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

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Albert Booker
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1 B Trees

2 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

3 Disk reads are done in blocks Example: One block read can retrieve 100 records

4 1,000,000 Records

5 Block 0 Block 1 Block Block ,000,000 Records

6 Block 0 Block 1 Block Block ,000,000 Records

7 10,000 Records Block 0 Block 1 Block Block ,000,000 Records

8 Block 0 Block 1 Block 99 10,000 Records Block 0 Block 1 Block Block ,000,000 Records

9 Block 0 Block 1 Block 99 10,000 Records Block 0 Block 1 Block Block ,000,000 Records

10 Block Records Block 0 Block 1 Block 99 10,000 Records Block 0 Block 1 Block Block ,000,000 Records

11 DEF: A B Tree of order m is an m way tree such that 1. All leaf nodes are at the same level. 2. All non leaf nodes (except the root) have at most m and at least m/2 children. 3. 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. 4. The root may have as few as 2 children unless the tree is the root alone.

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

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

DS L19: B Trees

DS L19: B Trees 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.in Slides