This Lecture. SOFT : BST, Priority Queue, Heap. Binary Search Trees. Recall binary trees. The Binary Search Tree. The Binary Search Tree 6/4/07

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1 SOFT : BST, Priority Queue, Heap Binary Search Trees Priority Queues This Lecture School of Information Technologies SOFT1002 The University of Sydney 1 2 Binary Search Trees Recall binary trees A binary search tree is a binary tree that you can use to search! If it is balanced then searching is O(log n) where n is the number of leaves have a left and right child, either or both of which can be null can have a root 3 4 The Binary Search Tree The Binary Search Tree A binary search tree is good for Searching for a particular item Each node n in a binary search tree satisfies the following properties n s value is greater than all values in its left subtree T L n s value is less than all values in its right subtree T R Both T L and T R are binary search trees Record could be A group of related items, called fields, that are not necessarily of the same data type Field A data element within a record A data item in a binary search tree has a specially designated search key A search key is the part of a record that identifies it within a collection of records KeyedItem class Contains the search key as a data field and a method for accessing the search key Must be extended by classes for items that are in a binary search tree 5 6 1

2 The Binary Search Tree Operations of the ADT binary search tree Insert a new item into a binary search tree Delete the item with a given search key from a binary search tree Retrieve the item with a given search key from a binary search tree Traverse the items in a binary search tree in preorder, inorder, or postorder Figure A binary search tree (This and subsequent figures from Carrano and Prichard, Algorithms for Operations on BSTs Since the binary search tree is recursive in nature, it is natural to formulate recursive algorithms for its operations A search algorithm search(bst, searchkey) Searches the binary search tree bst for the item whose search key is searchkey 2006) 7 8 Binary Search Tree: Insertion Binary Search Tree: Insertion insertitem(treenode, newitem) Inserts newitem into the binary search tree of which treenode is the root Figure 10.21c c) insertion at a leaf Figure 10.21a and 10.21b a) Insertion into an empty tree; b) search terminates at a leaf 9 10 Binary Search Tree: Deletion Binary Search Tree: Deletion Steps for deletion Use the search algorithm to locate the item with the specified key If the item is found, remove the item from the tree Three possible cases for node N containing the item to be deleted N is a leaf N has only one child N has two children Strategies for deleting node N If N is a leaf Set the reference in N s parent to null If N has only one child Let N s parent adopt N s child If N has two children Locate another node M that is easier to remove from the tree than the node N Copy the item that is in M to N Remove the node M from the tree

3 Binary Search Tree: Retrieval Binary Search Tree: Traversal Retrieval operation can be implemented by refining the search algorithm Return the item with the desired search key if it exists Otherwise, return a null reference Traversals for a binary search tree are the same as the traversals for a binary tree The inorder traversal of a binary search tree T will visit its nodes in sorted search-key order Reference-Based Implementation Efficiency of Search Operations BinarySearchTree Extends BinaryTreeBasis Inherits the following from BinaryTreeBasis isempty() makeempty() getrootitem() The use of the constructors TreeIterator Can be used with BinarySearchTree 15 The maximum number of comparisons for a retrieval, insertion, or deletion is the height of the tree The maximum and minimum heights of a binary search tree n is the maximum height of a binary tree with n nodes Figure A maximum-height binary tree with seven nodes 16 Efficiency of BST Operations Efficiency of BST Operations A full binary tree of height h 0 has 2 h 1 nodes The maximum number of nodes that a binary tree of height h can have is 2 h 1 The minimum height of a binary tree with n nodes is log 2 (n+1) The height of a particular binary search tree depends on the order in which insertion and deletion operations are performed Figure Counting the nodes in a full binary tree of height h Figure The order of the retrieval, insertion, deletion, and traversal operations for the reference-based implementation of the ADT binary search tree

