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1 Lzy Complton o Prtl Orr to t Smllst Ltt Krll Brtt 1, Ml Morvn 1, Lour Nourn 2 1 LITP/IBP - Unvrst Dns Drot Prs 7 Cs , pl Jussu Prs Cx 05 Frn. ml: (rtt,morvn)@ltp.p.r. 2 Dprtmnt 'Inormtqu Fonmntl LIRMM, Unvrst Montpllr II CNRS UMR C ru A, Montpllr Cx 5, Frn. ml: nourn@lrmm.r. Astrt. Ltt struturs r otn us n knowlg prossng, ut strtng rom prtl orr, omplton nto ltt poss ny prolms. W us som rnt rsults on ltt tory to propos n onln lgortm or ntly omputng o t smllst ltt ontnng gvn prtl orr, ll t Dkn-MNll omplton. By onln w mn tt t lmnts r nssry wn w omput t grtst lowr oun or t lst uppr oun o two lmnts tt o not lry xst. Ts rsult n us n knowlg prossng or mntnng t rry o typs n n on-ln son. Kywors: lgortm, prtl orr, ltt, Dkn-MNll omplton, nong. 1 Introuton n Motvtons Ltts r n mportnt lss o prtl orrs us o tr struturl proprts n tr ntrst n mny rs s knowlgs prossng wt t rry o typs [4,6,1,13,7]. T prolm o mnmlly xtnng prtl orr to t smllst ltt pprs n mny ppltons. Su n pplton s t rry o typs w s rt yl grp wr ntrprtton pns on ppltons (nrt, susum, unton,...). Mntnng t grp o typs s rul n knowlg prossng. Ts grp s usully ltt, ut t s rlz ynmlly,. wl prossng nw knowlg. Most o t tm, t pplton supposs tt only prtl grp s known n tt ts grp s omplt wn nssry y ng nw typ. Tt s, w r gvn som typs n w wnt to omput t gnrlzton or t splzton. But omputng t st o ll gnrl typs s not lwys prtl, sn som typs my not not sgnnt or ts pplton. Formlly, gvn prtl orr P = (X; E), w v to omput t Dkn- MNll omplton o P, not y DM(P ), w s t smllst ltt w ms P s suorr. For mor tls on Dkn-MNll omplton s Bout [3] n Dvy n Prstly [5]. Fgur 1 sows n xmpl

2 o su omplton. From Bout n Wll [3,13]w know tt DM(P) s somorp to t ltt o mxml ntns o t prtt orr (X; X; 6), n lso s somorp to t Glos ltt o (X; X; ). So wy to omput DM(P ) s smply to us lgortms or omputng t ltt o mxml ntns or t Glos ltt o gvn prtl orr. Tr r two nt lgortms to o ts [2,9]. In ts ppr, w prsnt n lgortm or on-ln omputng DM(P ), nmly t lzy Dkn-MNll omplton: W ynmlly omput DM(P ) y ng only lmnts wn w n tm s t lowr or uppr oun o two gvn lmnts. In Ston 3 w sow t mn onstrnt to rspt n orr to nsur orrt mnml onstruton. To o so, w propos n nong o P w gvs wy w s su n lmnt n n nt wy s on n rsult [1,11] n ltt tory. Fnlly, w prsnt n lgortm w s su n lmnt wt t us o tr. P DM(P) Fg. 1. Dkn-MNll omplton o post 2 Dntons n Nottons A prtl orr P = (X; P ) s rxv, ntsymmtr n trnstv nry rlton on st X. Two stnt lmnts x n y r s to omprl x P y or y P x. Otrws, ty r s nomprl, not y xjj P y. P s s prtt orr wn X = X1 [ X2 wt X1 \ X2 = ; n x P y mpls x 2 X1 n y 2 X2. A prtt orr s otn not y P = (X1; X2; P ). W n t ollowng sts or n lmnt x o X : P r(x) = y 2 Xjy < P xg t st o prssors o x, Su(x) = y 2 Xjx < P yg t st o sussors o x. A prtl orr Q = (X 0 ; Q ) s suorr o P X 0 X n t rlton n Q s t sm s t rlton n P on t lmnts o X 0. An lmnt z 2 X s n uppr oun o x; y 2 X x P z n y P z. Lt z n uppr oun o x n y, z s ll t lst uppr oun or t jon o x n y z P t, or ll uppr ouns t o x n y. T grtst lowr oun, or mt s n ully. W not y LUB(x; y) (rsp GLB(x; y)) t lst uppr oun (rsp t grtst lowr oun) o x n y.

