A simple algorithm to generate the minimal separators and the maximal cliques of a chordal graph

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1 A smpl lgortm to gnrt t mnml sprtors nd t mxml lqus o ordl grp Ann Brry 1 Romn Pogorlnk 1 Rsr Rport LMOS/RR Fbrury 11, 20 1 LMOS UMR CNRS 6158, Ensmbl Sntqu ds Cézux, F Aubèr, Frn, [email protected]

2 Abstrt W prsnt smpl und lgortm pross w uss tr LxBFS or MCS on ordl grp to gnrt t mnml sprtors nd t mxml lqus n lnr tm n sngl pss. Kywords: MCS, LxBFS, mnml sprtor, mxml lqu, ordl grp. 1

3 1 ntroduton Gnrtng t mnml sprtors nd t mxml lqus o ordl grp s smpl wt t lp o sp sr lgortms. Cordl grps (w r t grps wt no ordlss yl on our or mor vrts), wr rtrzd by Fulkrson nd Gross [7] s t grps or w on n rptdly nd smpll vrtx (.. vrtx wos ngborood s lqu) nd rmov t rom t grp, untl no vrtx s lt; ts pross, lld smpll lmnton, dns n ordrng α on t vrts lld prt lmnton ordrng (po). (α() dnots t vrtx brng numbr, nd α 1 (x) t numbr o vrtx x). At stp o t lmnton pross, nw trnstory grp s dnd. Ros [] sowd tt or ny gvn po α o ordl grp, ny mnml sprtor o t grp s dnd n t ours o t smpll lmnton pross s t trnstory ngborood o som vrtx. Morovr, t s sy to s tt n ny mxml lqu K, t vrtx y o smllst numbr by α dns K s ts trnstory losd ngborood. n t rst, w wll ll su vrts gnrtors, nd our gol wll b to dtt ts gnrtors ntly. W wll us grp sr lgortms LxBFS [10] nd MCS [11], bot tlord to gnrt po o ordl grp. T d bnd ts s qut smpl. Bot lgortms numbr t vrts rom n to 1 (n s t numbr o vrts o t grp). E vrtx brs lbl, w my b modd s t lgortm prods. At stp, nw vrtx s osn to b numbrd. t lbl o ts nw vrtx x s not lrgr tt t lbl o t prvous vrtx x +1, tn w know tt x gnrts mnml sprtor o t grp. T vrts o ts mnml sprtor r t lrdy numbrd ngbors o x. W lso know tt t prvously numbrd vrtx, x +1, gnrts mxml lqu: x +1, togtr wt ts lrdy numbrd ngbors (xludng x ), dn mxml lqu o t grp. Tus t lbls o lgortms LxBFS nd MCS nbl t usr to dtt ts gnrtors s soon s ty r numbrd. T mnml sprtors nd mxml lqus n b ound on-ln, wtout rqurng prlmnry pss o t lgortm to numbr ll t vrts. T orrspondng rsults v bn provd or MCS n two sprt pprs. Blr nd Pyton [5], wl studyng ow MCS dns lqu tr o ordl grp, sowd ow to gnrt t mxml lqus (w r t nods o t lqu tr), usng t MCS lbls. Kumr nd Mdvn [8] sowd ow to gnrt t mnml sprtors o ordl grp, gvn n MCS ordrng. 2

