Chapter 11: Indexing and Hashing - جامعة الملك سعود
INDEXING AND HASHING Database System Concepts - 6th Edition 11.1 Silberschatz, Korth and Sudarshan Basic Concepts Indexing mechanisms used to speed up access to desired data. E.g., author catalog in library Search Key - attribute to set of attributes used to look up records in a file. An index file consists of records (called index entries) of the form pointer
search-key Index files are typically much smaller than the original file Two basic kinds of indices: Ordered indices: search keys are stored in sorted order Hash indices: search keys are distributed uniformly across buckets using a hash function. Database System Concepts - 6th Edition 11.2 Silberschatz, Korth and Sudarshan Index Evaluation Metrics Access types supported efficiently. E.g.,
records with a specified value in the attribute or records with an attribute value falling in a specified range of values. Access time Insertion time Deletion time Space overhead Database System Concepts - 6th Edition 11.3
Silberschatz, Korth and Sudarshan Ordered Indices In an ordered index, index entries are stored sorted on the search key value. E.g., author catalog in library. Primary index: in a sequentially ordered file, the index whose search key specifies the sequential order of the file. Also called clustering index The search key of a primary index is usually but not necessarily the primary key. Secondary index: an index whose search key specifies an order different from the sequential order of the file. Also called
non-clustering index. Index-sequential file: ordered sequential file with a primary index. Database System Concepts - 6th Edition 11.4 Silberschatz, Korth and Sudarshan Dense Index Files Dense index Index record appears for every search-key value in the file. E.g. index on ID attribute of instructor relation Database System Concepts - 6th Edition 11.5 Silberschatz, Korth and Sudarshan Dense Index Files (Cont.) Dense index on dept_name, with instructor file sorted on
dept_name Database System Concepts - 6th Edition 11.6 Silberschatz, Korth and Sudarshan Sparse Index Files Sparse Index: contains index records for only some search-key values. Applicable when records are sequentially ordered on search-key To locate a record with search-key value K we: Find index record with largest search-key value < K
Search file sequentially starting at the record to which the index record points Database System Concepts - 6th Edition 11.7 Silberschatz, Korth and Sudarshan Sparse Index Files (Cont.) Compared to dense indices: Less space and less maintenance overhead for insertions and deletions. Generally slower than dense index for locating records. Good tradeoff: sparse index with an index entry for every block in file,
corresponding to least search-key value in the block. Database System Concepts - 6th Edition 11.8 Silberschatz, Korth and Sudarshan Secondary Indices Example Secondary index on salary field of instructor Index record points to a bucket that contains pointers to all the actual records with that particular search-key value. Secondary indices have to be dense Database System Concepts - 6th Edition 11.9 Silberschatz, Korth and Sudarshan Primary and Secondary Indices Indices offer substantial benefits when searching for records. BUT: Updating indices imposes overhead on database
modification --when a file is modified, every index on the file must be updated, Sequential scan using primary index is efficient, but a sequential scan using a secondary index is expensive Each record access may fetch a new block from disk Block fetch requires about 5 to 10 milliseconds, versus about 100 nanoseconds for memory access Database System Concepts - 6th Edition 11.10 Silberschatz, Korth and Sudarshan Multilevel Index If primary index does not fit in memory, access becomes expensive. Solution: treat primary index kept on disk as a sequential file
and construct a sparse index on it. outer index a sparse index of primary index inner index the primary index file If even outer index is too large to fit in main memory, yet another level of index can be created, and so on. Indices at all levels must be updated on insertion or deletion from the file. Database System Concepts - 6th Edition 11.11 Silberschatz, Korth and Sudarshan Multilevel Index (Cont.) Database System Concepts - 6th Edition
11.12 Silberschatz, Korth and Sudarshan Index Update: Deletion If deleted record was the only record in the file with its particular search-key value, the search-key is deleted from the index also. Single-level index entry deletion: Dense indices deletion of search-key is similar to file record deletion. Sparse indices if an entry for the search key exists in the index, it is deleted by replacing the entry in the index with the next search-key value in the file (in search-key order).
If the next search-key value already has an index entry, the entry is deleted instead of being replaced. Database System Concepts - 6th Edition 11.13 Silberschatz, Korth and Sudarshan Index Update: Insertion Single-level index insertion: Perform a lookup using the search-key value appearing in the record to be inserted. Dense indices if the search-key value does not appear in the index, insert it.
