Pattern Matching. a b a c a a b. a b a c a b. a b a c a b

Transcription

1 Pattern Matching a b a c a a b 1 a b a c a b a b a c a b

2 Strings A string is a seuence of characters Exampes of strings: n Python program n HTML document n DNA seuence n Digitized image An aphabet Σ is the set of possibe characters for a famiy of strings Exampe of aphabets: n ASCII n Unicode n {0, 1} n {A, C, G, T} 2014 Goodrich, Tamassia, Godwasser Pattern Matching Let P be a string of size m n A substring P[i.. j] of P is the subseuence of P consisting of the characters with ranks between i and j n A prefix of P is a substring of the type P[0.. i] n A suffix of P is a substring of the type P[i..m - 1] Given strings T (text) and P (pattern), the pattern matching probem consists of finding a substring of T eua to P Appications: n Text editors n Search engines n Bioogica research 2

3 Brute-Force Pattern Matching The brute-force pattern matching agorithm compares the pattern P with the text T for each possibe shift of P reative to T, unti either n a match is found, or n a pacements of the pattern have been tried Brute-force pattern matching runs in time O(nm) Exampe of worst case: n T = aaa ah n P = aaah n may occur in images and DNA seuences n unikey in Engish text Agorithm BruteForceMatch(T, P) Input text T of size n and pattern P of size m Output starting index of a substring of T eua to P or -1 if no such substring exists for i 0 to n - m { test shift i of the pattern } j 0 whie j < m T[i + j] = P[j] j j + 1 if j = m return i {match at i} ese break whie oop {mismatch} return -1 {no match anywhere} 2014 Goodrich, Tamassia, Godwasser Pattern Matching 3

4 Right to Left Matching Matching the pattern from right to eft For a pattern abc: T: bbacdcbaabcddcdaddaaabcbcb P: abc Worst case is sti O(n m) Pattern Matching 4

5 Bad Character Rue (BCR) (1) On a mismatch between the pattern and the text, we can shift the pattern by more than one pace. ddbbacdcbaabcddcdaddaaabcbcb acabc Pattern Matching 5

6 Bad Character Rue (BCR) (2) Preprocessing n A tabe, for each position in the pattern and a character, ast occurrence of the mismatched character in P preceding the mismatch. O(n Σ ) space. O(1) access time. a c a b c a b 4 4 c Pattern Matching 6

7 Bad Character Rue (BCR) (3) On a mismatch, shift the pattern to the right unti the first occurrence of the mismatched character in P. Sti O(n m) worst case running time: T: aaaaaaaaaaaaaaaaaaaaaaaaa P: abaaaa Pattern Matching 7

8 Good Suffix Rue (GSR) (1) We want to use the knowedge of the matched characters in the pattern s suffix. If we matched S characters in T, what is (if exists) the smaest shift in P that wi aign a sub-string of P of the same S characters? Pattern Matching 8

9 Good Suffix Rue (GSR) (2) Exampe 1 how much to move: T: bbacdcbaabcddcdaddaaabcbcb P: cabbabdbab cabbabdbab Pattern Matching 9

10 Good Suffix Rue (GSR) (3) Exampe 2 what if there is no aignment: T: bbacdcbaabcbbabdbabcaabcbcb P: bcbbabdbabc bcbbabdbabc Pattern Matching 10

11 Good Suffix Rue (GSR) (4) We mark the matched sub-string in T with t and the mismatched char with x 1. In case of a mismatch: shift right unti the first occurrence of t in P such that the next char y in P hods y x 2. Otherwise, shift right to the argest prefix of P that aigns with a suffix of t. Pattern Matching 11

12 Boyer Moore Agorithm Preprocess(P) k := n whie (k m) do n Match P and T from right to eft starting at k n If a mismatch occurs: shift P right (advance k) by max(good suffix rue, bad char rue). n ese, print the occurrence and shift P right (advance k) by the good suffix rue. Pattern Matching 12

13 Boyer-Moore Agorithm Demo GTTATAGCTGATCGCGGCGTAGCGGCGAA GTAGCGGCG Bad Character Rue shift by 7 Good Suffix Rue shift by 0 Pattern Matching 13

14 Boyer-Moore Agorithm Demo GTTATAGCTGATCGCGGCGTAGCGGCGAA GTAGCGGCG Bad Character Rue shift by 1 Good Suffix Rue shift by 3 (9-6) Pattern P GTAGCGGCG Prefixes of P G GT GTA GTAG GTAGC GTAGCG t t - GCG Find the ongest prefix of P which has t as the suffix. Pattern Matching 14

15 Boyer-Moore Agorithm Demo GTTATAGCTGATCGCGGCGTAGCGGCGAA GTAGCGGCG t Pattern P GTAGCGGCG Prefixes of P G GT GTA GTAG GTAGC GTAGCG t - GCGGCG GTAGCGG GTAGCGGC GTAGCGGCG Find the ongest prefix of P which has t as the suffix. Pattern Matching 15