4 Howvr, ltoug mnml sprtor nd mxml lqu gnrtor r tully dttd t t sm stp o t lgortm, ts rsults v not bn und. Morovr, [8] us po s nput, but t mnml sprtors s wll s t mxml lqus n b omputd durng t xuton, w my b n mportnt tur wn ndlng vry lrg grps, sn globl pr-numbrng o t vrts s not nssry or vn usul. LxBFS, s w wll sow, xbts t sm proprty s MCS rgrdng ts gnrtors. Our m s to prsnt smpl und sngl-pss lgortm w gnrts t mnml sprtors nd t mxml lqus o ordl grp. T ppr s orgnzd s ollows: n Ston 2, w gv som prlmnry rsults nd dntons. n Ston 3, w prsnt our mn torm, w dsrbs ow t gnrtors r dttd, nd dsuss ts proo or LxBFS. Ston 4 prsnts our lgortm nd provds n xmpl. 2 Prlmnrs n t rst, w wll onsdr ll grps to b onntd (or dsonntd grp, t prosss w dsrb n b ppld ndpndntly to onntd omponnt). A grp s dnotd G = (V, E), V = n, E = m. Vrts x nd y r djnt xy E. For X V, G(X) dnots t subgrp ndud by X. T ngborood N G (x) o vrtx x n grp G s N G (x) = {y x xy E} (subsrpts wll b omttd wn t s lr w grp w work on). T losd ngborood s N G [x] = {x N(x)}. T ngborood o vrtx st X V s N G (X) = x X N G (x) X. A subst X o vrts s lld modul t vrts o X sr t sm xtrnl ngborood: x X, N(x) X = N(X). A lqu s st o prws djnt vrts. A subst S o vrts o onntd grp G s lld sprtor G(V S) s not onntd. A sprtor S s lld n xy-sprtor x nd y l n drnt onntd omponnts o G(V S), mnml xy-sprtor S s n xy-sprtor nd no propr subst o S s n xy-sprtor. A sprtor S s mnml sprtor, tr s som pr {x, y} su tt S s mnml xy-sprtor. Altrntly, S s n mnml sprtor nd only G(V S) s t lst 2 onntd omponnts C 1 nd C 2 su tt N(C 1 ) = N(C 2 ) = S; su omponnts r lld ull omponnts o S n G, nd S s tn mnml xy-sprtor or ny {x, y} wt x C 1 nd y C 2. LxBFS [10] nd MCS [11] r bot lnr-tm sr lgortms w numbr t vrts rom n to 1, trby dnng po. E vrtx x brs lbl w orrsponds to t lst o numbrs o t ngbors o x wt 3

5 gr numbr tn tt o x (LxBFS) or to t rdnlty o ts lst (MCS). Bot lgortms r rlld blow. Algortm LxBFS (Lxogrp Brdt-Frst Sr)[10] nput : A grp G = (V, E). output: An ordrng α o V. ntlz ll lbls s t mpty strng; or n to 1 do Pk n unnumbrd vrtx v wos lbl s mxml undr lxogrp ordr; α() v ; or unnumbrd vrtx w djnt to v do ppnd to lbl(w); Algortm MCS (Mxml Crdnlty Sr)[11] nput : A grp G = (V, E). output: An ordrng α o V. ntlz ll lbls s 0; or n to 1 do Pk n unnumbrd vrtx v wt mxmum lbl; α() v ; or unnumbrd vrtx w djnt to v do lbl(w) lbl(w) + 1; 3 Mn Torm W wll now prsnt our mn torm, rom w our lgortm s drvd. W wll nd t ollowng dntons: Dnton 3.1 Gvn ordl grp G nd po α o G, or ny vrtx x, Mdj(x) s t st o ngbors o x wt numbr gr tn tt o x: Mdj(x) = {y N(x) α 1 (y) > α 1 (x)}. Dnton 3.2 Gvn ordl grp G nd po α o G, w wll ll: 4

