On the Characterization of Level Planar Trees by Minimal Patterns

Transcription

1 On t Crtriztion o Lvl Plnr Trs Miniml Pttrns Aljnro Estrll-Blrrm, J. Jos Fowlr, n Stn G. Koourov Drtmnt o Comutr Sin, Univrsit o Arizon {l,owlr}@mil.rizon.u, oourov@s.rizon.u Astrt. W onsir rtriztions o lvl lnr trs. Hl t l. [8] rtriz t st o trs tt r lvl lnr in trms o two miniml lvl non-lnr (MLNP) ttrns. Fowlr n Koourov [7] ltr rov tt t st o ttrns ws inomlt n two itionl ttrns. In tis r, w sow tt t rtriztion is still inomlt roviin nw MLNP ttrns not inlu in t rvious rtriztions. Morovr, w introu n itrtiv mto to rt n ritrr numr o MLNP ttrns, tus rovin tt t st o miniml ttrns tt rtrizs lvl lnr trs is ininit. 1 Introution An imortnt lition o utomti r rwin n oun in t lout o rs tt rrsnt irril rltionsis. Wn rwin rs in t -ln, tis trnslts to rstrit orm o lnrit wr t -oorint o vrt is ivn n t rwin loritm onl s t rom to oos t -oorint. Tis rstrit orm o lnrit is ll lvl lnrit, n ivn -oorint orrsons to lvl. Jünr, Lirt, n Mutzl [13] rovi linr-tim ronition loritm or lvl lnr rs. Tis loritm is s on t lvl lnrit tst ivn Ht n Pmmrju [9,10]. T loritm Ht n Pmmrju is s on t mor rstrit PQ-tr lvl lnrit tstin loritm o irris (lvlrso irt li rs in wi ll s r twn jnt lvls n ll t sour vrtis r on t urmost lvl) ivn Di Bttist n Nrlli in [3]. In t r, t utors lso rtriz su irris in trms o lvl non-lnr (LNP) ttrns. Jünr n Lirt [12] rovi linr-tim lvl lnr min loritm tt oututs st o linr orrins in t -irtion or t vrtis on lvl. Howvr, to otin strit-lin lnr rwin on ns to susuntl run n O( V ) loritm ivn Es t l. [4] wo monstrt tt vr lvl lnr min s strit-lin rwin, tou it m ruir onntil r. Hl t l. [8] us LNP ttrns to rovi st o miniml lvl non-lnr (MLNP) sur ttrns tt rtriz lvl lnr rs. Tis is t ountrrt or lvl rs to t rtriztion o lnr rs Kurtowsi [14] in trms o orin suivisions o K 5 n K 3,3.TwonwMLNP tr ttrns wr in [7] Tis wor ws suort in rt NSF rnt CCF D. Estin n E.R. Gnsnr (Es.): GD 2009, LNCS 5849, , Srinr-Vrl Brlin Hilr 2010

2 70 A. Estrll-Blrrm, J.J. Fowlr, n S.G. Koourov Fowlr n Koourov to t rvious st o ttrns ivn Hl t l. Intis r, w sow tt t rtriztion rmins inomlt roviin nw MLNP ttrns not inlu in t rvious rtriztions. Morovr, w introu n itrtiv mto to rt n ritrr numr o MLNP ttrns, tus rovin tt t st o miniml ttrns tt rtrizs lvl lnr trs is ininit. T stu o MLNP ttrns is motivt in rt t rolm o visulizin irril struturs. Suim t l. [15] sri wt s om t stnr rmwor or rwin irt li rs. In tis rmwor vrtis r ssin to lvls n tn on lvl vrtis r orr, wit t ovrll ol o minimizin t numr o rossins. Tr ists oo uristis n som t mtos s uon intr linr rorms (ILPs) to in oo orrs witin lvls [11]. Howvr, till t ssinmnt o vrtis to lvls is on wit t l o r lol otimiztions [2]. Unrstnin t unrlin ostrutions to lvl lnrit (su s MLNP ttrns) oul l to ttr solutions to t lvl ssinmnt st. Lvl lnrit is lso rlt to simultnous min [1]. In nrl, st o rstritions on t lout o on r m l in t lout o son r on t sm vrt st. Siill, wn min t wit lnr r, i t r n rwn on orizontl lvls, tn t t n rwn in -monoton sion witout rossins. Estrll-Blrrm t l. [6] rtriz t st o unll lvl lnr (ULP) trs on n vrtis tt r lvl lnr ovr ll ossil llins o t vrtis in trms o two orin trs: T 8 n T 9. A lvl non-lnr llin o T 9 wsustootinmlnp ttrns P 3 n P 4 in [7]; s Fi Prliminris A -lvl r G(V, E,φ)onn vrtis is irt r G(V, E) wit lvl ssinmnt φ : V {1,...,} su tt t inu rtil orr is strit: φ(u) <φ(v) or vr (u, v) E. A-lvl r is -rtit r in wi φ rtitions nto innnt sts V 1, V 2,..., V, wi orm t lvls o G. Alvl- j vrt v is on t j t lvl i φ(v) = j (i.. v ). In lvl r, n (u, v) issort i φ(v) = φ(u) + 1 wil s snnin multil lvls r lon.aror lvl r s onl sort s. An lvl r n m ror suiviin lon s into sort s. In tis r, lvl r is ror unlss stt otrwis. A lvl r G s lvl rwin i tr ists rwin su tt vr vrt in is l lon t orizontl lin l j = {(, j) } n t s r rwn s stritl -monoton ollins. T orr tt t vrtis o r l lon l j in lvl rwin o ror r inus mil o linr orrs lon t - irtion, wi orm linr min o G. A lvl rwin, n onsuntl its lvl min, is lvl lnr i it n rwn witout rossins. A lvl r G is lvl lnr i it mits lvl lnr min. T inition o lvl rwins llowin onl strit-lin smnts or s is uivlnt, ivn tt Es t l. [4] v sown tt vr lvl lnr r s strit-lin lnr rwin. A t is non-rtin orr sun o vrtis (v 1, v 2,...,v n )orn 1. A str wit n vrtis is tr wit on vrt o r n 1, ll t root, nn 1 vrtis o r 1. A sir is n ritrril suivi str, wr suiviin n

