Remember. 376a. Database Design. Also. B + tree reminders. Algorithms for B + trees. Remember

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1 376a. Database Design Dept. of Computer Science Vassar College Class 14 B + trees, multi-key indices, partitioned hashing and grid files B and B + -trees are used one implementation for dynamic, multilevel indices. B and B + -trees are one method for maintaining uniform access insert and delete. B-trees keep keys (data) in nodes and leaves B + -trees keep keys (data) only in leaves. Node size is selected to fit in a disk block (why?) 1 2 B + tree reminders Also Leaf nodes are usually linked together for fast sequential access. Each internal node is order p. Each internal node has the following structure <P 1, K 1, P 2, K 2, P q-1, K q-1, P q >. Subtree P i has values < K i-1 and [ K i Each node is at least ½ full. Each leaf node has the following structure <<K 1, Pr 1 >, <K 2,Pr 2 >..<K q-1, Pr q-1 > P next > K i is the key. K i < K i+1 <K i+2 Pr i is a pointer to the record or block containing the record. P next is a pointer to the next leaf node All leaf nodes are at the same level and are at least ½ full. 3 4 Algorithms for B + trees If K is a non-key field, P points to a list of records which share that key field. Search Insert Deletion 5 6 and Procedural Abstraction 1

2 Search Insert N r root block while (N is not a leaf node) { if x[k 1 then N r P 1 else if x > k q-1 then N rp q else search N for i s.t. k i-1 <x [ k i then N r Pi } // search internal nodes to leaf Search leaf node for key. If found, return record. Initially root node is also a leaf node. Insertion is linear with # of key slots. Only problem is on over flow. 7 Example Answer Insert 3, 7,, 9, 10, 4, 5, 6 into block with 4 slots. 3 *3 7 *7 * 9 *9 Split block, and propagate split to parent Insertion of 10 causes overflow 3 *3 7 *7 * 9 10 Continue On split, largest key from left block is propagated up to root. If # of cells in leaf node is even, split at n/2+1 otherwise split at (n+1)/2. DO NOT DELETE this value! All keys must be present in leaf nodes. Insert 4 3 *3 4 *4 7 *7 * and Procedural Abstraction 2

3 Continue Try insert Insert , 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6., 6.9 Notice how root node splits 7 *7 * 3 *3 4 *4 5 * Delete is a little more tricky Try delete 9 and 10 Delete from the leaf node and propagate up the tree. If it occurs in an internal node, replace it with the immediate left most node. If underflow, (# of nodes is less than required minimum), merge with left or right sibling node and redistribute so both are at least half full. (Try left first) Try delete 9, 10 Delete 9 causes underfull ripple Variations in B + trees But what happens B*-tree each node is at least 2/3rds full. All trees are determined by two factors Fill factor of internal nodes Fill factor of leaf nodes These two do not need to be the same. If the search key is not on a single attribute? 17 1 and Procedural Abstraction 3

4 Indices on multiple keys Set up keys on multiple attributes For example: EMPLOYEE(Ssn, Name, Age, Dno, Salary ) Find all employees in Dno=4 and Age=42 Neither Dno or age are keys in EMPLOYEE Several strategies for searching Assume index on Dno. Find employees with Dno=4 using index and select among them for Age=42. Assume index on Age. (Do same as above) If indices on both, find intersection between records from both. Only efficient if one of the two criterion limits number of records significantly Composite keys Partitioned hashing Keys containing multiple attributes. Ordered index on multiple attibutes. Assuming attributes are <A 1, A 2, A 3, A n > Attributes are ordered by left most attribute first, then next leftmost, etc. Numbers are increasing. Strings are lexicographically sorted. For example <Dno, Age>: <4, 34><4,52><5,19> Extension of static external hashing. Can only do equality testing. For a key with n-attributes Each attribute is hashed separately. The bucket address is formed by concatenating the results of the individual hashes. For example, for Age and Dno, Age can be hashed into 4 bits and Dno into 4 bits Partitioned hashing cont. Grid files Try h(4)=0x4 and h(59) =0xb Employees matching are in bin (0x4b) To find employees in Department #4, search buckets 0x4_ (16 buckets). Benefits: easily extends to n-attributes. Drawback: can t handle ranges on any attributes. Create an array with each different attribute assigned a different dimension. For example: if there are 5 attributes, use a 5 dimensional array. Each element in the array points to a bucket pool. Axes are normally scaled (otherwise, we need dense coverage.) and Procedural Abstraction 4

5 Grid files example Grid files Good for fixed data. < >60 1,2,7 3,4 5,10 6, Bucket Pool Problems Space for setup. Dynamic data requires frequent reorganization and maintenance Other kinds of indices Logical indices All previous examples assumed physical indices (pointers to physical disk blocks). Problems with physical indices: Reorg/Insert/Delete moves records. All pointers need to be revalidated. Need to traverse all other indices (secondary) and apply same fix. Provide another level of indirection. Associate <K, Kp> where K is the logical address and Kp is the physical address. Requires one more lookup operation but very useful when physical record address change frequently and Procedural Abstraction 5

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