6 mnml sprtor gnrtor ny vrtx x wt numbr by α, su tt Mdj(x ) s mnml sprtor o G nd lbl(x ) lbl(x +1 ), wr x +1 = α( + 1). mxml lqu gnrtor vrtx y su tt Mdj(y) {y} s mxml lqu o G. Our mn rsult s t ollowng: Torm 3.3 Lt α b po dnd by tr LxBFS or MCS, lt x b t vrtx o numbr, lt x +1 b t vrtx o numbr + 1. ) x s mnml sprtor gnrtor nd only lbl(x ) lbl(x +1 ). b) x +1 s mxml lqu gnrtor nd only lbl(x ) lbl(x +1 ) or = 1. W wll now dsuss t proo o Torm 3.3. W wll rst prsnt moplx lmnton, w s pross tt xplns ow bot MCS nd LxBFS sn t mnml sprtors nd t mxml lqus o ordl grp, tn prov Torm 3.3 or LxBFS. 3.1 Moplx lmnton W wll us t noton o moplx, ntrodud by [1]: Dnton 3.4 [1] A moplx s lqu X su tt X s modul nd N(X) s mnml sprtor. W xtnd ts dnton to lqu wos ngborood s mpty. A smpll moplx s moplx wos vrts r ll smpll. [1] provd: Proprty 3.5 [1] Any ordl grp w s not lqu s t lst two non-djnt smpll moplxs. From ts, ty drvd vrnt o t rtrzton o Fulkrson nd Gross or ordl grps by smpll lmnton o vrts: Crtrzton 3.6 [1] A grp s ordl nd only on n rptdly dlt smpll moplx untl t grp s lqu (w w wll ll t trmnl moplx ). W ll ts pross moplx lmnton. 5

7 Not tt moplx lmnton on ordl grp s spl s o smpll lmnton, sn t stp st o smpll vrts s lmntd. Not lso tt or onntd grp G, t trnstory lmnton grp obtnd t t nd o stp rmns onntd. Moplx lmnton dns n ordrd prtton (X 1, X 2,..., X k ) o t vrts o t grp nto t sussv moplxs w r dnd n t sussv trnstory lmnton grps. W wll ll ts prtton moplx ordrng. Our bs rsult s t ollowng: Torm Lt G b ordl grp, lt (X 1, X 2,..., X k ) b moplx ordrng o G. At stp < k o t lmnton pross ndng moplx X n trnstory grp G, N G (X ) s mnml sprtor o G. X N G (X ) s mxml lqu o G. T trmnl moplx X k s mxml lqu. Tr r no otr mnml sprtors or mxml lqus n G. To prov ts, w wll rst rll t rsult rom Ros []: Proprty 3.8 [] Lt G b ordl grp, lt α b po o G, lt S b mnml sprtor o G. Tn tr s som vrtx x su tt Mdj(x) = S. From Proprty 3.8, t s sy to ddu t ollowng wll-known proprty: Proprty 3. [] Lt G b ordl grp, lt S b mnml sprtor o G. Tn n vry ull omponnt C o S, tr s som vrtx x su tt S N(x) (su vrtx s lld onlun pont o C). Lt us now prov Torm 3.7. Proo: (o Torm 3.7) Lt us rst prov tt N G (X ) s mnml sprtor o G. n trnstory grp G, N G (X ) s by dnton mnml sprtor S o G, wt t lst two ull omponnts C 1 nd C 2, ontnng onlun pont, w w wll ll x 1 nd x 2. Suppos S s not mnml sprtor o G. Tn tr must b ordlss pt rom x 1 to x 2 n G w ontns no vrtx o S. Lt y b t rst vrtx o ts pt to b lmntd, t 1 T rsults provd n ts subston wr brly prsntd n [2]. 6

