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Comparison of B-Trees, AVL Trees, Red-Black Trees, and m-Way Search Trees, Slides of Computer Science

Alagappa UniversityComputer Science

Information on various search trees, including b-trees, avl trees, red-black trees, and m-way search trees. It discusses their properties, such as degree, height, and number of pairs, and compares their performance in terms of search time and disk access. It also includes examples of insertion and splitting processes.

Typology: Slides

2012/2013

Uploaded on 03/19/2013

dharmakeerti
dharmakeerti 🇮🇳

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B-Trees
•Large degree B-trees used to represent very large
dictionaries that reside on disk.
•Smaller degree B-trees used for internal-memory
dictionaries to overcome cache-miss penalties.
AVL Trees
•n = 230 = 109(approx).
•30 <= height <= 43.
•When the AVL tree resides on a disk, up to
43 disk access are made for a search.
•This takes up to (approx) 4seconds.
•Not acceptable.
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Download Comparison of B-Trees, AVL Trees, Red-Black Trees, and m-Way Search Trees and more Slides Computer Science in PDF only on Docsity!

B-Trees

  • Large degree B-trees used to represent very largedictionaries that reside on disk.
  • Smaller degree B-trees used for internal-memorydictionaries to overcome cache-miss penalties.

AVL Trees

• n = 2^30 = 10^9 (approx).

• 30 <= height <= 43.

• When the AVL tree resides on a disk, up to 43 disk access are made for a search.

• This takes up to (approx) 4 seconds.

• Not acceptable.

Red-Black Trees

• n = 2^30 = 10^9 (approx).

• 30 <= height <= 60.

• When the red-black tree resides on a disk,up to 60 disk access are made for a search.

• This takes up to (approx) 6 seconds.

• Not acceptable.

m-way Search Trees

• Each node has up to m – 1 pairs and m children.

• m = 2 => binary search tree.

Capacity Of m-Way Search Tree

h = 3 m = 2 7 m = 2008 * 10^6 - 1

h = 5 31 3.2 * 10^11 - 1

h = 7 127 1.28 * 10^16 - 1

Definition Of B-Tree

• Definition assumes external nodes(extended m-way search tree).

• B-tree of order m.

ƒ ƒ mNot empty-way search tree. => root has at least 2 children.

ƒ Remaining internal nodes (if any) have at least ceil(m/2) children.

ƒ External (or failure) nodes on same level.

2-3 And 2-3-4 Trees

• B-tree of order ƒ m-way search tree. m.

ƒ ƒ Not emptyRemaining internal nodes (if any) have at least => root has at least 2 children.

ƒ ceil(m/2)External (or failure) nodes on same level.^ children.

• 2-3 tree is B-tree of order 3.

• 2-3-4 tree is B-tree of order 4.

B-Trees Of Order 5 And 2

• B-tree of order ƒ m-way search tree. m.

ƒ Not empty => root has at least 2 children.

ƒ Remaining internal nodes (if any) have at least ceil(m/2) children.

ƒ External (or failure) nodes on same level.

• B-tree of order 2 -node though). 5 is 3-4-5 tree (root may be

• B-tree of order 2 is full binary tree.

Choice Of m

• Worst-case search time. ƒ (time to fetch a node + time to search node) * height

ƒ (a + b*m + c * log where a, b and c are constants. 2 m) * h

m

searchtime

Insert

splitting pass.^ Insertion into a full leaf triggers bottom-up node

Split An Overfull Node

• ai is a pointer to a subtree.

• pi is a dictionary pair.

m a 0 p 1 a 1 p 2 a 2 … pm am

ceil(m/2)-1 a 0 p 1 a 1 p 2 a 2 … pceil(m/2)-1 aceil(m/2)-

m-ceil(m/2) aceil(m/2) pceil(m/2)+1 aceil(m/2)+1 … pm am

• pparent.ceil(m/2) plus pointer to new node is inserted in

Insert

  • Insert a pair with key = 2.
  • New pair goes into a 3 -node.

Insert

  • Insert a pair with key = 2 plus a pointer into parent.

Insert

  • Now, insert a pair with key = 18.

Insert Into A Leaf 3-node

• Insert new pair so that theascending order. 3 keys are in

• Split the overflowed node.

• Insert middle key and pointer to new nodeinto parent.

Insert

  • Insert a pair with key = 18.

Insert

  • Now, insert a pair with key = 7.

Insert

  • Insert a pair with key = 6 plus a pointer into parent.

Insert

  • Insert a pair with key = 4 plus a pointer into parent.

Insert

  • Insert a pair with key = 8 plus a pointer into parent.
  • There is no parent. So, create a new root.