3 A non-mpty prtl orr P s ll ltt LUB(x; y) n GLB(x; y) xst or ll x; y 2 X. It s lr tt nt ltt s on mnml lmnt n on mxml lmnt not rsptvly y > n?. Sn w r lng wt lgortms t ltts w onsr r suppos to nt. Lt L = (X; L ) ltt. An lmnt x 2 X s s to jon-rrul x 6=? n x = LUB(y; z) mpls x = y or x = z or ll y; z 2 X. Mtrrul lmnts r n ully. W not t st o jon-rrul lmnts o L y J(L) n t st o mt-rrul lmnts y M(L). W not y 2 X t st o ll susts o X. Dnton 1. [10] Lt P = (X; P ) prtl orr. T Dkn-MNll omplton o P, not y DM(P ), s t smllst ltt ontnng P s suorr. 3 Dkn-MNll Complton o Prtl Orr Sn P s suorr o DM(P ), w v to lmnts n P n orr to omput DM(P ). A nv lgortm onssts n ng n lmnt n P wn t GLB o two lmnts x n y os not xst n P, wt x n y s sussors n P r(x) \ P r(y) s prssors. But ts lgortm os not omput t Dkn-MNll omplton s t ollowng xmpl llustrts t. Exmpl 2. Lt P t prtl orr n Fgur 2(). Suppos tt t pplton sks or t GLB o n. T nv lgortm wll n lmnt low n s n () us GLB(; ) os not lry xst. Lkws, t pplton sks or t GLB o n g, n lmnt wll low n g s n (). In ts wy, ts two lmnts r not n t Dkn-MNll omplton o P sown n (). Clrly ts lgortm nnot us. Lt L ltt n (L) t st o ll prtl orrs vng L s Dkn- MNll omplton. W know tt t prtl orr nu y J(L) [ M(L) s suorr o orr o (L) [5]. Consr t st o lmnts o L w r not rrul lmnts. E sust o ts st orrspons to st o lmnts tt w v to to J(L) [ M(L) to otn prtl orr o (L). Sn ll t susts o ts st orr y nluson orms ooln ltt, w otn n somorp ooln ltt wt t st (L) orr y suorr (. P P 0 P s suorr o P 0 ). Clrly, t > lmnt o (L) s L n t? s t prtl orr nu y J(L) [ M(L). Fgur 3 sows t st (L) or ltt L. In (DM(P )), ny pt rom prtl orr P to > o DM(P ) orrspons to wy o nrmntlly omputng DM(P ). T o o su wy n tkn rltvly to t pplton, or xmpl ng lmnts rom? to > on y lvl t stp. In t ollowng, w gv n lgortm tt ollows pt orng to qustons sk y t pplton. T lmnts r nssry wn w omput GLB or LUB tt os not lry xst.

4 g g ( ) ( ) g g ( ) ( ) Fg. 2. Crton o ummy vrts 4 Enong Prsrvng LUB n GLB W now rll n nong o prtl orr y sts, w llows us to vo t rton o xtr lmnts s sown n Fgur 2. Ts nong s s on rsult o Mrkowsky [11] on ltts, w pprs mpltly n At-K t l[1] ppr on ltts nong. Dnton 3. Lt T n L ltts. Lt x n y two lmnts o T. A unton ' : T! L s s to : 1. Mt-prsrvng '(GLB(x; y)) = GLB('(x); '(y)). 2. Jon-prsrvng '(LUB(x; y)) = LUB('(x); '(y)). Torm 4. [11] Lt L = (X; L ) ltt. Tn t mppng 1. ' : X! 2 J(L) wt '(x) = j 2 J(L) su tt j L xg s mt-prsrvng,. '(GLB(x; y)) = '(x) \ '(y). 2. : X! 2 M(L) wt (x) = m 2 M(L) su tt m L xg s jonprsrvng,. (LUB(x; y)) = (x) \ (y). Ts two nongs r optml (. t szs o ' n r mnml). Not tt or strutv ltt ' n r ot jon-prsrvng n mt-prsrvng[11]. Lmm 5. Lt L = (X; L ) ltt. Tn ny lmnt o L ut t? (rsp. >), s t LU B (rsp.glb) o jon-rrul (rsp. mt-rrul) lmnts.