3 On t Crtriztion o Lvl Plnr Trs Miniml Pttrns 71 () () Fi. 1. Oriinl MLNP ttrns P 1 in () n P 2 in () roos Hl t l (u, v) rls t wit nw vrt w n nw s (u, w) n(w, v). In r- sir, t root s r. A in-lin, not u v, is t rom vrt u to vrt v wit u v su tt intrnl vrt w tt lis lon t t s r 2. Lt φ(u v) not t st o lvls o t intrnl vrtis wr i φ(u v) j is sort-n or sin tt i φ(w) j or intrnl vrt w o t in-lin u v. Unlss stt otrwis w ssum tt φ(u) φ(u v) φ(v) or in-lin u v.alinin in,orsimlin, is sun o on or mor in-lins. Noti tt vrt in t intrstion o two ins is not onsir rossin twn t ins. In ll iurs, urv onntin two vrtis, rrsnts in. In lvl non-lnr r, ttrn is n ostrutin sur wit lvl ssinmnt tt ors rossin. Sin r w in rtiulr ttrns in trms o ins, t rrsnt st o rs wit similr rortis in trms o lvlin. A lvl non-lnr ttrn is miniml i t rmovl o n ritrr ms t ttrn lvl lnr. All t ttrns sri r (wit t tion o w tt r smmtril) v orrsonin orizontll li vrsion. 3 Prvious Wor 3.1 Crtriztion o Lvl Plnr Trs Hl t l. Hl t l. [8] in MLNP ttrns s ollows: Lt i n j t minimum n mimum lvl, rstivl, o n vrt in t ttrn. Lt vrt o r 3 wit tr sutrs wit t ollowin rortis: (i) sutr s t lst on vrt on ot trm lvls; (ii) sutr is itr in or it s two sutrs tt r ins; (iii) ll lvs r lot on trm lvls (n l is t onl vrt in its sutr on t trm lvl); n (iv) t sutrs tt r ins n v nonl vrtis on on trm lvl, lso v t lst on l vrt on t oosit trm lvl. Tn t istinuis two ttrns; P 1 wit on n trm lvl n P 2 wit on non-trm lvl (Hl t l. not tm T1 n T2). Fiur 1 sows P 1 n P 2. Noti tt ts ttrns r in in trms o sutrs. Tis imlis, or ml,

