Multidimensional Indexing
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Multidimensional indexes allow querying data across multiple dimensions. Grid files partition data into grid cells and store pointers to data within each cell, allowing quick retrieval of data matching queries on individual or ranges of dimensions. Partitioned hash functions map multiple dimensions to a single hash value, grouping related data. Tree structures like kd-trees and quad trees recursively split the data space, allowing efficient nearest neighbor and spatial queries. R-trees group regions hierarchically to support region queries on multidimensional data.
Multidimensional Indexing
- 1. Multidimensional Indexes Chapter 5
- 2. Outline • Applications • Hash-like structures • Tree-like structures • Bit-map indexes
- 3. Search Key • Search key is (F1, F2, ..Fk) – Separated by special markers. – Example • If F1=abcd and F2=123, the search key is “abcd#123”
- 4. Applications: GIS • Geographic Information Systems – Objects are in two dimensional space – The objects may be points or shapes – Objects may be houses, roads, bridges, pipelines and many other physical objects. • Query types – Partial match queries • Specify the values for one or more dimensions and look for all points matching those values in those dimensions. – Range queries • Set of shapes within the range – Nearest neighbor queries • Closest point to a given point – Where-am-I queries • When you click a mouse, the system determines which of the displayed elements you were clicking.
- 5. Data Cubes • Data exists in high dimensional space • A chain store may record each sale made, including – The day and time – The store in which the sale was made – The item purchased – The color of the item – The size of the item • Attributes are seen as dimensions multidimensional space, data cube. • Typical query – Give a class of pink shirts for each store and each month of 1998.
- 6. Multidimensional queries in SQL • Represent points as a relation Points(x,y) • Query – Find the nearest point to (10, 20) – Compute the distance between (10,20) to every other point.
- 7. Rectangles • Rectangles(id, x11, y11, xur, yur) • If the query is – Find the rectangles enclosing the point (10,20) • SELECT id FROM Rectangles WHERE x11 <= 10.0 AND y11 <= 20.0 AND xur >=10.0 AND yur >=20.0;
- 8. Data Cube • Fact table – Sales(day,store, item. color, size) • Query – Summarize the sale of pink shirts by day and store • SELECT day, store, COUNT(*) AS totalSales FROM Sales WHERE item=‘shirt’ AND color=‘pink’ GROUP BY day, store;
- 9. Executing range queries in Conventional Indexes • Motivation: Find records where DEPT = “Toy” AND SAL > 50k • Strategy 1: Use “Toy” index and check salary • Strategy II: Use “Toy” index and SAL index and intersect. – Complexity • Toy index: number of disk accesses = number of disk blocks • SAL index: number of disk accesses= number of records • So conventional indexes of little help.
- 10. Executing nearest-neighbor queries using conventional indexes • There might be no point in the selected range – Repeat the search by increasing the range • The closest point within the range might not be the closest point overall. – There is one point closer from outside range.
- 11. Other Limitations of Conventional Indexes • We can only keep the file sorted on only attribute. • If the query is on multiple dimensions – We will end-up having one disk access for each record • It becomes too expensive
- 12. Overview of Multidimensional Index Structures • Hash-table-like approaches • Tree-like approaches • In both cases, we give-up some properties of single dimensional indexes • Hash – We have to access multiple buckets • Tree – Tree may not be balanced – There may not exist a correspondence between tree nodes and disk blocks – Information in the disk block is much smaller.
