第09章文本处理TextProcessing.ppt

4/6/2019 1:42 PM,Text Processing,1,Chapter 9: Text Processing,4/6/2019 1:42 PM,Text Processing,2,Outline and Reading,Strings and Pattern Matching (§9.1) Tries (§9.2) Text Compression (§9.3) Optional: Text Similarity (§9.4). No Slides.,4/6/2019 1:42 PM,Text Processing,3,Texts & Pattern Matching,4/6/2019 1:42 PM,Text Processing,4,Strings,A string is a sequence of characters Examples of strings: Java program HTML document DNA sequence Digitized image An alphabet S is the set of possible characters for a family of strings Example of alphabets: ASCII Unicode 0, 1 A, C, G, T,Let P be a string of size m A substring Pi j of P is the subsequence of P consisting of the characters with ranks between i and j A prefix of P is a substring of the type P0 i A suffix of P is a substring of the type Pi m - 1 Given strings T (text) and P (pattern), the pattern matching problem consists of finding a substring of T equal to P Applications: Text editors Search engines Biological research,4/6/2019 1:42 PM,Text Processing,5,Brute-Force Algorithm,The brute-force pattern matching algorithm compares the pattern P with the text T for each possible shift of P relative to T, until either a match is found, or all placements of the pattern have been tried Brute-force pattern matching runs in time O(nm) Example of worst case: T = aaa ah P = aaah may occur in images and DNA sequences unlikely in English text,Algorithm BruteForceMatch(T, P) Input text T of size n and pattern P of size m Output starting index of a substring of T equal to P or -1 if no such substring exists for i 0 to n - m test shift i of the pattern j 0 while j m Ti + j = Pj j j + 1 if j = m return i match at i else break while loop mismatch return -1 no match anywhere,4/6/2019 1:42 PM,Text Processing,6,Boyer-Moore Heuristics,The Boyer-Moores pattern matching algorithm is based on two heuristics Looking-glass heuristic: Compare P with a subsequence of T moving backwards Character-jump heuristic: When a mismatch occurs at Ti = c If P contains c, shift P to align the last occurrence of c in P with Ti Else, shift P to align P0 with Ti + 1 Example,4/6/2019 1:42 PM,Text Processing,7,The Boyer-Moore Algorithm,Algorithm BoyerMooreMatch(T, P, S) L lastOccurenceFunction(P, S ) i m - 1 j m - 1 repeat if Ti = Pj if j = 0 return i match at i else i i - 1 j j - 1 else character-jump l LTi i i + m min(j, 1 + l) j m - 1 until i n - 1 return -1 no match ,4/6/2019 1:42 PM,Text Processing,8,Example,4/6/2019 1:42 PM,Text Processing,9,Analysis,Boyer-Moores algorithm runs in time O(nm + s) Example of worst case: T = aaa a P = baaa The worst case may occur in images and DNA sequences but is unlikely in English text Boyer-Moores algorithm is significantly faster than the brute-force algorithm on English text,4/6/2019 1:42 PM,Text Processing,10,The KMP Algorithm - Motivation,Knuth-Morris-Pratts algorithm compares the pattern to the text in left-to-right, but shifts the pattern more intelligently than the brute-force algorithm. When a mismatch occurs, what is the most we can shift the pattern so as to avoid redundant comparisons? Answer: the largest prefix of P0j that is a suffix of P1j,x,j,.,.,a,b,a,a,b,.,.,.,.,.,a,b,a,a,b,a,a,b,a,a,b,a,No need to repeat these comparisons,Resume comparing here,4/6/2019 1:42 PM,Text Processing,11,KMP Failure Function,Knuth-Morris-Pratts algorithm preprocesses the pattern to find matches of prefixes of the pattern with the pattern itself The failure function F(j) is defined as the size of the largest prefix of P0j that is also a suffix of P1j Knuth-Morris-Pratts algorithm modifies the brute-force algorithm so that if a mismatch occurs at Pj Ti we set j F(j - 1),4/6/2019 1:42 PM,Text Processing,12,The KMP Algorithm,The failure function can be represented by an array and can be computed in O(m) time At each iteration of the while-loop, either i increases by one, or the shift amount i - j increases by at least one (observe that F(j - 1) j) Hence, there are no more than 2n iterations of the while-loop Thus, KMPs algorithm runs in optimal time O(m + n),Algorithm KMPMatch(T, P) F failureFunction(P) i 0 j 0 while i 0 j Fj - 1 else i i + 1 return -1 no match ,4/6/2019 1:42 PM,Text Processing,13,Computing the Failure Function,The failure function can be represented by an array and can be computed in O(m) time The construction is similar to the KMP algorithm itself At each iteration of the while-loop, either i increases by one, or the shift amount i - j increases by at least one (observe that F(j - 1) j) Hence, there are no more than 2m iterations of the while-loop,Algorithm failureFunction(P) F0 0 i 1 j 0 while i 0 then use failure function to shift P j Fj - 1 else Fi 0 no match i i + 1,4/6/2019 1:42 PM,Text Processing,14,Example,4/6/2019 1:42 PM,Text Processing,15,Tries,4/6/2019 1:42 PM,Text Processing,16,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,4/6/2019 1:42 PM,Text Processing,17,Standard Trie (1),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 ,4/6/2019 1:42 PM,Text Processing,18,Standard Trie (2),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,4/6/2019 1:42 PM,Text Processing,19,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,4/6/2019 1:42 PM,Text Processing,20,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,4/6/2019 1:42 PM,Text Processing,21,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,4/6/2019 1:42 PM,Text Processing,22,Suffix Trie (1),The suffix trie of a string X is the compressed trie of all the suffixes of X,4/6/2019 1:42 PM,Text Processing,23,Suffix Trie (2),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,4/6/2019 1:42 PM,Text Processing,24,Encoding Trie (1),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 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,4/6/2019 1:42 PM,Text Processing,25,Encoding Trie (2),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,T2,4/6/2019 1:42 PM,Text Processing,26,Text Compression,4/6/2019 1:42 PM,Text Processing,27,Huffmans Algorithm,Given a string X, Huffmans 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,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(),4/6/2019 1:42 PM,Text Processing,28,Example,X = abracadabra,Frequencies,