16 Boyer-Moore Agorithm Demo GTTATAGCTGATCGCGGCGTAGCGGCGAA GTAGCGGCG Bad Character Rue shift by 3 Good Suffix Rue shift by 8 (9-1) t Pattern P GTAGCGGCG suffixes of t G CG GCG GCGG GCGGC GCGGCG Prefixes of P G GT GTA GTAG GTAGC GTAGCG Pattern Matching t - GCGGCG Find the ongest prefix of P which is a suffix of t. 16

17 Boyer-Moore Agorithm Demo GTTATAGCTGATCGCGGCGTAGCGGCGAA GTAGCGGCG Pattern Matching 17

18 Good Suffix Rue (GSR) (5) L(i) The biggest index j, such that j < n and prefix P[1..j] contains suffix P[i..n] as a suffix but not suffix P[i-1..n] i P G T A G C G G C G L(i) Pattern Matching 18

19 Good Suffix Rue (GSR) (6) (i) The ength of the ongest suffix of P[i..n] that is aso a prefix of P i P G T A G C G G C G (i) Pattern Matching 19

20 Good Suffix Rue (GSR) (7) Putting it together n If mismatch occurs at position n, shift P by 1 n If a mismatch occurs at position i-1 in P: w If L(i) > 0, shift P by n L(i) w ese shift P by n (i) n If P was found, shift P by n (2) Pattern Matching 20

21 Boyer-Moore Agorithm Anaysis Worst case n O(n+m) if the pattern does not occur in the text n O(nm) if the pattern does occur in the text n For patterns of sma size, the agorithm might not be efficient. Pattern Matching 21

22 Knuth-Morris-Pratt - Agorithm KMP Agorithm n compares the pattern to the text in eft-toright, but shifts the pattern more inteigenty than the brute-force agorithm. text t 1... t j-r+1... t j... pattern p 1... p r... pattern p 1... p r... pattern p 1... p r... 22

23 The KMP Agorithm Knuth-Morris-Pratt s agorithm compares the pattern to the text in eft-toright, but shifts the pattern more inteigenty than the brute-force agorithm. When a mismatch occurs, what is the most we can shift the pattern so as to avoid redundant comparisons? Answer: the argest prefix of P[0..j] that is a suffix of P[1..j].. a b a a b x..... a b a a b a No need to repeat these comparisons 2014 Goodrich, Tamassia, Godwasser Pattern Matching 23 j a b a a b a Resume comparing here

24 KMP Faiure Function Knuth-Morris-Pratt s agorithm preprocesses the pattern to find matches of prefixes of the pattern with the pattern itsef The faiure function F(j) is defined as the size of the argest prefix of P[0..j] that is aso a suffix of P[1..j] Knuth-Morris-Pratt s agorithm modifies the bruteforce agorithm so that if a mismatch occurs at P[j] T[i] we set j F(j - 1) j P[j] a b a a b a F(j) a b a a b x..... a b a a b a j a b a a b a F(j 1) 2014 Goodrich, Tamassia, Godwasser Pattern Matching 24

25 Computing the Faiure Function The faiure function can be represented by an array and can be computed in O(m) time At each iteration of the whieoop, either n i increases by one, or n the shift amount i - j increases by at east one (observe that F(j - 1) < j) Hence, there are no more than 2m iterations of the whie-oop Agorithm faiurefunction(p) F[0] 0 i 1 j 0 whie i < m if P[i] = P[j] {we have matched j + 1 chars} F[i] j + 1 i i + 1 j j + 1 ese if j > 0 then {use faiure function to shift P} j F[j - 1] ese F[i] 0 { no match } i i Goodrich, Tamassia, Godwasser Pattern Matching 25

26 The KMP Agorithm The faiure function can be represented by an array and can be computed in O(m) time At each iteration of the whieoop, either n i increases by one, or n the shift amount i - j increases by at east one (observe that F(j - 1) < j) Hence, there are no more than 2n iterations of the whie-oop Thus, KMP s agorithm runs in optima time O(m + n) Agorithm KMPMatch(T, P) F faiurefunction(p) i 0 j 0 whie i < n if T[i] = P[j] if j = m - 1 return i - j { match } ese i i + 1 j j + 1 ese if j > 0 j F[j - 1] ese i i + 1 return -1 { no match } 2014 Goodrich, Tamassia, Godwasser Pattern Matching 26

27 Exampe a b a c a a b a c c a b a c a b a a b b a b a c a b 7 a b a c a b j P[j] a b a c a b F(j) a b a c a b 13 a b a c a b a b a c a b Pattern Matching 27

28 Tries e i mi nimize ze mize nimize ze nimize ze

29 Preprocessing Strings Preprocessing the pattern speeds up pattern matching ueries n After preprocessing the pattern, KMP s agorithm performs pattern matching in time proportiona to the text size If the text is arge, immutabe 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 a the words in a text n A trie supports pattern matching ueries in time proportiona to the pattern size 2014 Goodrich, Tamassia, Godwasser Tries 29

30 Standard Tries The standard trie for a set of strings S is an ordered tree such that: n Each node but the root is abeed with a character n The chidren of a node are aphabeticay ordered n The paths from the externa nodes to the root yied the strings of S Exampe: standard trie for the set of strings S = { bear, be, bid, bu, buy, se, stock, stop } b s e i u e t a d y o r c p 2014 Goodrich, Tamassia, Godwasser Tries 30 k