Sparse indices if index stores an entry for each block of the file, no change needs to be made to the index unless a new block is created. If a new block is created, the first search-key value appearing in the new block is inserted into the index. Multilevel insertion and deletion: algorithms are simple extensions of the single-level algorithms Database System Concepts - 6th Edition 11.14 Silberschatz, Korth and Sudarshan Secondary Indices Frequently, one wants to find all the records whose values in a certain field (which is not the search-key of the primary index) satisfy some condition. Example 1: In the instructor relation stored sequentially by
ID, we may want to find all instructors in a particular department Example 2: as above, but where we want to find all instructors with a specified salary or with salary in a specified range of values We can have a secondary index with an index record for each search-key value Database System Concepts - 6th Edition 11.15 Silberschatz, Korth and Sudarshan B+-Tree Index Files B+-tree indices are an alternative to indexed-sequential files. Disadvantage of indexed-sequential files performance degrades as file grows, since many overflow blocks get created. Periodic reorganization of entire file is required.
Advantage of B+-tree index files: automatically reorganizes itself with small, local, changes, in the face of insertions and deletions. Reorganization of entire file is not required to maintain performance. (Minor) disadvantage of B+-trees: extra insertion and deletion overhead, space overhead. Advantages of B+-trees outweigh disadvantages B+-trees are used extensively Database System Concepts - 6th Edition 11.16 Silberschatz, Korth and Sudarshan Example of B+-Tree Database System Concepts - 6th Edition 11.17 Silberschatz, Korth and Sudarshan B+-Tree Index Files (Cont.)
A B+-tree is a rooted tree satisfying the following properties: All paths from root to leaf are of the same length Each node that is not a root or a leaf has between n/2 and n children. A leaf node has between (n1)/2 and n1 values Special cases: If the root is not a leaf, it has at least 2 children. If the root is a leaf (that is, there are no other nodes in the tree), it can have between 0 and (n1) values. Database System Concepts - 6th Edition 11.18 Silberschatz, Korth and Sudarshan B+-Tree Node Structure Typical node
Ki are the search-key values Pi are pointers to children (for non-leaf nodes) or pointers to records or buckets of records (for leaf nodes). The search-keys in a node are ordered K1 < K2 < K3 < . . . < Kn1 (Initially assume no duplicate keys, address duplicates later) Database System Concepts - 6th Edition 11.19 Silberschatz, Korth and Sudarshan Leaf Nodes in B+-Trees Properties of a leaf node: For i = 1, 2, . . ., n1, pointer Pi points to a file record with search-key value Ki, If Li, Lj are leaf nodes and i < j, Lis search-key values are less
than or equal to Ljs search-key values Pn points to next leaf node in search-key order Database System Concepts - 6th Edition 11.20 Silberschatz, Korth and Sudarshan Non-Leaf Nodes in B+-Trees Non leaf nodes form a multi-level sparse index on the leaf nodes. For a non-leaf node with m pointers: All the search-keys in the subtree to which P1 points are less than K1 For 2 i n 1, all the search-keys in the subtree to which Pi points have values greater than or equal to Ki1 and less than Ki
All the search-keys in the subtree to which Pn points have values greater than or equal to Kn1 Database System Concepts - 6th Edition 11.21 Silberschatz, Korth and Sudarshan Example of B+-tree B+-tree for instructor file (n = 6) Leaf nodes must have between 3 and 5 values ((n1)/2 and n 1, with n = 6). Non-leaf nodes other than root must have between 3 and 6 children ((n/2 and n with n =6). Root must have at least 2 children. Database System Concepts - 6th Edition 11.22 Silberschatz, Korth and Sudarshan
Observations about B+-trees Since the inter-node connections are done by pointers, logically close blocks need not be physically close. The non-leaf levels of the B+-tree form a hierarchy of sparse indices. The B+-tree contains a relatively small number of levels Level below root has at least 2* n/2 values Next level has at least 2* n/2 * n/2 values .. etc. If there are K search-key values in the file, the tree height is no more than logn/2(K)
thus searches can be conducted efficiently. Insertions and deletions to the main file can be handled efficiently, as the index can be restructured in logarithmic time (as we shall see). Database System Concepts - 6th Edition 11.23 Silberschatz, Korth and Sudarshan Queries on B+-Trees (Cont.) If there are K search-key values in the file, the height of the tree is no more than logn/2(K). A node is generally the same size as a disk block, typically 4 kilobytes
With 1 million search key values and n = 100 and n is typically around 100 (40 bytes per index entry). at most log50(1,000,000) = 4 nodes are accessed in a lookup. Contrast this with a balanced binary tree with 1 million search key values around 20 nodes are accessed in a lookup above difference is significant since every node access may need a disk I/O, costing around 20 milliseconds Database System Concepts - 6th Edition 11.24 Silberschatz, Korth and Sudarshan Updates on B+-Trees: Insertion 1. Find the leaf node in which the search-key value would appear 2. If the search-key value is already present in the leaf node 1.
