G. Gambosi (*), J. Ne~etgil (**), M. Talamo (*)

Transcription

1 EFFICIENT REPRESENTATION OF TAXONOMIES G. Gamosi (*), J. N~tgil (**), M. Talamo (*) (*) Istituto i Analisi i Sistmi Inormatica l C.N.R.~ Vial Manzoni 30, 00185, Roma,Italy (**) Charls Univrsity Malostransk~ Namst± 25, Pragu, Czchoslovakia ABSTRACT In this papr, th us o th concpts o imnsion an oolan imnsion o partial orrs is stui or what concrns th icint rprsntation o taxonomis. Unr this approach, lalings o th nos (linar xtnsions o partial orrs) ar us to answr icintly to such quris as "Is lmnt a rlat to lmnt?". INTRODUCTION Th icint rprsntation o taxonomis, i.. r-structur sts o concpts, rprsnts a rathr intrsting prolm, as or xampl in som sctors o Artiicial Intllignc, whr it turns out to al to answr quickly to such quris as "Is lmnt a inclu (riv) rom lmnt?". In orr to tackl this prolm in th cas o a static structur, th us o suital lalings o th nos (lmnts) sms to worth o invstigation. I w consir a taxonomy as a partial orr, th approach (xtnsivly stui in post thory) o using th concpt o linar xtnsions an o imnsion o posts [KT] maks it possil to trmin th numr o such lalings an, in som cass, to otain th laling thmslvs. Morovr, a gnralization o th concpt o imnsion, not as oolan imnsion, introuc an xtnsivly stui in th ull papr [GNT], sms to convnint to us in orr to Otain a ttr prormanc oth in trms o qury tim an o spac. In sction I, som initions ar givn togthr with th introuction o th concpt o oolan imnsion. This work was partially support y th ESPRIT Projct ALPES.

2 233 In sction 2, a laling o simpl taxonomis (root trs) is prsnt which allows an icint tratmnt o inclusion quris. In sction 3, such approach is xtn to two typs o gnraliz taxonomis. I. BASIC DEFINITIONS AND TERMINOLOGY Givn a partially orr st P = (N,<p), whr N is an orr rlation on st N, a linar xtnsion L (topological sorting) on P JR] is a total orring <L on N which ms P itsl, i.. such that or ach nl,n 2 c N, i n I <p n 2 thn n I <L n2" Th (orr) imnsion o P (im P) is in as th minimum numr o irnt linar xtnsions o P whos intrsction is ncssary to trmin P itsl. Hnc, im P is th minimum k or which thr xists a st i = {LI,L 2...,L k} o linar xtnsions o P such that n I <p n 2 i n I <Lin2 or ach i = 1,...,k (ig. I). In thir sminal papr on th imnsion thory o posts, Dushnik an Millr [DM] hav shown, among othrs, that th imnsion o an aritrary partial orr can unoun as n = INI tns to ininity an can grow as ast as O(n) (*) Not that, i im P = k, it is possil to rprsnt P introucing a k-laling on st N: hnc, or ach no n E N thr xist k lals ~i (n),~2(n),...,ik(n) an, givn nl,n 2 E N, n I <p n 2 i th ollowing oolan ormula is vrii: F = xiax2 A.. Ax k whr x i is tru i Zi(n I) < Zi(n2). P a c a L I L 2 c? c a Fig. t. im P = 2 (*) As usual, w will say that, givn two unctions (n),g(n),(n) = = O(g(n)) i thr xist c,n such that (n) < c.g(n) or n > n o o -- o

3 234 Morovr, givn a post P = (N,<p), th Hass iagram o P is in as a ag HD(P) = (N,A) such that: I. A is not transitiv, i.. i <nl,n 2 > 6 A, thn thr is no path o lngth gratr than two rom n I to n 2 in HD(P) (HD(P) is a transitiv ruction). 2. Th transitiv closur (*) HD(P)*= (N,A*) o HD(P) is isomorphic to P = (N,<p). In th ollowing, w will rr to posts only in trms o th associat Hass iagrams an o thir charactristics. Givn a post P = (N,<p) an a oolan ormula F in on a st {Xl,...,x k} o oolan varials, w will say that P is F-rprsntal i thr xists a st o k linar xtnsion L = {LI,...,L k} such that, givn any nl,n 2 E N, n I <p n 2 i F(Xl,...,x k) = tru, whr x i =tru i Zi(nl) < li(n2). In ig. 2 an xampl o a oolan ormula F an a P a c L I c L 2 L 3 L 4 c c a a F = (x I Ax2) A (x 3 Ax4) Fig. 2 post P which is F-prsntal is givn. Givn a post P, lt us not as oolan imnsion o P [GNT] (imbp) th minimum k or which thr xists a ormula F(Xl,..,x k) such that P is F-prsntal. Osrv that such inition inclus th usual inition o orr imnsion givn aov as th particular cas whr all ormulas ar o th orm (*) Givn a irct graph G = (N,A) th transitiv closur o G is in as th irct graph G* = (N,A*) such that, givn nl,n 2 E N, (nl,n 2 ) A* i thr xists a path rom n I to n 2 in G.

