Trees • Binary Trees • Newer Types of Tree Structures • M-Ary Trees

Data Organization and Processing Hierarchical Indexing (NDBI007) David Hoksza <a href="/goto?url=http://siret.ms.mff.cuni.cz/hoksza" target="_blank">http://siret.ms.mff.cuni.cz/hoksza </a>1Outline • prefix tree • … • Background • graphs • search trees • binary trees • m-ary trees • Newer types of tree structures • Typical tree type structures • B-tree • B+-tree • B*-tree 2Motivation • Similarly as in hashing, the motivation is to find record(s) given a query key using only a few operations • unlike hashing, trees allow to retrieve set of records with keys from given range • Tree structures use “clustering” to efficiently filter out non relevant records from the data set • Tree-based indexes practically implement the index file data organization • for each search key, an indexing structure can be maintained 3Drawbacks of Index(-Sequential) Organization • Static nature • when inserting a record at the beginning of the file, the whole index needs to be rebuilt • an overflow handling policy can be applied → the performance degrades as the file grows • reorganization can take a lot of time, especially for large tables • not possible for online transactional processing (OLTP) scenarios (as opposed to OLAP – online analytical processing) where many insertions/deletions occur 4Tree Indexes • Most common dynamic indexing structure for external memory • When inserting/deleting into/from the primary file, the indexing structure(s) residing in the secondary file is modified to accommodate the new key • The modification of a tree is implemented by splitting/merging nodes • Used not only in databases • NTFS directory structure is built over B+ tree 5Trees • root • Undirected graph without cycles • is the only node without parent – root of the hierarchy • we will be interested in rooted trees → one node designated as the root → orientation • leaves • nodes without children • inner nodes • nodes with children • Nodes are in parent-child relation • branch • every parent has a finite set of descendants (children nodes) • from a parent an edge points to child • path from a leaf to the root • every child has exactly one parent 6<ul style="display: flex;"><li style="flex:1">Tree height = 3 </li><li style="flex:1">Root </li></ul>Tree arity = 3 Tree Characteristics (1) D:0-H:3 width = 1 width = 2 level 0 level 1 • Tree arity/degree • maximum number of children of any node D:1-H:1 D:1-H:2 • Node depth • the length of the simplest path (number of edges) from the root node D:2-H:1 • depth(root) = 0 width = 5 • Tree depth level 2 level 3 • maximum of node depths D:2-H:0 • Tree level width = 2 • set of nodes with the same depth (distance from root) • Level width D:3-H:0 • number of nodes at given level • Node height Tree depth = 3 • number of edges on the longest downward simple path from Y to a leaf • Tree height • height of the root node 7Tree Characteristics (2) • Balanced (vyv ážený ) tree • a tree whose subtrees differ in height by no more than one and the subtrees are height-balanced as well • Unsorted tree • general tree – the descendants of a node are not sorted at all • Sorted tree • unlike general tree the order of children of each node matters • children of a node are sorted based on a given key 8Binary Tree • Each inner node contains at most two child nodes (tree arity = 2) • Characteristics • number of nodes of a perfect binary tree of height ℎ 풉#풏풐풅풆풔풑풆풓풇 = ퟐ풊 = ퟐ풉+ퟏ − ퟏ • Types • perfect (perfektní) binary tree – every non-leaf node has two child nodes • complete (komletní) binary tree – full tree except for the first nonleaf level where nodes are as far left as possible 풊=ퟎ • number of leaf nodes of a perfect binary tree of height ℎ #풍풆풂풇_풏풐풅풆풔풑풆풓풇 = ퟐ풉 9Binary Search Tree (BST) <ul style="display: flex;"><li style="flex:1">• Binary sorted tree where </li><li style="flex:1">• Searching a record with a key 푘 </li></ul>• each node is assigned a value 1. Enter the tree at the root level corresponding to one of the keys 2. Compare 푘 with the value 푣 of the • all nodes in the left subtree of a node 푋 correspond to keys with lower values than the value of 푋 • all nodes in the right subtree of a node 푋 correspond to keys with higher values than the value of 푋 current node 3. If 푘 = 푣 return the record corresponding to the current node key 4. If 푘 < 푣 descend to the left subtree 5. If 푘 > 푣 descend to the right subtree 6. Repeat