Overview Introduction to Algorithms Introduction to Algorithms � Review basic abstract data structures � Sets Data Structures Data Structures � Lists � Trees � Trees � Graphs CSE 680 � Review basic concrete data structures � Linked-List and variants Prof. Roger Crawfis � Trees and variants � Examine key properties � Examine key properties � Discuss usage for solving important problems (search, sort, selection). Sets and Multisets Set – Language Support g g pp � Common operations � .NET Framework Support � .NET Framework Support � Fixed Sets (C#,VB,C++,…) � Contains (search) � Is empty � IEnumerable interface � Size Size � Enumerate � ICollection interface � Dynamic Sets add: � Add � Java Framework Support pp � Remove Remo e � Other operations (not so common) � Collection � Intersection � Set � Union � Union � Sub-set � STD library (C++) � Note, these can, and probably should, be implemented � Set and Multiset classes and their iterators. statically (outside of the class).
List List – Language Support g g pp � Common Queries � Arrays – fixed size. Arrays fixed size. � Enumerate � .NET Framework Support (C#,VB,C++,…) � Number of items in the list � IList interface � Return element at index i . i R t l t t i d � List<T> class � Search for an item in the list (contains) � Java Framework Support pp � Common Commands � Common Commands � List interface � Add element � Set element at index i . � ArrayList<T> and Vector<T> classes � Remove element? � STD library (C++) � Insert before index i ? � std::vector<T> class. Concrete Implementations p Rooted Trees � A tree is a collection of nodes and � A tree is a collection of nodes and � Set � Set directed edges , satisfying the following � What might you use to implement a properties: p p concrete set? concrete set? � There is one specially designated node � What are the pro’s and con’s of each called the root , which has no edges approach? approach? pointing to it. i ti t it � List � Every node except the root has exactly one edge pointing to it edge pointing to it. � Other than arrays, could you implement a � Other than arrays could you implement a list with any other data structure? � There is a unique path (of nodes and edges) from the root to each node. g )
Basic Tree Concepts p Height and Level of a Tree g A � Height – # of edges on g g � Node – user-defined data structure that that contains pointers to data and pointers to other nodes: the longest path from the root to a leaf. � Root – Node from which all other nodes descend B C � Parent � Parent – has child nodes arranged in subtrees. has child nodes arranged in subtrees � Level – Root is at level � Child – nodes in a tree have 0 or more children. 0, its direct children are � Leaf – node without descendants D E at level 1, etc. t l l 1 t � Degree – number of direct children a tree/subtree has. � Recursive definition for height: height: F G 1+ max(height(T L ), height(T R )) H Rooted Trees: Example p Rooted Trees � If an edge goes from node a to node b , then a is � A is the root A is the root called the parent of b , and b is called a child of a . f b d b i ll d h ll d hild f A � D, E, G, H, J & � Children of the same parent are called siblings . K are leaves � If there is a path from a to b , then a is called an If there is a path from a to b , then a is called an � B is the parent � B is the parent ancestor of b , and b is called a descendent of a . B C of D, E & F � A node with all of its descendants is called a � D, E & F are subtree . subtree . siblings and siblings and F F children of B � If a node has no children, then it is called a leaf of D E G H the tree. � I, J & K are I subtree descendants of descendants of � If a node has no parent (there will be exactly one of � If a node has no parent (there will be exactly one of these), then it is the root of the tree. B � A & B are J K ancestors of I ancestors of I
Binary Trees y Binary Search Trees y � Intuitively, a binary tree is a tree in which each node has � A binary search tree is a binary tree in which each y y y y In other words can we In other words, can we no more than two children. node, n , has a value satisfying the following put non-hierarchical data into a tree. We properties: will study Binary � n s value is > all values in its left subtree, T L , � n ’s value is > all values in its left subtree T Search Trees later. � n ’s value is < all values in its right subtree, T R , and � T L and T R are both binary search trees . 21 John 3 34 (These two binary Brenda Peter trees are distinct .) 55 2 8 Amy Mary Tom 5 13 This term is Binary Trees y Binary Trees y ambiguous, some i di indicate that each t th t h node is either full or empty. � A binary tree is full if it has no missing nodes. � A binary tree of height h is complete if it is full down to y g y g p level h – 1 , and level h is filled from left to right. � It is either empty. � All nodes at level h – 2 and above have 2 children each, � Otherwise, the root’s subtrees are full binary trees � If a node at level h � If a node at level h – 1 has children, all nodes to its left 1 has children all nodes to its left of height h – 1. at the same level have 2 children each, and � If not empty, each node has 2 children, except the nodes at � If a node at level h – 1 has 1 child, it is a left child. level h which have no children. level h which have no children � Contains a total of 2 h+1 -1 nodes (how many leaves?)
