Tries
e mize
i nimize ze nimize
mi ze
nimize
ze
2004 Goodrich, Tamassia
Tries
Preprocessing Strings
Preprocessing the pattern speeds up pattern matching queries
After preprocessing the pattern, KMPs algorithm performs pattern matching in time proportional to the text size
If the text is large, immutable and searched for often (e.g., works by Shakespeare), we may want to preprocess the text instead of the pattern A trie is a compact data structure for representing a set of strings, such as all the words in a text
A tries supports pattern matching queries in time proportional to the pattern size
2004 Goodrich, Tamassia
Tries
Standard Tries
The standard trie for a set of strings S is an ordered tree such that:
Each node but the root is labeled with a character The children of a node are alphabetically ordered The paths from the external nodes to the root yield the strings of S
Example: standard trie for the set of strings
S = { bear, bell, bid, bull, buy, sell, stock, stop }
b e a r l l i d l l
Tries
s u y e l l c k t o p
3
2004 Goodrich, Tamassia
Analysis of Standard Tries
A standard trie uses O(n) space and supports searches, insertions and deletions in time O(dm), where:
n total size of the strings in S m size of the string parameter of the operation d size of the alphabet
b e a r l l i d l l
Tries
s u y e l l c k t o p
4
2004 Goodrich, Tamassia
Word Matching with a Trie
We insert the words of the text into a trie Each leaf stores the occurrences of the associated word in the text e
a r 6 l l 78
s e e s e e b i d h e a r
a a
b e a r ? b u l l ?
s e l b u y b i d l ?
s t o c k ! s t o c k !
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
s t o c k ! t h e
s t o c k ! s t o p !
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
b e l
69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88
b i d 47, 58 l l 30 u y 36
h e a r 69 e 0, 24 l 12 e l
s t o c k 17, 40, 51, 62 p 84
2004 Goodrich, Tamassia
Tries
Compressed Tries
A compressed trie has internal nodes of degree at least two It is obtained from standard trie by compressing chains of redundant nodes
b e a r l l i d l l
Tries
b e ar ll id ll u y ell
s to ck p
s u y e l l c k t o p
6
2004 Goodrich, Tamassia
Compact Representation
Compact representation of a compressed trie for an array of strings:
Stores at the nodes ranges of indices instead of substrings Uses O(s) space, where s is the number of strings in the array Serves as an auxiliary index structure
0 1 2 3 4 S[0] = S[1] = S[2] = S[3] = 0 1 2 3 S[4] = S[5] = S[6] = 0 1 2 3 S[7] = S[8] = S[9] =
s e e b e a r s e l l s t o c k
1, 0, 0
b u l l b u y b i d
h e a r b e l l s t o p
7, 0, 3 4, 1, 1 4, 2, 3 5, 2, 2
Tries
0, 0, 0 0, 1, 1 3, 1, 2 2, 2, 3 3, 3, 4 9, 3, 3
7
1, 1, 1 1, 2, 3 8, 2, 3
6, 1, 2
0, 2, 2
2004 Goodrich, Tamassia
Suffix Trie
The suffix trie of a string X is the compressed trie of all the suffixes of X
m i n i m i z e 0 1 2 3 4 5 6 7
e mize
i nimize ze nimize
mi ze
nimize
ze
2004 Goodrich, Tamassia
Tries
Analysis of Suffix Tries
Compact representation of the suffix trie for a string X of size n from an alphabet of size d
Uses O(n) space Supports arbitrary pattern matching queries in X in O(dm) time, where m is the size of the pattern Can be constructed in O(n) time
m i n i m i z e 0 1 2 3 4 5 6 7
7, 7 4, 7
2004 Goodrich, Tamassia
1, 1 2, 7 6, 7
Tries
0, 1 2, 7 6, 7
2, 7
6, 7
