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Indexing and Execution

Indexing and Execution. May 20 th , 2002. Indexing - Recap. Primary file organization: heap, sorted, hashed. Primary vs. secondary Clustered vs. unclustered Dense vs. sparse Composite search key indexes. Not all combinations are possible. Next: B+-trees. Tree-Based Indexes.

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Indexing and Execution

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  1. Indexing and Execution May 20th, 2002

  2. Indexing - Recap • Primary file organization: heap, sorted, hashed. • Primary vs. secondary • Clustered vs. unclustered • Dense vs. sparse • Composite search key indexes. • Not all combinations are possible. • Next: B+-trees.

  3. Tree-Based Indexes • ``Find all students with gpa > 3.0’’ • If data is in sorted file, do binary search to find first such student, then scan to find others. • Cost of binary search can be quite high. • Simple idea: Create an `index’ file. Index File kN k2 k1 Data File Page N Page 3 Page 1 Page 2 • Can do binary search on (smaller) index file!

  4. 40 51 63 20 33 46* 55* 40* 51* 97* 10* 15* 20* 27* 33* 37* 63* Tree-Based Indexes (2) index entry P K P P K P K m 2 0 1 m 1 2 Root

  5. Index Entries DataEntries B+ Tree: The Most Widely Used Index • Insert/delete at log F N cost; keep tree height-balanced. (F = fanout, N = # leaf pages) • Minimum 50% occupancy (except for root). Each node contains d <= m <= 2d entries. The parameter d is called the order of the tree. Root

  6. Example B+ Tree • Search begins at root, and key comparisons direct it to a leaf. • Search for 5*, 15*, all data entries >= 24* ... 30 24 13 17 39* 22* 24* 27* 38* 3* 5* 19* 20* 29* 33* 34* 2* 7* 14* 16*

  7. B+ Trees in Practice • Typical order: 100. Typical fill-factor: 67%. • average fanout = 133 • Typical capacities: • Height 4: 1334 = 312,900,700 records • Height 3: 1333 = 2,352,637 records • Can often hold top levels in buffer pool: • Level 1 = 1 page = 8 Kbytes • Level 2 = 133 pages = 1 Mbyte • Level 3 = 17,689 pages = 133 MBytes

  8. Inserting a Data Entry into a B+ Tree • Find correct leaf L. • Put data entry onto L. • If L has enough space, done! • Else, must splitL (into L and a new node L2) • Redistribute entries evenly, copy upmiddle key. • Insert index entry pointing to L2 into parent of L. • This can happen recursively • To split index node, redistribute entries evenly, but push upmiddle key. (Contrast with leaf splits.)

  9. Insertion in a B+ Tree Insert (K, P) • Find leaf where K belongs, insert • If no overflow (2d keys or less), halt • If overflow (2d+1 keys), split node, insert in parent: • If leaf, keep K3 too in right node • When root splits, new root has 1 key only (K3, ) to parent

  10. Insertion in a B+ Tree Insert K=19 10 15 18 20 30 40 50 60 65 80 85 90

  11. Insertion in a B+ Tree After insertion 10 15 18 19 20 30 40 50 60 65 80 85 90

  12. Insertion in a B+ Tree Now insert 25 10 15 18 19 20 30 40 50 60 65 80 85 90

  13. Insertion in a B+ Tree After insertion 10 15 18 19 20 25 30 40 50 60 65 80 85 90

  14. Insertion in a B+ Tree But now have to split ! 10 15 18 19 20 25 30 40 50 60 65 80 85 90

  15. Insertion in a B+ Tree After the split 10 15 18 19 20 25 30 40 50 60 65 80 85 90

  16. Deleting a Data Entry from a B+ Tree • Start at root, find leaf L where entry belongs. • Remove the entry. • If L is at least half-full, done! • If L has only d-1 entries, • Try to re-distribute, borrowing from sibling (adjacent node with same parent as L). • If re-distribution fails, mergeL and sibling. • If merge occurred, must delete entry (pointing to L or sibling) from parent of L. • Merge could propagate to root, decreasing height.

  17. Deletion from a B+ Tree Delete 30 10 15 18 19 20 25 30 40 50 60 65 80 85 90

  18. Deletion from a B+ Tree After deleting 30 May change to 40, or not 10 15 18 19 20 25 40 50 60 65 80 85 90

  19. Deletion from a B+ Tree Now delete 25 10 15 18 19 20 25 40 50 60 65 80 85 90

  20. Deletion from a B+ Tree After deleting 25 Need to rebalance Rotate 10 15 18 19 20 40 50 60 65 80 85 90

  21. Deletion from a B+ Tree Now delete 40 10 15 18 19 20 40 50 60 65 80 85 90

  22. Deletion from a B+ Tree After deleting 40 Rotation not possible Need to merge nodes 10 15 18 19 20 50 60 65 80 85 90

  23. Deletion from a B+ Tree Final tree 10 15 18 19 20 50 60 65 80 85 90

  24. Issues in Indexing • Multi-dimensional indexing: • how do we index regions in space? • Document collections? • Multi-dimensional sales data • How do we support nearest neighbor queries? • Indexing is still a hot and unsolved problem!

