String Matching - II. Patterns {he, she, his, hers} The KMP Algorithm. Theorem: At most 2N comparisons in total. The Aho-Corasick Algorithm (1986)

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1 Agoritm Dign nd Anyi Victor Admcik CS Spring 2014 Lctur 40 Apr 28, 2014 Crngi Mon Univrity String Mtcing - II T KMP Agoritm Torm: At mot 2N comprion in tot Morri Knut T Ao-Corick Agoritm (1986) Pttrn {,, i, r trt 0 1 r 2 8 * 9 i T goritm prproc t t of pttrn n 7 n 5 T Ao-Corick Agoritm W ti u t ongt uffix ru. If w fi on mking trnition from nod N to it cid, w trnition to nod M, wr t tring tt dfin M i t frtt nod (ongt prfix) from t root wic i o uffix of t tring w d mtcd wn w fid (rmoving t firt trnition). T ony diffrnc i tt intd of trvring ing tring ft-to-rigt w now v to trvr tri. T Rbin-Krp Agoritm (1981) T goritm u t id of ing 1

2 pttrn = 4848 T min id txt = W do not mtc tring gint givn pttrn, but rtr compr tir cod. T min id pttrn = 4848 % 71 = % 71 = % 71 = % 71 = 28 W rd t txt in t numbr of crctr qu to t ngt of t pttrn, comput it cod nd compr wit t pttrn cod. Wt i it compxity? M = pttrn.ngt() N = txt.ngt(); Computing cod How cn w gt from 145 to 456? W wi do ti by crting cin of oprtion Simir to brut-forc mtcing T ky id of improving t goritm i in computing cod in O(1) Rmov t ding digit, mutipy by b, dd ing digit. It tk O(1) to comput cod from t prviou vu. Exmp Exmp Givn: cod for Givn: cod for mod 41 = mod 41 = 36 Tk: comput cod for Tk: comput cod for Obrv, = ( *10 4 ) *

3 Exmp 17295%41 = [(31729%41-3*10 4 %41) *10 + 5]% %41 i rdy computd. 3*10 4 % 41 wi b prcomputd 17295%41 = [(36-29) * ]%41 = 75 % 41 = 34 Rbin-Krp formizd Lt P[1... m] b pttrn nd T[1... n] b txt. W dfin pttrn P = 10 m-1 P[1] + 10 P[m-1] + + P[m] nd ift in t txt: t = 10 m-1 T[+1] + 10 T[+m-1] + + T[+m] T vu t +1 cn b obtind from t by t +1 = (t - 10 m-1 T[+1]) 10 + T[+m+1] Exrci Hornr Ru W id 31729%41 i rdy computd How woud you comput it ft? x 4 + b x 3 + c x 2 + d x + = + x (d + x (c + x (b + x) Impmnttion pubic int rc(string T, String P){ int M = P.ngt(), N = T.ngt(); Impmnttion (cont.) if(1 == 2) rturn 0; int dm = 1, 1 = 0, 2 = 0; int q = ; /*pick it t rndom */ int d = 256; /* rdix */ for(int j = 1; j < M; j++) dm = (d*dm) % q; for(int j = 0; j < M; j++){ 1 = (1*d + P.crAt(j)) % q; 2 = (2*d + T.crAt(j)) % q; for(int i = M; i < N; i++) { 2 = 2 - T.crAt(i - M) * dm % q; 2 = (2*d + T.crAt(i)) % q; if(1 == 2) rturn i - M + 1; rturn -1; 3

4 F mtc TRIES = "rtriv" T == P mod q Wt do w do in c of f mtc? Wn w found mtc w cn cck t mtc by cr comprion. Frdkin (1960) Min id: bd on t digit of t ky! TRIES by t or Ec nod (or dg) i bd wit crctr Cidrn of nod r ordrd (pbticy) Pt from root to v yid input tring b y $ o r t ntin Appiction Auto comption Sp cckr Dt comprion Computtion bioogy Goog invrtd tb Nod Structur Oftn wtfu of pc bcu mny of t cid fid r nu. Poib nod rprnttion: Arry H Tb Linkd Lit Binry Tr 4

5 Src pubic boon find (TriNod nod, String ky) { if (ky.ngt()==0) rturn nod.iword(); cr c = ky.gtcr(0); String rt = ky.ubtring(1); TriNod cid = nod.gtcid(c); if(cid == nu) rturn f; Runtim rturn find (cid, rt); compxity -? Inrt pubic void inrt (TriNod nod, String ky) { if (ky.ngt()==0) nod.tword(tru); cr c = ky.gtcr(0); String rt = ky.ubtring(1); TriNod cid = nod.gtcid(c); if(cid == nu) { nod.tcid(nw TriNod(c), c); inrt (nwcid, rt); Runtim compxity -? inrt (cid, rt); Prfix Mtc Find tring trting wit b t y $ o r Advntg, rtiv to BST Advntg, rtiv to ing Src i ftr! It do not dpnd on t numbr of mnt in t tr. Tri p wit prfix-mtcing. No coiion. No function. Apbtic orting. How? 5

6 Comprd Tri Comprd Tri by t or Ec non-f nod (xcpt root) t t two cidrn Rpc cin of on-cid nod wit ing nod bd wit tring by t $ or Compct Tri (PATRICIA) A mor compct rprnttion of comprd tri Compct Tri (PATRICIA) A mor compct rprnttion of comprd tri by t by t Intgr indx $ or t $ or t o r (i, j, k) o r 6

Binary Search Trees. Definition Of Binary Search Tree. The Operation ascend() Example Binary Search Tree

Binary Search Trees. Definition Of Binary Search Tree. The Operation ascend() Example Binary Search Tree Binary Sar Trs Compxity O Ditionary Oprations t(), put() and rmov() Ditionary Oprations: ƒ t(ky) ƒ put(ky, vau) ƒ rmov(ky) Additiona oprations: ƒ asnd() ƒ t(indx) (indxd inary sar tr) ƒ rmov(indx) (indxd