31 Anaysis of Standard Tries A standard trie uses O(n) space and supports searches, insertions and deetions in time O(dm), where: n tota size of the strings in S m size of the string parameter of the operation d size of the aphabet b s e i u e t a d y o r c p 2014 Goodrich, Tamassia, Godwasser Tries 31 k

32 Appication of Tries A standard trie supports the foowing operations on a processed text in O(m) time, where m is the ength of the string. n word matching: find the first occurrence of word X in the text. n prefix matching: find the first occurrence of the ongest prefix of word X in the text. Tries 32

33 Word Matching with a Trie insert the words of the text into trie Each eaf is associated w/ one particuar word eaf stores indices where associated word begins ( see starts at index 0 & 24, eaf for see stores those indices) s e e a b e a r? s e s t o c k! s e e a b u? b u y s t o c k! b i d s t o c k! b i d s t o c k! h e a r t h e b e? s t o p! b h s e i u e e t a d y a e o r , r 69 0, 24 c 12 k p Goodrich, Tamassia, Godwasser Tries 17, 40, 33 51, 62

34 Compressed Tries A compressed trie has interna nodes of degree at east two It is obtained from standard trie by compressing chains of redundant nodes ex. the i and d in bid are redundant because they signify the same word ar e b id u y e s ck to p b s e i u e t a d y o r c p 2014 Goodrich, Tamassia, Godwasser Tries 34 k

35 Why Compressed Tries? A compressed trie storing a coection S of s strings from an aphabet of size d has the foowing properties n Every interna node of T has at east two chidren and at most d chidren n T has s eaf nodes n The number of interna nodes of T is O(s) Tries 35

36 Compact Representation Compact representation of a compressed trie for an array of strings: n n n 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 auxiiary index structure ar e b id hear u y s e e e ck to p S[0] = S[1] = S[2] = S[3] = s e e b e a r s e s t o c k S[4] = S[5] = S[6] = S[7] = b u b u y b i d S[8] = S[9] = h e a r b e s t o p 1,0,0 7,0,3 0,0,0 1,1,1 6,1,2 4,1,1 0,1,1 3,1,2 1,2,3 8,2,3 4,2,3 5,2,2 0,2,2 2,2,3 3,3,4 9,3,3 Tries 36

37 Insertion/Deetion in Compressed Trie f g or u one reat n r f g search ends here insert green or n u one r reat f g or u one re insert green n r at en Tries 37

38 Appication - Routing through Tries Internet Routers maintain a Trie (tabe) It is not a ookup n forwards packets to its neighbors using IP prefix matching rues ~ 2 9 ~ 1 ~ Tries 38

39 Suffix Trie The suffix trie of a string X is the compressed trie of a the suffixes of X Each eaf corresponds to a suffix of X m i n i m i z e e i mi nimize ze mize nimize ze nimize ze Goodrich, Tamassia, Godwasser Tries 39

40 Anaysis of Suffix Tries Compact representation of the suffix trie for a string X of size n from an aphabet of size d n n n Uses O(n) space Supports arbitrary pattern matching ueries 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 , 7 1, 1 0, 1 2, 7 7 4, 7 2, 7 6, 7 2, 7 6, , Goodrich, Tamassia, Godwasser Tries 40

41 Prefix Matching using Suffix Trie If two suffixes have a same prefix, then their corresponding paths are the same at their beginning, and the concatenation of the edge abes of the mutua part is the prefix. ize and imize m i n i m i z e e i mi 7 mize nimize ze nimize ze nimize ze 2 6 Tries 41

42 Suffix Trie Not a strings are guaranteed to have corresponding suffix trie. For exampe: xabxa bxa a bxa xa bxa bxa a $ bxa xa $ bxa Tries 42

43 Constructing a Suffix Trie S[1..n] is the string start with a singe edge for S enter the edges for the suffix S[i..n] where i goes from 2 to n n Starting at the root node find the ongest part from the root whose abe matches a prefix of S[i..n]. At some point, no further matches are possibe w If the point is at a node, then denote this node by w w If it is in the midde of an edge, then insert a new node caed w, at this point w create a new edge running from w to a new eaf abeed S[i..n] Tries 43

44 Exampe minimize minimize minimize minimize inimize minimize inimize minimize nimize minimize i minimize nimize mize nimize Tries 44

45 Exampe (2) minimize mize i nimize minimize nimize minimize mize i nimize m inimize ize nimize Tries 45

46 Exampe (3) m i n i m i z e e i mi 7 mize nimize ze nimize ze nimize ze 2 6 Tries 46

47 Constructing a Suffix Trie Compexity- O(n 2 ) S[1..n] is the string start with a singe edge for S enter the edges for the suffix S[i..n] where i goes from 2 to n n Starting at the root node find the ongest part from the root whose abe matches a prefix of S[i..n]. At some point, no further matches are possibe w If the point is at a node, then denote this node by w w If it is in the midde of an edge, then insert a new node caed w, at this point w create a new edge running from w to a new eaf abeed S[i..n] Tries 47