Add record to the file 2. If necessary add a pointer to the bucket. 3. If the search-key value is not present, then 1. add the record to the main file (and create a bucket if necessary) 2. If there is room in the leaf node, insert (key-value, pointer) pair in the leaf node 3. Otherwise, split the node (along with the new (key-value, pointer) entry) as discussed in the next slide. Database System Concepts - 6th Edition 11.25
Silberschatz, Korth and Sudarshan Updates on B+-Trees: Insertion (Cont.) Splitting a leaf node: take the n (search-key value, pointer) pairs (including the one being inserted) in sorted order. Place the first n/2 in the original node, and the rest in a new node. let the new node be p, and let k be the least key value in p. Insert (k,p) in the parent of the node being split. If the parent is full, split it and propagate the split further up. Splitting of nodes proceeds upwards till a node that is not full is found.
In the worst case the root node may be split increasing the height of the tree by 1. Result of splitting node containing Brandt, Califieri and Crick on inserting Adams Next step: insert entry with (Califieri,pointer-to-new-node) into parent Database System Concepts - 6th Edition 11.26 Silberschatz, Korth and Sudarshan B+-Tree Insertion B+-Tree before and after insertion of Adams Database System Concepts - 6th Edition 11.27 Silberschatz, Korth and Sudarshan B+-Tree Insertion B+-Tree before and after insertion of Lamport Database System Concepts - 6th Edition 11.28
Silberschatz, Korth and Sudarshan Insertion in B+-Trees (Cont.) Splitting a non-leaf node: when inserting (k,p) into an already full internal node N Copy N to an in-memory area M with space for n+1 pointers and n keys Insert (k,p) into M Copy P1,K1, , K n/2-1,P n/2 from M back into node N Copy Pn/2+1,K n/2+1,,Kn,Pn+1 from M into newly allocated node N
Insert (K n/2,N) into parent N Read pseudocode in book! Califieri Adams Brandt Califieri Crick Database System Concepts - 6th Edition Adams Brandt 11.29 Crick Silberschatz, Korth and Sudarshan Non-Unique Search Keys Alternatives to scheme described earlier Buckets on separate block (bad idea)
List of tuple pointers with each key Extra code to handle long lists Deletion of a tuple can be expensive if there are many duplicates on search key (why?) Low space overhead, no extra cost for queries Make search key unique by adding a record-identifier Extra storage overhead for keys
Simpler code for insertion/deletion Widely used Database System Concepts - 6th Edition 11.30 Silberschatz, Korth and Sudarshan B+-Tree File Organization Index file degradation problem is solved by using B+-Tree indices. Data file degradation problem is solved by using B+-Tree File Organization. The leaf nodes in a B+-tree file organization store records, instead of pointers.
Leaf nodes are still required to be half full Since records are larger than pointers, the maximum number of records that can be stored in a leaf node is less than the number of pointers in a nonleaf node. Insertion and deletion are handled in the same way as insertion and deletion of entries in a B+-tree index. Database System Concepts - 6th Edition 11.31 Silberschatz, Korth and Sudarshan B+-Tree File Organization (Cont.) Example of B+-tree File Organization
Good space utilization important since records use more space than pointers. To improve space utilization, involve more sibling nodes in redistribution during splits and merges Involving 2 siblings in redistribution (to avoid split / merge where possible) results in each node having at least 2n / 3 entries Database System Concepts - 6th Edition 11.32 Silberschatz, Korth and Sudarshan Other Issues in Indexing Record relocation and secondary indices If a record moves, all secondary indices that store record pointers have to be updated
Node splits in B+-tree file organizations become very expensive Solution: use primary-index search key instead of record pointer in secondary index Extra traversal of primary index to locate record Higher cost for queries, but node splits are cheap Add record-id if primary-index search key is non-unique Database System Concepts - 6th Edition 11.33 Silberschatz, Korth and Sudarshan Multiple-Key Access Use multiple indices for certain types of queries. Example:
select ID from instructor where dept_name = Finance and salary = 80000 Possible strategies for processing query using indices on single attributes: 1. Use index on dept_name to find instructors with department name Finance; test salary = 80000 2. Use index on salary to find instructors with a salary of $80000; test dept_name = Finance. 3. Use dept_name index to find pointers to all records pertaining to the Finance department. Similarly use index on salary. Take intersection of both sets of pointers obtained. Database System Concepts - 6th Edition 11.34 Silberschatz, Korth and Sudarshan Indices on Multiple Keys Composite search keys are search keys containing more than one attribute
E.g. (dept_name, salary) Lexicographic ordering: (a1, a2) < (b1, b2) if either a1 < b1, or a1=b1 and a2 < b2 Database System Concepts - 6th Edition 11.35 Silberschatz, Korth and Sudarshan Indices on Multiple Attributes Suppose we have an index on combined search-key (dept_name, salary). With the where clause where dept_name = Finance and salary = 80000 the index on (dept_name, salary) can be used to fetch only records that satisfy both conditions.
