String Searching. String Search. Applications. Brute Force: Typical Case

Transcription

1 String Serch String Serching String serch. Given pttern string p, find first mtch in text t. Model. Cn't fford to preprocess the text. Prmeters. N = length of text, M = length of pttern. typiclly N >> M Serch Pttern M = 6, N = 1 Serch Text i n h y s t c k i n Reference: Chpter 19, Algorithms in C, nd Edition, Roert Sedgewick. Roert Sedgewick nd Kevin Wyne Copyright 00 Applictions Brute Force: Typicl Cse Applictions.! Prsers.! Lexis/Nexis.! Spm filters.! Virus scnning.! Digitl lirries.! Screen scrpers.! Word processors.! We serch engines.! Nturl lnguge processing.! Crnivore surveillnce system.! Computtionl moleculr iology.! Feture detection in digitized imges. h y n e e d s n x 3 4

2 Brute Force Brute Force: Worst Cse Brute force: Check for pttern strting t every text position. pulic sttic int serch(string pttern, String text) { int M = pttern.length(); int N = text.length(); for (int i = 0; i < N - M; i++) { int j; for (j = 0; j < M; j++) { if (text.chrat(i+j)!= pttern.chrat(j)) rek; if (j == M) return i; return offset i if found return -1; return -1 if not found 6 Anlysis of Brute Force Screen Scrping Anlysis of rute force.! Running time depends on pttern nd text.! Worst cse: M N comprisons.! "Averge" cse: 1.1 N comprisons. (!)! Slow if M nd N re lrge, nd hve lots of repetition. Find current stock price of Google.! t.indexof(p): index of 1 st occurrence of pttern p in text t.! Downlod html from: Find first string delimited y <> nd </> ppering fter Lst Trde pulic clss StockQuote { pulic sttic void min(string[] rgs) { String nme = " + rgs[0]; In in = new In(nme); String input = in.redall(); int p = input.indexof("lst Trde:", 0); int from = input.indexof("<>", p); int to = input.indexof("</>", from); String price = input.sustring(from + 3, to); System.out.println(price); % jv StockQuote goog

3 Algorithmic Chllenges Theoreticl chllenge. Liner-time gurntee. Prcticl chllenge. Avoid ckup. fundmentl lgorithmic prolem Krp-Rin often no room or time to sve text Now is the time for ll people to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for mny good people to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for lot of good people to come to the id of their prty. Now is the time for ll of the good people to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for ech good person to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for ll good Repulicns to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for mny or ll good people to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for ll good Democrts to come to the id of their prty. Now is the time for ll people to come to the id of their prty. Now is the time for ll good people to come to the id of theirprty. Now is the time for mny good people to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for lot of good people to come to the id of their prty. Now is the time for ll of the good people to come to the id of their prty.now is the time for ll good people to come to the id of their ttck t dwn prty. Now is the time for ech person to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for ll good Repulicns to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for mny or ll good people to come to the id of their prty. Now is the time for ll good people to come to the id of their prty. Now is the time for ll good Democrts to come to the id of their prty. 9 Roert Sedgewick nd Kevin Wyne Copyright 00 Krp-Rin Rndomized Fingerprint Algorithm Computing the Hsh Function Ide: use hshing.! Compute hsh function for ech text position.! No explicit hsh tle: just compre with pttern hsh! Brute force. O(M) rithmetic ops per hsh. Fster method to compute hsh of djcent sustrings.! Use previous hsh to compute next hsh.! O(1) time per hsh, except first one. Ex. Hsh "tle" size = 97. Serch Pttern % 97 = 9 Ex. Pre-computed:! % 97 = 9! Previous hsh: 419 % 97 = 76! Next hsh: 196 % 97 =?? Serch Text % 97 = % 97 = % 97 = % 97 = % 97 = 9 Oservtion. key property of mod: cn mod out ny time! 196 % 97! (419 (4 * 10000)) * ! (76 (4 * 9 )) * ! 406!

