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Multi-way Search Trees: 2-3, 2-4, and B-trees

Learn about multi-way search trees, including 2-3 trees, 2-4 trees, and B-trees. Understand the concepts, insertion, and removal of elements in these trees.

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Multi-way Search Trees: 2-3, 2-4, and B-trees

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  1. CHAPTER 12: Multi-way Search Trees Java Software Structures: Designing and Using Data Structures Third Edition John Lewis & Joseph Chase Modified by Chuck Cusack, Hope College

  2. Chapter Objectives • Examine 2-3 and 2-4 trees • Introduce the generic concept of a B-tree • Examine some specialized implementations of B-trees

  3. Multi-way Search Trees • In a multi-way search tree, • each node may have more than two child nodes • there is a specific ordering relationship among the nodes • In this chapter, we examine three forms of multi-way search trees • 2-3 trees • 2-4 trees • B-trees

  4. 2-3 Trees • A 2-3 tree is a multi-way search tree in which each node has zero, two, or three children • A node with zero or two children is called a 2-node • A node with zero or three children is called a 3-node

  5. 2-3 Trees: 2-Nodes • A 2-node contains one element and either has no children or two children • Elements of the left sub-tree less than the element • Elements of the right sub-tree greater than or equal to the element

  6. 2-3 Trees: 3-Nodes • A 3-node contains two elements, one designated as the smaller and one as the larger • A 3-node has either no children or three children • If a 3-node has children then • Elements of the left sub-tree are less than the smaller element • The smaller element is less than or equal to the elements of the middle sub-tree • Elements of the middle sub-tree are less then the larger element • The larger element is less than or equal to the elements of the right sub-tree

  7. 2-3 Trees • All of the leaves of a 2-3 tree are on the same level • Thus a 2-3 tree maintains balance

  8. Inserting Elements into a 2-3 Tree • All insertions into a 2-3 tree occur at the leaves • The tree is searched to find the proper leaf for the new element • Insertion has three cases • Tree is empty • Create a new 2-node, insert the element into the 2-node, and make it the root. • Insertion point is a 2-node (next slide) • Insertion point is a 3-node (a few slides later)

  9. Inserting in a 2-node • Add the element to the leaf and make it a 3-node • E.g. insert 27:

  10. Inserting in a 3-node • The 3 elements (the two old ones and the new one) are ordered • The 3-node is split into two 2-nodes, one for the smaller element and one for the larger element • The middle element is promoted (or propagated) up a level. There are 2 cases: • The parent of the 3-node is a 2-node • The parent of the 3-node is a 3-node

  11. Insertion into 3-node (continued) • If the parent of the 3-node being split is a 2-node then it becomes a 3-node by adding the promoted element and references to the two resulting two nodes • E.g. Inserting 32:

  12. Insertion into 3-node (continued) • If the parent of the 3-node is itself a 3-node then it also splits into two 2-nodes and promotes the middle element again • E.g. inserting 25:

  13. Removing Elements from a 2-3 Tree • Removal of elements is also made up of three cases, based on the location of the element to be removed: • a leaf that is a 3-node • a leaf that is a 2-node • an internal node

  14. Case 1: 3-node leaf • The simplest case is that the element to be removed is in a leaf that is a 3-node • In this case the element is simply removed and the node is converted to a 2-node

  15. Case 2: 2-node leaf • This creates a situation called underflow • We must rotate the tree and/or reduce the tree’s height in order to maintain the properties of the 2-3 tree • This case can be broken down into four subordinate cases

  16. Case 2.1: 2-node leaf case 1 • The 2-node has a right child that is a 3-node • In this case, we rotate the smaller element of the 3-node around the parent

  17. Case 2.2: 2-node leaf case 2 • The underflow cannot be fixed through a local rotation but there are 3-node leaves in the tree • In this case, we rotate prior to removal of the element until the right child of the parent is a 3-node • Then we follow the steps for our previous case

  18. Case 2.3: 2-node leaf case 3 • None of the leaves are 3-nodes but there are 3-node internal nodes • In this case, we can convert an internal 3-node to a 2-node and rotate the appropriate element from that node to rebalance the tree

  19. Case 2.4: 2-node leaf case 4 • There not any 3-nodes in the tree • This case forces us to reduce the height of the tree • To accomplish this, we combine each the leaves with their parent and siblings in order • If any of these combinations produce more than two elements, we split into two 2-nodes and promote the middle element

  20. Case 4: internal node • As we did with binary search trees, we can simply replace the element to be removed with its inorder successor

  21. 2-4 Trees • 2-4 Trees are very similar to 2-3 Trees adding the characteristic that a node can contain three elements • A 4-node contains three elements and has either no children or 4 children • The same ordering property applies as 2-3 trees • The same cases apply to both insertion and removal of elements as illustrated on the following slides

  22. B-Trees • Both 2-3 trees and 2-4 trees are examples of a larger class of multi-way search trees called B-trees • We refer to the maximum number of children of each node as the order of a B-Tree • Thus 2-3 trees are 3 B-trees and 2-4 trees are 4 B-trees • B-trees of order m have the following properties:

  23. A B-tree of order 6

  24. Motivation for B-trees • To make the most efficient use of the relationship between main memory and secondary storage • Until now, we have assumed that an entire collection (data structure) exists in memory at once • What if the collection is too large to fit in memory? • B-trees were designed to flatten the tree structure and to allow for larger blocks of data that could then be tuned so that the size of a node is the same size as a block on secondary storage • This reduces the number of nodes and/or blocks that must be accessed, thus improving performance

  25. B*-trees • A variation of B-trees called B*-trees were created to solve the problem that the B-tree could be half empty at any given time • B*-trees have all of the properties as B-trees except: • in a B*-tree, each node has k children where (2m–1)/3 ≤ k ≤ m • (Recall that for a B-tree it was: m/2 ≤ k ≤ m) • This means that each non-root node is at least two-thirds full

  26. B+-trees • Another potential problem for B-trees is sequential access • B+-trees provide a solution to this problem by requiring that each element appear in a leaf regardless of whether it appears in an internal node • By requiring this and then linking the leaves together, B+-trees provide very efficient sequential access while maintaining many of the benefits of a tree structure

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