4 72 A. Estrll-Blrrm, J.J. Fowlr, n S.G. Koourov i i () () () V V () () () Fi. 2. (-) Fiv vritions o ttrn P 1 in ition to t on in Fi. 1(); () On vrition o ttrn P 2 in ition to t on in Fi. 1() tt sutr wit vrt o r 3 m rl t. Fowlr n Koourov, on t otr n, in t ttrns in trms o ts. Hn, to rorl omr t st o ttrns w n to onsir t irnt ss, or vritions, o t sutrs in P 1 n P 2. Hn, P 1 ls to vritions P1 A,...,PF 1 n P 2 ls to vritions P2 A n P2 B ; s Fi. 2. Noti tt wn in rs n trm lvl wit r-2 vrt, mor r-2 vrtis o t in n lso on t trm lvl. Tis is illustrt in Fi. 2() or t in wit son r-2 vrt. Hl t l. [8] sow tt ot o ts ttrns r miniml lvl non-lnr. 3.2 Crtriztion o Lvl Plnr Trs Fowlr n Koourov T two trs T 8 n T 9 wr sown to t onl ostrutions in t ontt o unll lvl lnrit or trs [6]. Howvr, s t tr T 9 os not mt n o t MLNP ttrns Hl t l. [8], nw ttrn P 3 ws roos [7]; s Fi. 3(). Not tt mtin T 9 wit itr o t rlir ttrns P 1 or P 2 woul imossil s ot P 1 n P 2 r s on ntrl vrt o r 3 (vrt in Fi. 1), wil T 9 n its mtin ttrn P 3 v ntrl vrt o r 4 (vrt in Fi. 3()). Yt notr ttrn P 4 n otin rom P 3 slittin vrt o r 4 su tt i < l φ() m < j into two vrtis o r 3 onnt t. In Fi. 3() vrt is rl in su tt l φ( ) m in Fi. 3(). Pttrns P 3 n P 4 wr to t rvious st o two ttrns (it vritions) to otin nw rtriztion onsistin o our ttrns (tn vritions). A st o

5 On t Crtriztion o Lvl Plnr Trs Miniml Pttrns () () () Fi. 3. Fowlr n Koourov nrliz t orin ULP tr T 9 in () to rou t MLNP ttrns P 3 in () n P 4 in () roo or t lim tt tis nw rtriztion is omlt ws m in [7], ut in t nt stion w sow tt t rtriztion rmins inomlt. 4 Nw Miniml Lvl Non-lnr Pttrns In tis stion, w sow tt t rtriztion o lvl lnr trs miniml ttrns is still inomlt. In St. 4.1, w sow tt tr r vritions o P 3 n P 4 tt wr not onsir. Tn in St. 4.2, w sri nw ttrn rviousl not onsir s it s vrt o r 5, wrs, ll o t rviousl nown MLNP ttrns v mimum r Vritions o Pttrns P 3 n P 4 T rvious rtriztion introus t nw ttrns P 3 n P 4. Just s wit t vritions o P 1 n P 2,irnt vritions o P 3 n P 4 n rou rlin som ins wit r-3 sirs. W sri ts vritions nt. Pttrn P3 A. Tis is t oriinl ttrn P 3; s Fi. 3(). Pttrn P3 B. Tis ttrn is similr to PA 3 ut rls t in su tt l φ() m, φ( ) = m, φ() = i, ni φ( ) m, wit r-3 sir root t n lvs,,n su tt l φ( ) m, φ() = φ( ) = m, φ() = i, nl φ( ) m; s Fi. 4(). Pttrn P C 3. Tis ttrn is similr to PA 3 ut rls t in, su tt l φ() m, φ() = l, φ() = j, nl φ( ) m wit r-3 sir root t n lvs,,nsu tt l φ( ) m, φ() = φ() = l, φ() = j, n l φ( ) m; s Fi. 4(). Pttrn P3 D. Tis ttrn ms ot rlmnts m ttrns PB 3 n PC 3 on P3 A su tt φ() = φ( ) = l, φ( ) = φ( ) = m, l φ() m, i φ( ) m, n l φ( ) j; s Fi. 4(). Pttrn P4 A. Tis is t oriinl ttrn P 4; s Fi. 3(). Pttrns P4 B, PC 4,nPD 4. Ts ttrns m nloous rlmnts on PA 4 s tosmp3 B, PC 3,nPD 3 on PA 3 ; s Fi. 4(-).

6 74 A. Estrll-Blrrm, J.J. Fowlr, n S.G. Koourov () () () () () () Fi. 4. (-) Vritions o ttrn P 3 (P B 3, PC 3,nPD 3 ); (-) Vritions o ttrn P 4 (P B 4, PC 4,n P D 4 ) T imortn o t nw vritions o P 3 n P 4 is tt t r t unmntl ssumtion m in t rl ttmts t rtriztions, nml tt in n miniml lvl non-lnr ttrn, lvs must li on trm lvls i or j. All o t nw ttrns v lvs on non-trm lvls. W omit t roos or t vritions o P 3 n P 4 s in t nt stion w ormll sow tt nw ttrn, P 5 wit non-trm lvs is MLNP. Morovr, in St. 5, w sow tt t st o MLNP ttrns or trs is not just missin w mor ttrns ut is tull ininit. 4.2 Nw Pttrn P 5 In tis stion, w sri nw ttrn P 5 n its vritions. T min rtristi o tis ttrn is t rsn o vrt wit r 5. Pttrn P5 A. Tis ttrn is r-5 sir, root t, wit two lvls l n m twn t trm lvls i n j su tt i < l <φ() m < j. Tr is in su tt φ() = i, in su tt φ() = j; in su tt φ() = m n φ() = l; in su tt φ() = l, φ( ) = m, φ() = i, nφ() = j; n in su tt l <φ() <φ(), φ() = j, φ() = i n l <φ( ) j; s Fi. 5(). Pttrn P5 B. Similr to PA 5 ut rls t in wit r-3 sir root t su tt l <φ( ) < m, wit,, n su tt l <φ() m, φ() = l, φ() = i, φ( ) = m, n tr is in