4 Jv Implementtion Krp-Rin: Flse Mtches pulic sttic int serch(string p, String t) { int M = p.length(), N = t.length(); int q = 83967; // tle size int d = 6; // rdix Flse mtch. Hsh of pttern collides with nother sustring.! 96 % 97 = 9! 936 % 97 = 9 int dm = 1; for (int j = 1; j < M; j++) dm = (d * dm) % q; int h1 = 0, h = 0; for (int i = 0; i < M; j++) { h1 = (h1*d + p.chrat(i)) % q; h = (h*d + t.chrat(i)) % q; if (h1 == h) return 0; // precompute d^(m-1) % q // hsh of pttern // hsh of text // mtch found How to choose modulus p.! p too smll " mny flse mtches.! p too lrge " too much rithmetic.! Ex: p = " void 3-it integer overflow. Ex: p = " void 64-it integer overflow. for (int i = M; i < N; i++) { h = (h + d*q - dm*t.chrat(i-m)) % q; // remove leftmost digit h = (h*d + t.chrat(i)) % q;m)) % q; // insert rightmost digit if (h1 == h) return i - M + 1; // mtch found return -1; // not found Theorem. If MN # 9 nd p is rndom prime etween 1 nd MN, then Pr[flse mtch] $.3/N. relies on prime numer theorem Krp-Rin: Rndomized Algorithms String Serch Implementtion Cost Summry Rndomized lgorithm. Choose tle size p t rndom to e huge prime. Monte Crlo version. Don't other checking for flse mtches.! Gurnteed to e fst: O(M + N).! Expected to e correct (ut flse mtch possile). Krp-Rin summry.! Crete fingerprint of ech sustring nd compre fingerprints.! Expected running time is liner.! Ide generlizes, e.g., to D ptterns. Ls Vegs version. Upon hsh mtch, do full compre; if flse mtch, try gin with new rndom prime.! Expected to e fst: O(M + N).! Gurnteed to e correct. Q. Would either version of Rin-Krp mke good lirry function? chrcter comprisons Implementtion Typicl Worst Brute 1.1 N M N Krp-Rin %(N) %(N) Serch for M-chrcter pttern in N-chrcter text ssumes pproprite model rndomized 1 16

5 How To Sve Comprisons Knuth-Morris-Prtt How to void re-computtion?! Pre-nlyze serch pttern.! Ex: suppose tht first chrcters of pttern p re ll 's. if t[0..4] mtches p[0..4], then t[1..4] mtches p[0..3] no need to check i = 1, j = 0, 1,, 3 sves 4 comprisons void these comprisons Roert Sedgewick nd Kevin Wyne Copyright Knuth-Morris-Prtt: DFA Simultion Knuth-Morris-Prtt: DFA Simultion KMP lgorithm. [over inry lphet]! Build DFA from pttern.! Run DFA on text. Serch Text Interprettion of stte i. Length of longest prefix of serch pttern tht is suffix of input string. Ex. End in stte 4 iff text ends in. Ex. End in stte iff text ends in (ut not or ). Serch Pttern ccept stte ccept stte 19 0

6 DFA Representtion KMP Algorithm DFA used in KMP hs specil property.! Upon chrcter mtch in stte j, go forwrd to stte j+1.! Upon chrcter mismtch in stte j, go ck to stte next[j]. Two key differences from rute force.! Text pointer i never "cks up."! Need to precompute next[] tle. Serch Pttern next only store this row int j = 0; for (int i = 0; i < N; i++) { if (t.chrat(i) == p.chrat(j)) j++; else j = next[j]; if (j == M) return i - M + 1; return -1; // mtch // mismtch // found // not found Simultion of KMP DFA (ssumes inry lphet) ccept stte 1 Knuth-Morris-Prtt: DFA Construction DFA Construction for KMP KMP lgorithm. [over inry lphet]! Build DFA from pttern.! Run DFA on text. Rule for creting next[] tle for pttern.! next[4]: longest prefix of tht is suffix of.! next[]: longest prefix of tht is suffix of. DFA construction for KMP. DFA uilds itself! Ex. Compute next[6] for pttern p[0..6] =.! Assume you know DFA for pttern p[0..] =.! Assume you know stte X for p[1..] =. X =! Updte next[6] to stte for. X + =! Updte X to stte for p[1..6] = X + = 3 compute y simulting on DFA

