Lecture 34. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 1

Transcription

1 Lecture 34 Log into Linux. Copy files on csserver from /home/hwang/cs215/lecture33/*.* In order to compile these files, also need bintree.h from last class. Project 7 posted. Due next week Friday, but want to talk about it today. No class next week on Friday (April 15) due to EECS senior project presentations. Next lecture will start next topic (Chapter 12.1). Questions? Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 1

2 Outline Huffman code trees Project 7 Binary search trees Example class: bag class Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 2

3 Huffman Code Trees Binary trees by themselves are not particularly useful. But as with arrays and linked lists, they can be used as an implementation technique for other data structures. The Huffman code tree in Project 7 is one such data structure. Since each code consists of two values (0 and 1), a binary tree can be used to represent each code as a path from the root to a leaf that contains the character encoded by the path. Class operations encode & decode messages using the constructed tree. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 3

4 bag Class Another data structure that can be represented by a binary tree is the bag class. Will use a binary tree in a particular way to make finding items in a bag more efficient than linear search. We first will concentrate on the storage rule and the basic algorithms for finding, inserting, and erasing data items. Then will look at the bag class operations Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 4

5 Binary Search Trees The storage rule we will use is called a binary search tree (BST). A BST is defined as a binary tree with the following ordering: For each node, the data values in the left subtree are less than or equal to the node value. For each node, the data values in the right subtree are greater than the node value. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 5

6 Binary Search Trees Here are some examples: Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 6

7 Finding Items Given root of BST on the right. How do we find 37? 37 < 50, go left 37 > 30, go right 37 > 35, go right 37 = 37, found item What about 58? < 58, go right 55 < 58, go right 58 < 60, go left null subtree, item not found 15 Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 7

8 Inserting Items Inserting items looks for the place the item would be if it were in the tree (i.e., use the find algorithm). First inserted item becomes root. Insert the following items: 35, 18, 25, 48, 72, 60. Tree on right shows first three inserts. Note that because of the ordering, an in-order print of the tree displays the values in ascending order Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 8

9 Erasing Items Three cases for erasing items. 50 Item is in a leaf, i.e., in a node with 0 children. E.g., 62. Just delete it; set child pointer to null. Item is in a node with 1 child. E.g. 25. Replace the node with its child and delete it. Item is in a node with 2 children. E.g., Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 9

10 Erasing Items Cannot just delete 50 or replace it with a child. Need to convert situation into case of one or two children. Replace the node value with its immediate predecessor, the largest value in the left subtree. How to find? Go left, then go right until reach last node in line, 37 in this example immediate predecessor of 50 Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 10

11 Erasing Items The immediate predecessor is chosen because it preserves the BST ordering. It will be at least as large as than any of the left subtree values, and it will be greater than any of the right subtree values. By definition, the node with the immediate predecessor will have no right child, so it may be deleted using Case 1 or Case 2 procedures Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 11

12 In-class Exercise Starting with an empty BST, draw the tree that results from inserting the following integers: 30, 67, 17, 47, 15, 25, 50, 45, 19, 27 Continuing with this tree, draw the result of deleting (in order): 50, 67, 17 Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 12

13 bag Class Definition Examine the bag template class definition in file bag6.h Has the same typedefs as the other bag classes. Otherwise operation prototypes are the exactly the same as the previous bag classes. Note that a bag only receives and returns data items; never pointers to the nodes. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 13

14 bag Class Attributes This bag class is implemented using a binary search tree. There is one attribute: root_ptr a pointer to the root node of the binary search tree that contains the bag items Note that the textbook once again chooses to compute the size of the data structure rather than keep track of the size as items are inserted and removed, making the size( ) operation O(n) rather than O(1). Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 14

15 bag Object Here is a way to think about how this works. bag<int> b; b.insert(15); b.insert(34); root_ptr Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 15

16 bag Class Usage Examine file bag6test.cpp This is another example of an interactive test program. Note that using the bag object is the same as previous usages. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 16

17 bag Class Implementation Examine the rest of file bag6.h In addition to the bag class operation implementations, there are several BST utility functions that encapsulate BST operations on BTNodes. Various parts are left as exercises, some of which will be completed today during class. As before, we will ignore any bad_alloc exceptions. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 17

18 Constructors, Destructor, Assignment operator= The default constructor simply sets root_ptr to null to indicate an empty tree. The copy constructor simply uses tree_copy( ) to make a copy of the source's tree. The destructor simply uses tree_clear( ) to destroy the tree's nodes. The assignment operator= makes a selfassignment test, then uses tree_clear( ) and tree_copy( ). Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 18

19 count( ) Since the storage representation is a BST, we can use an iterative algorithm to provide the count of a particular element. Since the left subtree contains items less than or equal to the node value, we need to find the first node that equals the target. Once there we need to count the node containing the target as we continue down the left subtree. Eventually, we get to a null child. Complete the count( ) function in bag6.h Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 19

20 insert( ) Special case: empty tree Create a node with the entry and make it the root node Otherwise, want to find the place the entry goes. Use a boolean flag done to know when to stop the loop that looks for this place. Need a pointer to the node where the new entry node is attached. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 20

21 insert( ) Basic find algorithm 1. Initialize cursor to root_ptr and done to false 2. While not done do 2.1. Check if entry <= cursor node value If so, check if left child exists Attach new node with entry as left child Set done to true Else Move cursor to left child 2.2. Else Do the same for the right child Complete the insert( ) function in bag6.h Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 21

22 erase_one( ), erase( ) The erasing algorithm for BSTs can be implemented directly, and is not too difficult. However, a parent pointer must be maintained in order to delete a node from a tree in a manner similar to singly-linked lists. The textbook presents an indirect method using recursion. Auxilliary functions, bst_remove( ) bst_remove_max( ), and bst_remove_all( ) are used. Completing these functions is left as an exercise for the curious. Wednesday, April 6 CS 215 Fundamentals of Programming II - Lecture 34 22