Tree Data Structures CSC 221

Transcription

1 Tree Data Structures CSC 221

2 BSTree Deletion - Merging template <class Type> // LOOK AT THIS PARAMETER!!! void BST<Type>::deleteByMerging(BSTNode<Type>* & nodepointer) { BSTNode<Type>* temp= nodepointer; } if (!nodepointer->right) nodepointer = nodepointer->leftchild; else if (!nodepointer->left) nodepointer = nodepointer->rightchild; // what about double 0 (no children case?) else { temp= nodepointer->leftchild; // move to largest on left while (temp->righchild!= 0) temp = temp->rightchild; temp->rightchild = nodepointer->rightchild; // do merging temp = nodepointer; // prep to delete nodepointer = nodepointer->leftchild; // set pointers appropriately to finalize } delete temp;

3 TreeSort: Insertion into a binary tree places a specific ordering on the elements. For the root, 30 Everything in the left subtree is < root 5 40 Everything in the right subtree is > root For each subtree, Everything on the left < subtree root, Everything on the right is > subtree root

4 TreeSort: Theoretically, should be able to construct an ordering of all elements from the tree: Generate an array of size equal to number of elements in tree Root goes in middle of array Left subtree fills in left half of array Right subtree fills in right half of array < > 30 And Recurse <5 5 >5 30 <40 40 >40

5 TreeSort: Extracting ordered array from binary tree: Perform in-order traversal (LVR) Ensures will visit all smaller items first and larger items last 30 LVR Ordering: 2,5,15,30,40,35,

6 TreeSort: Analysis of TreeSort: Given an array of size n, have to build binary a tree with n- elements Requires N insertions Given a binary tree with n-elements, have to traverse tree in LVR order to extract sorted order Construction: O(n * log 2 n) if balanced O(n * n) if not balanced Traversal: O(n) anytime Average Case: O(n log 2 n), Worst Case: O(n 2 )

7 TreeSort: Very similar to quicksort! Same average case [O(n log n)] and worst case [O(n 2 )] times Roots of binary search tree subnodes are the pivots Place data smaller than pivot on left of pivot (leftchild), place larger data on right of pivot (rightchild) The better the pivot is, the more balanced the tree is (same for quicksort recursion) Nearly sorted/already sorted data leads both to trouble: Bad partitioning for quicksort, Bad construction for treesort

8 Rank Information Often times when working with lists of data, interested in rank information: What is the largest item? What is the smallest? What is the median? What is the fifth smallest item? Largest and smallest are trivial [O(n)] What if want to ask alot of questions about rank or want to know about something other than largest smallest?

9 Rank Information Sorting approach to rank information: Sort the list Return list[rankofinterest] O(n log n) [sort] + O(1) [value retrieval] If using dynamic data, may not have the array to work with instead a linked list would be more likely

10 Rank Information Linked List Approach Sort list Assuming sort for linked lists Traverse list to find rankofinterest element O(n log n) [assuming sort can be written and is effective] + O(rankOfInterest) [traversal] Can handle dynamic data, but slower!

11 Rank Information Binary Tree Approach: Insert into binary tree Inorder traversal up until rankofinterest node (goes through in sorted order) O(n log n) [building tree] + O(rankOfInterest) [traversal] Same cost as linked list approach (probably easier since don t have to write sort for linked lists).

