- A compressed trie
is a trie with
one
additional rule:
- Each internal node has ≥ 2 children
Such compacted trie is als known as:
- Patricia tries or
Radix tries
Wikipedia page: click here
- In order to enforce the
above rule, the labels
are generalized:
- Each node is labeled
with a string (multiple characters)
(The label used to be a single character)
- Each node is labeled
with a string (multiple characters)
- Redundant node:
- An internal node
v is
redundant if
- Node v is
not the
root node,
and
- Node v has 1 child node
- Node v is
not the
root node,
and
Example:
- An internal node
v is
redundant if
- Redundant chain of edges:
- A chain of edges:
-
(v0,v1),
(v1,v2),
...,
(vk−1,vk)
is redundant if:
- Nodes
v1, v2, ..., vk−1
are
redundant
- Nodes v0 and vk are not redundant
- Nodes
v1, v2, ..., vk−1
are
redundant
Example:
- A chain of edges:
- Compression algorithm:
- Replace:
- a redundant chain or edges (v0,v1), (v1,v2), ..., (vk−1,vk)
by one edge:
- (v0 ,vk)
- Replace:
- the label vk
by the label:
- v1 v2 ... vk
Example:
- Before compression:
- After compression:
- Replace:
- Recall how the
standard trie is
implemented:
- The compressed trie
uses strings as
keys:
- Find a node that
has only one child node:
Result:
- Find a node that
has only one child node:
- Another example:
- Find a node that
has only one child node:
Result:
- Find a node that
has only one child node:
- Labels used in Compressed trie:
- variable length
strings
Example:
Fact:
- Variable length data items is a pain to store...
- variable length
strings
- Solution:
- Convert the
variable length string
representation to a
fixed length
string representation.
- The fixed length
string representation method:
- Store the keys
is a separate array of strings
- Store athe following
triplet to
represent a substring
of one of the keys:
- ( keyword_index, start_char_pos, end_char_pos )
- Store the keys
is a separate array of strings
- Convert the
variable length string
representation to a
fixed length
string representation.
- Example:
- Compressed trie with
Variable
length
string labels:
- Compressed trie with
Fixed
length
string labels:
Here is the complete picture (with full nodes):
- Compressed trie with
Variable
length
string labels:
- Properties:
- Each internal node has
≥ 2 children
and
≤ |Σ| children
- A compressed trie T storing
s strings (keys)
has:
s
external nodes
- A compressed trie T storing
s strings (keys)
has:
O(s)
total number of nodes
- Because a compressed trie is
in the worst case
comparable to a
binary tree
- In a binary tree:
- # external nodes = # internal nodes + 1
- Hence:
- # internal nodes in a compress trie
< s
- Total # nodes < 2 × s
- # internal nodes in a compress trie
< s
- Because a compressed trie is
in the worst case
comparable to a
binary tree
- Each internal node has
≥ 2 children
and
≤ |Σ| children