- 13. Hash like structures: GRID Files • Each dimension, grid lines partition the space into stripes, – Points that fall on a grid line will be considered to belong to the stripe for which that grid line is lower boundary – The number of grid lines in each dimension may vary. – Space between grid lines in the same dimension may also vary
- 14. Grid Index Key 2 X1 X2 …… Xn V1 V2 Key 1 Vn To records with key1=V3, key2=X2 14
- 15. CLAIM • Can quickly find records with – key 1 = Vi ∧ Key 2 = Xj – key 1 = Vi – key 2 = Xj 15
- 16. CLAIM • Can quickly find records with – key 1 = Vi ∧ Key 2 = Xj – key 1 = Vi – key 2 = Xj • And also ranges…. – E.g., key 1 ≥ Vi ∧ key 2 < Xj 16
- 17. But there is a catch with Grid Indexes! • How is Grid Index stored on disk? V1 V2 V3 Like Array... X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4 17
- 18. But there is a catch with Grid Indexes! • How is Grid Index stored on disk? V1 V2 V3 Like Array... X1 X2 X3 X4 X1 X2 X3 X4 X1 X2 X3 X4 Problem: • Need regularity so we can compute position of Vi,Xj entry 18
- 19. Solution: Use Indirection X1 X2 X3 Buckets -- V1 -- -- V2 -- -- -- V3 *Grid only -- V4 -- -- contains pointers to buckets -- -- -- -- Buckets -- -- 19
- 20. With indirection: • Grid can be regular without wasting space • We do have price of indirection 20
- 21. Can also index grid on value ranges Salary Grid 0-20K 1 20K-50K 2 50K- 3 8 1 2 3 Linear Scale Toy Sales Personnel 21
- 22. Grid files Good for multiple-key search + Space, management overhead - (nothing is free) Need partitioning ranges that evenly - split keys 22
- 24. Partitioned hash function Idea: 010110 1110010 Key1 Key2 h1 h2 24
- 25. EX: h1(toy) =0 000 h1(sales) =1 001 h1(art) =1 010 . 011 . h2(10k) =01 100 h2(20k) =11 101 h2(30k) =01 110 h2(40k) =00 111 . . Fred,toy,10k,Joe,sales,10k Insert Sally,art,30k 25
- 26. EX: h1(toy) =0 000 h1(sales) =1 001 Fred h1(art) =1 010 . 011 . h2(10k) =01 100 h2(20k) =11 101 JoeSally h2(30k) =01 110 h2(40k) =00 111 . . Fred,toy,10k,Joe,sales,10k Insert Sally,art,30k 26
- 27. h1(toy) =0 000 Fred h1(sales) =1 001 JoeJan h1(art) =1 010 Mary . 011 . h2(10k) =01 100 Sally h2(20k) =11 101 h2(30k) =01 110 TomBill h2(40k) =00 111 Andy . . • Find Emp. with Dept. = Sales ∧ Sal=40k 27
- 28. h1(toy) =0 000 Fred h1(sales) =1 001 JoeJan h1(art) =1 010 Mary . 011 . h2(10k) =01 100 Sally h2(20k) =11 101 h2(30k) =01 110 TomBill h2(40k) =00 111 Andy . . • Find Emp. with Dept. = Sales ∧ Sal=40k 28
- 29. h1(toy) =0 000 Fred h1(sales) =1 001 JoeJan h1(art) =1 010 Mary . 011 . h2(10k) =01 100 Sally h2(20k) =11 101 h2(30k) =01 110 TomBill h2(40k) =00 111 Andy . . • Find Emp. with Sal=30k 29
- 30. h1(toy) =0 000 Fred h1(sales) =1 001 JoeJan h1(art) =1 010 Mary . 011 . h2(10k) =01 100 Sally h2(20k) =11 101 h2(30k) =01 110 TomBill h2(40k) =00 111 Andy . . • Find Emp. with Sal=30k look here 30
- 31. h1(toy) =0 000 Fred h1(sales) =1 001 JoeJan h1(art) =1 010 Mary . 011 . h2(10k) =01 100 Sally h2(20k) =11 101 h2(30k) =01 110 TomBill h2(40k) =00 111 Andy . . • Find Emp. with Dept. = Sales 31
- 32. h1(toy) =0 000 Fred h1(sales) =1 001 JoeJan h1(art) =1 010 Mary . 011 . h2(10k) =01 100 Sally h2(20k) =11 101 h2(30k) =01 110 TomBill h2(40k) =00 111 Andy . . • Find Emp. with Dept. = Sales look here 32
- 33. Tree-like Structures • Multiple-key indexes • Kd-trees • Quad trees • R-trees
- 34. Multiple-key indexes • Several attributes representing dimensions of data points • Multiple key index is an index of indexes in which the nodes at each level are indexes for one attribute.
- 35. Strategy: • Multiple Key Index One idea: I2 I1 I3 35
- 36. Example 10k 15k Example Art 17k Record Sales 21k Toy Dept 12k Name=Joe Index 15k DEPT=Sales 15k SAL=15k 19k Salary Index 36
- 37. For which queries is this index good? Find RECs Dept = “Sales” SAL=20k Find RECs Dept = “Sales” SAL 20k Find RECs Dept = “Sales” Find RECs SAL = 20k 37
- 38. KD-trees • K dimensional tree is generalizing binary search tree into multi-dimensional data. • A KD tree is a binary tree in which interior nodes have an associated attribute “a” and a value “v” that splits data into two parts. • The attributes at different levels of a tree are different, and levels rotating among the attributes of all dimensions.
