Outline and Reading
Standard tries (§9.2.1)
Compressed tries (§9.2.2)
Suffix tries (§9.2.3)
Huffman encoding tries (§9.3.1)
Tries
e
mize
i
nimize
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mi
ze
nimize
nimize
ze
ze
Tries
1
Preprocessing Strings
After preprocessing the pattern, KMP’s algorithm performs
pattern matching in time proportional to the text size
Tries
The standard trie for a set of strings S is an ordered tree such that:
!
!
!
A tries supports pattern matching queries in time
proportional to the pattern size
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2
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
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
!
Tries
Standard Trie (1)
Preprocessing the pattern speeds up pattern matching
queries
!
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3
S = { bear, bell, bid, bull, buy, sell, stock, stop }
b
e
i
a
l
r
l
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s
d
u
l
y
l
e
t
l
o
l
Tries
c
k
p
4
Standard Trie (2)
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 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
s
i
a
l
r
l
u
d
l
y
e
t
l
o
a
l
r
l
l
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c
p
k
Tries
5
id
ll
a
l
r
l
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d
u
ell
ll
y
to
!
ck
p
!
l
s e e
a
b u l
y
l
e
t
l
o
l
Tries
s t o c k !
l ?
b u y
s t o c k !
b i d
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
b
h
i
u
y
l
d
47, 58
e
t
e
a
36
l
s
e
l
0, 24
r
30
o
p
c
l
69
12
84
k
17, 40,
51, 62
Tries
6
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
0 1 2 3 4
u
l
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
78
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] =
s
i
s e l
Compact representation of a compressed trie for an array of strings:
s
!
e
b
e
b e a r ?
Compact Representation
b
ar
a
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
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Compressed Trie
A compressed trie has
internal nodes of degree
at least two
It is obtained from
standard trie by
compressing chains of
“redundant” nodes
l
6
s e e
c
k
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, 0, 0
1, 1, 1
p
1, 2, 3
7
6, 1, 2
8, 2, 3
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0 1 2 3
b u l l
b u y
0, 0, 0
7, 0, 3
4, 1, 1
4, 2, 3
0, 1, 1
5, 2, 2
Tries
0, 2, 2
3, 1, 2
2, 2, 3
3, 3, 4
9, 3, 3
8
Suffix Trie (1)
Suffix Trie (2)
Compact representation of the suffix trie for a string X of size n
from an alphabet of size d
The suffix trie of a string X is the compressed trie of all the
suffixes of X
Uses O(n) space
Supports arbitrary pattern matching queries in X in O(dm) time,
where m is the size of the pattern
!
m i n i m i z e
0 1 2 3 4 5 6 7
!
m i n i m i z e
0 1 2 3 4 5 6 7
e
i
mi
nimize
ze
7, 7
mize
nimize
ze
nimize
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ze
4, 7
Tries
9
Encoding Trie (1)
!
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
00
010
011
10
11
a
b
c
d
e
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a
Tries
2, 7
0, 1
6, 7
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2, 7
2, 7
6, 7
6, 7
Tries
10
Encoding Trie (2)
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
!
1, 1
d
b
c
e
Given a text string X, we want to find a prefix code for the characters
of X that yields a small encoding for X
Frequent characters should have long code-words
Rare characters should have short code-words
!
!
Example
X = abracadabra
T1 encodes X into 29 bits
T2 encodes X into 24 bits
!
!
!
T1
T2
c
d
a
11
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b
r
a
b
c
Tries
r
d
12
Huffman’s Algorithm
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
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Example
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.minKey()
T1 ← Q.removeMin()
f2 ← Q.minKey()
T2 ← Q.removeMin()
T ← join(T1, T2)
Q.insert(f1 + f2, T)
return Q.removeMin()
Tries
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
13
b
2
c
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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
14