Tries

1
Tries

• When searching for the name Smith in a phone
  book, we first locate the group of names
  starting with S, then within those we search
  for m, etc.
• Idea Perform a search based on a prefix of the
  key, rather than a comparison of the whole key.
• The branching is determined by a partial key
  value.
• Each node branches out to as many nodes as there
  are characters.

2
Tries

• A Trie is a multi-way tree for storing strings in
  which
• there is one node for every common prefix
• the strings are stored in the leaves
• The order of the trie is m, where m is the size
  of the alphabet.

3
Tries

• Example The set of strings "mat", "mad, "am",
  "bad" stored in a BST and in a trie

m
mat
a
b
am
mad
m
a
a
am
bad
d
t
d
mat
mad
bad
4
Tries

• Advantage
• The height of the tree depends on the length of
  the keys
• A trie can be used to store very large sets but
  the height (and therefore the search time) is
  very short.
• Applications
• spell checker
• web search engine (search by index word)
• network router (search by IP address)

5
Example (slightly compressed...)
Alphabet A, P, S, T Words A, APT, AT,
PAT, PASS, PAST, PS, SAP, SAT, TAP
6
Tries

• Observation Many nodes have few non-null
  pointers.
• We would like to save space
• Idea 1 Briandais tree
• Store only the pointers that are used
• Maintain all siblings in a linked list
• Disadvantages
• the list needs to be traversed linearly
• we still use extra space for the pointers

7
Tries

• Observation Many nodes have few non-null
  pointers.
• We would like to save space
• Idea 2 Compressed Trie
• Interleave the arrays in the nodes
• Minor disadvantage
• an unsuccessful search may take more steps to end.

8
Compressed Tries
A
P
S

T
node 1
?
?
?
A
P
S

T
A
P
S

T
node 3
?
?
?
?
?
node 2
A
AT
PS
PA
APT
node 1 node 2 node 3
p1 stands for pointer to node 1
p2
p3
?
A
APT
AT
PA
PS
9
PATRICIA trees

• Observation
• Sometimes the actual set of keys is a small
  subset of the potential set of keys.
• This may result in a large number of nodes that
  only have one descendant.
• We would like to save some space
• Idea
• Make the tree more compact by collapsing long
  chains.
• The resulting tree is called a PATRICIA tree

Practical Algorithm To Retrieve Information
Coded In Alphanumeric
10
PATRICIA trees

• Idea
• Collapse chains of nodes that have only one child
• For each branch indicate how many characters
  should be skipped (i.e. what the length of the
  collapsed chain is)

11
Patricia tree example