8 stp j < ; y must b smpll n G j, but ts s mpossbl, sn y s two non-djnt ngbors on t pt. Tus vry N G (X ) ( < k) s mnml sprtor o G, nd by Proprty 3.8, ll t mnml sprtors o G v tus bn nountrd. Lt us now prov tt X N G (X ) s mxml lqu o G, by nduton on. Clrly, X 1 N G (X 1 ) s mxml lqu o G. Lt us xmn X N G (X ): t s mxml lqu o G nd tus lqu o G. Suppos t s not mxml lqu o G, nd lt y b vrtx blongng to lrgr lqu o G ontnng X N G (X ). Sn N G (X ) s mnml sprtor o G, y must blong to t sm ull omponnt o N G (X ) s t vrts o X. But tn y blongs to t sm mxml lqu modul s X, ontrdton. T trmnl moplx X k s by dnton lqu; suppos t s not mxml lqu: tn t n b ugmntd wt som vrtx y, w blongs to som moplx X. By Torm 3.7, S = N G (X ) s mnml sprtor o G. Lt C 1,..., C t b t ull omponnts o S n G, nd lt z b t onlun pont blongng to t moplx o gst numbr X j. n G j, S nnot b mnml sprtor, sn ll t otr ull omponnts o S v bn lmntd rom G j. Tror, X j N Gj (X j ) must b t trmnl moplx, ontrdton. Not tt or gvn ordl grp G, tr my b mny drnt moplx ordrngs, but tr s lwys t sm numbr o moplxs, sn ts s t numbr o mxml lqus o t grp. Wt ny moplx ordrng (X 1, X 2,..., X k ), w n ssot po α by prossng t moplxs rom 1 to n nd gvng onsutv numbrs to t vrts o gvn moplx. Usng α, w n dn t mnml sprtors nd mxml lqus s vrtx ngboroods: Proprty 3.10 Lt G b ordl grp, lt (X 1, X 2,..., X k ) b moplx ordrng o G, lt α b po ssotd wt ts moplx ordrng. Tn n t ours o moplx lmnton, t stp, prossng moplx X, T vrtx x o moplx X wos numbr s t smllst by α dns mxml lqu {x} Mdj(x) o G. T vrtx y o moplx X wos numbr s t gst by α dns mnml sprtor Mdj(y) o G. 7

9 3.2 Gnrtors o LxBFS LxBFS dns moplx ordrng nd n ssotd po: [1] sowd tt LxBFS lwys numbrs s 1 vrtx blongng to moplx (w w wll ll X 1 ). Ty lso provd tt t vrts o X 1 rv onsutv numbrs by LxBFS. Ts proprts r tru t stp o LxBFS n t trnstory lmnton grp. Tror, LxBFS dns moplx lmnton (X 1,..., X k ), by numbrng onsutvly t vrts o X 1, tn numbrng onsutvly t vrts o X 2, nd so ort: Torm 3.11 n ordl grp, LxBFS dns moplx ordrng. Not tt t s sy to ddu rom [5] tt MCS lso dns moplx ordrng. Exmpl 3.12 Fgur 1 sows t numbrs nd lbls o LxBFS xuton on ordl grp. moplx mnml sprtor mxml lqu X 1 = {g} {b, } {b,, g} X 2 = {} {b,, d} {, b,, d} X 3 = {d, } {b, } {b,, d, } X 4 = {} {b, } {b,, } X 5 = {b,, } - {b,, } 2 [6,5,4] [] 8 [8,7] 6 b [7,6] 5 [6,5] 4 d 7 [8] g 1 [6,5] [6,5,4] 3 Fgur 1: Numbrs nd lbls o LxBFS xuton on ordl grp. Sn t vrts o ny trnstory moplx X r numbrd onsutvly, t lbls wll nrs w.r.t. lxogrp ordr untl t ntr moplx s bn numbrd, nd tn wll stop nrsng. Mor prsly, wn 8