5 L : Fg. 3. St (L) or ltt L In t ollowng, w sow ow to omput t jon-rrul n mtrrul lmnts o DM(P ) not y J(P ) n M(P ) rom t prtl orr P. Ts n on wtout omputng DM(P ). For tt, w n to ntrou t ollowng notons: Lt P = (X; P ) prtl orr. W omput t ollowng prtt orr B = (X; X 0 ; B ), wt X 0 opy o X n x B y x s n X, y s n X 0 n x 6 P y. Dnton 6. [12] A prtt orr B = (X; X 0 ; P ) s s to ru : C0 : For ll x 2 X, or ll y 2 X 0, w v Su B (x) 6= ;, n P r S B (y) 6= ; k C1 : For ll x 2 X, or ll x1; : : : ; x k 2 Xnxg, w v Su B (x) 6= S =1 Su B(x ) C2 : For ll y 2 X 0, or ll y1; : : : ; y k 2 X 0 k nyg, w v P r B (y) 6= =1 P r B(y ) Conton C1 s not rspt n t ollowng ss: ) Tr xst x n x 0 n X su tt x 6= x 0 n Su B (x) = Su B (x 0 ) ) Tr xst x n X n x1; : : : ; x k n Xnxg wt k > 1 su tt Su B (x) = S k =1 Su B(x ) T symmtrl rmrk n m or C2: ') Tr xst y n y 0 n X 0 su tt y 6= y 0 n P r B (y) = P r B (y 0 ) ') Tr xst y n X 0 n y1; : : : ; y k n X 0 nyg wt k > 1 su tt P r B (y) = S k =1 P r B(y ) So w n trnsorm prtt orr B = (X; X 0 ; B ) n ru prtt orr Bp(P ) = (J; M; B ) y ltng sngltons (C0), ltng on o t two lmnts x or x 0 (rtrrly) n s ), ltng x n s ), ltng on o t two lmnts y or y 0 (rtrrly) n s 0 ), ltng y n s 0 ).

6 It n sown tt J (rsp. M) s t st o jon-rrul (rsp. mtrrul) lmnts o DM(P ). Exmpl 7. Fgur 4 llustrts t nongs or t prtl orr o Fgur 2: () t prtt orr (X; X 0 ; 6 P ), () ts ruton Bp(P ) = (J; M; 6 P ) wr s lt us P r() = P r() [ P r() n s lt us P r() = ;, () t nongs ' n o t lmnts n P. g g ( ) g M(P) J(P) ( ) ' g g g g g ; () Fg. 4. Enongs ' n Now w n pply Torm 4 to v t jon-prsrvng n mt-prsrvng nong or DM(P ) usng J(P ) n M(P ). Sn P s suorr o DM(P ) tn w no only lmnts or P n t otrs n otn usng GLB or LUB sk y t pplton. T ollowng orollry s rt onsqun o Torm 1. Corollry 8. T nongs ' n omput y Algortm 1 r rsptvly mt-prsrvng n jon-prsrvng. W r now gong to sow ow to omput t o o t GLB o two gvn lmnts o P n tn ow to o ts normton to otn t no lmnt(. to sr '?1 ). W suss only GLB, or LUB n otn ully. Lmm 9. Lt x n y two lmnts o prtl orr P. Tn '(GLB(x; y)) = '(x) \ '(y), urtrmor GLB(x; y) = '?1 ('(x) \ '(y)). T omplxty o t lgortm omputng GLB(x; y) pns on t t strutur us or storng t o '. To v goo omplxty, w us t tr t strutur ntrou n [8] or n optml rprsntton o strutv ltt. Ts tr llows us to v st tm omplxty to omput '?1.

7 Algortm 1: Jon-prsrvng n mt-prsrvng nongs Dt : A prtl orr P = (X; P ) Rsult : T mt-prsrvng nong ' : P! 2 J(P ). t jon-prsrvng nong : P! 2 M(P ). gn Comput B = (X; X 0 ; 6 P ); Comput t ruton Bp(P ) = (J; M; B) o B. n Enong ll lmnts n P g or ll x 2 X o '(x) = S j2j; jp x jg; (x) = S m2m; mp x mg; 5 Algortm In ts ston, w gv n lgortm s on tr t strutur to omput GLB(x; y). To o so, onsr prtl orr P = (X; P ) n t nong ' o P omput prvously. For lmnt x 2 P, w suppos tt '(x) s sort orng to topologl sortng o J(P ). Lt us n t tr t strutur, not y T ' or t nong '. { Lt Root ' t root o T ' (ts orrspons to t mxml lmnt n DM(P )). { E no o t tr orrspons to n lmnt o DM(P ). { Evry g o t tr s ll y lmnts o J(P ), su tt t unon o lls rom ny no x to Root ' orrspons '(x) sort orng to. { T sons o ntrnl no r sort orng to?1 Not tt or strutv ltt, vry g s ll y on lmnt o J(P ). Most oprtons on ts tr r sr n [8]. W us ts tr to stor t o ': or o '(x) o n lmnt x, w v to sr t tr n rursv wy to vry x s n t tr. I not, w n sly t. Exmpl 10. Fgur 5 sows T ' or t nong o t prtl orr o Fgur 2, wt ' = < < < < <. Lt us now sr t omputng o GLB(x; y) usng t tr. Exmpl 11. Consr t prtl orr n Fgur 2. I t pplton sks or GLB(; ), Algortm 2 omputs '(GLB(; )) = '() \ '() = w s n t tr o t nong ' s Fgur 6(). Tn, nw lmnt pprs n P (not xpltlly n P ut rprsnt y ts o) s n Fgur 6(). Now, t pplton sks or GLB(; g), Algortm 2 omputs '(GLB(; g)) = '() \ '(g) = w lry xsts n T '.

