CPSC-608 Database Systems Fall 2010 Instructor: Jianer Chen Office: HRBB 315C Phone: Notes #7

B+Trees Support fast search Support range search Support dynamic changes Could be either dense or sparse A B+tree node: 2 pnpn knkn k2k2 p1p1 k1k1 p0p0 p2p2 ……

Root B+Tree Examplen=

Sample non-leaf to keysto keysto keys to keys < 5757  k<8181  k<95 

Sample leaf node: From non-leaf node to next leaf in sequence To record with key 57 To record with key 81 To record with key 85 5

Size of nodes:n+1 pointers n keys FIXED 6

Don’t want nodes to be too empty Use at least Non-leaf:  (n+1)/2  pointers Leaf:  (n+1)/2  pointers to data 7

Full nodeMin. node Non-leaf Leaf n=

B+tree rulestree of order n (1) All leaves at same lowest level (balanced tree) (2) Pointers in leaves point to records except for “sequence pointer” 9

(3) Number of pointers/keys for B+tree Non-leaf (non-root) n+1n  (n+1)/ 2   (n+1)/ 2  – 1 Leaf (non-root) n+1n Rootn+1n21 Max Max Min Min ptrs keys ptrs  data keys  (n+ 1) / 2  10 could be 1

Search in a B+tree Start from the root Search in a leaf block May not have to go to the data file 11

Pseudo Code for Search in a B+tree Search(ptr, k); \\ search a record of key value k in the subtree rooted at *ptr Case 1. *ptr is a leaf IF (k = k i ) for a key k i in *ptr THEN return(p i ); ELSE return(Null); Case 2. *ptr is not a leaf find a key k i in *ptr such that k i ≤ k < k i+1 ; return(Search(p i, k)); 12

Insert into B+tree (a) simple case –space available in leaf (b) leaf overflow (c) non-leaf overflow (d) new root 13

Pseudo Code for Insertion in a B+tree Insert(ptr, (k,p), (k',p')); \\ p is a pointer to a data record with key value k, which are inserted into the subtree rooted at \\ *ptr; p' is a pointer to a new brother of *ptr, if created, in which k' is the smallest key value; Case 1. *ptr is a leaf IF there is room in *ptr, THEN insert (k,p) into *ptr; return(k'=0, p'=Null); ELSE re-arrange the content in *ptr and (k,p) into (p 0, k 1, p 1,..., k n+1, p n+1 ); leave (p 0, k 1,..., p r-2, k r-1, p r-1 ) in *ptr; \\ r =  (n+1)/2  create a new leaf q; put (p r, k r+1,..., p n, k n+1, p n+1 ) in *q; return( k'= k r, p' = q ); Case 2. *ptr is not a leaf find a key k i in *ptr such that k i ≤ k < k i+1 ; Insert(p i, (k,p), (k",p")); IF (p" = Null) THEN return(k'=0, p'=Null); ELSE IF there is room in *ptr, THEN insert (k",p") into *ptr; return(k'=0, p'=Null); ELSE re-arrange the content in *ptr and (k",p") into (p 0, k 1, p 1,..., k n+1, p n+1 ); leave (p 0, k 1,..., p r-2, k r-1, p r-1 ) in *ptr; \\ r =  (n+1)/2  create a new leaf q; put (p r, k r+1,..., p n, k n+1, p n+1 ) in *q; return( k'= k r, p' = q ). 14

(a) Insert key = 32 n=

(a) Insert key = 32 n=

(b) Insert key = 7 n=

(b) Insert key = 7 n=

(c) Insert key = 160 n=

(c) Insert key = 160 n=

(d) New root, insert 45 n=

(d) New root, insert 45 n= new root

(a) Simple case (no example) (b) Coalesce with neighbor (sibling) (c) Re-distribute keys (d) Cases (b) or (c) at non-leaf Deletion from B+tree 23

(b) Coalesce with sibling –Delete n=4 24

(b) Coalesce with sibling –Delete n=

(c) Redistribute keys –Delete n=4 26

(c) Redistribute keys –Delete n=

(d) Non-leaf coalesce –Delete 37 n= new root 28

B+tree deletions in practice –Often, coalescing is not implemented –Too hard and not worth it! 29

Variation on B+tree: B-tree (no +) Idea: –Avoid duplicate keys –Have record pointers in non-leaf nodes 30

to record to record to record with K1 with K2 with K3 to keys to keys to keys to keys k3 K1 P1K2 P2K3 P3 31

B-tree examplen=

B-tree examplen= sequence pointers not useful now! 33

Tradeoffs: B-trees have faster lookup than B+trees  in B-tree, non-leaf & leaf different sizes  in B-tree, insertion and deletion more complicated  B+trees preferred! 34