CSE Data Structures and Introduction to Algorithms... In Java! Instructor: Fei Wang. Binary Search Trees. CSE2100 DS & Algorithms 1

Transcription

1 CSE 2100 Data Structures and Introduction to Algorithms...! In Java!! Instructor: Fei Wang! Binary Search Trees 1

2 R-10.6, 10.7!!! Draw the 11-entry hash table that results from using the hash function (3i+5) mod 11, to hash the keys 12,44,13,88,23,94,11,39,20,16 and 5, assuming collisions are handled by chaining or linear probing. 2

3 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

4 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

5 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

6 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

7 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

8 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

9 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

10 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

11 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

12 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

13 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

14 Chaining (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

15 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

16 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

17 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

18 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

19 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

20 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

21 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

22 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

23 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

24 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

25 Linear Probing (3i+5) mod 11 12,44,13,88,23,94,11,39,20,16 and

26 Exercise An anagram is a type of word play, the result of rearranging the letters of a word or phrase to produce a new word or phrase, using all the original letters exactly once; for example, the word anagram can be rearranged into nag-a-ram.! Given an array of strings, return all groups of strings that are anagrams. Note: All inputs will be in lower-case. 26

27 Create a HashMap:! Method key: String (after chars are sorted);! value: The specific corresponding string.! Then, iterate through the HashMap. If there is a key with more than two values, we add them into the result container. 27

28 Solution public ArrayList<String> anagrams(string[] strs) { ArrayList<String> res = new ArrayList<String>(); Hashtable<String, LinkedList<String>> hash = new Hashtable<String, import java.util.comparator; import java.util.*; import java.util.hashtable; import java.util.linkedlist;! public class Solution { public String sortchars(string s) { char[] content = s.tochararray(); Arrays.sort(content); return new String(content); }! } } /* Group words by anagram */ for (String s : strs) { String key = sortchars(s); if (!hash.containskey(key)) { hash.put(key, new LinkedList<String>()); } LinkedList<String> anagrams = hash.get(key); anagrams.push(s); } for (String key : hash.keyset()) { LinkedList<String> list = hash.get(key); if (list.size() > 1) { for (String t : list) { res.add(t); } } } return res; 28

29 Binary Trees! A binary tree is a tree with the following properties: " Each internal node has at most two children (exactly two for proper binary trees) " The children of a node are an ordered pair! We call the children of an internal node left child and right child! Alternative recursive definition: a binary tree is either " a tree consisting of a single node, or " a tree whose root has an ordered pair of children, each of which is a binary tree! Applications: D " arithmetic expressions " decision processes " searching B H E I A F C G 29

30 Full Binary Tree 8 full binary tree: a binary tree is which each node was exactly 2 or 0 children 30

31 Complete Binary Tree complete binary tree: a binary tree in which every level, except possibly the deepest is completely filled. At depth n, the height of the tree, all nodes are as far left as possible 31

32 Balance a" b" c" d" e" f" g" h" i" j" A balanced binary tree d" c" b" f" a" e" g" h" i" j" An unbalanced binary tree! A binary tree is balanced if every level above the lowest is full (contains 2 n nodes)! In most applications, a reasonably balanced binary tree is desirable 32

33 Sorted Binary Trees! A binary tree is sorted if every node in the tree is larger than (or equal to) its left descendants, and smaller than (or equal to) its right descendants! Equal nodes can go either on the left or the right (but it has to be consistent)

34 BST Representation! Represented by a linked data structure of nodes.! root(t) points to the root of tree T.! Each node contains fields:» key» left pointer to left child: root of left subtree.» right pointer to right child : root of right subtree.» p pointer to parent. p[root[t]] = NIL (optional). 34

35 BST Property! Stored keys must satisfy the binary search tree property.» y in left subtree of x, then key[y] key[x].» y in right subtree of x, then key[y] key[x]

36 BST Examples m g q b k x e

37 Searching a BST! Describe an algorithm for searching a binary search tree.! Try searching for the value 31, then 6.! What is the maximum number of nodes you would need to examine to perform any search? overall root

38 BST Examples m g q b k x e

39 In-Order Traversal visit left subtree visit node visit right subtree What does this guarantee with a BST? In order listing:

40 Search! To search for a key k, we trace a downward path starting at the root! The next node visited depends on the comparison of k with the key of the current node! If we reach a leaf, the key is not found! Example: get(4):! Call TreeSearch(4,root)! The algorithms for nearest neighbor queries are similar Algorithm TreeSearch(k, v) if T.isExternal (v) return v if k < key(v) return TreeSearch(k, left(v)) else if k = key(v) return v else { k > key(v) } return TreeSearch(k, right(v)) 2 <! >! =! 9 40

41 Insertion! To perform operation put(k, o), we search for key k (using TreeSearch)! Assume k is not already in the tree, and let w be the leaf reached by the search! We insert k at node w and expand w into an internal node! Example: insert 5 2 <! >! w >! w

42 Deletion! To perform operation remove(k), we search for key k! Assume key k is in the tree, and let let v be the node storing k! If node v has a leaf child w, we remove v and w from the tree with operation removeexternal(w), which removes w and its parent! Example: remove w 2 >! <! v

43 Deletion! We consider the case where the key k to be removed is stored at a node v whose children are both internal! we find the internal node w that follows v in an inorder traversal 1 2 z 3 w v ! we copy key(w) into node v! we remove node w and its left child z (which must be a leaf) by means of operation removeexternal(z)! Example: remove v

44 Exercise!!!! Validate if a binary tree is a binary search tree 44

45 BST Definition!! The left subtree of a node contains only nodes with keys less than the node's key. The right subtree of a node contains only nodes with keys greater than the node's key. Both the left and right subtrees must also be binary search trees. 45

46 JAVA Code 1 public boolean isvalidbst(treenode root) { 2 return isbst(root, Integer.MIN_VALUE, Integer.MAX_VALUE); 3 } 4 5 public boolean isbst(treenode node, int low, int high){ 6 if(node == null) 7 return true; 8 9 if(low < node.val && node.val < high) 10 return isbst(node.left, low, node.val) && isbst(node.right, node.val, high); 11 else 12 return false; 13 }! 46