The Constrained Longest Common Subsequence Problem. Rotem.R and Rotem.H

Transcription

1 T Constrn Lonst Common Susqun Prolm Rotm.R n Rotm.H

2 Prsntton Outln. LCS Alortm Rmnr Uss o LCS Alortm T CLCS Prolm Introuton Motvton For CLCS Alortm T CLCS Prolm Nïv Alortm T CLCS Alortm A Dynm Prormmn Alortm Tm Anlyss

3 LCS Alortm Rmnr Bor w wll t to know our prolm lts rmmr LCS prolm n ts ynm soluton: T Prolm: Gvn squns X=<x,..,xm> n Y=<y,..,yn>, n ommon susqun wos lnt s mxmum. Exmpl: Sprntm n tournmnt sktll Prntn nort roln snoynk prnt nrn s

4 LCS Alortm Rmnr Soluton: Lt Z = <z,..., z k > ny LCS o X n Y.. I x m = y n, tn z k = x m = y n n Z k s n LCS o X m n Y n.. I x m y n, tn tr z k x m n Z s n LCS o X m n Y.. or z k y n n Z s n LCS o X n Y n X: Y:

5 LCS Alortm Rmnr Soluton: Lt Z = <z,..., z k > ny LCS o X n Y.. I x m = y n, tn z k = x m = y n n Z k s n LCS o X m n Y n.. I x m y n, tn tr z k x m n Z s n LCS o X m n Y.. or z k y n n Z s n LCS o X n Y n. Cs : X m = Y n X: Y:

6 LCS Alortm Rmnr Soluton: Lt Z = <z,..., z k > ny LCS o X n Y.. I x m = y n, tn z k = x m = y n n Z k s n LCS o X m n Y n.. I x m y n, tn tr z k x m n Z s n LCS o X m n Y.. or z k y n n Z s n LCS o X n Y n. Cs : X m = Y n X: Y: LCS(X m, Y n )

7 LCS Alortm Rmnr Soluton: Lt Z = <z,..., z k > ny LCS o X n Y.. I x m = y n, tn z k = x m = y n n Z k s n LCS o X m n Y n.. I x m y n, tn tr z k x m n Z s n LCS o X m n Y.. or z k y n n Z s n LCS o X n Y n. Cs : X m Y n X: Y: m

8 LCS Alortm Rmnr Soluton: Lt Z = <z,..., z k > ny LCS o X n Y.. I x m = y n, tn z k = x m = y n n Z k s n LCS o X m n Y n.. I x m y n, tn tr z k x m n Z s n LCS o X m n Y.. or z k y n n Z s n LCS o X n Y n. Cs : X m Y n X: Y: m LCS(X m, Y n )

9 LCS Alortm Rmnr Soluton: Lt Z = <z,..., z k > ny LCS o X n Y.. I x m = y n, tn z k = x m = y n n Z k s n LCS o X m n Y n.. I x m y n, tn tr z k x m n Z s n LCS o X m n Y.. or z k y n n Z s n LCS o X n Y n. Cs : X m Y n X: Y: m LCS(X m, Y n )

10 Uss o LCS Alortm Wy mt w wnt to solv t lonst ommon susqun prolm? Tr r svrl motvtn ppltons. Fl omprson: T Unx prorm "" s us to ompr two rnt vrsons o t sm l, to trmn wt ns v n m to t l. It works y nn lonst ommon susqun o t lns o t two ls; ny ln n t susqun s not n n, so wt t splys s t rmnn st o lns tt v n. In ts nstn o t prolm w soul tnk o ln o l s n snl omplt rtr n strn.

11 Uss o LCS Alortm

12 Uss o LCS lortm Srn rsply: Mny txt tors lk "ms" sply prt o l on t srn, uptn t srn m s t l s n. For slow ln trmnls, ts prorms wnt to sn t trmnl s w rtrs s possl to us t to upt ts sply orrtly. It s possl to vw t omputton o t mnmum lnt squn o rtrs n to upt t trmnl s n sort o ommon susqun prolm (t ommon susqun tlls you t prts o t sply tt r lry orrt n on't n to n).