Using separate indices in less efficient we may fetch many records (or pointers) that satisfy only one of the conditions. Can also efficiently handle where dept_name = Finance and salary < 80000 But cannot efficiently handle where dept_name < Finance and balance = 80000 May fetch many records that satisfy the first but not the second condition Database System Concepts - 6th Edition 11.36 Silberschatz, Korth and Sudarshan Static Hashing
A bucket is a unit of storage containing one or more records (a bucket is typically a disk block). In a hash file organization we obtain the bucket of a record directly from its search-key value using a hash function. Hash function h is a function from the set of all search-key values K to the set of all bucket addresses B. Hash function is used to locate records for access, insertion as well as deletion. Records with different search-key values may be mapped to the same bucket; thus entire bucket has to be searched sequentially to locate a record.
Database System Concepts - 6th Edition 11.37 Silberschatz, Korth and Sudarshan Example of Hash File Organization Hash file organization of instructor file, using dept_name as key (See figure in next slide.) There are 10 buckets, The binary representation of the ith character is assumed to be the integer i. The hash function returns the sum of the binary representations of the characters modulo 10 E.g. h(Music) = 1 h(History) = 2
h(Physics) = 3 h(Elec. Eng.) = 3 Database System Concepts - 6th Edition 11.38 Silberschatz, Korth and Sudarshan Example of Hash File Organization Hash file organization of instructor file, using dept_name as key (see previous slide for details). Database System Concepts - 6th Edition 11.39 Silberschatz, Korth and Sudarshan Hash Functions Worst hash function maps all search-key values to the same bucket; this makes access time proportional to the number of search-key values in the file.
An ideal hash function is uniform, i.e., each bucket is assigned the same number of search-key values from the set of all possible values. Ideal hash function is random, so each bucket will have the same number of records assigned to it irrespective of the actual distribution of search-key values in the file. Typical hash functions perform computation on the internal binary representation of the search-key. For example, for a string search-key, the binary representations of all the characters in the string could be added and the sum modulo the number of buckets could be returned. . Database System Concepts - 6th Edition 11.40 Silberschatz, Korth and Sudarshan
Handling of Bucket Overflows Bucket overflow can occur because of Insufficient buckets Skew in distribution of records. This can occur due to two reasons: multiple records have same search-key value chosen hash function produces non-uniform distribution of key values Although the probability of bucket overflow can be reduced, it cannot be eliminated; it is handled by using overflow buckets.
Database System Concepts - 6th Edition 11.41 Silberschatz, Korth and Sudarshan Handling of Bucket Overflows (Cont.) Overflow chaining the overflow buckets of a given bucket are chained together in a linked list. Above scheme is called closed hashing. An alternative, called open hashing, which does not use overflow buckets, is not suitable for database applications. Database System Concepts - 6th Edition 11.42 Silberschatz, Korth and Sudarshan
Hash Indices Hashing can be used not only for file organization, but also for indexstructure creation. A hash index organizes the search keys, with their associated record pointers, into a hash file structure. Strictly speaking, hash indices are always secondary indices if the file itself is organized using hashing, a separate primary hash index on it using the same search-key is unnecessary. However, we use the term hash index to refer to both secondary index structures and hash organized files. Database System Concepts - 6th Edition 11.43
Silberschatz, Korth and Sudarshan Example of Hash Index hash index on instructor, on attribute ID Database System Concepts - 6th Edition 11.44 Silberschatz, Korth and Sudarshan Deficiencies of Static Hashing In static hashing, function h maps search-key values to a fixed set of B of bucket addresses. Databases grow or shrink with time. If initial number of buckets is too small, and file grows, performance will degrade due to too much overflows.
If space is allocated for anticipated growth, a significant amount of space will be wasted initially (and buckets will be underfull). If database shrinks, again space will be wasted. One solution: periodic re-organization of the file with a new hash function Expensive, disrupts normal operations Better solution: allow the number of buckets to be modified dynamically. Database System Concepts - 6th Edition 11.45 Silberschatz, Korth and Sudarshan Comparison of Ordered Indexing and Hashing Cost of periodic re-organization
Relative frequency of insertions and deletions Is it desirable to optimize average access time at the expense of worst-case access time? Expected type of queries: Hashing is generally better at retrieving records having a specified value of the key. If range queries are common, ordered indices are to be preferred In practice: PostgreSQL supports hash indices, but discourages use due to poor performance Oracle supports static hash organization, but not hash indices
SQLServer supports only B+-trees Database System Concepts - 6th Edition 11.46 Silberschatz, Korth and Sudarshan
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