Algorithm Design (5) Text Search

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1 Algorithm Design (5) Text Serch Tkshi Chikym School of Engineering The University of Tokyo Text Serch Find sustring tht mtches the given key string in text dt of lrge mount Key string: chr x[m] Text Dt: chr y[n] Function to e defined is: Given key chrcter string x nd text dt y Return the indices of the occurrences of x ppering in y, or Return 1 if x does not pper in x. Text serch lgorithms cn e pplied to ny serch prolems requiring exct susequence mtches. A Nïve Algorithm Compring ll the sustrings of the text dt with the key string Key length Text length int serch(chr key[], int m, chr text[], int n) { for (int pos = 0; pos+m <= n; pos++) { for (int j = 0; j < m; j++) { if (key[ j ]!= text[ pos+j ]) goto MISMATCH; } return pos; MISMATCH: ; } return 1; } How the Nïve Algorithm Works Text dt c c c c c Key string to serch The worst cse complexity is (mn) For exmple, think of finding in 1

2 Knuth-Morris-Prtt Algorithm The nïve method does not mke use of the informtion otined y prtil mtches Couldn t tht informtion utilized? A shift mount cn e cndidte only if the key grees with the prtilly mtched prt of the key This cn e decided only from the key string, without knowing the text dt text: key: key: These prts within the key should gree Prtil Agreements within the Key c c c A disgreement found c c c Prtil greement Never mtch This shift mount might mke mtch c c c c c c c c c c c c c c c Shift Amounts tht Potentilly Give Mtches When the first k chrcters gree nd the k+1 th chrcter disgrees, shifting j chrcters more mkes the key mtch with the text only when: The first k j chrcters re the sme s the til of the prtil mtch, nd The k j + 1 th chrcter is different from the unmtched chrcter For ech k, compute the smllest of such j in dvnce k y: x: j Computtionl Complexity of Knuth-Morris-Prtt Chrcter in the text dt re inspected only once Mtches of the first k-j chrcters re known Serch itself hs the complexity of O(n) The tle of shift mounts with potentil mtch cn e uilt in lmost the sme wy: O(m) The totl complexity is O(m+n) Lessons lerned Do not wste informtion otined Approprite preprocessing cn e the key of efficiency improvements 2

3 Boyer-Moore Method Motivtion Cn t we shift the pttern more t time? Method Try mtching ckwrds from the til of the key Two fctors in deciding shift mounts Alredy mtched sustring of the key should mtch t the new position (good suffix) Ide common with Knuth-Morris-Prtt The chrcter tht cused the mismtch should mtch with the new position (d symol) Good Suffix Rule tht utlizes prtil mtch c c c c c c This prtil mtch succeeds c At lest this mount of shift is required for possile mtch Good Suffix Rule: Potentil Shift Amounts On mismtch fter chrcters of mtch, new shift mount should mke those chrcters gree, nd th chrcters disgree. This depends only on the key, not on the text Bd Symol Rule tht Utilizes the Mismtch c c c c c This hs mtching possiility This is known to cuse mismtch gin Shifting should mke the mismtched chrcter mtch At lest this mount of shift is required 3

4 Bd Symol Rule tht Utilizes the Mismtch c c c c If the mismtched chrcter does not pper in the key, shift the key so tht the key do not overlp tht chrcter Utilizing Both Good Suffix nd Bd Symol The good suffix shift mount depends on the position of mismtch The d symol shift mount depends on the chrcter tht cuses the mismtch A tle tht considers oth fctors is two dimensionl with size ( is the size of the lphet) If is to lrge, uilding two seprte tles ( ) nd use the mximum of the two is not d choice The Glil Rule: A Further Improvement on Boyer-Moore Utilize prtil mtch info to ccelerte mtching c c c c c c c c c Comprison of this prt cn e skipped Computtion Complexity of Boyer-Moore Method Without the Glil Rule, the worst cse is O(mn) with multiple occurrences of the key Finding in, for exmple Unlike Knuth-Morris-Prtt, the sme text my e inspected mny times The Glil Rule improves it to O(m+n) Finding in is O(n/m) The repeted s cn e skipped efficiently Empiriclly, fster thn Knuth-Morris-Prtt in most cses 4

5 Wht We Cn Lern from the Boyer-Moore Method Common with Knuth-Morris-Prtt Do not wste the informtion otined Approprite preprocessing cn e the key of efficiency improvements Specific to Boyer-Moore Try thinking the other wy round Mismtch informtion found close to the til my llow lrger shift Strt mtching from the til nd proceed in the reverse order Boyer-Moore-Horspool Algorithm Mtching tried from the til, s with Boyer-Moore Insted of using unmtched symol, use the symol t the til of the text to determine the shift mount c c c c c c Preprocessing mkes tle of rightmost occurrences of symols, except for one t the til Used in plce of the d symol of Boyer-Moore Merits nd Demerits of Horspool Algorithm Simple! Efficient with lrge vriety of chrcters If the sme chrcter does not pper frequently, lrge shift t time is possile If the mismtched chrcter does not pper in the key, the shift mount cn e m, in which cse only n/m comprisons re needed The worst cse complexity is O(mn) Hshing Hsh vlue: A single integer vlue computed for possily complicted dt The hsh vlue is the sme for the sme dt The sme hsh vlue doesn t men the sme dt Different hsh vlues men different dt Comprison of hsh vlues is much more efficient thn comprison of the dt themselves Mismtches cn e often told with low cost With detection of mismtches mde efficient, costly full comprisons re needed less often 5