steps 2-5 until a leaf is reached. If the condition in step 3 is not met at any level, no record with the key 푘 is present • when using BST for storing records, the records can be stored directly in the nodes or addressed from them 10 Redundant vs. Non-Redundant BST <ul style="display: flex;"><li style="flex:1">Non-Redundant BST </li><li style="flex:1">Redundant BST </li></ul><ul style="display: flex;"><li style="flex:1">• Records stored in (addressed </li><li style="flex:1">• Records stored in (addressed </li></ul><ul style="display: flex;"><li style="flex:1">from) both inner and leaf nodes </li><li style="flex:1">from) the leaves </li></ul>• Structure of inner and leaf nodes differ 8 8<ul style="display: flex;"><li style="flex:1">4</li><li style="flex:1">9</li></ul>12 412 9 2 7<ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">7</li><li style="flex:1">9</li></ul><ul style="display: flex;"><li style="flex:1">7</li><li style="flex:1">8</li></ul>11 <ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">4</li></ul>Applications of Binary Trees (not related to storing data records) • Expression trees • leaves = variables • inner nodes = operands • Huffman coding • leaves = data • coding along a branch leading to give leaf = leaf’s binary representation • Query optimizers in DBMS • query can be represented by an algebraic expression which can be in turn represented by a binary tree 12 M-ary Trees (1) • Motivation • binary trees are not suitable for magnetic disks because of their height – indexing 16 items would need an index of height 4 → up to 4 disk accesses (potentially in different locations) to retrieve one of the 16 records • log2 1000 = 10 , log2 1,000,000 = 20 • But increasing arity leads to decreasing the tree height 13 M-ary Trees (2) • Generalization of binary search trees • A binary tree is a special case of an m-ary tree for 푚 = 2 • only m-ary trees with 푚 > 2 are usually considered as true m−ary trees • unlike binary tree, m-ary tree contains 풎 − ퟏ discriminators (keys) in every node • every node 푁 points to up to 풎 child nodes (subtrees) • every subtree of 푁 contains records with keys restricted by the pair of discriminators of 푁 between which the subtree is rooted • the leftmost subtree contains only records with keys having lower values than all the discriminators in 푁 • the rightmost subtree contains only records with keys having higher values than all the discriminators in 푁 14 M-ary Trees (3) 풎 = ퟒ Data could be stored directly in the node as well. But it is not usual in real-world database environments. disk block 푘1 <ul style="display: flex;"><li style="flex:1">푘2 </li><li style="flex:1">푘3 </li></ul>data1 data3 data2 subtree with all keys 푘1 < 푘푖 < 푘2 subtree with all keys 푘2 < 푘푖 < 푘3 subtree with all keys subtree with all keys 푘3 < 푘푖 푘푖 < 푘1 15 Characteristics of M-Ary Trees (1) • Maximum number of nodes • Maximum number of records/keys • tree height ℎ • 푙푒푣푒푙0 = 푚0 = 1, 푙푒푣푒푙1 = • every node contains up to 푚 − 1 푚1, 푙푒푣푒푙2 = 푚2, … 푙푒푣푒푙ℎ = 푚ℎ keys/records 푚ℎ+1 − 1 ℎ풎풉+ퟏ − ퟏ × 푚 − 1 = 풎풉+ퟏ − ퟏ 푚 − 1 푚푖 = 풎 − ퟏ 푖=0 • 푚 = 3, ℎ = 3 → #푟푒푐표푟푑푠 ≤ 34 − 1 = 80 • 푚 = 100, ℎ = 3 → #푟푒푐표푟푑푠 ≤ 1004 − 1 = 100,000,000 16 Characteristics of M-Ary Trees (2) • Minimum height 풉 = 퐥퐨퐠풎 풏 • Maximum height 풏풉 ≐ 풎• maximum height of the tree corresponds to the maximum number of disk operations needed to fetch a record → search complexity 푶(풏) • minimum height of the tree corresponds to the minimum number of disk operations needed to fetch a record → search complexity 푶(퐥퐨퐠퐦 풏) • The challenge is to keep the complexity logarithmic, that is to keep the tree more or less balanced 17 B-tree • Bayer & McCreight, 1972 • B-tree is a sorted balanced m-ary (not binary) tree with additional constraints restricting the branching in each node thus causing the tree to be reasonably “wide” • Inserting or deleting a record in B-tree causes only local changes and not rebuilding of the whole index 18 B-tree definition (1) • B-trees are balanced m-ary trees fulfilling the following conditions: 1. The root has at least two children unless it is a leaf. 풎2. Every inner node except of the root has at least and at most 풎 ퟐchildren. 풎3. Every node contains at least − ퟏ and at most 풎 − ퟏ (pointers ퟐto) data records. 4. Each branch has the same length. 