12 Rank Information: Binary Tree Approach: Add a new variable to each node in the tree leftsize = indicates number of elements in nodes left subtree + self In node constructor set left size to 1 (for self) Insert elements into binary tree As pass by parent nodes in searching for appropriate place, store references to each parent node in an auxiliary list If do insertion, update each parent s leftsize value If don t insert (non-unique), no updates for leftsize Search by rank using traditional binary tree search on leftsize value Function on next slide

13 Rank Information: template <class Type> BinaryTreeNode<Type>* BinarySearchTree<Type>:: search(int rank) { BinaryTreeNode<Type>* current = root; while (current) { if (k == current->leftsize) return current; else if (rank < current->leftsize) current = current->leftchild; else { rank = rank leftsize; current = current->rightchild;} } }

14 Rank Information: Example leftsize values: 4 What is 2nd element? 2 John Mike Thomas 2 Rank 2 < leftsize(mike) [4] Move to root->leftchild Rank 2 == leftsize(john) [2] Georgia Kylie Tyler Shelley Real Ranks for Data [First is rank 1, Last is 7]: Georgia, John, Kylie, Mike, Shelley, Thomas, Tyler Return John Node What is 5 th element? Rank 5 > leftsize(mike) [4] Move to root->rightchild Rank = 5-4 = 1 < leftsize(thomas) [2] Move to leftchild of Thomas Rank == leftsize(shelley) [1] Return Shelley Node

15 Rank Information: Analysis Searching (traversal) is now bounded by the height of the tree On average O(log n) Building tree was O(n log n), but we added more work Original n log n comes from n insertions, log n cost each Now have to update parents leftsize values However, maximum number of parents = height of tree = on average log n So the cost for a single insertion is now just 2 log n, and all insertions costs are still bounded by O(n log n) So for dynamic data, can do rank information in: O(n log n) [building] + O(log n) [searching] Better than approaches that sort and traverse to rank position

16 Phonebooks, Again! How do we use a phonebook? Search for a phone number Retrieve information associated with a phone number William Turkett : , West 240

17 Dictionary Datastructures There are a lot of everyday tools that have this type of structure: Dictionary: Word:Meaning Thesaurus: Word:Synonyms Unix password file: LoginName:Password What are these essentially? Mapping a specific key to a value/set of values

18 Dictionary Datastructures Key: A unique identifier Values: Information that is associated with that unique identifier Address Book: Key William Turkett Tyler Hogate Values turketwh@wfu.edu hogatb2@wfu.edu

19 Compiler Symbol Tables Stores association between symbols in a program and information about that symbol Key: Variable/Function Identifier Values: Initial Value, Lines of code using identifier Identifier: Initial Value: Lines of Interest top 0x2345 1,3,5,17 size 0 5,7,9

20 Symbol Table Operations Operations of interest for a symbol table : Determine if a name is in the table Retrieve the values associated with a name Modify attributes associated with a name Insert a new name and attributes Delete a name and its attributes

21 Symbol Table Definition template <class Key, Value> class SymbolTable { public: SymbolTable(int size = defaultsize); bool isin(key key); Value* find(key key); void insert(key key, Value value); void delete(key key); private:?? }

22 Symbol Table Definition SymbolTable(int size = defaultsize) // Create an empty symbol table with capacity size bool isin(key key) // if key is in the symbol table, return true; else return false; Value* find(key key) // if key is in the symbol table, return a pointer to the corresponding value; else return 0

23 Symbol Table Definition void insert(key key, Value value) // if key is in the symbol table, replace its existing value with the new parameter value; else insert the pair (key, value) into the symbol table void delete(key key) // if key is in the symbol table, delete the pair (key, value) from the symbol table

24 Symbol Table Operations Note that modification of a value is really just re-inserting with a new value! So, three essential operations performing: Search, Insert, and Delete Really need to have these operations be performed efficiently

25 Symbol Table Operations Best performance so far for search, insert, and delete: O (log n) Binary Search Trees, if remain well balanced We have at least been told that there are trees which can guarantee O(log n) height, so that sounds good. Not good enough! It would be really nice to have search, insert, and delete of near constant time - O(1)

26 Hashing All previous search methods have used identifier comparisons to perform a search Is Turkett < Hogate? Is 5 > 3? Hashing relies on a formula called the hash function to do searching. Hash(Turkett) =? Hash(Hogate) =? Ideally, hash function output can provide the exact location to go to for retrieving values.