13 T CLCS Prolm Introuton Suppos w wnt to qunty t smlrty o S n S. W my sy tt tr smlrty s sn LCS(S,S ) s strn o lnt S : S : Howvr, ts msurmnt s not ststory w know tt t susqun pprs n ot strns n soul onsr or t smlrty msurmnt.

14 T CLCS Prolm Introuton To tk t susqun nto ount, w sk or t lonst (not nssrly unqu) su squn tt s susqun o ot S n S n ontns s susqun S : S : In ts s, ot n o lnt r LCSs unr ts onstrnt, n t smlrty msurmnt oms.

15 Motvton : Som Trmnoloy Frst RNs (Ronuls): lns llulr RNA tt s no lonr rqur. ply ky rols n t mturton o RNA moluls, su s mssnr RNAs.

16 Motvton T orns o CLCS ly n onormts, wr squns ontnn DNA n protns r ompr. In t lssl LCS, w look or t lonst susqun n somtms t otn rsult s o lttl olol vlu. Ts s us y t t tt w v no posslty to us ny pror, olol, knowl o t squns. For xmpl (Tn t l.): In t lnmnt o RNs squns, t s known tt squns ontn tr tvst rsus, Hs(H), Lyn(K), Hs(H), tt r ssntl or RNA rn. Tror, olosts otn r ntrst only n t susqun n w ts tr rsus our n t vn orr (t squn HKH s onstrn).

17 T CLCS Prolm Nïv Alortm Gvn strns S= n, S= m n P=p p p r wr r mn(m,n), t CLCS prolm or S n S wt rspt to P s to n lonst ommon susqun o S n S tt ontns P. Nïv Alortm: For vry susqun o S or S tt ontns t susqun P, k wtr t s susqun o S or S. Rturn lonst su susqun. Tm: O(mx(m,n) mn(n,m) ). mn(n,m) susquns o S or S. E susqun tks O(n) or O(m) tm to sn or P. It tks O(m) or O(n) to k wtr tt squn pprs n S or S.

18 T CLCS Prolm Intuton Smplton: Look t t lnt o CLCS(S, S, P) n xtn t l. to n CLCS(S, S, P) tsl S : S : P:

19 T CLCS Prolm Intuton Smplton: Look t t lnt o CLCS(S, S, P) n xtn t l. to n CLCS(S, S, P) tsl S : S : P: LCS I S [] = S []!= P[k], w my sy tt It woul tk prt n ormn CLCS(S, S, P) So woul ny otr rtr tt ontruts to t lnt o CLCS(S, S, P)

20 T CLCS Prolm Intuton Strty: Consr rkponts orrsponn to mtn rtrs n S n S. Us LCS n CLCS rursvly S : S : P:? LCS

21 T CLCS Prolm Intuton Strty: Consr rkponts orrsponn to mtn rtrs n S n S. Us LCS n CLCS rursvly. W rk pont s ttr? S : S : P: CLCS LCS LCS All possl omntons o rkponts must tkn nto ount to W orms t optml soluton.

22 T CLCS Prolm Dntons Lt S[x] t rtr t poston x n strn S. Lt S[x..y] t sustrn o S rom postons x to y x y, n n mpty strn otrws. Lt L(x,y,x',y') t lnt o LCS(S[x..x ], S[y..y']) x<x' n, n y<y' m, n otrws. For k r, n, m, lt L k (,) t lnt o CLCS (S[..], S[..], P[..k]) S[]=S[]=P[k], n n otrws. It s sy to know tt: L (,)=L(,,,)+ S[]=S[]=P[], n n otrws.