8 g Algortm 2: GLB(x; y) Fg. 5. Tr o t nong ' Dt : A prtl orr P = (X; P ); Two lmnts x n y o P ; T root T '; Rsult : GLB(x; y) gn o = '(x) \ '(y) ; tr xst z n T ' wt '(z) = o tn Rturn( z); n ls GLB(x; y) o not xst Hr t progrm sks t usr GLB(x; y) s sgnton I ys w nw lmnt z n T '. Corollry 12. Lt P = (X; P ) prtl orr. { T ntl omputton o t nongs ' n T ' rqurs O(jJ(P )j jxj) tm. { T omputton o GLB rqurs O(jJ(P )j). O ours, t s sy to moy t lgortm to xpltly nw lmnt to P. Ts nturlly nrss t omplxty o t GLB omputton to O(jJ(P )j + jxj) T LU B oprton n on ully. For smultnous omputton o GLB n LUB w must o som upt on T ' wn ng nw lmnt tr LUB n v vrs. 6 Conluson In ts ppr w v sr nw lgortm to omput t Dkn MNll omplton o prtl orr, s on GLB n LU B oprtons. Ts sms to ntrstng or ppltons n knowlgs prossng.

9 g g ( ) ( ) Fg. 6. Computton o GLB(; ) Furtrmor our lgortm n us to omput t Glos ltt or t onpt ltt rom Wll trmnologs. Our rgumnts or ts r s on quvlns twn Dkn MNll Complton, Glos ltt n t ltt o Mxml ntns [12]. Rmrk 13. Ts lgortm ws mplmnt unr Mpl lngug (ttp://wwwltp.p.r:= rtt/). Rrns 1. H. At-K, R. Boyr, P. Lnoln, n R. Nsr. Ent mplmntton o ltt oprtons. ACM Trnstons on Progrmmng Lnggs n Systms, 11(1):115{ 146, jnury J.P. Bort. Clul prtqu u trlls gllos 'un orrsponn. In Mt. S. Hum, 96, pgs 31{47, A. Bout. Cogs t mnsons rltons nrs. Annls o Dsrt Mtmts 23, Orrs: Dsrpton n Rols, (M. Pouzt, D. Rr s), Yvs Csu. Ent nlng o multpl nrtn rrs. In OOP- SLA'93, pgs 271{287, B. A. Dvy n H. A. Prstly. Introuton to ltts n orrs. Cmrg Unvrsty Prss, son ton, R. Gon n H. Ml. Bulng n mntnng nlyss-lvl lss rrs usng glos ltts. In OOPSLA'93, pgs 394{410, M. H n L. Nourn. Bt-vtor nong or prtlly orr sts. In Pro. o Intrntonl Worksop ORDAL`94, tor, Orrs, Algortms n Appltons, numr 831, pgs 1{12, Lyon, Frn, July Sprngr Vrlg. 8. M. H n L. Nourn. Tr strutur or strutv ltts n ts ppltons. TCS, 165(2):391{405, otor C. Jr, G.-V. Journ, n J.-X. Rmpon. Computng on-ln t ltt o mxml ntns o posts. Tnl rport, IRISA, Rnns, Frn, Frury To ppr n Orr. 10. H.M. MNll. Prtlly orr sts. Trns. Amr. So., 42:416{460, G. Mrkowsky. T rprsntton o posts n ltts y sts. Algr Unvrsls, 11:173{192, 1980.

10 12. M. Morvn n L. Nourn. Smpll lmnton sms, xtrml ltts n mxml ntn ltts. Orr 13, No 2:159{173, R. Wll. Rstruturng ltt tory: An ppro s on rrs o ontxts. n Orr sts,i. Rvl, Es. NATO ASI No 83, Rl, Dort, Holln, pgs 445{470, 1982.

Lecture 20: Minimum Spanning Trees (CLRS 23)

Lecture 20: Minimum Spanning Trees (CLRS 23) Ltur 0: Mnmum Spnnn Trs (CLRS 3) Jun, 00 Grps Lst tm w n (wt) rps (unrt/rt) n ntrou s rp voulry (vrtx,, r, pt, onnt omponnts,... ) W lso suss jny lst n jny mtrx rprsntton W wll us jny lst rprsntton unlss