6 Rin-Krp Method Use hshing for comprison of chrcter strings Hshing costs should e mde lower thn O(m), which is needed for full comprison Use the notion of differences to mke hshing more efficient Compute the hsh vlue of only from the hsh vlue of nd chrcters t nd Hsh Function Computed from Differences If, for exmple, the hsh vlue of string strting t k is defined y: ( ) m 1 m j hk yk jd 1 mod q j0 then, using the difference eqution h( k 1) cn compute h(k+1) from h(k) with constnt cost Any definitions of hsh functions with constnt computtion cost will do m1 h( k) d y[ k] d y[ k m] mod q Computtionl Complexity of Rin-Krp Method If hshing works s expected, O(m+n) With mny hsh collisions, tht is, when the hsh vlue is the sme for mny different dt, O(mn) Hsh Tles: Serch Tles with Hshing Generl principles in lgorithm design Minimize the costs for frequent cses Almost ll the comprisons result in mismtch, which should e mde efficient Considering the differences my mke the computtion more efficient Hsh 6

7 Generliztion of Rin-Krp Method Serching multiple key strings t time Any of given strings with the sme length re to e found Build hsh tle nd register entries corresponding to ech of the keys Building the tle costs O(mk) Hsh vlues of the text sustrings cn e computed with constnt costs s with Rin-Krp Comprison cost for one text sustring remins constnt with no hsh collisions The totl cost is O(n+mk) Building Indexes Prepring n index for the text dtse Effective for repeted serch in the sme dtse The cost of uilding the index my e well compensted y the improved serch efficiency Effective lso when response time is criticl The index uilding cost cn e justified y quicker response Typicl Serch Opertions For key chrcter string, Does it pper in the dtse or not? How mny times does it pper? Where in the dtse does it pper? Which is the longest sustring of the given key ppering in the dtse? Trie A serch tree with key elements ssocited with rnches (for informtion retrievl) The prefix tree for set of words tme, tem, te, mte nd mt m e t e m te m t e Finding the word te mt tme tem mte 7

8 Trie Compred with Ordered Serch Tree Serch cn e fster When the numer of the registered strings is n, the length of the string to find is m, nd the chrcter set size is s, serch costs re: O(sm) vs. O(log n) Serch for short string in lrge dtse cn e fst If the chrcter set size is very lrge (s with Chinese or Jpnese), serch my ecome slower Insertion nd deletion hs the sme complexity s finding Suffix Tree for Dtse Index Tries cn e used s indexes for text dtses All sustrings s the keys Strt positions of sustrings s vlues A suffix tree index for text dtse nn n 6 n n 0 n n n 2 Findig n 4 Compressed Suffix Tree With Suffix Tree 0 n 1 n n n Finding n 6 Pths with no rnches re compressed into one Strings ssocited with rcs re ctully pir of the position in the DB nd the length The numer of leves is the sme s the dtse size Memory Size is O(n) Does key exist in the DB? Trcing down from the root O(m) Where does it pper? Trcing down to node nd look elow For k occurrences O(m+k) How mny times does it pper? Count leves elow the node found O(m+k) It is lso possile to record the numer of leves elow in tree nodes O(m) 8

9 Building Suffix Tree Nïve Method: Inserting suffixes one y one There re n suffixes The insertion cost is 1 through n O(n 2 ) in totl, which is unerly lrge Ukkonen s method cn reduce it to O(n) E. Ukkonen. Constructing Suffix Trees On-Line in Liner Time. Proc. IFIP 12th World Computer Congress, A lrge mount of memory is required, however. 10 to 20 times the size of the dtse string Suffix Automton A finite utomton tht recognizes suffixes The smllest of the mny different utomt with the sme function Suffix tree with the identicl sutrees merged An utomton recognizing suffixes of Chrcteristics of Suffix Automton The numer of sttes is O(n) Building cost is O(n log k) where k is the chrcter set size Computtionl complexity with suffix utomt Whether key string of length m exists: O(m) Counting numer of occurrences of key: O(n) Suffix Arry A sorted tle of ll suffixes Entries re ctully strt positions in the originl text 9

10 Serch in Suffix Arry Serch for string of length m Comprison cost is O(m) With n entries, inry serch requires O(log n) repetitions The totl cost is thus O(m log n) An extr dt structure with informtion of longest common prefixes speeds up comprison nd, the cost cn e reduced to O(m + log n) Building Suffix Arry In nïve wy: Cost of one comprison is O(n) Numer of comprisons is O(n log n) The totl cost is O(n 2 log n) Cn e improved to O(n log n) Chrcteristics of Suffix Arrys Simple, esy to understnd Fits in reltively smll memory: O(n) Serch is not s fst s suffix trees O(n log n) vs. O(m) Summry A vriety of lgorithms for text serch A vriety of devices for speeding up serch Thinking of differences: Roin-Krp Preprocessing cn decrese the cost Preprocessing on the key string serched for Knuth-Morris-Prtt, Boyer-Moore, Horspool Informtion otined in filed serch cn e utilized Preprocessing on the text dt: Prepre indexes Suffix Tree, Suffix Automton, Suffix Arry 10

11 Topic of Next Week Dt Compression Representing some dt with smller mount of memory Lossy vs. Lossless Whether or not some informtion in the dt cn e neglected or not The min topic will e lossless compression lgorithms 11

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