10 runnng LxBFS (numbrng t vrts rom n to 1), s long s t lbls nrs, w r dnng moplx, X. Wn t lbls stop nrsng (wn numbrng vrtx y), tn w v strtd nw moplx X 1 w wll ontn y. Usng Proprty 3.10, w n ddu tt LxBFS gnrts t mnml sprtors nd t mxml lqus ordng to Torm 3.3. Tus t s sy, usng LxBFS ordrng, to dn wt sngl pss t mnml sprtors o G t mxml lqus o G t orrspondng moplx ordrng 4 Algortm W n now drv rom Torm 3.3 gnrlzd lgortm to gnrt t mnml sprtors nd t mxml lqus o ordl grp. n t lgortm blow, t s onsdrd tt tr LxBFS or MCS s usd. Us to nrmnt t lbl o y s trnsltd s dd to lbl o y or LxBFS nd s dd 1 to t lbl o y or MCS. T lbls r ll onsdrd ntlzd t t bgnnng, s or LxBFS nd s 0 or MCS. G NUM G NUM +{x } s sortnd or V NUM V NUM +{x }; G NUM G(V NUM ). n t sm son, G ELM G ELM {x } s sortnd or V ELM V ELM {x } ; G ELM G(V ELM ) ; Symbol + dnots dsjont unon. Not tt or LxBFS, x s mnml sprtor gnrtor, tn t lbl o x gvs t sprtor drtly., or nstn, lbl(x ) = (, 7, 6), tn t mnml sprtor w x gnrts wll b {x, x 7, x 6 }.

11 Algortm Mnsps-Mxlqus nput : A ordl grp G = (V, E). output: St S o mnml sprtors o G ; St S o mnml sprtor gnrtors o G ; St K o mxml lqus o G ; St K o mxml lqu gnrtors o G ; nt: G NUM G( ); G ELM G ; S ; S ; K ; K ; or =n downto 1 do Coos vrtx x o G ELM o mxmum lbl ; G NUM G NUM + {x } ; n nd lbl(x ) λ tn //x s mn. sp. gnrtor nd x +1 s mx. lqu gnrtor S S + {x } ; S S {N GNUM (x )} ; K K + {x +1 } ; K K + {(N GNUM (x ) {x }}; λ lbl(x ) ; or y N GELM (x ) do Us to nrmnt t lbl o y ; G ELM G ELM {x } ; K K + {x 1 } ; K K + {N G (x 1 )} ; T omplxty o t bov lgortm s t sm s or LxBFS or MCS, w s lnr (O(n + m)) tm. Exmpl 4.1 Fgur 4 blow gvs stp-by-stp xmpl usng MCS. St o mnml sprtors: S = {{}, {g, }, {d}}. St o mnml sprtor gnrtors: S = {d,, }. St o mxml lqus: K = {{, b, }, {, d, g, }, {g,, }, {d,, }}. St o mxml lqu gnrtors: K = {,,, }. Wn prossng vrtx x 6, x 6 = d s dnd s mnml sprtor gnrtor or {} nd tus x 7 = s mxml lqu gnrtor or {, b, }. Wn prossng vrtx x 3, x 3 = s dnd s mnml sprtor gnrtor or {g, }, so x 4 = s mxml lqu gnrtor, or {, d, g, }. Wn prossng vrtx x 2, x 2 = s dnd s mnml sprtor gnrtor or {d}, so x 3 = s mxml lqu gnrtor or {g,, }. 10

12 At t nd, x 1 = s dnd mxml lqu gnrtor, or {d,, }. b d 8 b d 7 8 b d g g g Prossng Prossng 8 Prossng b d b d b d 6 g 5 g 5 g 4 Prossng 6 Prossng 5 Prossng b d 6 5 g b d g b d g Prossng 3 Prossng 2 Prossng 1 3 Fgur 2: A stp by stp xuton usng MCS. T dsd vrtx numbrs rprsnt t mnml sprtor gnrtors, nd t rld vrtx sts r t mnml sprtors. 4.1 Multplty o t gnrtd mnml sprtors A mnml sprtor my b gnrtd svrl tms, dpndng on t numbr o ull omponnts t dns: Proprty 4.2 Lt α b po dnd by LxBFS or MCS, lt S b mnml sprtor, lt k b t numbr o ull omponnts o S. Tn S s k 1 gnrtors by α. 11