23 T CLCS Alortm Lmm: For k r, n n m, L k (,) = Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. S : x S : y P: CLCS LCS Proo: Suppos tt S[]=S[]=P[k], k>. Lt x n y su tt S[x]=S[y]=P[k ] wr x< n y<. Tr xsts onstrn ommon susqun o S[..] n S[.. ] wt lnt L k (x,y)+l(x+,y+,, )+. Sn L k (, ) s t lonst lnt o onstrn ommon susqun or S[..] n S[.. ] wt rspt to P[..k], w v: L k (,) Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}

24 T CLCS Alortm Lmm: For <=k<=r, <=<=n n <=<=m, L k (,) = Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. S : x S : y P: CLCS LCS Assum L k (,) > Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+} n lt x n y su tt S[x ]=S[y ]=P[k] n n optml soluton s st wt lnt Lk(,). Tn L k (,) = CLCS(S [..x ],S [..y ],P[..k])+LCS(S [(x +)..()],S [(y +)..()])+ > Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+} But: L k (x,y ) CLCS(S[..x ],S[..y ],P[..k]) L(x +,y +,,) LCS(S[(x +)..()],S[(y +)..()]) Tror: L k (x,y )+L(x +,y +,,)+ > Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+} w s ontrton.

25 T CLCS Alortm Lmm: T lnt o ls, G, or strns S n S wt rspt to strn P s: G =mx <=<=n,<=<=m {L r (,)+L(+,+,n,m)} wr n, m n r r t lnts o S, S n P, rsptvly. Proo: Lt S =S $, S =S $ n P =P $, wr $ s not prsnt n ny o S, S, P. Lt L (x,x,y,y ) t lnt o ls(s [x..x ],S [y..y ]) Lt L k (,) t lnt o ls(s [..],S [..],P [..k]) S []=S []=P [k] n n otrws. By t rsult o Lmm, w v t ollown: L k (,) = Lt G not ls(s,s,p). It ollows tt G = G <=x<x <=n+, <=y <y<=m+ <=k <=r+, <= <=n+, <= <=m+ Mx <=x<,<=y< {L k (x,y)+l (x+,y+,,)+}, S []=S []=P [k] n, otrws <=k<=r+, <=<=n+, <=<=m+ S [n+]=s [m+]=p [r+]=$ G = L r+ (n+,m+)=mx <=x<n+,<=y<m+ {L r (x,y)+l (x+,y+,n,m)+} S () [..n(m)]=s () [..n(m)], P[..r]=P [..r] L r (x,y)=l r (x,y) n L(x+,y+,n,m)=L (x+,y+,n,m) G = L r+ (n+,m+) = Mx <=x<=n,<=y<=m {L r (x,y)+l(x+,y+,n,m)+} <=x<=n <=y<=m Fnlly, w onlu tt: G = G = Mx <=x<=n,<=y<=m {L r (x,y)+l(x+,y+,n,m)}.

26 A Rursv Alortm Lmms & v totr rursv lortm: Lmm: lnt o CLCS(S,S,P)=mx <=<=n,<=<=m {L r (,)+L(+,+,n,m)} wr (n,m,r)=lnt(s,s,p). CLCS (S,S,P): (n,m,r) ln(s,s,p) G n or =..n, =..m: rs CLCSR(S [..],S [..],P) + LCS(S [+.n],s [+..m]) G < rs tn G rs Lmm: L k (,) = Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. CLCSR (S,S,P): // nor s s (,,k) ln(s,s,p) L[,,k] n I S []=S []=P[k]: or x=.., y=..: rs CLCSR(S [..x],s [..y],p[..k]) + LCS(S [x+..],s [y+..])+ L[,,k] < rs tn L[,,k] rs rturn L[,,k]

27 A Dynm Prormmn Alortm Ovrlppn suprolms: S : S : P: S[]=S[]=P[] n (,) xpt (,), (,) CLCSR (S [..],S [..],P[..]): n (,)!= (,), (,) CLCSR (S [..],S [..],P[..]): n (,)!= (,), (,) LCS(S [..],S [..]) // s s LCS(S [..],S [..]) // s s Ovrlppn LCS su prolms: Us prprossn. Ovrlppn CLCS su prolms my lso our: Us DP l.

