CS 234. Module 6. October 16, CS 234 Module 6 ADT Dictionary 1 / 33

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1 CS 234 Module 6 October 16, 2018 CS 234 Module 6 ADT Dictionary 1 / 33

2 Idea for an ADT Te ADT Dictionary stores pairs (key, element), were keys are distinct and elements can be any data. Notes: Tis is similar, but not identical, to te Pyton data type dictionary. Tis is similar to te textbook s ADT Map. ADT Map is restricted to te case were keys are orderable. Tis is inspired by, but not identical to, a language dictionary, were keys are words and elements are etymology, pronunciation, and definitions. Te term Dictionary is used trougout te literature. Restrictions to consider: Keys are orderable. CS 234 Module 6 ADT Dictionary 1 / 33

3 ADT Dictionary Data: (key, element pairs) were keys are distinct but not necessarily orderable, and elements are any data. Preconditions: For all D is a dictionary, k is a key, and e is an element. Postconditions: Mutation by Add (add/replace item wit key k) and Delete (delete item wit key k, if any). Name Create() IsEmpty(D) LookUp(D, k) Add(D, k, e) Delete(D, k) Returns a new empty dictionary true if empty, else false item wit key k if any, else false CS 234 Module 6 ADT Dictionary 2 / 33

4 Array and linked implementations Customizing Array or linked list, items not ordered Add, Delete require LookUp (for add, to see key not already tere) LookUp O(n) worst case Special case of orderable data Store in increasing order of key Costs still depend on LookUp Recall binary searc from CS 116 CS 234 Module 6 ADT Dictionary 3 / 33

5 Data structure: binary searc tree (BST) A binary searc tree (or BST) is a binary tree tat satisfies te binary searc tree property, tat is, for every node n in te tree, for k te key stored in te node: Te values stored in te left subtree of n are smaller tan k. Te values stored in te rigt subtree of n are greater tan k. Note: We can generalize tis to allow repeated values, using and > or < and. CS 234 Module 6 ADT Dictionary 4 / 33

6 Customizing ADT Binary Tree to implement BST Additional data: Store bot keys and elements at nodes. Notes: Only keys are ordered using te binary searc tree property. Our illustrations sow keys only. Modified ADT Binary Tree operations: AddNode(B, p, k, e, side) Additional ADT Binary Tree operations: KeyAtNode(B, n) ElementAtNode(B, n) StoreInNode(B, n, k, e) - canges wat is stored in a node CS 234 Module 6 ADT Dictionary 5 / 33

7 Implementing ADT Dictionary operations LookUp(D, k) Compare, coose direction (use LeftCild, RigtCild) Find element or failure (its empty pointer) Add(D, k, e) First execute LookUp(D, k) If ends in empty (side = left or rigt) pointer of p, ten AddNode(B, p, node, side) Analysis of operations Θ(1) for eac node examined: ow many in total? Θ(eigt of tree) Θ(n) in worst case If nice tree, worst case is Θ(log n) Lower bound of Ω(logn) ( in n node tree, some node must be at least log n away from te root) CS 234 Module 6 ADT Dictionary 6 / 33

8 Implementing Delete(D, k) Find te node n containing te key Case 1: n as no cildren: delete n Case 2: n as only one cild c: make c into te cild of n s parent, ten delete n Case 3: n as two cildren: Find te inorder successor s of n. Replace n s (key, element) pair wit s s (key, element) pair. Delete te node containing n (Case 1 or 2). CS 234 Module 6 ADT Dictionary 7 / 33

9 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 8 / 33

10 Finding an inorder successor Guide students: Were is it? (Answer, leftmost node in rigt subtree) How can you find it? (Go rigt, ten always to te left) Cost of finding it? (Worst case: eigt of tree) Can you use te inorder predecessor? (Yes.) Were is it? (Rigtmost node in left subtree) Will it always ave one cild? (Yes, same argument, mirror image.) CS 234 Module 6 ADT Dictionary 9 / 33

