CS 234. Module 6. October 25, CS 234 Module 6 ADT Dictionary 1 / 22

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1 CS 234 Module 6 October 25, 2016 CS 234 Module 6 ADT Dictionary 1 / 22

2 Case study Problem: Find a way to store student records for a course, wit unique IDs for eac student, were records can be accessed, added, or deleted. CS 234 Module 6 ADT Dictionary 1 / 22

3 ADT Dictionary Data: (key, element pairs) were keys are distinct but not necessarily orderable, and elements are any data. Operations: CreateDict() IsEmptyDict(D) LookUpDict(D, key) AddToDict(D, key, element) DeleteFromDict(D, key) Note: AddToDict and DeleteFromDict bot require a use of LookUpDict. Optional operation for restriction wit orderable keys: ClosestKeyBefore(k) CS 234 Module 6 ADT Dictionary 2 / 22

4 Additional operations needed to store values KeyAtNode(B, node) ElementAtNode(B, node) StoreInNode(B, node, key, value) CS 234 Module 6 ADT Dictionary 3 / 22

5 Implementing DeleteFromDict(D, key) Find te node n containing te key Case 1: n as no cildren: delete n Case 2: n as only one cild c: delete n, make c into te cild of n s parent Case 3: n as two cildren: Find te inorder successor s of n. Replace n s key wit s s key. Delete te node containing n. CS 234 Module 6 ADT Dictionary 4 / 22

6 Binary searc tree deletion, illustrated Case 1: 12, 29, 35, 50 Case 2: 10, 15, 27 Case 3: 20, ten 27 (Case 2) Case 3: 30, ten 35 (Case 1) Case 3: 40, ten 50 (Case 1) CS 234 Module 6 ADT Dictionary 5 / 22

7 AVL trees A node is balanced if te difference in eigt of left and rigt subtrees is at most 1. A tree satisfies te eigt-balance property if every node is balanced. An AVL tree is a eigt-balanced BST. CS 234 Module 6 ADT Dictionary 6 / 22

8 AVL tree example Balance: 0 if even, 1 if left iger, -1 if rigt iger CS 234 Module 6 ADT Dictionary 7 / 22

9 AVL tree AddToDict Te pivot node is te lowest unbalanced node in te tree after a new node as been inserted. A rotation on te pivot node is a rearrangement of subtrees tat rebalances te tree witout violating binary searc order. Algoritm: Find leaf to insert as in BST. Trace pat from leaf to root to identify te pivot node. Rebalance by executing a rotation on te pivot node. CS 234 Module 6 ADT Dictionary 8 / 22

10 AVL AddToDict case 1 D DB DB D E C A C E A Insertion in A causes imbalance at D. Letters sow BST order preserved. Balance at B means A is +1 and C is. Imbalance at D means E is. CS 234 Module 6 ADT Dictionary 9 / 22

11 AVL AddToDict case 2 F DB D D G B F A C E A C E G Insertion at C or E causes imbalance at F. Balance at B means A is and one of C and E is, te oter -1. Imbalance at F means G is. CS 234 Module 6 ADT Dictionary 10 / 22

12 AVL AddToDict case 3 B D D B +1 A C +1 C A C E E Insertion at E causes imbalance at B. Balance at D means C is. Imbalance at B means A is. CS 234 Module 6 ADT Dictionary 11 / 22

13 AVL AddToDict case 4 B F D A D B F G C E A C E G Insertion at C or E causes imbalance at B. Balance at D means C and E bot -1 or. Balance at F means G is. Imbalance at B means A is. CS 234 Module 6 ADT Dictionary 12 / 22

14 AVL tree DeleteFromDict Algoritm: Delete as in BST. Trace up to find first unbalanced node z. y = cild of z wit greater eigt x = cild of y wit smaller eigt (maybe bot same) Rebalance using rotation ***Keep tracing up to see if anoter unbalanced node iger.*** CS 234 Module 6 ADT Dictionary 13 / 22

15 Multiway searc trees Properties: Ordered tree. Eac internal node as at least 2 cildren. A d-node (wit d cildren) as d-1 (key,element) pairs, wit keys in order k 1,...,k d 1. Any item (k,e) in te subtree of v rooted at te it cild of v as k i 1 k k i (view k 0 = and k d = ). CS 234 Module 6 ADT Dictionary 14 / 22

16 (2,3) tree (2,3) tree definition Eac internal node as eiter one key and two cildren or two keys and tree cildren. All leaves are at te same dept and ave one or two keys CS 234 Module 6 ADT Dictionary 15 / 22

17 (2,3) tree AddToDict If tere are too many values in a node, ten overflow as occurred. Te split operation splits a node into two and rearranges values as needed. Basic idea of operation: Searc leads to a leaf. Add item to leaf. If overflow, split leaf. If splitting te leaf leads to overflow in te parent, ten te parent may need to be split as well. Splitting can propagate up to te root. CS 234 Module 6 ADT Dictionary 16 / 22

18 Basic idea of split Node wit tree keys becomes two nodes, one wit smallest and one wit largest. Middle key is sent up to te parent. If te node being split is te parent, te tree now as a new root and is one taller. CS 234 Module 6 ADT Dictionary 17 / 22

19 (2,3) tree AddToDict single split Add CS 234 Module 6 ADT Dictionary 18 / 22

20 (2,3) tree AddToDict cascading split Add CS 234 Module 6 ADT Dictionary 19 / 22

21 (2,3) tree DeleteFromDict If tere are too few values in a node, ten underflow as occurred. Te fuse operation fuses two nodes into one and rearranges values as needed. Basic idea of operation: Searc leads to a leaf or internal node. If internal node, like in BST swap wit inorder successor. Remove item from leaf. If underflow, need to fuse nodes. If fusing leads to underflow in te parent, ten te parent and its sibling may need to be fused as well. Fusing can propagate up to te root. CS 234 Module 6 ADT Dictionary 20 / 22

22 Definitions concerning memory Te memory ierarcy consists of different type of memory, were te size of eac type increases as te access speed decreases: registers cace (multiple levels) main memory secondary storage tertiary storage A page is a fixed-size cunk of data tat is moved between levels. A paging algoritm is used to determine wic page(s) to move out wen a new page is moved in. CS 234 Module 6 ADT Dictionary 21 / 22

23 B-tree B-tree of order d: d cosen so tat d data items fill a page. Root as at most d cildren. Oter nodes ave at least d/2 and at most d cildren. Heigt is O(log d/2 n) and Ω(log d n). Operations: d +1 cildren split into nodes wit (d +1)/2 and (d +1)/2 items. d/2 1 cildren joined to become a node of size at most d. Space usage as low as 50%, average 69%. CS 234 Module 6 ADT Dictionary 22 / 22