1. B-Trees and Sorting Zachary G. Ives April 12, 2019
University of Pennsylvania
CIS 550 – Database & Information Systems
April 12, 2019

2. Speeding Operations over Data
Three general data organization techniques:
Indexing
Sorting
Hashing

3. B+ Tree: The DB World’s Favorite Index
Insert/delete at log F N cost
(F = fanout, N = # leaf pages)
Keep tree height-balanced
Minimum 50% occupancy (except for root).
Each node contains d <= m <= 2d entries. d is called the order of the tree.
Supports equality and range searches efficiently.
Index Entries
(Direct search)
Data Entries
("Sequence set") 9

4. Example B+ Tree
Search begins at root, and key comparisons direct it to a leaf.
Search for 5*, 15*, all data entries >= 24* ...
Root 13 17 24 30 2* 3* 5* 7*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
Based on the search for 15*, we know it is not in the tree! 10

5. 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 = page = Kbytes
Level 2 = pages = Mbyte
Level 3 = 17,689 pages = 133 MBytes

6. Inserting Data into a B+ Tree
Find correct leaf L.
Put data entry onto L.
If L has enough space, done!
Else, must split L (into L and a new node L2)
Redistribute entries evenly, copy up middle key.
Insert index entry pointing to L2 into parent of L.
This can happen recursively
To split index node, redistribute entries evenly, but push up middle key. (Contrast with leaf splits.)
Splits “grow” tree; root split increases height.
Tree growth: gets wider or one level taller at top. 6

7. Inserting 8* into Example B+ Tree
Observe how minimum occupancy is guaranteed in both leaf and index pg splits.
Recall that all data items are in leaves, and partition values for keys are in intermediate nodes
Note difference between copy-up and push-up. 12

8. Inserting 8* Example: Copy up
Root 13 17 24 30 2* 3* 5* 7*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
Want to insert here; no room, so split & copy up: 8*
Entry to be inserted in parent node. 5
(Note that 5 is copied up and
continues to appear in the leaf.) 2* 3* 5* 7* 8*

9. Inserting 8* Example: Push up
Need to split node & push up
Root 13 17 24 30 5 2* 3*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39* 5* 7* 8*
Entry to be inserted in parent node.
(Note that 17 is pushed up and only
appears once in the index. Contrast
this with a leaf split.) 17 5 13 24 30

10. Deleting Data 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, merge L and sibling.
If merge occurred, must delete entry (pointing to L or sibling) from parent of L.
Merge could propagate to root, decreasing height. 14

11. B+ Tree Summary
B+ tree and other indices ideal for range searches, good for equality searches.
Inserts/deletes leave tree height-balanced; logF N cost.
High fanout (F) means depth rarely more than 3 or 4.
Almost always better than maintaining a sorted file.
Typically, 67% occupancy on average.
Note: Order (d) concept replaced by physical space criterion in practice (“at least half-full”).
Records may be variable sized
Index pages typically hold more entries than leaves 23

12. Other Kinds of Indices Multidimensional indices Text indices
R-trees, kD-trees, …
Text indices
Inverted indices
Structural indices
Object indices: access support relations, path indices
XML and graph indices: dataguides, 1-indices, d(k) indices
Describe parent-child, path relationships

13. Speeding Operations over Data
Three general data organization techniques:
Indexing
Sorting
Hashing

14. Technique II: Sorting Pass 1: Read a page, sort it, write it
Can use a single page to do this!
Pass 2, 3, …, etc.:
Requires a minimum of 3 pages
INPUT 1
OUTPUT
INPUT 2
Disk
Main memory buffers
Disk 5

15. Two-Way External Merge Sort
Divide and conquer: sort into subfiles and merge
Each pass: we read & write every page
If N pages in the file, we need: dlog2(N)e + 1
passes to sort the data, yielding a cost of:
2Ndlog2(N)e + 1
3,4
6,2
9,4
8,7
5,6
3,1 2
Input file
PASS 0
3,4
2,6
4,9
7,8
5,6
1,3 2
1-page runs
PASS 1
4,7
2,3
1,3
2-page runs
4,6
8,9
5,6 2
PASS 2
2,3
4,4
1,2
4-page runs
6,7
3,5
8,9 6
PASS 3
1,2
2,3
3,4
8-page runs
4,5
6,6
7,8 9 6

16. General External Merge Sort
How can we utilize more than 3 buffer pages?
To sort a file with N pages using B buffer pages:
Pass 0: use B buffer pages. Produce dN / Be sorted runs of B pages each
Pass 2, …, etc.: merge B-1 runs
INPUT 1
. . .
INPUT 2
. . .
. . .
OUTPUT
INPUT B-1
Disk
Disk
B Main memory buffers 7