7 On t Crtriztion o Lvl Plnr Trs Miniml Pttrns 75 () () () Fi. 5. Pttrns P A 5, PB 5,nPC 5 su tt φ() = l wr l φ( ) m, l φ( ) m, n i φ( ) m ; s Fi. 5(). Pttrn P C 5. Similr to PA 5 ut rls t in wit r-3 sir root t su tt φ() = l wit lvs,, n ; s Fi. 5(). In t ollowin two lmms w sow tt tis nw ttrn is MLNP. Lmm 1. Pttrn P 5 is lvl non-lnr. Proo. W sow tt P5 A is lvl non-lnr (t ss or PB 5 n PC 5 r similr). First noti tt to voi rossin wit in, ll t vrtis o t in must li to t rit o t in wil ll t vrtis o t in must li to t lt, or vi vrs; s Fi. 5(). Assum w.l.o.. tt lis to t lt n lis to rit o in (s in Fi. 5()). Now osrv tt in orr to voi rossin o in wit ins, or, t in must li twn ins n or li twn ins n. Howvr, in t irst s rossin will our wit in (sin φ() <φ() nφ() φ( ) m) n in t ltr s rossin will our wit in. Lmm 2. T rmovl o n in ttrn P 5 ms it lvl lnr. Proo. W onsir t irnt ss o rmovl rom t ins in P5 A (PB 5 n P C 5 r similr): s 1) I n is rmov rom in, tn t rossin wit in is voi wn is to t rit o s in Fi. 6(). s 2) I n is rmov rom ins or, tn ll t vrtis in t in (t ) n to t lt or to t rit o in

8 76 A. Estrll-Blrrm, J.J. Fowlr, n S.G. Koourov () () () () Fi. 6. Dirnt ss o rmovin n (ott) rom ttrn P A 5 wr in n on t otr si voiin t rossin s in Fi. 6(). s 3) I n is rmov rom in, tn ins n n on t sm si wit rst to. Tus voiin t rossin wit in ; s Fi. 6(). s 4) I n is rmov rom in, tn in n li to t rit o in s in Fi. 6(). W now us Lmms 1 n 2 to sow tt P 5 is in MLNP. Torm 1. P 5 is miniml lvl non-lnr ttrn or trs. Proo. B Lmm 1, P 5 is lvl non-lnr n Lmm 2, P 5 is miniml. Minimlit lso imlis tt P 5 os not ontin n MLNP ttrn s sur. Morovr, ttrn P 5 os not mt n o t rvious ttrns ivn tt vrt s r 5, wil ll o t rviousl nown ttrns v mimum r 4. In tis stion, w v sown tt nw ttrn P 5 is MLNP. Howvr,P 5 is not t onl ttrn missin rom rlir rtriztions. Nw ttrns P 6,...,P 11 r sown lon wit tir vritions in [5]. T roos o lvl non-lnrit n minimlit o ts ttrns r similr to t on ivn or P 5. Tus, inst o rovin tt o ts ttrns is MLNP, w sri onstrutiv mto or nrtin n ininit numr o istint MLNP ttrns in t nt stion.

