Download Trie and Search Trees

Trie and Search Trees
Dr. Andrew Wallace PhD BEng(hons) EurIng
andrew.wallace@cs.umu.se
Overview
• Trie
• Binary search trees
Trie
• Special type of tree
• retrieval
• Dynamic sets
• Key is a string
• Root node is an empty string
Trie
• Anna : data
• Andrew : data
• Alexandra : data
• Alwen : data
• Bertil : data
• Bridget : data
• Beryl : data
A
N
L
W
B
E
E
R
N
A
Data
Trie Implementation
• As a table
• 2 x 2 array
• One columns per letter in the alphabet, n
• One row per node, m
• log2 m to represent the data
Trie Implementation
• As a linked list
• Each node contains
• A letter
• Link to child
• de la Brandais tree
Trie operations
• Insert child
• Delete child
• Child
• Look up
Trie implementation
• Auto correct
• File structures
• DNA sequencing
• Data compression
• LZ78
• Huffman encoding
Trie implementation
• LZ78
• Lossless data compression
• Dictionary based
• Random access
Trie implementation
• She sells sea shells
S
E
H
_
E
_
S
L
A
L
H
E
L
L
S
Step
Phrase
Output
1
S
0, S
2
H
0, H
3
E
0, E
4
_
0, _
5
SE
1, E
6
L
0, L
7
LL
6, L
8
S_
1, _
9
A
0, A
10
_SHELLS
4, SHELLS
Trie
Encode(n)
Dictionary  empty
Prefix  empty
DictionaryIndex  1
while(n is not empty)
Char  next character in n
if(Prefix + Char exists in the Dictionary)
Prefix  Prefix + Char
else
if(Prefix is empty)
CodeWordForPrefix  0
else
CodeWordForPrefix  DictionaryIndex for Prefix
Output: (CodeWordForPrefix, Char)
insertInDictionary( ( DictionaryIndex , Prefix + Char) )
DictionaryIndex++
Prefix  empty
Huffman coding
• Frequency encoding
13:*
5:1
Frequency
Value
5
1
8
2
10
3
15
4
20
5
8:2
Huffman coding
23:*
10:3
Frequency
Value
10
3
13
*
15
4
20
5
13:*
Huffman coding
Frequency
Value
15
4
20
5
23
*
35:*
15:3
20:*
Huffman coding
Frequency
Value
23
*
35
*
48:*
35:4
23:5
Huffman coding
48:*
23:*
35:*
Frequency
Value
48
*
Huffman coding
48:*
• To encode:
• Right = 1 and left = 0
• Example
•
•
•
•
•
1 = 110
2 = 111
3 = 10
4 = 00
5 = 01
35:*
15:4
23:*
20:5
10:3
13:*
5:1
8:2
Huffman coding
• Applications
• Zip file compression
• Jpeg
• PNG
Binary Search trees
• Each node has max two child nodes
• Relationship between child nodes
• Nodes key is larger than all the nodes in the left sub tree
• Nodes key is smaller than all the nodes in the right sub tree
10
8
5
18
9
12
23
Binary Search trees
• Binary search tree and a binary tree
• In a binary search tree all nodes much have a label
• Delete and Insert needs a label as does create tree for the root
• Delete can break the tree
• Fix it downwards
• Insert must insert in sorted order
Binary Search trees
• Operations
• Search
• Insert
• Delete
10
Binary Search trees
8
• Search
• Fast if tree is ordered
• Does node have value x? Where is 12?
• Search left if node value is greater than x
• Search right if node value is less than x
• How long will it take?
• Tree is complete
• O(log n)
5
18
9
12
23
Binary Search trees
• Searching a tree
TreeSearch(x, k)
if x = NULL or k = key[x]
then return x
if k key[x]
then return TreeSearch(left[x], k)
else return TreeSearch(right[x], k)
Binary Search trees
• Insert
• Keep the tree complete
• Check left sub tree. If it is full, insert the value in the higher tree
and move old down the tree
10
9
9
6
5
18
8
10
10
Binary Search trees
TreeInsert(T,z)
y ← NULL
x ← NULL
while x = NULL
do y ← x
if key[z] < key[x]
then x ← left[x]
else x ← right[x]
p[z] ← y
if y = NULL
then root[T] = z
else if key[z] < key[y]
then left[y] ← z
else right[y] ← z
Binary Search trees
10
• Delete a node
• Case 1
8
18
• No sub trees
• Case 2
• One sub tree
• Case 3
• Two sub trees
5
9
12
Binary Search trees
• Case 2
10
• Move sub tree to the delete node’s position
• Delete 18
• Move 12 upwards
8
5
12
18
9
12
Binary Search trees
• Case 3
•
•
•
•
Delete the node
Promote the lowest value in the right sub tree
Delete 10
Promote 12
12
10
8
5
18
9
12
Binary Search trees
TreeDelete(T, z)
if left[z] = NULL or right[z] = NULL
then y ← TreeSuccessor(z)
if left[y] = NULL
then x ← left[y]
else x ← right[y]
if x = NULL
then p[x] ← p[y]
if p[y] = NULL
then root[T] ← x
else if y = left[p[y]]
then left[p[y]] ← x
else right [p[y]] ← x
if y = z
then key[z] ← key[y]
return y
Binary Search trees
• Applications
• Construction of a dictionary
• In order travers gives a sorted sequence
• Router tables in networking
Binary Search trees
• Balancing a binary search tree
• Rotations
• Self-balancing
• Performed at key times
Binary Search trees
• Binary tree extensions
• Quad tree
• As a binary tree but based on 4 instead of 2
• Breaks up a 2D region into 4 parts
• Handy for collision detection
Questions?