7 DFA Construction for KMP DFA Construction for KMP DFA construction for KMP. DFA uilds itself! DFA construction for KMP. DFA uilds itself! Ex. Compute next[7] for pttern p[0..7] =.! Assume you know DFA for pttern p[0..6] =.! Assume you know stte X for p[1..6] =. X = 3! Updte next[7] to stte for. X + = 4! Updte X to stte for p[1..7] = X + = Crucil insight.! To compute trnsitions for stte n of DFA, suffices to hve: DFA for sttes 0 to n-1 stte X tht DFA ends up in with input p[1..n-1]! To compute stte X' tht DFA ends up in with input p[1..n], it suffices to hve: DFA for sttes 0 to n-1 stte X tht DFA ends up in with input p[1..n-1] 6 DFA Construction for KMP: Implementtion Optimized KMP Implementtion Build DFA for KMP.! Tkes O(M) time.! Requires O(M) extr spce to store next[] tle. Ultimte serch progrm for pttern.! Specilized C progrm.! Mchine lnguge version of C progrm. int X = 0; int[] next = new int[m]; for (int j = 1; j < M; j++) { if (p.chrat(x) == p.chrat(j)) { // chr mtch next[j] = next[x]; X = X + 1; else { next[j] = X + 1; X = next[x]; // chr mismtch DFA Construction for KMP (ssumes inry lphet) int kmperch(chr t[]) { int i = 0; s0: if (t[i++]!= '') goto s0; s1: if (t[i++]!= '') goto s0; s: if (t[i++]!= '') goto s; s3: if (t[i++]!= '') goto s0; s4: if (t[i++]!= '') goto s0; s: if (t[i++]!= '') goto s3; s6: if (t[i++]!= '') goto s; s7: if (t[i++]!= '') goto s4; return i - 8; ssumes pttern is in text (o/w use sentinel) next[] 7 8

8 KMP Over Aritrry Alphet KMP Over Aritrry Alphet DFA for ptterns over ritrry lphet &.! For ech chrcter in lphet, determine next stte.! Lookup tle requires O(M & ) spce. NFA for ptterns over ritrry lphet &.! Red new chrcter only upon success (or filure t eginning).! Reuse current chrcter upon filure nd follow ck. Ex. DFA for pttern c. cn e expensive if & = Unicode lphet Ex. NFA for pttern c. 0 ' 1 ' ' 3 ' 4 ' ' 3 ' 1 ' 0 ' 1 ' ' 3 text =, c mismtch mismtch c mismtch, c c c c c mismtch mismtch 9 30 String Serch Implementtion Cost Summry History of KMP KMP nlysis.! NFA simultion requires t most N comprisons. dvnces $ N retrets $ dvnces! NFA construction tkes %(M) time nd spce.! Good efficiency for ptterns nd texts with much repetition. History of KMP.! Inspired y esoteric theorem of Cook tht sys liner time lgorithm should e possile for -wy pushdown utomt.! Discovered in 1976 independently y two theoreticins nd hcker. Knuth: discovered liner time lgorithm Prtt: mde running time independent of lphet Morris: trying to uild n editor nd void nnoying uffer for string serch chrcter comprisons Implementtion Brute Typicl 1.1 N Worst M N Krp-Rin %(N) %(N) KMP 1.1 N N Serch for M-chrcter pttern in N-chrcter text ssumes pproprite model rndomized Resolved theoreticl nd prcticl prolems.! Surprise when it ws discovered.! In hindsight, seems like right lgorithm. 31 3

9 Right-to-Left Scnning Boyer-Moore Right-to-left scnning.! Find offset i in text y moving left to right.! Compre pttern to text y moving j right to left. h i c k o r y, d i c k o r y, d o c k,. Bo Boyer J. Strother Moore Roert Sedgewick nd Kevin Wyne Copyright Bd Chrcter Rule Bd Chrcter Rule Bd chrcter rule. Bd chrcter rule.! Use right-to-left scnning. right! Use right-to-left scnning. right! Upon mismtch of text chrcter c, increse offset so tht chrcter c in pttern lines up with text chrcter c.! Precompute right[c] = rightmost occurrence of c in pttern. c k l o 3 4 1! Upon mismtch of text chrcter c, increse offset so tht chrcter c in pttern lines up with text chrcter c.! Precompute right[c] = rightmost occurrence of c in pttern. c k l o * -1 * -1 h i c k o r y d i c k o r y d o c k. h i c k o r y, d i c k o r y, d o c k,. 3 36

