1. Multidimensional Data Many applications of databases are ``geographic'' = 2­dimensional data. Others involve large numbers of dimensions. Example: data about sales. - A sale is described by (store, day, item, color, size, etc.). Sale = point in 5­dim space. - A customer is described by (age, salary, pcode, marital­status, etc.). Typical Queries Range queries: ``How many customers for gold jewelry have age between 45 and 55, and salary less than 100K?'' Nearest neighbor : ``If I am at coordinates (x,y), what is the nearest McDonalds.'' They are expressible in SQL. Do you see how?

2. SQL Range queries: ``How many customers for gold jewelry have age between 45 and 55, and salary less than 100K?'‘ SELECT * FROM Customers WHERE age>=45 AND age<=55 AND sal<100; Nearest neighbor : ``If I am at coordinates (a,b), what is the nearest McDonalds.'‘ Suppose we have a relation Points(x,y,name) SELECT * FROM Points p WHERE p.name=‘McDonalds’ AND NOT EXISTS ( SELECT * FROM POINTS q WHERE (q.x-a)*(q.x-a)+(q.y-b)*(q.y-b) < (p.x-a)*(p.x-a)+(p.y-b)*(p.y-b) AND q.name=‘McDonalds’ );

3. Big Impediment For these types of queries, there is no clean way to eliminate lots of records that don't meet the condition of the WHERE­clause. An Approach Index on attributes independently. - Intersect pointers in main memory to save disk I/O.

4. Attempt at using B-trees for MD-queries Database = 1,000,000 points evenly distributed in a 1000×1000 square. Stored in 10,000 blocks (100 recs per block) B-tree indexes on x and on y Range query {(x,y) : 450  x  550, 450  y  550} 100,000 pointers (i.e. 1,000,000/10) for the x range, and same for y 10,000 pointers for answer (found by pointer intersection) Retrieve 10,000 records. If they are stored randomly we need to do 10,000 I/O’s. Add here the cost of B-Trees: Root of each B-tree in main memory Suppose leaves have avg. 200 keys  500 disk I/O in each B-tree to get pointer lists  1000 + 2(for intermediate B-tree level) disk I/O’s Total 11,002 disk I/O’s more than sequential scan of file = 10,000 I/O’s.

5. Nearest Neighbor query using B-trees Turn NN to (10,20) into a range-query {(x,y):10-d  x  10+d, 20-d  y  20+d } Possible problem: No point in the selected range The closest point inside may not be the answer Solution: re-execute range query with slightly larger d

6. NN-queries, example Same relation Points and its indexes on x and y as before, and Query: NN to (10,20) Choose d = 1  range-query = {(x,y): 9  x  11, 19  y  21} 2000 points in [9,11], same in [19,21]  each dimension = 10+1 I/O’s to get pointers (+1 is because points with x=9 may not start just at the beginning of the leaf) With an extra I/O for the intermediate node for each index  24 + 1 disk I/O’s to get the answer, assuming 1 of the 4 points is the answer, which we can determine by their coordinates, prior to getting the data blocks holding the points However, if d is too small, we have to run another range query with a larger d

7. Grid files (hash-like structure) Data: (25,60) (45,60) (50,75) (50,100) (50,120) (70,110) (85,140) (30,260) (25,400) (45,350) (50,275) (60,260) Divide data into stripes in each dimension Rectangle in grid points to bucket Example: database records (age,salary) for people who buy gold jewelry.

8. Grid file

9. Operations Lookup Find coordinates of point in each dimension --- gives you a bucket to search. Nearest Neighbor Lookup point P. Consider points in that bucket. Problem: there could be points in adjacent buckets that are closer. Problem: there could be no points at all in the bucket: widen search? Range Queries Ranges define a region of buckets. Buckets on border may contain points not in range. Example: 35 < age <= 45; 50 < salary <= 100. Queries Specifying Only One Attribute Problem: must search a whole row or column of buckets.

10. Insertion Use overflow buckets, or split stripes in one or more dimensions Insert (52,200). Split central bucket, for instance by splitting central salary stripe The blocks of 3 buckets are to be processed. In general the blocks of n buckets are to be processed during a split. n is the number of buckets in the chosen direction

11. Insertion Insert (52,200). Split central bucket, for instance by splitting central salary stripe (One possibility)

12. Grid files Advantages Good for multiple-key search Supports Partial Match, Range Queries, NN queries Disadvantages Space management overhead Need partitioning ranges that evenly split keys Possibility of overflow buckets for insertion

13. Partitioned hashing I If we hash the concatenation of several keys then such a hash table cannot be used in queries specifying only one dimension (key). A preferable option is to design the hash function so it produces some number of bits, say k. These k bits are divided among n attributes. I.e. the hash function h is a concatenation of n hash functions, one for each dimensional attribute. h = (h 1, …, h n ) the bucket where to put a tuple (v 1, …, v n ) is computed by concatenating the bit sequences h 1 (v 1 )…h n (v n ).

14. Partitioned hashing II Example: Gold jewelry with first bit = age mod 2 bits 2 and 3: salary mod 4 Works well for: partial match (i.e. just an attribute specified) Bad for: range nearest neighbor queries

15. Partitioned hashing III Partial match query –specifying only the value of a: compute h age (a), which could be, say 1. Then, locate all the relevant buckets, which are from 100 to 111. –specifying only the value of salary: compute h salary (s), which could be, say 10. Then, locate the relevant buckets, which are 010 and 110.

16. Grid files vs. partitioned hashing If many dimensions  many empty cells in grid. While partitioned hashing is OK. Both support exact and partial match queries. Grid files good for range and Nearest Neighbor queries, while partitioned hashing is not at all.