- 39. Interesting application: • Geographic Data y DATA: X1,Y1, Attributes x X2,Y2, Attributes ... 39
- 40. Queries: • What city is at Xi,Yi? • What is within 5 miles from Xi,Yi? • Which is closest point to Xi,Yi? 40
- 41. Example i a e d h b n f l o c j g k m 41
- 42. Example i a e d h b 10 20 n f l o c j g k m 10 20 42
- 43. Example a 40 i e d h 30 b 10 20 n f 20 l o c 10 j g 25 15 35 20 m k 10 20 43
- 44. Example a 40 i e d h 30 b 10 20 n f 20 l o c 10 j g 25 15 35 20 m k 10 20 5 15 15 44
- 45. Example a 40 i e d h 30 b 10 20 n f 20 l o c 10 j g 25 15 35 20 m k 10 20 5 h i g f d e c a b 15 15 j k l m n o 45
- 46. Example a 40 i e d h 30 b 10 20 n f 20 l o c 10 j g 25 15 35 20 m k 10 20 5 h i g f d e c a b 15 15 • Search points near f • Search points near b j k l m n o 46
- 47. Queries • Find points with Yi 20 • Find points with Xi 5 • Find points “close” to i = 12,38 • Find points “close” to b = 7,24 47
- 48. Quad trees • Divides multidimensional space into quadrants and divides the quadrants same way if they have too many points. • If the number of points in a square – fits in a block, it is a leaf node – no longer fits in a block, it becomes an interior node, four quadrants are its children.
- 50. R-tree • Captures the spirit of B-tree for multidimensional data. • Represents a collection of regions by grouping them into a hierarchy of larger regions. • Data is divided into regions. • Interior node is corresponds to interior region – region can be of any shape • Rectangular is popular – Children corresponds to sub-regions.
- 51. R-tree
- 52. Bitmap indexes • Assume that records have permanent numbers • A bit-map index is a collection of bit vectors of length n, one for each value may appear in the field F. • The vector for value v has 1 in position i if the i’th record has v in field F, and it has 0 there if not. • Example for F and G fields: – (30, foo), (30,bar), (40, baz), (50, foo), (40, bar), (30, baz) – Bit index for F, each of 6 bits. For 30, it is 110001, for 40, it is 001010, and for 50, it is 000100. – Bit index for G also have three vectors. For foo it is, 100100, for bar it is 010010, and for baz it is 001001.
- 53. Bit map indexes: Partial match • Bit maps allow answering of partial match queries quickly and efficiently. • Example: – Movie(title, year, length, studioName) – SELECT title – FROM Movie – WHERE studioName= ‘Disney’ and Year=1965; • If we have bitmap for studioName and year, then intersection or AND operation will give the result. • Bit vectors do not occupy much space.
- 54. Bitmap indexes: range queries • Example: consider the gold jewelry data of twelve points – 1 (25,60), 2(45,60), 3(50,75), 4(50,100), 5(50,120), 6(70, 110), 7(85,140), 8(30,260), 9(25,400), 10(45,350), 11(50,275), 12(60,260) • Age has seven different values – 25(100000001000), 30(000000010000), 45(010000000100), 50(001110000010), 60(000000000001), 70(000001000000) – 85(000000100000) • Salary has 10 different values – 60(000000000000), 75(001000000000), 100(000100000000) – 110(000001000000), 120(000010000000), 140(000000100000) – 260(000000010001), 275(000000000010), 350(000000000100) – 400(000000001000),
- 55. Example continued • Find the jewelry buyers with an age range 45-55 and salary in the range 100-200 • Find the bit vectors of for the age values in the range and take OR – 010000000100 (for 45) and 001110000010 (for 50) – Result: 011110000110 • Find the bit vectors of salaries between 100 and 200 thousand. – There are four: 100,110,120, and 140, their bitwaise OR is 000111100000 • Take AND of both bit vectors – 000110000000 – Find two records (50,100) and (50,120) are in the range.
- 56. Compressed bitmaps • If number of different values is large, then number of 1 is rare. • Run-length coding is used – Sequence of 0’s followed by 1. – Example: 000101 is two runs, 3 and 1. the binary representation is 11 and 1. So it is decoded as 111. • To save space, the bitmap indexes tend to consist of vectors with very few 1’s are compressed using run-length coding.
- 57. Finding bit vectors • Use any index technique to find the values. • From the values to bit vectors. • B-tree is a good choice.