Science Outline
Computer 331
Binary Search Trees Insertion and Deletion
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BST Insertion BST Deletion Case 1 Case 2 Case 3 Case 4 Complexity Discussion References
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Mike Jacobson
Department of Computer Science University of Calgary
Lecture #13
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Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
1 / 17
Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
2 / 17
BST Insertion
BST Insertion
Insertion: An Example
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A Recursive Insertion Algorithm
// Non-recursive public function calls recursive worker function public void insert(E key, V value) { root = insert(root, key, Value); } protected bstNode<E,V> insert(bstNode<E,V> T, E newKey, V newValue) { if (T == null) else if (newKey.compareTo(T.key) < 0) else if (newKey.compareTo(T.key) > 0) else return T; }
Computer Science 331 Lecture #13 3 / 17 Mike Jacobson (University of Calgary) Computer Science 331 Lecture #13 4 / 17
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10
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Idea: Nodes Visited (inserting 9): Start at 6 : Next node Next node Next node
Mike Jacobson (University of Calgary)
BST Insertion
BST Insertion
Analysis: Partial Correctness
Prove that insert is partially correct for all trees T of height h. Base cases are correct (by inspection): empty tree replaced by new node containing newKey and newValue if T.key == newKey, a KeyFoundExcpetion is thrown Assume that the algorithm is correct for all trees of height h 1 : if newKey < T.key, key/value inserted in left subtree if newKey > T.key, key/value inserted in right subtree in either case, algorithm is called recursively on a subtree of height at most h 1 and new subtree is correct by assumption the new T is still a BST, because both children are BSTs and the new element was added to the correct subtree
Mike Jacobson (University of Calgary) Computer Science 331 Lecture #13 5 / 17
Termination and Bound on Running Time
Let hi denote the height of the subtree with root x at level i of the recursion. Consider the function f (i) = hi + 1 :
Worst case running time is (h) :
Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
6 / 17
BST Deletion
BST Deletion
Case 1
Deletion: Four Important Cases
First Case: Key Not Found
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10 1 5 7
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Idea: Key is/has . . .
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Not Found (Eg: Delete 8) At a Leaf (Eg: Delete 7) One Child (Eg: Delete 10) Two Children (Eg: Delete 6)
Computer Science 331 Lecture #13 7 / 17
Nodes Visited (delete 8): Start at 6 : Next node Next node Next node
Mike Jacobson (University of Calgary)
Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
8 / 17
BST Deletion
Case 1
BST Deletion
Case 2
Algorithm and Analysis
protected bstNode<E,V> delete(bstNode<E,V> T, E key) { if (T != null) { if (key.compareTo(T.key) < T.left 0) = delete(T.left, key); else if (key.compareTo(T..key) > 0) T.right = delete(T.right,key); else if ... // found node with given key } else throw new KeyNotFoundException(); return T; }
Second Case: Key is at a Leaf
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Idea: Nodes Visited (delete 7): Start at 6 : Next node Next node
Mike Jacobson (University of Calgary) Computer Science 331 Lecture #13 10 / 17
Correctness and Efciency For This Case: tree is not modied if key is not found (base case will be reached) worst-case cost (h) (same as search)
Mike Jacobson (University of Calgary) Computer Science 331 Lecture #13 9 / 17
BST Deletion
Case 2
BST Deletion
Case 3
Algorithm and Analysis
Third Case: Key is at a Node with One Child
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Extension of Algorithm: else if ()
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Correctness and Efciency For This Case: Idea: Nodes Visited (delete 10): Start at 6 : Next node
Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
11 / 17
Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
12 / 17
BST Deletion
Case 3
BST Deletion
Case 4
Algorithm and Analysis
Fourth Case: Key is at a Node with Two Children
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Extension of Algorithm: else if (T.left == null) else if (T.right == null) Correctness and Efciency For This Case: Idea: Nodes Visited (delete 6): Start at 6 :
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Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
13 / 17
Mike Jacobson (University of Calgary)
Computer Science 331
Lecture #13
14 / 17
BST Deletion
Case 4
Complexity Discussion
Algorithm and Analysis
Extension of Algorithm: else {
More on Worst Case
All primitive operations (search, insert, delete) have worst-case compl...