Lecture 19 Tries Linked List - PowerPoint Presentation

Slide1

Lecture 19

Tries

Slide2

Linked List

Stack

Queue

Heap

Priority Queue

Hash Table

BST

AVL

Multiway

Red-

Black

Algorithm Analysis Tools

Slide3

Is finding a pattern same as finding a word in the dictionary?

Slide4

How fast can you search for a word in a set of words?

Slide5

How to make ordered tree for strings?

Slide6

Standard Tries

Except root, each node is labeled with a characterThe children of a node are alphabetically orderedThe path from root to external nodes yields the string

Slide7

S = { bear, bell, bid, bull, buy, sell, stock, stop }

Slide8

Insert

Assumption: no word is prefix of other word!What if this is not the case?

Slide9

Time complexity

n total size of the strings in Sm size of given string d size of the alphabet

s number of words

Slide10

Search?

O(m)

Slide11

Insert

& delete?O(md)

Slide12

Word Matching with a

Trie

s

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b

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?

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b

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a

Slide13

How many nodes do we need?

Can we reduce the number of nodes?

Slide14

Compressed Tries

Slide15

Compressed Tries

A compressed trie has internal nodes of degree at least twoIt is obtained from standard trie by compressing chains of “redundant” nodes

A redundant node is the one that has only one child

Slide16

How many nodes do I have in a compressed

Trie?

Slide17

Compressed

Trie PropertiesEvery internal node of T has at least two children and at most d childrenT has s leaf nodesThe number of internal nodes of T is O(s)

Slide18

1,0,0

1,1,1

4

,1,1

0

,1,1

3

,1,2

0

,0,0

7,0,3

6,1,2

1,2,3

8,2,3

4,2,3

5,2,2

0,2,2

2,2,3

3,3,4

9,3,3

hear

e

ll

e

Slide19

Compact Representation of Tries

Stores at the nodes ranges of indices instead of substringsUses O(s) space, where s is the number of strings in the arrayServes as an auxiliary index structure

Slide20

Insertion/Deletion in Compressed

Trie

f

g

reat

one

u

or

n

r

insert green

f

g

reat

one

u

or

n

r

insert

gre

en

search ends

here

f

g

re

one

u

or

n

r

at

en

Slide21

Suffix

Trie

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Slide22

Suffix

TrieNot all strings are guaranteed to have corresponding suffix trie.For example: xabxa

a

bxa

bxa

xa

bxa

a

xa

bxa

bxa

bxa

$

$

Slide23

Pattern Matching Using Suffix

Trie

Compressed Trie

Auxiliary Trie

Slide24

How much space do we need to make a suffix

trie of n character string?

Slide25

To

Trie or Not to Trie?

What would you use to store phonebook?

Slide26

Constructing a Suffix

TrieS[1..n] is the stringStart with a single edge for SEnter

the edges for the suffix S[i..n] where i goes from 2 to nStarting at the root node find the longest part from the root whose label matches a prefix of S[i..n]. At some point, no further matches are possibleIf the point is at a node, then denote this node by wIf it is in the middle of an edge, then insert a new node called w, at this pointCreate

a new edge running from w to a new leaf labeled S[i..n]

Slide27

Example

minimize

minimize

m

inimize

minimize

inimize

m

i

nimize

m

in

imize

minimize

inimize

nimize

minimize

nimize

m

ize

nim

ize

i

Slide28

Example

m

in

imize

minimize

nimize

m

ize

nim

ize

i

m

ini

mize

nimize

m

ize

nim

ize

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inimize

ize

m

Slide29

Example

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