Post on 02-Jan-2016
description
transcript
O(lg n) Search Tree
Tree T is a search tree made up of n elements: x0 x1 x2 x3 … xn-1
No function (except transverse) takes more than O(lg n) in the worst case.
Functions:createEmptyTree() returns a newly created empty binary treedelete(T, p) removes the node pointed to by p from the tree Tinsert(T, x) returns T with x added in the proper location search(T, key) returns a pointer to the node in T that has a key that
matches key returns null if x is not found traverse(T) prints the contents of T in orderisEmptyTree(T) returns true if T is empty and false if it is not
Homework 5
• Describe how to implement a search tree that has a worst time search, insert, and delete time of no more than O(lg n). This tree should have no number of element limit.
• Do the six Search Tree functions.• Discuss how you would implement this if there
was a maximum to the number of elements.
AVL Tree
54
30
25 7910
5 60
86
72
84
210
+1
+1
0
-1
0
AVL Tree
• The five functions are the same.
• Except that the tree needs to be rebalanced after insertion or deletion.
• Keep track of the path used to insert/delete.
• Balance starting at the parent of the leaf inserted or deleted.
• Work up to the root.
Insertion Case 1
0
Insertion Case 1
-1
Insertion Case 2
+1
Insertion Case 2
0
Insertion Case 3
+1
Insertion Case 3
+2
Deletion Case 1
-1
Deletion Case 1
0
Deletion Case 2
0
Deletion Case 2
+1
Deletion Case 3
+1
Deletion Case 3
+2
Single Rotation
+2
+1
Double Rotation
+2
-1
Single Rotation
+2
+1
Single Rotation
+2
+1
A
B
Single RotationA B
Single RotationA B
+2 +1
Single Rotation
0 0
A B
Single Rotation
+2
+1
Single Rotation
0
+1
Single Rotation
0 +1
Single Rotation
0
+1
Single Rotation
0
+1
Single Rotation
0
0
Double Rotation
+2
-1
Double Rotation
+1
0
0
Double Rotation1
+2
-1
-1
A
B
C
Double Rotation1
+2 -1-1
A B C
Double Rotation1
+2
-1
-1
A
B
C
Double Rotation1
+2
-1
-1
A
B
C
Double Rotation1
+1
+1
-1
A
B
C
Double Rotation1
+1
+1-1
A
B C
Double Rotation1
+1
+1-1
A
B C
Double Rotation1
+1
+1
-1
A
B
C
Double Rotation1
+2
+1
+1
A
B
C
Double Rotation1
0
+1
+1
A
B
C
Double Rotation1
0
+1
+1
A B
C
Double Rotation1
0
+1
+1
A
B
C
Double Rotation1
0
+1
0
A
B
C
Double Rotation1
0+1
0
A
B
C
Double Rotation2
+2
-1
+1
A
B
C
Double Rotation2
+2
-1
+1
A
B
C
Double Rotation2
+2
0
+1
A
B
C
Double Rotation2
+2
0+1
A
B C
Double Rotation2
+2
0
+2
A
B
C
Double Rotation2
-1
0
+2
A
B
C
Double Rotation2
-1
0
+2
A B
C
Double Rotation2
-1
0
0
A
B
C
Double Rotation2
-10
0
A
B
C
Binary Search Tree -- Array
10 112 3 4 5 6 7 8 9 10
2 * i + 1 is the left child2 * i + 2 is the right child
Binary Search Tree -- Array
10 112 3 4 5 6 7 8 9 10
2 * i + 1 is the left child2 * i + 2 is the right child
Binary Search Tree -- Array
10 112 3 4 5 6 7 8 9 10
2 * i + 1 is the left child2 * i + 2 is the right child
Binary Search Tree -- Array
• Advantages– fast– can access the parent from a child
• Disadvantages– fixed size– standard AVL rotates greater than O(lg n)
Priority Queue
A Priority Queue Set S is made up of n elements: x0 x1 x2 x3 … xn-1
Functions:
createEmptySet() returns a newly created empty priority queue
findMin(S) returns the minimum node with respect to ordering
insert(S, x) returns S with x added
deleteMin(S) returns S with the minimum node removed
isEmptySet(S) returns true if S is empty and false if it is not
Homework 6
• Describe how to implement a priority queue that has a worst case findMin in O(1) time and insert and deleteMin in no more than O(lg n) time. You can assume that n is always less than 128. In other words, there is a max of 127 elements that can be stored in the queue.
• Do the five Priority Queue functions.
Priority Queue -- Lists
• Ordered Array – Find in constant time. – Insert and delete in O(n).
• Unordered Array– Insert in constant time. – Find and delete in O(n).
Priority Queue -- Lists
• Ordered Linked List– Find and delete in constant time. – Insert in O(n).
• Unordered Linked List– Insert in constant time. – Find and delete in O(n).
Priority Queue Trees
• Binary Search Tree– Find can be more than O(lg n) if out of balance.– Insert and delete can be more than O(lg n).
• AVL Tree– Find is O(lg n) time.– Insert and delete are O(lg n).
Priority Queue Trees
• AVL Tree with pointer to smallest– Find is O(1) time.– Insert and delete are O(lg n).– Works, but is way too complicated for the task
• We need a simpler solution
Homework 6
• Describe how to implement a priority queue that has a worst case findMin in O(1) time and insert and deleteMin in no more than O(lg n) time. You can assume that n is always less than 128. In other words, there is a max of 127 elements that can be stored in the queue.
• Do the five Priority Queue functions.