4 235 = A x. F(xl... Xk) 1<i<k l This maks it possil to riv immiatly th ollowing rsult: FACT For any post P = (N,<p), imb(p) ~ im(p). This asy inquality is complmnt y th ollowing thorm Thorm I. For vry n thr xists P n such that imb(p n) = 4 an im(p n) = n Thorm I is as on th stanar xampl on Fig ROOTED TREES Lt us not a post T = (N,< T) as a root tr i its Hass iagram HD(T) is a root tr. Hnc, T is a root tr i: I. Thr xists r 6 N such that r <T n or any n N-{r} k 2. Thr xists a partition o N-{r} = U Ni, N. A Nj = i=i l i i ~ j, such that, noting y T i = (Ni,<T) th supost inuc k l y st Ni, <T = ( u <T. ) w {<r,n > E <Tin N}, i.. nos in ii=i l rnt susts Ni,N j o N-{r} ar not rlat in T. 3. Each T i = (Ni,<T.) is a root tr. From [TM] sion im T < 2. l it is trivial to riv that a root tr T has imn- As wll known, simpl taxonomis can mol as root trs: hnc, y proviing two suital lalings LI,L 2 o th ntitis con- sir in th taxonomy, it will possil to answr quris o th typ "Is ntity a inclu in ntity?" in tim T(n) = 0(I) an spac S(n) = 0(I) or ach ntity. In orr to intiy such two lalings, lt us not as 0 i, i E N, th total orr in y an aritrary prmutation Si o sons o no i in HD(T) = (N,A), S i = {n NI(i,n > 6 A}. on th st Thn, th lals i (n),i2(n) o a no n N can riv as: I. Z1(n) is th rank o no n in th total orr trmin y a Dpth-First travrsal o T, provi that, or ach i N, th orr o application o th DFS to th st o sutrs inuc y th st

5 236 o i is xactly O i- 2. Z2(n) is th rank o no n in th total orr riv y a DFS travrsal o T, provi that, or ach i N, th orr o applica-! tion o th DFS to th st o sutrs inuc y S i is O i, orring otain y rvrsing O i (ig. 3). th I-I Fig. 3 Th ollowing thorm hols: Thorm 2. Givn a root tr T = (N,< T) an th two lalings LI,L 2 ovr N in aov, th ollowing hols: or ach nl,n 2 E N, n I is an ancstor o n 2 (i.. n1< T n 2) i ~i(ni) < 11 (n 2) an 9~2(nl)<i2(n2). Proo (i). Lt us assum w.l.o.g, that no n I is not an ancstor o no n 2. Lt us not as n 3 th Lowst Common Ancstor o n I an n 2 in T an lt r(n I) (r(n2)) th son o n 3 which is root o th sutr containing n I (n 2). Two cass ar possil: I. r(nl) < r(n 2) in On3 : hnc, sinc r(n 2) < r(n I) in O'n3, a. ~2(n2) < i2(r(nl)) y th DFS proprtis. Z2(r(n I)) < Z2(nl) thn, i 2(n 2) < ~2(nI) an th conition is not vrii. 2. r(n 2) < r(nl) in On3 thn: a. ~1(n2) < Zl(r(nl))

6 237, Z1(r(nl)) < Z1(nl) hnc, it(n2) < i1(nl) an th conition is not vrii again. (Only i). By th proprtis o DFS travrsal, oth i1(i)<i1(j) an i2(i) < i2(j), whr i, j N an i <T j (i is th root o a sutr containing j). D 3. GENERALIZED TAXONOMIES In orr to rprsnt sts o taxonomis in on th sam st o lmnts in orr to answr icintly to quris o th typ "Is class a inclu y class?" w n to rprsnt th'corrsponing structurs (gnralizations o trs) in such a way to answr icintly to such quris as "Is thr any path rom no a to no?". In th ollowing w will consir two typs o gnraliz taxonomis: local root trs an la-analgamat orsts, an will show how, unr gnral conition, it is possil to rprsnt such taxonomis icintly. Loca~ root trs Local root trs ar an xtnsion o th inition o root trs introuc, among othrs, to rprsnt mor complx taxonomis, whr a nw taxonomy is in among th lmnts sons o th sam lmnt. It is possil to in a k-local root tr rcursivly as: - A root tr is a l-local root tr - A k-local root tr T = (N,<T) is a post such that th corrsponing Hass Diagram HD(T) = (N,A) is a igraph such that A = A I u A~, A I A A~ whr: I. Th igraph RT = (N,A I) is th H.D. o a l-local root tr. 2. Lt us not as S i th st o sons o no i E N in RT. Morovr, lt us not as A~ (i) ~ A~ th sust o A~ inuc y S i on A~. Thn: ien - A~ (i) n A~ (j) i i ~ j, i,j E N. - (Si, E~ (i)) is a t-local root tr with t < k an thr xists at last on no n N such that (Sn,E 4 (n)) is a (k-1)-local