Binary Trees y Complete & Balanced Trees p � A binary tree is balanced if the difference in height b between any node’s left and right subtree is ≤ 1. d ’ l f d i h b i 1 Complete and Balanced � Note that: � A full binary tree is also complete. � A complete binary tree is not always full. � A complete binary tree is not always full � Full and complete binary trees are also balanced. Not Balanced, Why? � Balanced binary trees are not always full or complete. Binary Tree: Pointer-Based Binary Tree: Table-Based Representation Representation Representation Representation Basic Idea: struct TreeNode; // Binary Tree nodes are struct ’s typedef string TreeItemType;// items in TreeNodes are string ’s d f i T I T // i i T N d i ’ � Instead of using pointers to the left and class BinaryTree right child of a node, use indices into an { private: private: array of nodes representing the binary tree. array of nodes representing the binary tree TreeNode *root; // pointer to root of Binary Tree � Also, use variable free as an index to the }; first position in the array that is available for first position in the array that is available for struct TreeNode // node in a Binary Tree: y a new entry. Use either the left or right { // place in Implementation file TreeItemType item; child indices to indicate additional, available TreeNode *leftChild; // pointer to TreeNode’s left positions. positions child child TreeNode *rightChild; // pointer to TreeNode’s right child � Together, the list of available positions in }; the array is called the free list . y
Binary Tree: Table-Based Binary Tree: Table-Based Representation Representation Representation Representation root root Index Item Left Right Index Item Left Right * Mary Added under Nancy. Child Child Child Child Child Child Child Child 0 0 0 Jane 1 2 0 Jane 1 2 1 Bob 3 4 1 Bob 3 4 free free Jane Jane 2 Tom 5 -1 2 Tom 5 -1 6 7 3 Alan -1 -1 3 Alan -1 -1 Bob Bob Tom Tom Bob Bob Tom Tom 4 4 Ellen Ell -1 1 -1 1 4 4 Ellen Ell -1 1 -1 1 5 Nancy -1 -1 5 Nancy 6 -1 6 ? -1 7 6 6 Mary Mary -1 1 -1 1 Alan Ellen Nancy Alan Ellen Nancy 7 ? -1 8 7 ? -1 8 8 ? -1 9 8 ? -1 9 Mary 9 . . . . . . . . . 9 . . . . . . . . . Binary Tree: Table-Based Binary Tree: Table-Based Representation Representation Representation Representation root Index Item Left Right * Ellen deleted. const int MaxNodes = 100; // maximum size of a Binary Tree Child Child Child Child typedef string TreeItemType; t d f t i T It T // it // items in TreeNodes are string’s i T N d t i ’ 0 0 Jane 1 2 struct TreeNode // node in a Binary Tree { 1 Bob 3 -1 free Jane TreeItemType item; TreeItemType item; 2 Tom 5 -1 int leftChild; // index of TreeNode’s left child 4 int rightChild; // index of TreeNode’s right child 3 Alan -1 -1 }; Bob Bob Tom Tom 4 4 ? ? -1 1 7 7 class BinaryTree l Bi T { 5 Nancy 6 -1 private: TreeNode node[MaxNodes]; 6 6 Mary Mary -1 1 -1 1 int root; // index of root of Binary Tree Alan Nancy 7 ? -1 8 int free; // index of free list, linked by rightChild }; 8 ? -1 9 Mary 9 . . . . . . . . .
Level Ordering g Array-Based Representation y p 1 1 1 1 2 3 2 3 4 5 6 7 4 5 6 7 Let i , 1 < i < n , be the number assigned to an element of a 1 2 3 6 complete binary tree. complete binary tree. Array-Based Representation y p Array-Based Representation y p � Array-based representations allow for � Array based representations allow for efficient traversal. Consider the node at index i index i. � Left Child is at index 2i+1. � Right Child is at index 2i+2 � Right Child is at index 2i+2. � Parent is at floor ( (i-1)/2 ).
Recommend
More recommend
