 00 to 20 bitmapvectors (0 for y and 0 for z) The readaccess for a point ( day out of 00) query needs however 2 read operations instead of just Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 4.2 Bitmap indexes Rangeencoded bitmap indexes: Persons born between March and August For normal encoded bitmap indexes read vectors Dec Nov Oct Sep Aug Jul Jun Mai Apr Mar Feb Jan Person A A 0 A 9 A 8 A 7 A A 5 A 4 A 3 A 2 A A Idea: set the bits of all bitmap vectors to if they are higher or equal to the given value Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Bitmap indexes Query: Persons born between March and August So persons which didn t exist in February, but existed in August! Just 2 vectors read: ((NOT A ) AND A 7 ) Dec Nov Oct Sep Aug Jul Jun Mai Apr Mar Feb Jan Person A A 0 A 9 A 8 A 7 A A 5 A 4 A 3 A 2 A A Rangeencoded bitmap indexes If the query is limited only on one side, (e.g., persons born in or after March), vector is enough (NOT A ) For point queries, 2 vector reads are however necessary! E.g., persons born in March: ((NOT A ) AND A 2 ) Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 3 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig Advantages Bitmap indexes are great for indexing the dimensions Fully indexing tables with a traditional Btrees can be expensive  the indexes can be several times larger than the data Bitmap indexes are typically only a fraction of the size of the indexed data in the table. They reduced response time for large classes of ad hoc queries bring dramatic performance gains even on hardware with a relatively small number of CPUs or a small amount of memory Summary BTrees are not fit for multidimensional data RTrees MBR as geometry to build multidimensional indexes Operations: select, insert, overflow problem, node splitting, delete Inefficient because they allow overlapping between neighboring MBRs R + trees  improve the search performance Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 5 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig
12 Summary UBTrees Reduce multidimensional data to one dimension in order to use BTree indexes ZRegions, ZCurve, use the advantage of bit operations to make optimal jumps Bitmap indexes Great for indexing tables with setlike attributes e.g., Gender: Male/Female Operations are efficient and easy to implement (directly supported by hardware) Multi component reduce the storage while range encoded allow for fast range queries Next lecture Optimization Partitioning Joins Materialized Views Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 7 Data Warehousing & OLAP WolfTiloBalke InstitutfürInformationssysteme TU Braunschweig 8 2
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