Preprocessing Strings
Preprocessing the pattern speeds up pattern matching
queries
Tries

e
i
mize
mi
nimize
ze
nimize
nimize
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
ze
ze

Tries
© 2004 Goodrich, Tamassia
1
Standard Tries


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
S = { bear, bell, bid, bull, buy, sell, stock, stop }
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l
r
l
© 2004 Goodrich, Tamassia
d
2
A standard trie uses O(n) space and supports
searches, insertions and deletions in time O(dm),
where:
s
i
a
Tries
© 2004 Goodrich, Tamassia
n total size of the strings in S
m size of the string parameter of the operation
d size of the alphabet
Example: standard trie for the set of strings
e
A tries supports pattern matching queries in time
proportional to the pattern size
Analysis of Standard Tries
The standard trie for a set of strings S is an ordered tree such that:

After preprocessing the pattern, KMP’s algorithm performs
pattern matching in time proportional to the text size
u
l
y
l
e
t
l
o
l
Tries
b
c
k
e
p
3
i
a
l
r
l
© 2004 Goodrich, Tamassia
s
d
u
l
y
l
e
t
l
o
l
Tries
c
k
p
4
Word Matching with a Trie
insert the words
of the text into
trie
Each leaf is
associated w/ one
particular word
leaf stores indices
where associated
word begins
(“see” starts at
index 0 & 24, leaf
for “see” stores
those indices)
s e e
a
b e a r ?
l
r
l
6
78
s t o c k !
s e e
a
b u l
b i d
l ?
b u y
s t o c k !
s t o c k !
b i d
s t o c k !
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68
h e a r
t h e
b e l
l ?
s t o p !
69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88
h
u
y
l
d
e
l
r
30
69
e
t
e
a
36
47, 58
l
o
0, 24
p
c
l
12
k
5
17, 40,
51, 62
Tries
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
0 1 2 3
S[4] =
S[1] =
s e e
b e a r
S[2] =
s e l l
S[3] =
s t o c k
S[0] =
a
l
r
l
S[7] =
S[5] =
S[8] =
h e a r
b e l l
S[6] =
b i d
S[9] =
s t o p
1, 2, 3
8, 2, 3
4, 1, 1
© 2004 Goodrich, Tamassia
4, 2, 3
0, 1, 1
5, 2, 2
Tries
u
ell
ll
y
to
ck
p
s
0, 2, 2
d
l
e
t
l
o
y
l
l
c
p
k
Tries
6
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
i
mi
nimize
ze
0, 0, 0
7, 0, 3
6, 1, 2
ll
u
© 2004 Goodrich, Tamassia
e
1, 1, 1
i
0 1 2 3
b u l l
b u y
1, 0, 0
id
Suffix Trie
Compact representation of a compressed trie for an array of strings:

e
s
84
Compact Representation

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s
e
© 2004 Goodrich, Tamassia

A compressed trie has
internal nodes of degree at
least two
It is obtained from standard
trie by compressing chains of
ar
“redundant” nodes
ex. the “i” and “d” in “bid”
are “redundant” because
they signify the same word
b
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
i
a
l
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
b
e
s e l
Compressed Tries
mize
3, 1, 2
2, 2, 3
3, 3, 4
nimize
ze
nimize
ze
9, 3, 3
7
© 2004 Goodrich, Tamassia
Tries
8
Analysis of Suffix Tries
Encoding Trie (1)
Compact representation of the suffix trie for a string
X of size n from an alphabet of size d
A code is a mapping of each character of an alphabet to a binary
code-word
A prefix code is a binary code such that no code-word is the prefix
of another code-word
An encoding trie represents a prefix code
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
1, 1
4, 7
2, 7

0, 1
6, 7
2, 7
2, 7
6, 7
9
Frequent characters should have short code-words
Rare characters should have long code-words
Example
X = abracadabra
T1 encodes X into 29 bits
T2 encodes X into 24 bits



T1
T2
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d
a
© 2004 Goodrich, Tamassia
b
r
a
b
c
Tries
011
10
11
a
b
c
d
e
a
Tries
d
b
c
e
10
Huffman’s Algorithm
Given a text string X, we want to find a prefix code for the characters
of X that yields a small encoding for X

010
© 2004 Goodrich, Tamassia
Encoding Trie (2)

00
6, 7
Tries
© 2004 Goodrich, Tamassia
Each leaf stores a character
The code word of a character is given by the path from the root to
the leaf storing the character (0 for a left child and 1 for a right child
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d
11
Given a string X,
Huffman’s algorithm
construct a prefix
code the minimizes
the size of the
encoding of X
It runs in time
O(n + d log d), where
n is the size of X
and d is the number
of distinct characters
of X
A heap-based
priority queue is
used as an auxiliary
structure
© 2004 Goodrich, Tamassia
Algorithm HuffmanEncoding(X)
Input string X of size n
Output optimal encoding trie for X
C  distinctCharacters(X)
computeFrequencies(C, X)
Q  new empty heap
for all c  C
T  new single-node tree storing c
Q.insert(getFrequency(c), T)
while Q.size() > 1
f1  Q.min()
T1  Q.removeMin()
f2  Q.min()
T2  Q.removeMin()
T  join(T1, T2)
Q.insert(f1 + f2, T)
return Q.removeMin()
Tries
12
Example
11
a
5
2
a
b
c
d
r
5
2
1
1
2
c
1
d
1
b
2
6
a
X = abracadabra
Frequencies
c
b
2
c
© 2004 Goodrich, Tamassia
d
b
2
r
2
a
5
c
2
d
r
2
r
6
2
a
5
4
a
5
Tries
c
4
d
b
r
4
d
b
r
13