1. Tries
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Text Compression
Efficiently encode a string X into a smaller string Y.
Save memory and/or bandwidth.
Huffman encoding
Compute frequency f(c) for each character c.
Encode characters with bit strings.
Use short strings for frequent characters.
No string is a prefix for another string.
Compute an optimal encoding expressed as a binary tree.
Huffman Encoding 1

2. Encoding Tree Example a b c d e
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Encoding Tree Example
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 tree represents a prefix code.
Each external node stores a character.
The code word of a character is given by the path from the root to the external node storing the character (0 for a left child and 1 for a right child). a b c d e 00
010
011 10 11 a b c d e
Huffman Encoding 2

3. Encoding Tree Optimization
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Encoding Tree Optimization
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 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 c a r d b T1 a c d b r T2
Huffman Encoding 3

4. Tries
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Huffman’s Algorithm
Algorithm HuffmanEncoding(X)
Input string X of size n
Output optimal encoding trie for X
C  distinctCharacters(X)
computeFrequencies(C, X)
Q  empty heap
for all c  C
T  new leaf 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()
Constructs a prefix code the minimizes the encoding size of a string X.
Runs in time O(nd log d), where n is the size of X and d is the number of characters in X.
A heap-based priority queue is used as an auxiliary structure. 4

5. Example X = abracadabra Frequencies a b c d r 5 2 1 c a b d r 2 4 6 11
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Example c a b d r 2 4 6 11
X = abracadabra
Frequencies a b c d r 5 2 1 c a b d r 2 5 4 6 c a r d b 5 2 1 c a r d b 2 5 c a b d r 2 5 4
Huffman Encoding 5

6. Tries
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Larger Huffman Tree
Huffman Encoding 6

7. Pattern Matching Pattern matching: find a pattern in a text.
Tries
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Pattern Matching
Pattern matching: find a pattern in a text.
The text is large, immutable, and often searched.
Preprocess the text to make search faster.
A trie is a compact data structure for representing a set of strings, such as all the words in a text.
Search time is linear in pattern size.
Tries 7

8. Tries The trie for a set of strings S is an ordered tree.
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Tries
The trie for a set of strings S is an ordered tree.
Each node but the root is labeled with a character.
The children of a node are alphabetically ordered.
The paths from the root to the external nodes yield the strings of S.
Example: S = { bear, bell, bid, bull, buy, sell, stock, stop }
Tries 8

9. Tries
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Analysis of Tries
A trie uses O(n) space and supports search, insertion, and deletion in time O(dm).
n total length of the strings in S
m length of the query string
d size of the alphabet
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10. Word Matching with a Trie
Tries
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Word Matching with a Trie
insert the words of the text into a trie.
A leaf stores the indices where a word begins.
see starts at index 0 & 24. Its leaf stores those indices. s e b a r ? l t o c k ! u y i d h p 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 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 a e b l s u t
0, 24 o c i r 6 78 d
47, 58 30 y 36 12 k
17, 40,
51, 62 p 84 h 69 10
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11. Tries
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Compressed Tries
A compressed trie has internal nodes of degree at least two.
It is obtained by compressing chains of redundant nodes.
The “i” and “d” in “bid” are redundant because they signify the same word.
Tries 11

12. Compact Representation
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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.
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13. Tries
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Suffix Trie
The suffix trie of a string X is the compressed trie of the non- null suffixes of X.
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14. Analysis of Suffix Tries
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Analysis of Suffix Tries
Compact representation of the suffix trie for a string X of length 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 length of the pattern.
Can be constructed in O(n) time (algorithm not covered).
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15. Huffman Encoding Revisited
Tries
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Huffman Encoding Revisited
Huffman encoding trees are tries.
Each leaf stores a character instead of a string.
The strings are generated by the algorithm, not input. a b c d e 00
010
011 10 11 a b c d e
Tries 15