EECE 330 Chapter, Fall 2011

Indexing Goals: – Store large files – Support multiple search keys – Support efficient insert, delete, and range queries Terms(1) Entry sequenced file: records ordered by time of insertion. – Search with sequential search Index file: stores pointers to actual records. – Could be organized with a tree or other data structure. Terms(2) Primary Key: A unique identifier for records. May be inconvenient for search. example: ID Secondary Key: An alternate search key, often not unique for each record. Often used for search key. example: last name Linear Indexing Linear index: Index file organized as a simple sequence of key/record pointer pairs with key values are in sorted order. Linear indexing is good for searching variable-length records. Linear Indexing (2) If the index is too large to fit in main memory, a second-level index might be used. Secondary Indices Secondary indices, such as last names, point to primary indices, which in turn point to full records. Multiple records with same secondary indices duplicate values within the index One solution is the 2-dimensional linear index Barakat AA10 AB12 AB39 FD37 Ayyash AX33 AX35 VC40 Karaki XA20 Secondary Indices (2) With 2-dimensional linear index, there is no duplication and the cost of insertion/deletion is reduced; A new row must be added when a new secondary key is added Array size is fixed which poses an upper limit There is wasted space (empty table cells) Alternative: Inverted Lists Inverted Lists Ayyash Barakat Karaki AA10 AB12 AB39 FD37 AX33 AX35 VC40 XA20 Secondary Key Primary Key Ayyash 0 Barakat 6 Karaki 3 AA10 AX35 VC40 XA20 AB12 AB39 AX33 FD37 Secondary Key Primary Key 4 2 5 7 1 Index Next Better Approach Tree Indexing (1) Linear index is poor for insertion/deletion. Tree index can efficiently support all desired operations: – Insert/delete – Multiple search keys (multiple indices) – Key range search Tree Indexing (2) Difficulties when storing tree index on disk: – Tree must be balanced. – Each path from root to leaf should cover few disk pages. 2-3 Tree (1) A 2-3 Tree has the following properties: 1. A node contains one or two keys 2. Every internal node has either two children (if it contains one key) or three children (if it contains two keys). 3. All leaves are at the same level in the tree, so the tree is always height balanced. The 2-3 Tree has a search tree property analogous to the BST. 2-3 Tree (2) The advantage of the 2-3 Tree over the BST is that it can be updated at low cost. 2-3 Tree Insertion (tree grows upward) -Find the leaf node that would contain the key (if it were in the tree) -If that node (L) contains 1 value The new key can just be added without any modifications -Else More space must be created: - Split this node (L) into two nodes - Now we have 3 keys: The new key and the two keys of L - Left most node receives smallest key - New node receives largest key - Middle key is passed up to the parent with a pointer to new node - If parent has only one key promoted key gets added to parent - Else li d d i 2-3 Tree Insertion (1) Insert 14 Case 1: available slot 2-3 Tree Insertion (2) Insert 55 55 should go here Case 2: full leaf node, available slot in parent 2-3 Tree Insertion (3) Insert 19 in tree – 2 slides earlier Split the node 20|21 and promote 20: Parent (23|30) is full  split and promote 23 Parent (18|33) is full  split and promote 23: No Parent  Create one B-Trees (1) The B-Tree is an extension of the 2-3 Tree. The B-Tree is now the standard file organization for applications requiring insertion, deletion, and key range searches. B-Trees (2) 1. B-Trees are always balanced. 2. B-Trees keep similar-valued records together on a disk page, which takes advantage of locality of reference. 3. B-Trees guarantee that every node in the tree will be full at least to a certain minimum percentage. This improves space efficiency while reducing the typical number of disk fetches necessary during a search or update operation. B-Tree Definition A B-Tree of order m has these properties: – The root is either a leaf or has two children. – Each node, except for the root and the leaves, has between m/2 and m children. – All leaves are at the same level in the tree, so the tree is always height balanced. A B-Tree node is usually selected to match the size of a disk block. – A B-Tree node could have hundreds of children. B-Tree Search (1) Search in a B-Tree is a generalization of search in a 2-3 Tree. 1. Do binary search on keys in current node. If search key is found, then return record. If current node is a leaf node and key is not found, then report an unsuccessful search. 2. Otherwise, follow the proper branch and repeat the process. B+-Trees The most commonly implemented form of the B- Tree is the B+-Tree. Internal nodes of the B+-Tree do not store record -- only values to guide the search. Leaf nodes store records or pointers to records. A leaf node may store more or less records than an internal node stores values. B+-Tree Example B+-Tree Insertion B+-Tree Deletion (1) B+-Tree Deletion (2) B+-Tree Deletion (3) B-Tree Space Analysis (1) B+-Trees nodes are always at least half full. The B*-Tree splits two pages for three, and combines three pages into two. In this way, nodes are always 2/3 full. Asymptotic cost of search, insertion, and deletion of nodes from B-Trees is (log n). – Base of the log is the (average) branching factor of the tree. B-Tree Space Analysis (2) Example: Consider a B+-Tree of order 100 with leaf nodes containing 100 records. 1 level B+-tree: 2 level B+-tree: 3 level B+-tree: 4 level B+-tree: Ways to reduce the number of disk fetches: – Keep the upper levels in memory. – Manage B+-Tree pages with a buffer pool.