19 B-tree definition (2) 5. The nodes have following organization: 풑ퟎ, 풌ퟏ, 풑ퟏ, 풅ퟏ , 풌ퟐ, 풑ퟐ, 풅ퟐ , … , , 풌풏, 풑풏, 풅풏 , 풖 • where 풑ퟎ, 풑ퟏ, … , 풑풏 are pointers to the child nodes, 풌ퟏ, 풌ퟐ, … , 풌풏 are keys, 풅ퟏ, 풅ퟐ, … , 풅풏 are associated data, 풖 is unused space and the records 풌풊, 풑풊, 풅풊 are sorted in increasing order with respect to the keys and 풎− ퟏ ≤ 풏 ≤ 풎 − ퟏ ퟐ6. If a subtree 푼 풑풊 corresponds to the pointer 푝푖 then: i. ∀풌 ∈ 푼 풑풊−ퟏ : 풌 < 풌풊 ii. ∀풌 ∈ 푼 풑풊 : 풌 > 풌풊 20 Example of a B-Tree • Keys: 25, 48, 27, 91, 35, 78, 12, 56, 38, 19 48 19 25 <ul style="display: flex;"><li style="flex:1">27 </li><li style="flex:1">78 </li></ul><ul style="display: flex;"><li style="flex:1">91 </li><li style="flex:1">12 </li><li style="flex:1">35 </li><li style="flex:1">38 </li><li style="flex:1">56 </li></ul>21 Redundant B-Tree • Redundant vs. non-redundant B-trees • the presented definition introduced the non-redundant B-tree where each key value occurred just once in the whole tree • redundant B-trees store the data values in the leaves and thus have to allow repeating of keys in the inner nodes • restriction 6(i) is modified so that • ∀풌 ∈ 푼 풑풊−ퟏ : 풌 ≤ 풌풊 • moreover, the inner nodes do not contain pointers to the data records 22 B-Tree implementations • Nodes vs. pages/blocks • usually one page/block contains one node • Existing DBMS implementations • one page usually takes 8KB • redundant B-trees • inner nodes contain only pairs 푘1, 푝1 → inner nodes and leaf nodes differ in structure and size • data are not stored in the indexing structure itself but addressed from the leaf nodes • if the key is of type 64-bit integer (8B) and the DMBS runs on a 64-bit OS (8B pointers) with 8KB blocks → (푘1, 푝1) = 16퐵 → one page can accommodate 8192/16 = 512 pairs → arity of the B-tree is then about 500 23 B-Tree - Search • Searching a (non-redundant) tree 푇 for a record with key 푘 • Enter the tree in the root node. • If the node contains a key 풌풊 such that 풌풊 = 풌 return the data associated with 풅풊. • Else if the node is leaf, return NULL. • Else find lowest 풊 such that 풌 < 풌풊 and set 풋 = 풊 − ퟏ. If there is no such ꢀ set 풋 as the rightmost index with existing key. • Fetch the node pointed to by 풑풋. • Repeat the process from step 2. 24 Updating a B-Tree • The logarithmic complexity is ensured by the condition that every node has to be at least half full • Inserting • finding a leaf where the new record should be inserted • when inserting into a not yet full node no splitting occurs • when inserting into a full node, the node is split in such a way that the two resulting nodes are at least half full • splitting a node increments the number of subtrees of the parent node and thus possibly causes splitting of the parent node → split cascade • Deleting • when deleting a record from a node more than half full, no reorganization happens • deleting in a half full node induces merging of the neighboring nodes • merging decreases the number of child nodes in the parent node thus possibly causing merging of the parent node→ merge cascade 25 Inserting into a B-tree (1) • Insert into a (non-redundant) tree 푇 a record 푟 with key 푘 • If the tree is empty, allocate a new node, insert the key 푘 and (pointer to record) 푟 and return. • Else find the leaf node 푳 where the key 풌 belongs. • If 푳 is not full insert 풓 and 푘 into 퐿 in such a position that the keys are sorted and return. • Else create a new node 푳′. • Leave lower half records (all the items from 퐿 plus 푟) in 푳 and the higher half records into 푳′ except of the item with the middle key 푘′. a) If 퐋 is the root, create a new root node, move the record with key 푘′ to the new root and point it to 푳 and 푳′ and return. b) Else move the record with key 푘′ to the parent node 푷 into appropriate position based on the value 푘′ and point the “left” pointer to 퐋 and the “right” pointer to 푳′. • If 푷 overflows, repeat step 5 for 푃 else return. 26 Inserting into a B-tree (2) 27 27 Insert 91 48 25 <ul style="display: flex;"><li style="flex:1">48 </li><li style="flex:1">91 </li><li style="flex:1">25 </li></ul>35 falls into the node (48,91) → (35, 48, 91) → middle key (48) moves to the parent node 27 35 48 Insert 35 <ul style="display: flex;"><li style="flex:1">25 </li><li style="flex:1">91 </li></ul><ul style="display: flex;"><li style="flex:1">퐿</li><li style="flex:1">퐿′ </li></ul>27 Inserting into a B-tree (3) <ul style="display: flex;"><li style="flex:1">27 </li><li style="flex:1">48 </li></ul>Insert 78,12

Trees • Binary Trees • Newer Types of Tree Structures • M-Ary Trees

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