Tries 07/28/16 11:04 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

Encoding Tree Example a b c d e Tries 07/28/16 11:04 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

Encoding Tree Optimization Tries 07/28/16 11:04 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

Tries 07/28/16 11:04 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

Example X = abracadabra Frequencies a b c d r 5 2 1 c a b d r 2 4 6 11 Tries 07/28/16 11:04 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

Tries 07/28/16 11:04 Larger Huffman Tree Huffman Encoding 6

Pattern Matching Pattern matching: find a pattern in a text. Tries 07/28/16 11:04 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

Tries The trie for a set of strings S is an ordered tree. 07/28/16 11:04 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

Tries 07/28/16 11:04 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 Tries 9

Word Matching with a Trie Tries 07/28/16 11:04 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 Tries

Tries 07/28/16 11:04 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

Compact Representation Tries 07/28/16 11:04 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. Tries 12

Tries 07/28/16 11:04 Suffix Trie The suffix trie of a string X is the compressed trie of the non- null suffixes of X. Tries 13

Analysis of Suffix Tries 07/28/16 11:04 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). Tries 14

Huffman Encoding Revisited Tries 07/28/16 11:04 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