CS 225 Data Structures and Software Principles

1
CS 225 Data Structures and Software Principles

• Section 12
• Tries

2
Agenda

• Tries
• Basics
• Jasons Code
• Patricia Trees
• De La Briandais Trees
• Hybrid Trees

3
Tries

• Data structure optimized for lookups on a key
  that can be decomposed into characters
• Take into account overlapping strings
• Regular expression search
• Pattern searches in images
• Non-compact tries
• De La Briandais Trees, Patricia Trees

4
Trie Representation

• Tries are represented using a tree of arrays
• For a character set of size k, the corresponding
  Trie structure is a (k1)-ary tree

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Tries

• The i-th character (starting at 0) in the data
  corresponds to the node at depth i
• Need a mapping of character to an index in the
  array
• The extra one cell in the array is used to hold a
  null character represented by ?
• Points to a leaf
• Ideally, no need to store key in a leaf, since it
  is completely determined by path followed
• Info stored at the leaf
• Spend only constant time at each level

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Trie Example
Words in Trie
raft star start stir
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Tries

• Running time of Find operation O(L) where L is
  the length of the string we are looking for
• Unique trie for any set of search keys
• Advantage NOT dependent on the number of strings
  we have in the Trie structure
• Disadvantage memory waste
• 27 cell array, one per character needed for
  Strings
• Space (k1) nodes sizeof(pointer)

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Jasons CodeTrieNode Data

• TrieNode
• int nodeLevel // level of the node
• bool isLeaf
• ArrayltTrieNodegt subtries // array of ptrs to
  nodes
• String key // string key in leaf nodes
  Etype storedInfo

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Patricia Trees

• Acronym Practical Algorithm To Retrieve
  Information Coded In Alphanumeric
• Trick only allocate arrays that make a
  decision
• Do not store nodes with only one non-NULL cell
• Store in each node the index of the character
  position on which it discriminates
• Tradeoff Less space required, but more work for
  Insert and Remove
• Key no longer uniquely determined by path
• Now we must store keys in the leaf

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Patricia Tree Example
Words in Trie
raft star start stir
a b c
r
s
z
0

a
i
raft
2
skipt
t
stir
4
skipr
star
start
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One (Patricia) Tree Application

• Communication Networks (CS 438)
• Task of efficiently finding the longest match
  between an IP address and variable-length
  prefixes in a forwarding table (due to CIDR)
• Given packet for 128.174.5.130, where would it go?

Net/MaskLength NextHop
128.174.0.0/16 A
128.0.0.0/1 B
64.0.0.0/2 C
0.0.0.0/2 D
1000 0000 1011 1100
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De La Briandais Trees

• Trick convert arrays in Trie to sparse arrays
• Allocate space only for used cells in the arrays
• At each level we now have a linked list
• Array cells are now nodes that not only point
  down, but to the next used character on that
  level
• Advantage can save much space good when the
  linked lists are not long
• Disadvantage search is now dependent on k
  (alphabet size)

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de la Briandais Tree Example
Words in Trie
raft star start stir
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Hybrid Structures

• Patricia/de la Briandais
• Uses both optimizations
• We eliminate all one node linked lists in the de
  la Briandais tree
• Trie/Patricia/de la Briandais
• Highly optimized data structure
• Some levels have arrays and others have linked
  lists