DS L19: B Trees

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Indian Institute of Science Bangalore, India भ रत य व ज ञ न स स थ न ब गल र, भ रत Department of Computational and Data Sciences DS286 2016-10-21 L19: B Trees Yogesh Simmhan simmhan@cds.iisc.ac.

1 Indian Institute of Science Bangalore, India भ रत य व ज ञ न स स थ न ब गल र, भ रत Department of Computational and Data Sciences DS L19: B Trees Yogesh Simmhan simmhan@cds.iisc.ac.in Slides courtesy: Department of Computational and Data Science, IISc, 2016 This work is licensed under a Creative Commons Attribution 4.0 International License Copyright for external content used with attribution is retained by their original authors CDS Department of Computational and Data Sciences

2 CDS.IISc.ac.in Department of Computational and Data Sciences Searching External Storage Main memory (RAM) is fast, but has limited capacity Different considerations for in-memory vs. on-disk data structures for search Problem: Database too big to fit memory Disk reads are slow Example: 1,000,000 records on disk Binary search might take 20 disk reads log2(1m) ~= Oct-16 2

3 CDS.IISc.ac.in Department of Computational and Data Sciences Searching External Storage But disks are accessed block at a time by OS Blocks are typically 1KiB 4KiB in size Can span multiple sectors on HDD Access time per block ~12ms for HDD <1ms for SSD Say 1KiB block, 100B per record 10,000 blocks for 1M records 21-Oct-16 3

4 CDS.IISc.ac.in Department of Computational and Data Sciences Storing & Searching on External Storage 21-Oct-16 4

5 CDS.IISc.ac.in Department of Computational and Data Sciences Storing & Searching on External Storage 21-Oct-16 5

6 CDS.IISc.ac.in Department of Computational and Data Sciences Storing & Searching on External Storage 21-Oct-16 6

7 CDS.IISc.ac.in Department of Computational and Data Sciences Storing & Searching on External Storage 21-Oct-16 7

8 CDS.IISc.ac.in Department of Computational and Data Sciences Storing & Searching on External Storage 21-Oct-16 8

9 CDS.IISc.ac.in Department of Computational and Data Sciences B-Trees Data structures for external memory, not main memory Goal is to reduce number of block accesses, not number of comparisons B-Trees Proposed by R. Bayer and E. M. McCreigh in Bayer, Balanced, Bushy, Boeing trees? Different from binary trees NOTE: In-memory data structure will be better than on-disk milliseconds vs. nano seconds So in-memory binary tree will be better than on-disk B Tree But on-disk B Tree better than on-disk binary tree 21-Oct-16 9

10 CDS.IISc.ac.in Department of Computational and Data Sciences B-Trees Definition: A B-Tree of order m is an m-way tree with All leaf nodes are at the same level. All non-leaf nodes (except the root) have at most m and at least m/2 children. The number of keys is one less than the number of children for non-leaf nodes and at most m-1 and at least m/2 for leaf nodes. The root may have as few as 2 children unless the tree is the root alone. 21-Oct-16 10

11 CDS.IISc.ac.in Department of Computational and Data Sciences B Tree of order 5 A B- Tree of order 5 is an 5-way tree such that All leaf nodes are at the same level. All non- leaf nodes (except the root) have at most 5 and at least 2 children The number of keys is one less than the number of children for non-leaf nodes and at most 4 and at least 2 for leaf nodes The root may have as few as 2 children unless the tree is the root alone. 21-Oct-16 11

12 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P

13 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P A B F G

14 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P A B F G K

15 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F A B G K

16 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F A B D G H K M

17 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F A B D G H J K M

18 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F J A B D G H K M

19 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F J A B D E G H I K M R S

20 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F J A B D E G H I K M R S X

21 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F J R A B D E G H I K M S X

22 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P F J R A B C D E G H I K M S X

23 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P C F J R A B D E G H I K M S X

24 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P C F J R A B D E G H I K L M N S T U X

25 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P C F J R A B D E G H I K L M N P S T U X

26 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P C F J M R A B D E G H I K L N P S T U X

27 CreaTng a B- tree of order 5 A G F B K D H M J E S I R X C L N T U P J C F M R A B D E G H I K L N P S T U X

28 DeleTng Nodes Delete E from leaf node J C F M R A B D E G H I K L N P S T U X

29 DeleTng Nodes Delete E J C F M R A B D G H I K L N P S T U X

30 DeleTng Nodes Borrow from a neighbor J C G M R A B D F H I K L N P S T U X

31 DeleTng Nodes Delete F but can t borrow from a neighbor J C G M R A B D H I K L N P S T U X

32 DeleTng Nodes Combine and push the problem up one level J C M R A B D G H I K L N P S T U X

33 DeleTng Nodes Can t borrow so combine C J M R A B D G H I K L N P S T U X

34 DeleTng Nodes Delete M from non- leaf node Note: immediate predecessor in non- leaf Is always in a leaf. C J M R A B D G H I K L N P S T U X

35 DeleTng Nodes Delete M from non- leaf node Overwrite M with immediate predecessor C J L R A B D G H I K N P S T U X

36 DeleTng Nodes Borrow from a neighbor C I L R A B D G H J K N P S T U X

37 CDS.IISc.ac.in Department of Computational and Data Sciences Practical considerations Since each node correspond to a block/page, their size is usually a power of 2. The number of records in a node is therefore usually even. The number of children (the order of the tree) is therefore usually odd. B -trees

38 CDS.IISc.ac.in Department of Computational and Data Sciences Efficiency of B-trees If a B-tree has order d, then each node (apart from the root) has at least d/2 children. So the depth of the tree is at most log d/2 (size)+1. In the worst case have to make d-1 comparisons in each node (linear search, but d-1 is a constant factor). Fewer disk accesses than for a binary tree. Each node could correspond to one block of data (plus addresses of successor nodes). B -trees

39 CDS.IISc.ac.in Department of Computational and Data Sciences Variant of B-trees Only leaves contain data Non-leaves contain only keys and block numbers Access speed is higher because each block can hold more block numbers. Programming is more complicated because there are two kinds of nodes: leaves and non-leaves. B -trees

40 CDS.IISc.ac.in Department of Computational and Data Sciences Indexing Index: list of key/block pairs. If small enough, could be kept in the main memory. If the index is too large to be loaded in the main memory, it is often kept in a B-tree. Multiple index files: if different kinds of search have to be performed, can keep one index for every set of keys. B -trees

41 CDS.IISc.ac.in Department of Computational and Data Sciences 21-Oct

42 CDS.IISc.ac.in Department of Computational and Data Sciences Tasks Self study Read: B Trees (online sources) Finish Assignment 4 by Wed Oct 26 (75 points) Make progress on CodeChef (100 points) 21-Oct-16 17

43 CDS.IISc.ac.in Department of Computational and Data Sciences Questions? Department of Computational and Data Science, IISc, 2016 This work is licensed under a Creative Commons Attribution 4.0 International License Copyright for external content used with attribution is retained by their original authors 21-Oct Department of Computational and Data Sciences

Problem: Data base too big to fit memory Disk reads are slow. Example: 1,000,000 records on disk Binary search might take 20 disk reads

B Trees Problem: Data base too big to fit memory Disk reads are slow Example: 1,000,000 records on disk Binary search might take 20 disk reads Disk reads are done in blocks Example: One block read can