4 Efficiency of BST Operations Priority Queues The efficiency of the binary search tree implementation of the ADT table is related to the tree s height Height of a binary search tree of n items Maximum: n Minimum: log 2 (n + 1) Height of a binary search tree is sensitive to the order of insertions and deletions Variations of the binary search tree Can retain their balance despite insertions and deletions A priority queue Orders its items by a priority value The first item removed is the one having the highest priority value Operations of the ADT priority queue Create an empty priority queue Determine whether a priority queue is empty Insert a new item into a priority queue Retrieve and then delete the item in a priority queue with the highest priority value Possible implementations Priority Queue as a variation of an ADT Table Sorted linear implementations Appropriate if the number of items in the priority queue is small Array-based implementation Maintains the items sorted in ascending order of priority value Reference-based implementation Maintains the items sorted in descending order of priority value Figure 11.7a and 11.7b Some implementations of the ADT priority queue: a) array based; b) reference based The ADT Priority Queue: A Variation of the ADT Table Possible implementations (Continued) Binary search tree implementation Appropriate for any priority queue A heap is a complete binary tree That is empty or Whose root contains a search key greater than or equal to the search key in each of its children, and Whose root has heaps as its subtrees Figure 11.7c Some implementations of the ADT priority queue: c) binary search tree

5 Maxheap A heap in which the root contains the item with the largest search key Minheap A heap in which the root contains the item with the smallest search key Pseudocode for the operations of the ADT heap createheap() // Creates an empty heap. heapisempty() // Determines whether a heap is empty. heapinsert(newitem) throws HeapException // Inserts newitem into a heap. Throws // HeapException if heap is full. heapdelete() // Retrieves and then deletes a heap s root // item. This item has the largest search key : An Array-based Implementation of a Heap : heapdelete Data fields items: an array of heap items size: an integer equal to the number of items in the heap Step 1: Return the item in the root Results in disjoint heaps Figure 11.8 A heap with its array representation Figure 11.9a a) Disjoint heaps : heapdelete : heapdelete Step 2: Copy the item from the last node into the root Results in a semiheap Step 3: Transform the semiheap back into a heap Performed by the recursive algorithm heaprebuild Figure 11.9b b) a semiheap Figure Recursive calls to heaprebuild

6 Efficiency heapdelete is O(log n) : heapdelete : heapinsert Strategy Insert newitem into the bottom of the tree Trickle new item up to appropriate spot in the tree Efficiency: O(log n) Heap class Represents an array-based implementation of the ADT heap Figure Deletion for a heap Figure Insertion into a heap A Heap Implementation of the ADT Priority Queue A Heap Implementation of the ADT Priority Queue Priority-queue operations and heap operations are analogous The priority value in a priority-queue corresponds to a heap item s search key PriorityQueue class Has an instance of the Heap class as its data field A heap implementation of a priority queue Disadvantage Requires the knowledge of the priority queue s maximum size Advantage A heap is always balanced Finite, distinct priority values A heap of queues Useful when a finite number of distinct priority values are used, which can result in many items having the same priority value ort Strategy Transforms the array into a heap Removes the heap's root (the largest element) by exchanging it with the heap s last element Transforms the resulting semiheap back into a heap Efficiency Compared to mergesort Both heapsort and mergesort are O(n * log n) in both the worst and average cases Advantage over mergesort ort does not require a second array Compared to quicksort Quicksort is the preferred sorting method Figure a) The initial contents of anarray; b) anarray s corresponding binary tree Figure ort partitions an array into two regions ort

7 Summary Summary Binary Search Trees (BSTs) can be used to find items in a collection in O(log n) time on average insertion works by searching for the item; always insert at a leaf deletion works by searching; three cases in-order traversal visits nodes in ascending order of key value have worst-case efficiency of O(n) if trees very unbalanced Priority Queue keeps items with a priority stored in increasing or decreasing order can serve item with highest priority can be implemented as a heap are binary trees with nice balance properties insert and delete are O(log n) can be used as a priority queue can be used to sort items (details not part of this unit) Tutorial and Lab Lab information This week in your tutorial you should answer the questions on BSTs ask your tutor about things you are unsure of This week in the lab you must demonstrate your DSP3 run the main method show insertion and deletion of Students there will be a list of test data available on the unit website. You can construct the following Student objects and grades: [ Student ] [ Grade ] Forename Surname SID Grade Susan Smith D Carl Jones B+ Susan Smith B+ Alex Yang A David Jones A- The first three columns are information for the Student object; you should then insert the Students in your Map collection with their grades. Your main method should print out a list of students with their grades, in ascending order as described in the text Next time Review! I will quickly go over the material covered in the unit. 41 7