Introduction to Data Management. Lecture 15 (More About Indexing)
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1 Introduction to Data Management Lecture 15 (More About Indexing) Instructor: Mike Carey Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 1 Announcements v HW s and quizzes: HW #6 due (at 6 pm) We re now back in a Friday HW groove Beware of (real!) late penalties again v This week s meetings Discussion & quiz (were) as usual v Today s lecture plan: Deeper dive into B+ tree indexes Overview of (static) hash-based indexes (More on indexes in CS122C if desired J) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 2
2 B+ Tree: Most Widely Used Index! v Insert/delete at log F N cost; keep tree heightbalanced. (F = fanout, N = # leaf pages) v Minimum 50% occupancy (except for root). Each node contains d <= m <= 2d entries. The (mythical) d is called the order of the B+ tree. v Supports equality and range-searches efficiently. Index Entries (Direct search) Data Entries ("Sequence set") Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 3... Example B+ Tree v Search begins at root, and key comparisons direct the search to a leaf (as in ISAM). v Ex: Search for 5*, 15*, all data entries >= 24*,... Root * 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39* Based on the search for 15*, we know it is not in the tree! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 4
3 Example B+ Tree (a clarification) v Search begins at root, and key comparisons direct the search to a leaf (as in ISAM). v Ex: Search for 5*, 15*, all data entries >= 24*,... Root P All pointers here are page IDs! P1 P2 P3 P4 P5 2* 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39* Based on the search for 15*, we know it is not in the tree! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 5 Inserting a Data Entry into a B+ Tree v Find correct leaf L (by searching for the new k). v Put new data entry (k*, a.k.a. (k, I(k)) in leaf L. If L has enough space, done! (Most likely case!) Else, must split L (into L and a new node L2) Redistribute entries evenly and copy up middle key. Insert new index entry pointing to L2 into parent of L. v This can happen recursively. To split an index node, redistribute entries evenly but push up the middle key. (Contrast with leaf splits!) v Splits grow tree; root split increases its height. Tree growth: gets wider or one level taller at top. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 6
4 Inserting 8* into Example B+ Tree v Observe how minimum occupancy is guaranteed in both leaf and index pg splits. v Note difference between copyup and push-up; be sure you understand the reasons for this! 2* 3* 5* 7* 8* Entry to be inserted in parent node. (Note that 5 is s copied up and continues to appear in the leaf.) Entry to be inserted in parent node. (Note that 17 is pushed up and only appears once in the index. Contrast this with a leaf split.) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 7 Example B+ Tree After Inserting 8* * 3* 5* 7* 8* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39* v Notice that root was split, leading to increase in height. v In this example, could avoid split by redistributing entries; however, not usually done in practice. (Q: Why is that?) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 8
5 Deleting a Data Entry from a B+ Tree v Start at root, find leaf L where entry belongs. v Remove the entry. If L is still at least half-full, done! If L has only d-1 entries, Try to redistribute, borrowing from sibling (adjacent node with same parent as L). If re-distribution fails, merge L and sibling. v If merge occurred, must delete search-guiding entry (pointing to L or sibling) from parent of L. v Merge could propagate to root, decreasing height. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 9 Example Tree After (Inserting 8*, Then) Deleting 19* and 20* * 3* 5* 7* 8* 14* 16* 22* 24* 27* 29* 33* 34* 38* 39* v Deleting 19* is easy. v Deleting 20* is done with redistribution. Notice how middle key is copied up. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 10
6 ... And Then Deleting 24* v Must merge. v Observe toss of index entry (on right), and pull down of index entry (below) * 27* 29* 33* 34* 38* 39* * 3* 5* 7* 8* 14* 16* 22* 27* 29* 33* 34* 38* 39* Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 11 Example of Non-leaf Redistribution v New/different example B+ tree is shown below during deletion of 24* v In contrast to previous example, can redistribute entry from left child of root to right child * 3* 5* 7* 8* 14* 16* 17* 18* 20* 21* 22* 27* 29* 33* 34* 38* 39* Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 12