  25. Indexing Exercise #1 • The Purchase table: • date, buyer, seller, store, product • Reports are generated once a day. • What’s the best file-organization/indexing strategy?

  26. Indexing Exercise #2 • Airline database: Reservations table -- • flight#, seat#, date#, occupied, customer-id

  27. Indexing Exercise #3 • Web log application: load all the logs every night into a database. • Generate reports every day (for curious professors).

  28. Query Execution Query update User/ Application Query compiler Query execution plan Execution engine Record, index requests Index/record mgr. Page commands Buffer manager Read/write pages Storage manager storage

  29. Query Execution Plans SELECT S.sname FROM Purchase P, Person Q WHERE P.buyer=Q.name AND Q.city=‘seattle’ AND Q.phone > ‘5430000’ buyer  City=‘seattle’ phone>’5430000’ • Query Plan: • logical tree • implementation choice at every node • scheduling of operations. (Simple Nested Loops) Buyer=name Person Purchase (Table scan) (Index scan) Some operators are from relational algebra, and others (e.g., scan, group) are not.

  30. The Leaves of the Plan: Scans • Table scan: iterate through the records of the relation. • Index scan: go to the index, from there get the records in the file (when would this be better?) • Sorted scan: produce the relation in order. Implementation depends on relation size.

  31. How do we combine Operations? • The iterator model. Each operation is implemented by 3 functions: • Open: sets up the data structures and performs initializations • GetNext: returns the the next tuple of the result. • Close: ends the operations. Cleans up the data structures. • Enables pipelining! • Contrast with data-driven materialize model. • Sometimes it’s the same (e.g., sorted scan).

  32. Implementing Relational Operations • We will consider how to implement: • Selection ( ) Selects a subset of rows from relation. • Projection ( ) Deletes unwanted columns from relation. • Join ( ) Allows us to combine two relations. • Set-difference Tuples in reln. 1, but not in reln. 2. • UnionTuples in reln. 1 and in reln. 2. • Aggregation (SUM, MIN, etc.) and GROUP BY

  33. Schema for Examples Purchase (buyer:string, seller: string, product: integer), Person (name:string, city:string, phone: integer) • Purchase: • Each tuple is 40 bytes long, 100 tuples per page, 1000 pages (i.e., 100,000 tuples, 4MB for the entire relation). • Person: • Each tuple is 50 bytes long, 80 tuples per page, 500 pages (i.e, 40,000 tuples, 2MB for the entire relation).

  34. SELECT * FROM Person R WHERE R.phone < ‘543%’ Simple Selections • Of the form • With no index, unsorted: Must essentially scan the whole relation; cost is M (#pages in R). • With an index on selection attribute: Use index to find qualifying data entries, then retrieve corresponding data records. (Hash index useful only for equality selections.) • Result size estimation: (Size of R) * reduction factor. More on this later.

  35. Using an Index for Selections • Cost depends on #qualifying tuples, and clustering. • Cost of finding qualifying data entries (typically small) plus cost of retrieving records. • In example, assuming uniform distribution of phones, about 54% of tuples qualify (500 pages, 50000 tuples). With a clustered index, cost is little more than 500 I/Os; if unclustered, up to 50000 I/Os! • Important refinement for unclustered indexes: 1. Find sort the rid’s of the qualifying data entries. 2. Fetch rids in order. This ensures that each data page is looked at just once (though # of such pages likely to be higher than with clustering).

  36. Two Approaches to General Selections • First approach:Find the most selective access path, retrieve tuples using it, and apply any remaining terms that don’t match the index: • Most selective access path: An index or file scan that we estimate will require the fewest page I/Os. • Consider city=“seattle AND phone<“543%” : • A hash index on city can be used; then, phone<“543%” must be checked for each retrieved tuple. • Similarly, a b-tree index on phone could be used; city=“seattle” must then be checked.

  37. Intersection of Rids • Second approach • Get sets of rids of data records using each matching index. • Then intersect these sets of rids. • Retrieve the records and apply any remaining terms.

  38. Implementing Projection SELECTDISTINCT R.name, R.phone FROM Person R • Two parts: (1) remove unwanted attributes, (2) remove duplicates from the result. • Refinements to duplicate removal: • If an index on a relation contains all wanted attributes, then we can do an index-only scan. • If the index contains a subset of the wanted attributes, you can remove duplicates locally.