11 Heigt of a binary searc tree Everyting depends on te eigt worst case Θ(n) best case Θ(log n) Goals: Maintain Θ(log n) eigt Implement Θ(log n) worst case time for LookUp(D, k), Add(D, k, e), and Delete(D, k) Can we maintain te minimum eigt at all times? Too costly to maintain, too inflexible. Idea: Find close enoug to minimum to work. CS 234 Module 6 ADT Dictionary 10 / 33

12 Data structure: AVL tree Key ideas: 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. To determine balance, store te eigt of eac node. For calculations, consider an absent subtree to ave eigt -1. Maintain balance and running times as follows: Any AVL tree wit n nodes as eigt Θ(logn). (Proof omitted.) Add preserves te eigt-balance property, worst case in Θ(log n). Deletion preserves te eigt-balance property, worst case in Θ(log n). CS 234 Module 6 ADT Dictionary 11 / 33

13 AVL tree example Heigts are stored in nodes. Balance: 0 if left and rigt subtrees ave te same eigt, 1 if te left subtree is one iger, -1 if te rigt subtree is one iger CS 234 Module 6 ADT Dictionary 12 / 33

14 Customizing ADT Binary Tree to implement AVL tree Since an AVL tree is a special type of BST, make all te same modifications as needed to implement a BST using te ADT Binary Tree, and ten also te ones below. Additional ADT Binary Tree operations: Heigt(B, n) - looks up eigt at node n ReplaceHeigt(B, n, ) - resets te eigt at node n to Customize eac data structure for ADT Binary Tree: Add a field for eac node storing te eigt of te node. CS 234 Module 6 ADT Dictionary 13 / 33

15 Implementing Add(D, k, e) in an AVL tree 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, updating eigts and cecking balance. Te first imbalanced node encountered (if any) is te pivot node. Rebalance by executing a rotation on te pivot node. Observations: A rotation entails updating a constant number of pointers. At most one rotation is needed to rebalance te tree. To ceck: Binary searc order is preserved Heigt balance property is preserved. CS 234 Module 6 ADT Dictionary 14 / 33

16 Rebalancing Possible cases for imbalance at a pivot node: 1. Left cild s left subtree is too ig 2. Left cild s rigt subtree is too ig 3. Rigt cild s rigt subtree is too ig (symmetrical to 1) 4. Rigt cild s left subtree is too ig (symmetrical to 2) In all upcoming slides: Letter labels sow order of values. Left image sows te portion of te tree rooted at te pivot node before rotation. Rigt image sows te replacement subtree after rotation. Te comments explain wy te eigts in te left image must be te way tey are. CS 234 Module 6 ADT Dictionary 15 / 33

17 Implementing Add(D, k, e), Case 1 D DB DB D E C A C E A 1. New node added in A eigt of A increased to make D te pivot node. 2. E is eigt + D is pivot node subtree rooted at B is eigt (1) + (2) A is eigt B is balanced C is eigt. CS 234 Module 6 ADT Dictionary 16 / 33

18 Implementing Add(D, k, e), Case 2 F DB D D G B F A C E A C E G 1. New node added in rigt subtree of B eigt of eiter C or E increased to make F te pivot node. 2. G is eigt + F is pivot node subtree rooted at B is eigt (1) + (2) exactly one of C and E is eigt. 4. (3) + D is balanced exactly one of C and E is eigt B is balanced A is eigt. CS 234 Module 6 ADT Dictionary 17 / 33

19 Implementing Add(D, k, e), Case 3 B D D B +1 A C +1 C A C E E 1. New node added in E eigt of E increased to make B te pivot node. 2. A is eigt + B is pivot node subtree rooted at D is eigt (1) + (2) E is eigt D is balanced C is eigt. CS 234 Module 6 ADT Dictionary 18 / 33

20 Implementing Add(D, k, e), Case 4 B F D A D B F G C E A C E G 1. New node added in left subtree of D eigt of eiter C or E increased to make B te pivot node. 2. A is eigt + B is pivot node subtree rooted at F is eigt (1) + (2) exactly one of C and E is eigt. 4. (3) + D is balanced exactly one of C and E is eigt F is balanced G is eigt. CS 234 Module 6 ADT Dictionary 19 / 33