7 After Redistribution v Intuitively, entries are redistributed by pushing through the splitting entry in the parent node * 3* 5* 7* 8* 14* 16* 17* 18* 20* 21* 22* 27* 29* 33* 34* 38* 39* Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 13 Bulk Loading of a B+ Tree v If we have a large collection of records, and we want to create a B+ tree on some field, doing so by repeatedly inserting records is very slow. v Bulk Loading can be done much more efficiently! v Initialization: Sort all data entries, insert pointer to first (leaf) page in a new (root) page. Root Sorted pages of data entries; not yet in B+ tree 3* 4* 6* 9* 10* 11* 12* 13* 20* 22* 23* 31* 35* 36* 38* 41* 44* Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 14
8 Bulk Loading (Contd.) v Index entries for leaf pages always entered into rightmost index page just above leaf level. When this fills up, it splits. (A split may go up the right-most path to the root.) 6 Root 12 Data entry pages not yet in B+ tree 3* 4* 6* 9* 10* 11* 12* 13* 20* 22* 23* 31* 35* 36* 38* 41* 44* Root Data entry pages not yet in B+ tree v Much faster than repeated inserts! * 4* 6* 9* 10* 11* 12* 13* 20* 22* 23* 31* 35* 36* 38* 41* 44* Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 15 A Note on B+ Tree Order v (Mythical) order (d) concept replaced by physical space criterion in practice ( at least half-full ). Index pages can typically hold many more entries than leaf pages. Variable sized records and search keys mean that different nodes will contain different numbers of entries. Even with fixed length fields, multiple records with the same search key value (duplicates) can lead to variable-sized data entries. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 16
9 (Page Implementation Details) Q: What if you were to open up a B+ Tree page? Control info (e.g., level, # children, free space offset) Search key array (with possible on-page indirection for variablelength data, using offsets), or key/data array for non-leaf vs. leaf pages, respectively 0 Child pointer array, where pointer Level (1) = page id on disk! NumChildren (3) 8... Free offset (40) P3 Ralph Timothy Key 0 offset (32) Key 1 offset (37) 20 Child 1 page id (P0) Child 2 page id (P1) P0 P1 P2 Child 3 page id (P2) 32 Leaf 1 Leaf 2 Leaf 3 Key 0 ( Ralph ) not to 37 } scale... Key 1 ( Timothy ) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 17 A Brief Aside: Hash-Based Indexes v As for any index, 3 alternatives for data entries k*: Data record with key value k <k, rid of data record with search key value k> <k, list of rids of data records with search key k> Choice is orthogonal to the indexing technique! v Hash-based indexes are fast for equality selections. Cannot support range searches. v Static and dynamic hashing techniques exist; trade-offs similar to ISAM vs. B+ trees. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 18
10 Static Hashed Indexes v # primary pages fixed, allocated sequentially, never de-allocated; overflow pages if needed. v h(k) mod N = bucket (page) to which data entry with key k belongs. (N = # of buckets) h(k) mod N k Ex: Using N = 8 and h(k) = k (a bad h(k)...!) h N-1 0*, 24*, -- 9*, 17*, 81* 25*, --, -- 26*, 42*, -- 15*, 71*, 7* Primary bucket pages (Ex: Find data with key=25 à 25*) 23*, 31*, -- Overflow pages Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 19 Static Hashed Indexes (Cont d.) v Buckets contain data entries (like for ISAM or B+ trees) very similar to what we just looked at. v Hash function works on search key field of record r. Must distribute values over range 0... M-1. h(key) = (a * key + b) usually works fairly well. a and b are constants; lots known about how to tune h. v Long overflow chains can develop and degrade performance. Extendible and Linear Hashing: Dynamic techniques to fix this problem. (Take CS122c!) Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 20
11 Indexing Summary v Tree-structured indexes are ideal for rangesearches, also good for equality searches. v ISAM is a static structure. (Prehistoric B+ Tree!) Only leaf pages modified; overflow pages needed. Overflow chains can degrade performance unless size of data set and data distribution stay constant. v B+ tree is a dynamic structure. Inserts/deletes leave tree height-balanced; log F N cost. High fanout (F) means depth rarely more than 3 or 4. If K* s are data records, use keys (vs. RIDs) elsewhere! Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 21 Indexing Summary (Cont d.) v Bulk loading can be much faster than repeated inserts for creating a B+ tree on a large data set. v Most widely used index in database management systems, and outside DBMSs as well, because of its versatility. Also the most optimized (e.g., for loading, locking, recovery, incremental loads,...). v Other database indexes to be aware of: Hash-based (for exact match queries). R tree (for spatial indexing and queries). Inverted keyword (for text indexing and queries). Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 22
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