  39. Equality Joins With One Join Column SELECT * FROM Person R, Purchase S WHERE R.name=S.buyer • R S is a common operation. The cross product is too large. Hence, performing R S and then a selection is too inefficient. • Assume: M pages in R, pR tuples per page, N pages in S, pS tuples per page. • In our examples, R is Person and S is Purchase. • Cost metric: # of I/Os. We will ignore output costs.

  40. Simple Nested Loops Join For each tuple r in R do for each tuple s in S do if ri == sj then add <r, s> to result • For each tuple in the outer relation R, we scan the entireinner relation S. • Cost: M + (pR * M) * N = 1000 + 100*1000*500 I/Os: 140 hours! • Page-oriented Nested Loops join: For each page of R, get each page of S, and write out matching pairs of tuples <r, s>, where r is in R-page and S is in S-page. • Cost: M + M*N = 1000 + 1000*500 (1.4 hours)

  41. Index Nested Loops Join foreach tuple r in R do foreach tuple s in S where ri == sj do add <r, s> to result • If there is an index on the join column of one relation (say S), can make it the inner. • Cost: M + ( (M*pR) * cost of finding matching S tuples) • For each R tuple, cost of probing S index is about 1.2 for hash index, 2-4 for B+ tree. Cost of then finding S tuples depends on clustering. • Clustered index: 1 I/O (typical), unclustered: up to 1 I/O per matching S tuple.

  42. Examples of Index Nested Loops • Hash-index on name of Person (as inner): • Scan Purchase: 1000 page I/Os, 100*1000 tuples. • For each Person tuple: 1.2 I/Os to get data entry in index, plus 1 I/O to get (the exactly one) matching Person tuple. Total: 220,000 I/Os. (36 minutes) • Hash-index on buyer of Purchase (as inner): • Scan Person: 500 page I/Os, 80*500 tuples. • For each Person tuple: 1.2 I/Os to find index page with data entries, plus cost of retrieving matching Purchase tuples. Assuming uniform distribution, 2.5 purchases per buyer (100,000 / 40,000). Cost of retrieving them is 1 or 2.5 I/Os depending on clustering.

  43. . . . Block Nested Loops Join • Use one page as an input buffer for scanning the inner S, one page as the output buffer, and use all remaining pages to hold ``block’’ of outer R. • For each matching tuple r in R-block, s in S-page, add <r, s> to result. Then read next R-block, scan S, etc. Join Result R & S Hash table for block of R (k < B-1 pages) . . . . . . Output buffer Input buffer for S

  44. Sort-Merge Join (R S) i=j • Sort R and S on the join column, then scan them to do a ``merge’’ on the join column. • Advance scan of R until current R-tuple >= current S tuple, then advance scan of S until current S-tuple >= current R tuple; do this until current R tuple = current S tuple. • At this point, all R tuples with same value and all S tuples with same value match; output <r, s> for all pairs of such tuples. • Then resume scanning R and S.

  45. Cost of Sort-Merge Join • R is scanned once; each S group is scanned once per matching R tuple. • Cost: M log M + N log N + (M+N) • The cost of scanning, M+N, could be M*N (unlikely!) • With 35, 100 or 300 buffer pages, both Person and Purchase can be sorted in 2 passes; total: 7500. (75 seconds).

  46. Original Relation Partitions OUTPUT 1 1 2 INPUT 2 hash function h . . . B-1 B-1 B main memory buffers Disk Disk Partitions of R & S Join Result Hash table for partition Ri (k < B-1 pages) hash fn h2 h2 Output buffer Input buffer for Si B main memory buffers Disk Disk Hash-Join • Partition both relations using hash fn h: R tuples in partition i will only match S tuples in partition i. • Read in a partition of R, hash it using h2 (<> h!). Scan matching partition of S, search for matches.

  47. Cost of Hash-Join • In partitioning phase, read+write both relations; 2(M+N). In matching phase, read both relations; M+N I/Os. • In our running example, this is a total of 4500 I/Os. (45 seconds!) • Sort-Merge Join vs. Hash Join: • Given a minimum amount of memory both have a cost of 3(M+N) I/Os. Hash Join superior on this count if relation sizes differ greatly. Also, Hash Join shown to be highly parallelizable. • Sort-Merge less sensitive to data skew; result is sorted.

  48. How are we doing?

  49. Double Pipelined Join (Tukwila) Hash Join • Partially pipelined: no output until inner read • Asymmetric (inner vs. outer) — optimization requires source behavior knowledge Double Pipelined Hash Join • Outputs data immediately • Symmetric — requires less source knowledge to optimize

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