13 Proo: n t ours o smpll lmnton pross on smpll vrts, bor ny vrtx s o mnml sprtor S n b lmntd, ll t ull omponnts o S (xpt on) must v bn lmntd, ls s nnot b smpll, s s ss t onlun ponts o rmnng ull omponnt. T lst vrtx lmntd rom ts ull omponnts gnrts S. Wn ll ull omponnts (xpt on) v bn lmntd, S s no longr mnml sprtor. Tror S s k ull omponnts, t s gnrtd xtly k 1 tms. Not tt [8] gvs n O(n + mlogn) pross to nd t multplty o mnml sprtor (wt MCS). Howvr, ty lso prov tt wt LxBFS, gvn t mnml sprtor gnrtor o lowst numbr w gnrts S, wtn t subst o mnml sprtor gnrtors w dn mnml sprtor o sz S, ll t gnrtors o S r onsutv. n t ours o n xuton o LxBFS, on n stor t lbls o t gnrtors n drnt lsts ordng to t sz o t mnml sprtor ty gnrt. Tn on n run troug lst, tstng or onsutv ourrns o gvn mnml sprtor. Ts n b don n lnr tm, s by Torm 3.7, t sum o t szs o t mnml sprtors s lss tn m. 5 Conluson n ts ppr, w prsnt smpl nd optml pross w gnrts t mnml sprtors nd mxml lqus o ordl grp n sngl pss o tr LxBFS or MCS, wtout rqurng t prlmnry omputton o po. Toug bot LxBFS nd MCS yld n optml lnr-tm pross or ts problm, t s mportnt to not tt ty dn drnt st o pos o ordl grp [4], nd xbt drnt lol bvors [3]. t my b mportnt to us on or t otr, dpndng on t ntndd pplton. W wll nd ts ppr by notng tt two rntly ntrodud sr lgortms, LxDFS nd MNS [6] l to bv n t sm son. A ountr-xmpl or MNS s gvn blow (T MNS lbls r t sts o gr-numbrd ngbors, omprd by st nluson): wn vrtx 2 s numbrd, ts lbl s not grtr tn tt o vrtx 3; owvr, Mdj(3) = {3, 4} s not mxml lqu.. 12

14 {4} 3 {5} {5} 1 {4,3} Fgur 3: MNS dos not gnrt t mxml lqus. Rrns [1] A. Brry, J.-P. Bordt Sprblty gnrlzs Dr s torm. Dsrt Appld Mtmts, 84(1-3):43 53, 18. [2] A. Brry, J.-P. Bordt Moplx lmnton ordrngs. Eltron Nots n Dsrt Mtmts, 8:6, 20. [3] A. Brry, J. R. S. Blr, J.-P. Bordt, nd G. Smont. Grp xtrmts dnd by sr lgortms. Rsr rport, LMOS, RR-07-05, 2007, to ppr n Algortms. [4] A. Brry, R. Krugr, G. Smont Gnvèv Smont. Mxml lbl sr lgortms to omput prt nd mnml lmnton ordrngs. SAM J. Dsrt Mt., 23(1): , 200. [5] J. R. S. Blr nd B. W. Pyton. An ntroduton to ordl grps nd lqu trs. Grp Tory nd Sprs Mtrx Computton. [6] D. G. Cornl, R. Krugr A und vw o grp srng. SAM J. Dsrt Mt., 22(4): , [7] D. R. Fulkrson nd O. A. Gross. ndn mtrs nd ntrvl grps. P Journl o Mtmts, 28: , 16. [8] P. Srnvs Kumr nd C. E. Vn Mdvn. Mnml vrtx sprtors o ordl grps. Dsrt Appld Mtmts, 8(1-3): , 18. [] D. J. Ros. Trngultd grps nd t lmnton pross. Journl o Mtmtl Anlyss nd Appltons,. [10] D. J. Ros, R. E. Trjn, nd G. S. Lukr. Algortm spts o vrtx lmnton on grps. SAM J. Comput., 5(2): ,

15 [11] R. E. Trjn nd M. Ynnkks. Smpl lnr-tm lgortms to tst ordlty o grps, tst ylty o yprgrps, nd sltvly rdu yl yprgrps. SAM J. Comput., 13(3):566 57,