1、,B+-Trees and Static Hashing,Chapter 9 and 10.1,Introduction Indexing Techniques,As for any index, 3 alternatives for data entries k*:Data record with key value kHash-based indexes are best for equality selections. Cannot support range searches. B+-trees are best for sorted access and range queries.
2、,Static Hashing,# primary pages fixed, allocated sequentially, never de-allocated; overflow pages if needed. h(k) mod M = bucket to which data entry with key k belongs. (M = # of buckets),h(key) mod N,h,key,Primary bucket pages,Overflow pages,2,0,N-1,Static Hashing (Contd.),Buckets contain data entr
3、ies. Hash fn works on search key field of record r. Must distribute values over range 0 . M-1. h(key) = (a * key + b) usually works well. a and b are constants; lots known about how to tune h. Long overflow chains can develop and degrade performance. Two approaches: Global overflow area Individual o
4、verflow areas for each bucket (assumed in the following) Extendible and Linear Hashing: Dynamic techniques to fix this problem.,Range Searches,Find all students with gpa 3.0 If data is in sorted file, do binary search to find first such student, then scan to find others. Cost of binary search can be
5、 quite high. Simple idea: Create an index file.,Can do binary search on (smaller) index file!,Page 1,Page 2,Page N,Page 3,Data File,k2,kN,k1,Index File,B+ Tree: The Most Widely Used Index,Insert/delete at log F N cost; keep tree height-balanced. (F = fanout, N = # leaf pages) Minimum 50% occupancy (
6、except for root). Supports equality and range-searches efficiently.,Example B+ Tree (order p=5, m=4),Search begins at root, and key comparisons direct it to a leaf (as in ISAM). Search for 5*, 15*, all data entries = 24* .,Based on the search for 15*, we know it is not in the tree!,Root,16,22,29,2*,
7、3*,5*,7*,14*,16*,19*,20*,22*,24*,27*,29*,33*,34*,38*,39*,7,p=5 because tree can have at most 5 pointers in intermediate node; m=4 because at most 4 entries in leaf node.,B+ Trees in Practice,Typical order: 200. Typical fill-factor: 67%. average fanout = 133 Typical capacities: Height 4: 1334 = 312,9
8、00,700 records Height 3: 1333 = 2,352,637 records Can often hold top levels in buffer pool: Level 1 = 1 page = 8 Kbytes Level 2 = 133 pages = 1 Mbyte Level 3 = 17,689 pages = 133 MBytes,Inserting a Data Entry into a B+ Tree,Find correct leaf L. Put data entry onto L. If L has enough space, done! Els
9、e, must split L (into L and a new node L2) Redistribute entries evenly, copy up middle key. Insert index entry pointing to L2 into parent of L. This can happen recursively To split index node, redistribute entries evenly, but push up middle key. (Contrast with leaf splits.) Splits “grow” tree; root
10、split increases height. Tree growth: gets wider or one level taller at top.,Inserting 8* into Example B+ Tree,Observe how minimum occupancy is guaranteed in both leaf and index pg splits. Note difference between copy-up and push-up; be sure you understand the reasons for this.,2*,3*,5*,7*,8*,5,Entry
11、 to be inserted in parent node.,(Note that 5 is,continues to appear in the leaf.),s copied up and,appears once in the index. Contrast,Example B+ Tree After Inserting 8*,Notice that root was split, leading to increase in height.,In this example, we can avoid split by re-distributing entries; however,
12、 this is usually not done in practice.,2*,3*,Root,16,22,29,14*,16*,19*,20*,22*,24*,27*,29*,33*,34*,38*,39*,8,3,7*,5*,8*,Deleting a Data Entry from a B+ Tree,Start at root, find leaf L where entry belongs. Remove the entry. If L is at least half-full, done! If L has only d-1 entries, Try to re-distri
13、bute, borrowing from sibling (adjacent node with same parent as L). If re-distribution fails, merge L and sibling. If merge occurred, must delete entry (pointing to L or sibling) from parent of L. Merge could propagate to root, decreasing height.,Example Tree After (Inserting 8*, Then) Deleting 19*
14、and 20* .,Deleting 19* is easy. Deleting 20* is done with re-distribution. Notice how middle key is copied up.,2*,3*,Root,16,29,14*,16*,33*,34*,38*,39*,8,3,7*,5*,8*,22*,24*,24,27*,29*,. And Then Deleting 24*,Must merge. Observe toss of index entry (on right), and pull down of index entry (below).,29
15、,22*,27*,29*,33*,34*,38*,39*,2*,3*,7*,14*,16*,22*,27*,29*,33*,34*,38*,39*,5*,8*,Root,29,8,3,16,Example of Non-leaf Re-distribution,Tree is shown below during deletion of 24*. (What could be a possible initial tree?) In contrast to previous example, can re-distribute entry from left child of root to
16、right child.,Root,8,3,16,18,21,29,After Re-distribution,Intuitively, entries are re-distributed by pushing through the splitting entry in the parent node. It suffices to re-distribute index entry with key 20; weve re-distributed 17 as well for illustration.,14*,16*,33*,34*,38*,39*,22*,27*,29*,17*,18
17、*,20*,21*,7*,5*,8*,2*,3*,Root,8,3,16,29,18,21,Clarifications B+ Tree,B+ trees can be used to store relations as well as index structures In the drawn B+ trees we assume (this is not the only scheme) that an intermediate node with q pointers stores the maximum keys of each of the first q-1 subtrees i
18、t is pointing to; that is, it contains q-1 keys. Before B+-tree can be generated the following parameters have to be chosen (based on the available block size; it is assumed one node is stored in one block): the order p of the tree (p is the maximum number of pointers an intermediate node might have
19、; if it is not a root it must have between round(p/2) and p pointers) the maximum number m of entries in the leaf node can hold (in general leaf nodes (except the root) must hold between round(m/2) and m entries) Intermediate nodes usually store more entries than leaf nodes,Summary B+ Tree,Most wide
20、ly used index in database management systems because of its versatility. One of the most optimized components of a DBMS. Tree-structured indexes are ideal for range-searches, also good for equality searches (log F N cost). Inserts/deletes leave tree height-balanced; log F N cost. High fanout (F) means depth rarely more than 3 or 4. Almost always better than maintaining a sorted file Self reorganizing (dynamic data structure) Typically 67%-full pages at an average,