28 A Dynm Prormmn Alortm Apply prprossn st or trmnn L(x,y,x,y ) n O() tm. Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = S : S : LCS:

29 A Dynm Prormmn Alortm Apply prprossn st or trmnn L(x,y,x,y ) n O() tm. Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = S : S : LCS:

30 A Dynm Prormmn Alortm Apply prprossn st or trmnn L(x,y,x,y ) n O() tm. Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = S : S : LCS:

31 A Dynm Prormmn Alortm Apply prprossn st or trmnn L(x,y,x,y ) n O() tm. Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = S : S : LCS: E ntry [,] n t lookup tl vn y pplyn t prprossn st, stors Mtrx, M,, o sz (n +) (m +) su tt t vlu o M, [u, v] s t lnt o LCS(S [.. + u ],S [..+ v ]), wr <=u<=n + n <=v<=m +.

32 A Dynm Prormmn Alortm Apply prprossn st or trmnn L(x,y,x,y ) n O() tm. Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = S : S : LCS: Prprossn: For n n m w omput mtrx M o sz (n+)x(m+) to stor LCS(S [..+u],s [..+v]) wr u n+ n v m+. E su mtrx n omput n O((n+)(m+)). All mtrs M n omput n Σ <=<=n Σ <=<=m O((n+)(m+))=O(m n )

33 A Dynm Prormmn Alortm Comput D mtrx, ottomup, to n L k (,). Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = 9 7 S : S : P: L (,): s s: L (,)= or ll (,) xpt tt: L (,)=L(,,,)+= L (,)=L(,,,)+=

34 A Dynm Prormmn Alortm Comput D mtrx, ottomup, to n L k (,). Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = 9 7 S : S : P: L (,): L (,)= or ll (,) xpt tt: L (,)=mx {L (,)+L(,,,), L (,)=L(,,,)}+= L (,)=mx {L (,)+L(,,,), L (,)=L(,,,)}+=

35 A Dynm Prormmn Alortm Comput D mtrx, ottomup, to n L k (,). Lmm: Mx <=x<,<=y< {L k (x,y)+l(x+,y+,,)+}, S []=S []=P[k] ; n, otrws. L k (,) = 9 7 S : S : P: L (,): L (,)= or ll (,) xpt tt: L (9,)=mx {L (,)+L(7,,,7), L (,)+L(7,7,,7)}+=

36 A Dynm Prormmn Alortm Comput D mtrx, ottomup, to n L k (,). Lmm: G =mx <=<=n,<=<=m {L r (,)+L(+,+,n,m)} 9 7 S : S : P: L (,) L (,) L (,) By t rsult o Lmm: G =mx{l (9,)+L(,9,,)}=7

37 Tm Anlyss Prprossn: For n n m w omput mtrx M o sz (n+)x(m+) to stor LCS(S [..+u],s [..+v]) wr u n+ n v m+. E su mtrx n omput n O((n+)(m+)). All mtrs M n omput n Σ <=<=n Σ <=<=m O((n+)(m+)) = O(m n ) Alortm: Atr Prprossn stp, t vlu o L(x,y,x,y ) n omput n O() tm y tl lookup or M xy [x x,y y]. Aorn to t ormulton n Lmm, L k (,) n oun n O() tm L k s known. Tn, L k n otn n O(Σ <=<=n,<=<=m ())=O(n m ) tm ll L k s n O(rn m ) tm. Morovr, t tks O(mn) to omput G orn to Lmm wn L r s Known. Tror, w onlu tt t CLCS prolm n solv n O(rn m ) tm n totl.