21 Implementing Delete(D, k) in an AVL tree Algoritm: Delete as in BST. Trace pat from site of deletion to root, updating eigts and cecking balance. Rebalance by executing a rotation on te first imbalanced node encountered (if any). Continue tracing te pat to te root, updating eigts and cecking balance, pivoting again as often as needed. Cost of Delete(D, k): Rotation may result in a subtree wic is sorter. Repeated rotation may be necessary. At worst one rotation per level in te tree. Worst case Θ(log n) rotations at Θ(1) cost eac. CS 234 Module 6 ADT Dictionary 20 / 33

22 Generalizing BSTs Idea: To reduce eigt, wy not ave nodes wit more keys and more cildren? less tan 10 between 10 and 35 between 35 and 60 between 60 and 87 greater tan 87 Multiway searc tree Analogy of perfect tree now as eigt log d n. For d a constant, still Θ(logn). For d not a constant, non-constant processing of a node. New idea: Make all leaves be te same distance from te root, but allow variation in te number of keys per node. Slack allows some flexibility and some ease of updates. CS 234 Module 6 ADT Dictionary 21 / 33

23 Data structure: (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. Keys in te left subtree are smaller tan te first key, in te middle subtree are between te first and second keys, and in te rigt subtree are greater tan te second key Any (2,3) tree storing n data items as eigt Θ(logn). CS 234 Module 6 ADT Dictionary 22 / 33

24 Customizing ADT Ordered Tree to implement BST Additional data: Store up to two (key, element) pairs and up to tree cildren at nodes. Modified ADT Ordered Tree operations: AddNode(O, p, key, element, s) Additional ADT Ordered Tree operations: KeyOneAtNode(O, n) ElementOneAtNode(O, n) KeyTwoAtNode(O, n) ElementTwoAtNode(O, n) StoreInNode(O, n, key1, e1, key2, e2) CS 234 Module 6 ADT Dictionary 23 / 33

25 Implementing LookUp(D, k) in a (2,3) tree Notes: Similar to searc in a binary searc tree, comparing searc key to keys stored in a node and coosing a subtree to searc. Eac unsuccessful searc leads to a leaf. Worst-case cost is in Θ(log n) (eigt of tree). CS 234 Module 6 ADT Dictionary 24 / 33

26 Implementing Add(D, k, e,) in a (2,3) tree 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. Basic idea of split: Node wit tree keys becomes two nodes, one wit smallest key and one wit largest key. Middle key is sent up to te parent. If te node being split was te root, te tree now as a new root. CS 234 Module 6 ADT Dictionary 25 / 33

27 Implementing Add(D, k, e), single split Add CS 234 Module 6 ADT Dictionary 26 / 33

28 Implementing Add(D, k, e), cascading split Add CS 234 Module 6 ADT Dictionary 27 / 33

29 Implementing Delete(D, k) in a (2,3) tree 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 28 / 33

30 More data structures for ADT Dictionary Customizing trees: Use internal nodes for searc; store all values in te leaves. Oter examples include: red-black trees splay trees skip lists (combining tree and list ideas) CS 234 Module 6 ADT Dictionary 29 / 33

31 A little more information about 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 All you need to know for tis course is tat tere is a ierarcy. Simplify to two levels: Main memory External memory (e.g. disks, cloud) CS 234 Module 6 ADT Dictionary 30 / 33

32 Impact on data structure design Moving data between levels: Processing takes place in main memory. Data to be processed is moved into main memory as it is needed, and moved out to make room for oter data to be processed. Data is moved between levels in fixed-sized cunks (pages). Tere are various paging algoritms used to determine wic page(s) to move out wen a new page is moved in. Bottom line: For small amounts of data, only main memory needs to be considered. For large amounts of data, take into account page size. Te number of accesses to external memory may dominate te cost of an algoritm. CS 234 Module 6 ADT Dictionary 31 / 33

33 Data structure: B-tree B-tree of order d: Generalization of a (2,3) tree. Coose d so tat d data items fill a page. Te root as at most d cildren. Oter nodes ave at least d/2 and at most d cildren. 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. Variants: B+-trees: all data appears in te leaves, treaded; increase space use by being judicious about splits. A B+-tree of order 256 olding 4 million keys uses at most 3 disk accesses. CS 234 Module 6 ADT Dictionary 32 / 33