7 238 root tr (ig. 4). Fig. 4 It is asy to not that th transitiv closur o a local root tr T is qual to th transitiv closur o a root tr: that immiatly implis that, i T is a local root tr, th transitiv closur o T can rprsnt y a coupl o lalings. Such lalings can immiatly riv applying th laling procur or root trs prsnt aov to th root tr T' otain rom T lting all transitiv arcs, i.. all arcs < i,j ) 6 A such that thr xists a path rom i to j o lngth gratr than two. La amalgamat orsts Dinition. Lt P = (N,<p) a post: givn a minimal lmnt x E N lt I(x) = {y N Ix <py}. W say that P is a la amalgamat orst i: I. Evry I(x) is a root tr 2. Dnoting as L(x) C I(x) th st o lavs (maximal lmnts) in I(x), I(x) N I(y) H L(x) A L(y) (x,y EN), i.. or any two nos x,y, I(x) an I(y) intrscts only in a st o lavs (ig. 5). Lt us now prov th ollowing thorm Thorm. Lt P = (N,<p) a tr amalgamation, thn imb(p) ~ 2., proviing that all in-grs o nos in N ar oun y. Proo. Lt us irst not that, givn a maximal lmnt x E N, or ach y <p x thr xists a uniqu maximal chain containing oth x an y. Furthrmor, i C an C' ar maximal chains which intrsct in a

8 239 Fig. 5 non maximal lmnt y, thn C ~ C' ~ <r E Nlr <p y}. Lt us consir a -coloring o th maximal chains in such a way that two chains with th sam color o not intrsct in a la. Thn~ th union o chains with th sam color turns out to a orst, i.. a post whos orr imnsion is 2. Lt us not as I,k ' i2, k th two lals rlativ to color k: thn it is asy to not that th oolan unction ~(x I I,x2 2,...,Xlt, t s x2,) = (Xl, 1A Xl,2) V (x2,1a x2,2) V... (X,l,X,2) (whr xi, j =tru i Zi,j(n I) < ii, j(n2), nl,n 2 E N) scris post P, hnc imb(p) < 2.. D Corollary. It is possil to answr to inclusion quris on la-amalga- mat orsts in tim O(k) an spac O(k.n), whr k is th numr o la-amalgamat trs. Proo. Drivs immiatly rom th thorm aov. D y ~ L}. Lt us not as L th st o lavs an as R = {xen i~ x,y) EA, In th cas whr also th ollowing hypothsis is vrii: 3. Wx ~ R, outgr(x) is a constant it is morovr possil to stat th ollowing thorm Thorm. Givn a la-amalgamat orstf which vriis conition 3 aov, thn imbf = O(h), whr h = max hight(x). xen Corollary. Givn a la-amalgamat orst F which vriis conition 3, thr xists a ata structur which maks it possil to answr to

9 240 inclusion quris on F in tim T(n) = O(h), whr h = max hight(x), an spac S(n) = O(n.h). xen From th rsults aov it ollows that, givn a la-amalgamat orst F, it is possil to manag th prolm o answring to inclusion quris in tim min(h,k) an spac min(h.n,k.n). This rsult is strongly pnnt rom th us o oolan imnsion; it is possil in act to stat th ollowing thorm. Thorm. Thr xists a la-amalgamat orst F which vriis conition 3 aov such that im F = ~(n). CONCLUSION Th concpts o orr imnsion an oolan imnsion o posts hav n appli to th rprsntation o (gnralization o) taxonomis, in orr to icintly manag inclusion quris. This work is part o a mor gnral stuy o th concpt o oolan imnsion o posts vlop in th ull papr. REFERENCES [DM] [GNAT] [KT] [R] [TM] B. Dushnik, E. Millr: Partially orr st~, Amr. J. Math. 63 (1941), G. Gamosi, J. N~tril, M. Talamo: On locally prsnt posts, sumitt to "Thortical Computr Scinc". D. Klly, W.T. Trottr: Dimnsion thory or orr sts in Orr sts, (I. Rival.), D. Ril, Dorrcht, I. Rival: Linar xtnsions o init orr sts, Annals o Discrt Math. 23 (1984), W.T. Trottr, J.I. Moor: Som thorms on graphs an posts, Discrt Math. 15 (1976),