Dictionaries CS 105. 10/02/05 L7: Dictionaries Slide 2 Copyright 2005, by the authors of these...
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Transcript of Dictionaries CS 105. 10/02/05 L7: Dictionaries Slide 2 Copyright 2005, by the authors of these...
Dictionaries
CS 105
L7: DictionariesSlide 2
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Definition
The Dictionary Data Structure structure that facilitates searching objects are stored with search keys;
insertion of an object must include a key searching requires a key and returns the
key-object pair removal also requires a key
Need an Entry interface/class Entry encapsulates the key-object pair
(just like with priority queues)
L7: DictionariesSlide 3
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Sample Applications
An actual dictionary key: word object: word record (definition,
pronunciation, etc.) Record keeping applications
Bank account records (key: account number, object: holder and bank account info)
Student records (key: id number, object: student info)
L7: DictionariesSlide 4
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Dictionary Interface
public interface Dictionary{ public int size(); public boolean isEmpty(); public Entry insert( int key, Object value )
throws DuplicateKeyException;public Entry find( int key );
// return null if not found public Entry remove( int key )
// return null if not found;}
L7: DictionariesSlide 5
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Dictionary details/variations
Key types For simplicity, we assume that the keys are ints But the keys can be any kind of object as long as
they can be ordered (e.g., string and alphabetical ordering)
Duplicate entries (entries with the same key) may be allowed Our textbook calls the data structure that does
not allows duplicates a Map, while a Dictionary allows duplicates
For purposes of this discussion, we assume that dictionaries do not allow duplicates
L7: DictionariesSlide 6
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Dictionary Implementations
Unordered list (section 8.3.1) Ordered table (section 8.3.3) Binary search tree (section 9.1)
L7: DictionariesSlide 7
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Unordered list
Strategy: store the entries in the order that they arrive O( 1 ) insert operation
Can use an array, ArrayList, or linked list Find operation requires scanning the list
until a matching key value is found Scanning implies an O( n ) operation
Remove operation similar to find operation Entries need to be adjusted if using array/ArrayList O( n ) operation
L7: DictionariesSlide 8
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Ordered table
Idea: if the list was ordered by key, searching is simpler/easier
Just like for priority queues, insertion is slightly more complex Need to search for proper position of
element -> O( n ) Find: don’t do a linear scan; instead,
do a binary search Note: use array/ArrayList; not a linked
list
L7: DictionariesSlide 9
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Binary search
Take advantage of the fact that the elements are ordered
Compare the target key with middle element to reduce the search space in half
Repeat the process until the element is found or search space reduces to 1
Arithmetic on array indexes facilitate easy computation of middle position Middle of S[low] and S[high] is S[(low+high)/2] Not possible with linked lists
L7: DictionariesSlide 10
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Binary Search Algorithm
Algorithm BinarySearch( S, k, low, high )
if low > high then return null; // not foundelse mid (low+high)/2 e S[mid]; if k = e.getKey() then return e; else if k < e.getKey() then return BinarySearch( S, k, low, mid-1 ) else return BinarySearch( S, k, mid+1, high )
array of Entries target key
BinarySearch( S, someKey, 0, size-1 );
L7: DictionariesSlide 11
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Binary Search Algorithm
42 5 7 8 9 12 14 17 19 22 25 27 28 33 37
low mid high
find(22)
mid = (low+high)/2
L7: DictionariesSlide 12
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Binary Search Algorithm
42 5 7 8 9 12 14 17 19 22 25 27 28 33 37
highlow mid
find(22)
mid = (low+high)/2
L7: DictionariesSlide 13
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
low
Binary Search Algorithm
42 5 7 8 9 12 14 17 19 22 25 27 28 33 37
midhigh
find(22)
mid = (low+high)/2
L7: DictionariesSlide 14
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
low=mid=high
Binary Search Algorithm
42 5 7 8 9 12 14 17 19 22 25 27 28 33 37
find(22)
mid = (low+high)/2
L7: DictionariesSlide 15
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Time complexity of binary search
Search space reduces by half until it becomes 1
n -> n/2 -> n/4 -> … -> 1 log n steps
Find operation using binary search isO( log n )
L7: DictionariesSlide 16
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Time complexity
Operation insert()
find() remove()
Unsorted List O( 1 ) O( n ) O( n )
Ordered Table
O( n ) O( log n )
O(n )
L7: DictionariesSlide 17
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Binary Search Tree (BST)
Strategy: store entries as nodes in a tree such that an inorder traversal of the entries would list them in increasing order
Search, remove, and insert are allO( log n ) operations All operations require a search that
mimics binary search: go to left or right subtree depending on target key value
L7: DictionariesSlide 18
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Traversing a BST Insert, remove, and find operations all
require a key First step involves checking for a matching
key in the tree Start with the root, go to left or right child
depending on key value Repeat the process until key is found or a null
child is encountered (not found) For insert operation, duplicate key error occurs if
key already exists Operation is proportional to height of tree
( usually O(log n ) )
L7: DictionariesSlide 19
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Insertion in BST (insert 78)44
17 88
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28
65 97
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80
L7: DictionariesSlide 20
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
78
Insertion in BST44
17 88
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65 97
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80
L7: DictionariesSlide 21
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
78
Removal from BST (Ex. 1)44
17 88
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65 97
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w
z
Remove 32
L7: DictionariesSlide 22
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
78
Removal from BST (Ex. 1)44
17 88
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65 97
8254
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L7: DictionariesSlide 23
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
28
29
78
Removal from BST (Ex. 1)44
17 88
65 97
8254
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80
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17
L7: DictionariesSlide 24
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
78
Removal from BST (Ex. 2)44
17 88
32
28
65 97
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wRemove
65
L7: DictionariesSlide 25
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
78
Removal from BST (Ex. 2)44
17 88
32
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65 97
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54
L7: DictionariesSlide 26
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Removal from BST (Ex. 2)44
17 88
32
28
65 97
8254
29
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88
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82
w
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8080
76
L7: DictionariesSlide 27
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Time complexity for BSTs
O( log n ) operations not guaranteed since resulting tree is not necessarily “balanced”
If tree is excessively skewed, operations would be O( n ) since the structure degenerates to a list
Tree could be periodically reordered to prevent skewedness
L7: DictionariesSlide 28
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Time complexity (average case)
Operation insert() find() remove()
Unsorted List
O( 1 ) O( n ) O( n )
Ordered Table
O( n ) O( log n )
O(n )
BST O( log n )
O( log n )
O( log n )
L7: DictionariesSlide 29
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
Time complexity (worst case)
Operation insert() find() remove()
Unsorted List
O( 1 ) O( n ) O( n )
Ordered Table
O( n ) O( log n )
O(n )
BST O( n ) O( n ) O( n )
L7: DictionariesSlide 30
Copyright 2005, by the authors of these slides, and Ateneo de Manila University. All rights reserved 10/02/05
About BSTs
AVL tree: BST that “self-balances” Ensures that after every operation, the
difference between the left subtree height and the right subtree height is at most 1
O( log n ) operation is guaranteed Many efficient searching methods are
variants of binary search trees Database indexes are B-trees (number of
children > 2, but the same principles apply)