10 Bd Chrcter Rule: Jv Implementtion Bd Chrcter Rule: Anlysis pulic sttic int serch(string pttern, String text) { int M = pttern.length(), N = text.length(); int[] right = new int[6]; for (int c = 0; c < 6; c++) right[c] = -1; for (int j = 0; j < M; j++) right[pttern.chrat(j)] = j; Bd chrcter rule nlysis.! Highly effective in prctice, prticulrly for English text: O(N / M).! Tkes ((MN) time in worst cse. int i = 0; // offset rightmost occurrence of c in pttern while (i < N - M) { int skip = 0; for (int j = M-1; j >= 0; j--) { if (pttern.chrat(j)!= text.chrat(i + j)) { skip = Mth.mx(1, j - right[text.chrat(i + j)]); rek; d chrcter rule if (skip == 0) return i; // found i = i + skip; return -1; Strong Good Suffix Rule Boyer-Moore Strong good suffix rule. [ KMP-like suffix rule]! Right-to-left scnning.! Suppose text mtches suffix t of pttern ut mismtches in previous chrcter c.! Find rightmost copy of t in pttern whose preceding letter is not c, nd shift; if no such copy, shift M positions. t = "" c = '' Boyer-Moore.! Right-to-left scnning.! Bd chrcter rule.! Strong good suffix rule. Boyer-Moore nlysis. lwys tke est of two shifts! O(N / M) verge cse if given letter usully doesn't occur in string. time decreses s pttern length increses suliner in input size!! At most 3N comprisons to find mtch. x x x x x x x?????? x x x x x x x x x c d d x c d d string good suffix rule: cn skip over this since we lredy know d doesn't mtch Boyer-Moore in the wild. Unix grep, emcs. d chrcter rule: skip only 1 position 39 40

11 String Serch Implementtion Cost Summry Boyer-Moore nd Alphet Size Boyer-Moore spce requirement. %(M + & ) Implementtion Brute Typicl 1.1 N Worst M N Krp-Rin %(N) %(N) KMP 1.1 N N Boyer-Moore N / M 3N Serch for M-chrcter pttern in N-chrcter text ssumes pproprite model rndomized Big lphets.! Direct implementtion my e imprcticl, e.g., UNICODE.! Fix: serch one yte t time. Smll lphets.! Loses effectiveness when & is too smll, e.g., DNA.! Fix: group chrcters together, e.g.,, c, Finding All Mtches Multiple String Serch Krp-Rin. Cn find ll mtches in O(M + N) expected time using Muthukrishnn vrint. Knuth-Morris-Prtt. Cn find ll mtches in O(M + N) time vi simple modifiction. Multiple string serch. Serch for ny of k different ptterns.! Nïve KMP: O(kN + M M k ).! Aho-Corsick: O(N + M M k ).! Ex: screen out dirty words from text strem. 6 ccept stte serch pttern: 8 9 Boyer-Moore. Cn find ll mtches in O(M + N) time using Glil vrint. 6 7 or or 43 44

12 Spm Filtering Tip of the Iceerg Spm filtering. Identify ptterns indictive of spm.! PROFITS! AMAZING! GUARANTEE! herl Vigr! There is no ctch.! This is one-time miling.! This messge is sent in complince with spm regultions.! You're getting this messge ecuse you registered with one of our mrketing prtners. Wildcrds / chrcter clsses.! O(M + N) time using O(M + & ) extr spce.! Ex: PROSITE ptterns for computtionl iology. Approximte string mtching: llow up to k mismtching chrs.! Ex: fix trnsmission errors in signl processing.! Ex: recover from typing or spelling errors in informtion retrievl. Edit-distnce: llow up to k edits.! Recover from mesurement errors in computtionl iology Jv String Lirry String Serch Summry Jv String lirry hs uilt-in string serching.! t.indexof(p): index of 1 st occurrence of pttern p in text t.! Cvet: it's rute force, nd cn tke ((MN) time. pulic sttic void min(string[] rgs) { int n = Integer.prseInt(rgs[0]); String s = ""; for (int i = 0; i < n; i++) s = s + s; String pttern = s + ""; String text = s + s; System.out.println(text.indexOf(pttern)); n n+1 Ingenious lgorithms for fundmentl prolem. Rin-Krp.! Esy to implement, ut usully worse thn rute-force.! Extends to more generl settings (e.g., D serch). Knuth-Morris-Prtt.! Quintessentil solution to theoreticl prolem.! Extends to more generl settings (e.g., multiple string serch). Boyer-Moore.! Simple ide leds to drmtic speedup for long ptterns.! Running time depends on lphet size.! Need to twek for smll or lrge lphets. Q. Why does Jv string lirry use rute force? 47 48