CPSC-608 Database Systems Fall 2011 Instructor: Jianer Chen Office: HRBB 315C Phone: Notes #10

secondary storage (disks) in tables (relations) database administrator DDL language database programmer DML (query) language DBMS file manager buffer manager main memory buffers index/file manager DML complier DDL complier query execution engine transaction manager concurrency control lock table logging & recovery graduate database

The Main Purpose of Index Structures 3

Speedup the search process 4 index σ a=6 (R) blocks contianing the desired tuples quickly figure out disks

The Main Purpose of Index Structures Speedup the search process 5 index σ a=6 (R) blocks contianing the desired tuples quickly figure out disks otherwise have to scan the entire R

The Main Purpose of Index Structures Speedup the search process 6 index σ a=6 (R) blocks contianing the desired tuples quickly figure out disks otherwise have to scan the entire R But also need to handle dynamic changes of R

B+Trees Support fast search Support range search Support dynamic changes Could be either dense or sparse dense: pointers to all records sparse: one pointer per block A B+tree node (of order n): 7 pnpn knkn k2k2 p1p1 k1k1 p0p0 p2p2 ……

root B+Tree Exampleorder n =

Sample non-leaforder n = 3 to keys to keysto keys to keys < 5757  k <  k < 95 

Sample leaf nodeorder n = 3 10 To record With key 57 To record With key 81 To record With key From non-leaf node To next leaf in sequence (in our algorithm, assume this is p 0 )

A B+ Tree of order n Each node has:n+1 pointers n keys These are fixed 11

Don’t want nodes to be too empty Use at least Non-leaf:  (n+1)/2  pointers (to children) Leaf:  (n+1)/2  pointers to data (plus a “sequence pointer” to the next leaf) 12

Full nodeMin. node Non-leaf Leaf Example (B+ tree of order n=3)

B+tree rules Rule 1. All leaves are at same lowest level (balanced tree) Rule 2. Pointers in leaves point to records except for “sequence pointer” 14

Rule 3. Number of pointers/keys for B+tree Non-leaf (non-root) n+1 n  (n+1)/ 2   (n+1)/ 2  – 1 Leaf (non-root) n+1 n Rootn+1 n 21 Max Max Min Min ptrs keys ptrs  data keys  (n+ 1) / 2  15 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 16

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 \\ p n is the sequence pointer IF (k = k i ) for a key k i in *ptr THEN return(p i-1 ); 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)); 17

Range Search in B+tree To research all records whose key values are between k 1 and k 2 : 18

Range Search in B+tree To research all records whose key values are between k 1 and k 2 : Step 1. Call Search(ptr, k 1 ); 19

Range Search in B+tree To research all records whose key values are between k 1 and k 2 : Step 1. Call Search(ptr, k 1 ); Step 2. Follow the sequence pointers until the search key value is larger than k 2. 20

Insert into B+tree (a) simple case (space available in leaf) (b) leaf overflow (c) non-leaf overflow (d) new root 21

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 = (p 0, k 1, p 1,..., k i, p i ) is a leaf \\ p 0 is the sequence pointer IF (i < n) THEN insert (k,p) into *ptr; return(k'=0, p'=Null); ELSE re-arrange (k 1, p 1 ),..., (k n, p n ), and (k, p) into (k 1, p 1,..., k n+1, p n+1 ); create a new leaf q; put (p 0, k r+1, p r+1, …, k n+1, p n+1 ) in *q; \\ r =  (n+1)/2  put (q, k 1, p 1,..., p r-2, k r, p r ) in *ptr; \\ p 0 and q are sequence pointers in *q and *ptr, resp. IF *ptr is the root THEN create a new root with children ptr and q (and key k r ) ELSE return( k‘ = k r+1, 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; IF *ptr is the root THEN create a new root with children ptr and q (and key k r ) ELSE return( k'= k r, p' = q ); 22

(a) Insert key = 32 order n=

(a) Insert key = 32 order n= < 100

(a) Insert key = 32 order n= < ≥ 30

(a) Insert key = 32 order n= < ≥ 30 There is room for 32

(a) Insert key = 32 order n=

(b) Insert key = 7 order n=

(b) Insert key = 7 order n=

(b) Insert key = 7 order n=

(b) Insert key = 7 order n=

(b) Insert key = 7 order n=

(c) Insert key = 160 order n=

(c) Insert key = 160 order n=

(c) Insert key = 160 order n=

(c) Insert key = 160 order n=

(c) Insert key = 160 order n=

(d) New root, insert 45 order n=

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

(a)Simple case (b)Re-distribute keys among siblings (c) Coalesce with a sibling and delete a pointer from its father (d) Cases (b) or (c) at non-leaf Deletion from B+tree 40

(a) Simple case –Delete order n=4 41

(a) Simple case –Delete order n=4 42

(b) Coalesce with sibling –Delete order n=4 43

(b) Coalesce with sibling –Delete order n=

(c) Redistribute keys –Delete order n=4 45

(c) Redistribute keys –Delete order n=

(d) Non-leaf coalesce –Delete 37 order n= new root 47

B+tree deletions in practice –Often, coalescing is not implemented –Too hard and not worth it! 48

Variation on B+tree: B-tree (no “+”) Idea: –Avoid duplicate keys –Have record pointers in non-leaf nodes 49

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 50

B-tree examplen=

B-tree examplen= sequence pointers not useful now! 52

B-tree examplen= sequence pointers not useful now! 53

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! 54