Friday Four Square! 4:15PM, Outside Gates
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1 Binary Search Trees
2 Friday Four Square! 4:15PM, Outside Gates
3 Implementing Set On Monday and Wednesday, we saw how to implement the Map and Lexicon, respectively. Let's now turn our attention to the Set. Major operations: Insert Remove Contains
4 An Inefficient Implementation We could implement the Set as a list of all the values it contains. To add an element: Check if the element already exists. If not, append it. To remove an element: Find and remove it from the list. To see if an element exists: Search the list for the element.
5 Using Hashing If we have a hash function for the elements being stored, we can implement a Set using a hash table. What is the expected time to insert a value? Answer: O(1). What is the expected time to remove a value? Answer: O(1). What is the expected time to check if a value exists? Answer: O(1). However, writing a good hash function for a set of elements can be very tricky!
6 Using Tries If our keys are strings, we can store the set using a trie. Looking up or inserting a string with L letters takes time O(L). Doesn't work for arbitrary values.
7 An Entirely Different Approach
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34 Binary Search Trees The data structure we have just seen is called a binary search tree (or BST). Uses comparisons between elements to store elements efficiently. No need for a complex hash function, or the ability to work one symbol at a time.
35 The Intuition
36 The Intuition
37 The Intuition
38 The Intuition
39 The Intuition
40 The Intuition Values less than two Values greater than two
41 The Intuition Values less than two Values greater than two
42 The Intuition
43 The Intuition Values less than -1 Values greater than -1
44 The Intuition Values less than -1 Values greater than -1
45 The Intuition
46 Tree Terminology As with a trie, a BST is a collection of nodes. The top node is called the root node. Nodes with no children are called leaves
47 A Recursive View of BSTs
48 A Recursive View of BSTs
49 A Recursive View of BSTs
50 Implementing Lookups
51 Inserting into a BST
52 Inserting into a BST
53 Inserting into a BST
54 Inserting into a BST
55 Inserting into a BST
56 Inserting into a BST
57 Let's Code it Up!
58 Insertion Order Matters Suppose we create a BST of numbers in this order: 4, 2, 1, 3,, 5,
59 Insertion Order Matters Suppose we create a BST of numbers in this order: 1, 2, 3, 4, 5,,
60 Tree Terminology The height of a tree is the number of nodes in the longest path from the root to a leaf
61 Tree Terminology The height of a tree is the number of nodes in the longest path from the root to a leaf
62 Efficiency of Insertion What is the big-o complexity of adding a node to a tree? Depends on the height of a tree! Worst-case: have to take the longest path down to find where the node goes. Time is O(h), where h is the height of the tree.
63 Tree Heights What are the maximum and minimum heights of a tree with n nodes? Maximum height: all nodes in a chain. Height is O(n). Minimum height: Tree is as complete as possible. Height is O(log n)
64 Tree Heights What are the maximum and minimum heights of a tree with n nodes? Maximum height: all nodes in a chain. Height is O(n). Minimum height: Tree is as complete as possible. Height is O(log n)
65 Keeping the Height Low There are many modifications of the binary search tree designed to keep the height of the tree low (usually O(log n)). A self-balancing binary search tree is a binary search tree that automatically adjusts itself to keep the height low. Details next time.
66 Walking Trees
67 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
68 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
69 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
70 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
71 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
72 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
73 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
74 Walking a BST One advantage of a BST is that elements are stored in sorted order. We can iterate over the elements of a BST in sorted order by walking the tree recursively
75 Tree Traversals There are three general types of tree traversals: Preorder: Visit the node, then visit the children. Inorder: Visit the left child, then the node, then the right child. Postorder: Visit the children, then visit the node.
76 Walking a Tree Inorder Preorder Postorder
77 Getting Rid of Trees
78 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
79 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
80 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
81 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
82 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
83 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
84 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
85 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
86 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
87 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
88 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
89 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
90 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
91 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them
92 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. 4 7
93 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. 4 7
94 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. 4 7
95 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. 4
96 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. 4
97 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. 4
98 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. 4
99 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them.
100 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them.
101 Freeing a Tree Once we're done with a tree, we need to free all of its nodes. As with a linked list, we have to be careful not to use any nodes after freeing them. This is done as follows: Base case: There is nothing to delete in an empty tree. Recursive step: Delete both subtrees, then delete the current node. What kind of tree traversal is this?
102 Range Searches We can use BSTs to do range searches, in which we find all values in the BST within some range. For example: If values in a BST are dates, can find all events that occurred within some time window. If values in a BST are samples of a random variable, can find everything within one and two standard deviations above the mean.
103 The Intuition
104 The Intuition
105 The Intuition
106 The Intuition
107 The Intuition
108 The Intuition
109 The Intuition
110 The Intuition
111 The Logic Base case: The empty tree has no nodes within any range. Recursive step: If this node is below the lower bound, recursively search the right subtree. If this node is above the upper bound, recursively search the left subtree. If this node is within bounds: Search the left subtree. Add this node to the output. Search the right subtree.
112 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
113 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
114 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
115 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
116 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
117 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
118 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
119 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
120 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
121 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find.
122 Complexity of Range Searches How do we get a runtime for a range search? Depends on how many nodes we find. If there are k nodes within the range, we do at least O(k) work finding them. In addition, we have two border sets of nodes that are immediately outside that range. Each set has size O(h), where h is the height of the tree. Total work done is O(k + h). This is an output-sensitive algorithm.
123 Next Time Fun With Data Structures. Balanced binary search trees. Ternary search trees. DAWGs.
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