17. Cost of External Merge Sort
Number of passes: 1+dlogB-1 dN / Bee
Cost = 2N * (# of passes)
With 5 buffer pages, to sort 108 page file:
Pass 0: d108/5e = 22 sorted runs of 5 pages each (last run is only 3 pages)
Pass 1: d22/4e = 6 sorted runs of 20 pages each (final run only uses 8 pages)
Pass 2: d6/4e = 2 sorted runs, 80 pages and 28 pages
Pass 3: Sorted file of 108 pages 8

18. Speeding Operations over Data
Three general data organization techniques:
Indexing
Sorting
Hashing

19. Technique 3: Hashing
A familiar idea, which we just saw for hash files:
Requires “good” hash function (may depend on data)
Distribute data across buckets
Often multiple items in same bucket (buckets might overflow)
Hash indices can be built along the same lines as what we discussed
The difference: they may be unclustered as well as clustered
Types:
Static
Extendible (requires directory to buckets; can split)
Linear (two levels, rotate through + split; bad with skew)
We won’t get into detail because of time, but see text

20. Making Use of the Data + Indices: Query Execution
Query plans & exec strategies
Basic principles
Standard relational operators
Querying XML

21. Making Use of the Data + Indices: Query Execution
Query plans & exec strategies
Basic principles
Standard relational operators
Querying XML

22. Query Plans Data-flow graph of relational algebra operators
Typically: determined by optimizer
Join PressRel.Symbol = EastCoast.CoSymbol
Join PressRel.Symbol = Clients.Symbol
Project CoSymbol
Select Client = “Atkins”
SELECT *
FROM PressRel p, Clients C
WHERE p.Symbol = c.Symbol AND c.Client = ‘Atkins’
AND c.Symbol IN
(SELECT CoSymbol FROM EastCoast)
Scan
PressRel
Scan
Clients
Scan EastCoast

23. Iterator-Based Query Execution
Execution begins at root
open, next, close
Propagate calls to children
May call multiple child nexts
Efficient scheduling & resource usage
Can you think of alternatives and their benefits?
Join PressRel.Symbol = EastCoast.CoSymbol
Join PressRel.Symbol = Clients.Symbol
Project CoSymbol
Select Client = “Atkins”
Scan
PressRel
Scan
Clients
Scan EastCoast

24. Execution Strategy Issues
Granularity & parallelism:
Pipelining vs. blocking
Materialization
Join PressRel.Symbol = EastCoast.CoSymbol
Join PressRel.Symbol = Clients.Symbol
Project CoSymbol
Select Client = “Atkins”
Scan
PressRel
Scan
Clients
Scan EastCoast

25. Basic Principles
Many DB operations require reading tuples, tuple vs. previous tuples, or tuples vs. tuples in another table
Techniques generally used:
Iteration: for/while loop comparing with all tuples on disk
Index: if comparison of attribute that’s indexed, look up matches in index & return those
Sort/merge: iteration against presorted data (interesting orders)
Hash: build hash table of the tuple list, probe the hash table
Must be able to support larger-than-memory data

26. Basic Operators One-pass operators: Multi-pass operators: Scan Select
Project
Multi-pass operators:
Join
Various implementations
Handling of larger-than-memory sources
Semi-join
Aggregation, union, etc.

27. 1-Pass Operators: Scanning a Table
Sequential scan: read through blocks of table
Index scan: retrieve tuples in index order
May require 1 seek per tuple! When?
Cost in page reads – b(T) blocks, r(T) tuples
b(T) pages for sequential scan
Up to r(T) for index scan if unclustered index
Requires memory for one block

28. 1-Pass Operators: Select (s)
Typically done while scanning a file
If unsorted & no index, check against predicate:
Read tuple
While tuple doesn’t meet predicate
Return tuple
Sorted data: can stop after particular value encountered
Indexed data: apply predicate to index, if possible
If predicate is:
conjunction: may use indexes and/or scanning loop above (may need to sort/hash to compute intersection)
disjunction: may use union of index results, or scanning loop

29. 1-Pass Operators: Project (P)
Simple scanning method often used if no index:
Read tuple
While tuple exists
Output specified attributes
Duplicate removal may be necessary
Partition output into separate files by bucket, do duplicate removal on those
If have many duplicates, sorting may be better
If attributes belong to an index, don’t need to retrieve tuples!