9 On t Crtriztion o Lvl Plnr Trs Miniml Pttrns () () Fi. 7. Constrution o nw ttrn (P A 4 ) 1 in () rom ttrn P A 4 in () 5 Ininit Miniml Lvl Non-lnr Pttrns Our ro or rtin nw MLNP ttrns is to t nown ttrn s s n tn rt sur o t ttrn min moiitions on t lvlin su tt t nw ttrn os not stritl ontin t rvious on. Hr w us P4 A ut t mto lis to otr ttrns s wll. Tirststistomoott 0 = su tt φ() = φ() = i <φ() <φ( ) <φ() = j s in Fi. 7() in orr to t nw t 1 = su tt φ( 1 ) = φ( 1 ) = i 1, φ( 1 ) = j, φ( 1 ) = j + 1, n φ( 1 ) = j + 2 s in Fi. 7(). T son st is to 1 to P4 A mrin vrtis 1 n rtin nw vrt o r 3 tt ts t l o. Tis nw lvl ssinmnt rts two nw trm lvls i 1nj + 2. W omlt t onstrution o t nw ttrn movin vrtis,, n to t nw trm lvls, siill, w st φ() = i 1nφ() = φ() = j + 2. W now nrliz t rvious onstrution to n ritrr numr o itrtions. W not t ttrn rt t itrtion t rom ttrn P s (P) t. Tus, t oriinl P4 A is (P4 A) 0 n t ttrn rt in Fi. 7() is (P4 A) 1. T vrtis in t ttrn r ll in t sm w, or ml 0 =. Tror, in orr to rt nw ttrn (P4 A) t+1 rom ttrn (P4 A) t, w irst o t t t = t t t t t to t nw t t+1 = t+1 t+1 t+1 t+1 t+1 su tt φ( t+1 ) = φ( t+1 ) = i t 1, φ( t+1 ) = j + 2t, φ( t+1 ) = j + 2t + 1, n φ( t+1 ) = j + 2t + 2. W tn mr t+1 wit t to otin t nw t+1. Finll, w st t lvls s φ() = i t 1, n φ() = φ() = j + 2t + 2; s Fi. 8. In t nt lmm w sow tt ttrn, (P4 A) t, nrt wit t rvious mto is lvl non-lnr. Lmm 3. Pttrn (P A 4 ) t or t 0, is lvl non-lnr. Proo. W us inution on t, t numr o itrtions in t nrtion mto. T s s is t = 0; tis is t oriinl ttrn P 4 wi is rovn to lvl non-lnr in t rtriztion Fowlr n Koourov [7]. W now ssum tt (P A 4 ) t is lvl

10 78 A. Estrll-Blrrm, J.J. Fowlr, n S.G. Koourov -t t t -t-1 Vi-t t t t+1 t+1 t t t t +2t t +2t t+1 t+1 +2t+2 t+1 () () Fi. 8. Constrution o nw ttrn (P A 4 ) t+1 in () rom ttrn (P A 4 ) t in () non-lnr in orr to rov tt (P4 A) t+1 is lvl non-lnr. Tt is, w sow tt t moiitions m to (P4 A) t to otin (P4 A) t+1 o not t t lvl non-lnrit o t nw ttrn. Clrl, t ition o vrtis n s nnot t t lvl non-lnrit o tr, n t ition o t t t+1 os not m t ttrn lvl lnr. Morovr, sin t ins n in (P4 A) t r ontin in t ins n o (P4 A) t+1, t n on t lvls o n r siml ition o vrtis n s tt nnot t t lvl non-lnrit o t ttrn. Finll, w onsir t n o lvl o vrt. Noti tt t rossin twn t in n t in in (P4 A) t nnot voi in (P4 A) t+1 wit t n o lvl o. Tis is us s is mov to t lvl o t in t+1 t+1 t+1 ls n nloous rol in t ttrn (P4 A) t+1 tt t in t ls in t ttrn (P4 A) t. Tt is, t ition o t in t+1 t+1 to t ttrn (P4 A) t+1 rvnts t swit o si o t in in orr to voi t rossin wit s tis will rou rossin wit t in t+1 t+1 (s in Fi. 9()). Tror, inution t ttrn (P4 A) t is lvl non-lnr or ll non-ntiv intrs t 0. W nt sow t minimlit o t ttrns nrt wit t mto ov. Lmm 4. T rmovl o n in (P A 4 ) t or n t 0, ms it lvl lnr. Proo. W onsir t ss o rmovl in (P A 4 ) t. s 1) I n is rmov rom t in, tn t slintrstion is voi s in Fi. 9(). s 2) I n is rmov rom t in, tn t in n us t to voi t rossin s in Fi. 9().

11 On t Crtriztion o Lvl Plnr Trs Miniml Pttrns 79 s 3) I n is rmov rom t in α α or α α or n α = 0,...,t, tn in n us t to rwn twn t ins α α n α α s in Fi. 9() or twn α n α. s 4) I n is rmov rom t ins α α or α α or n α = 0,...,t, tn t in n intrn sis wit t in α α i α = t s in Fi. 9(). Wn α<t, ll t ins β β β β β or β = α + 1,...,t r mov lon wit t in α α. Wit t lst two lmms w now sow tt ttrn nrt wit t itrtiv mto sri in tis stion is MLNP. Torm 2. Pttrn (P A 4 ) t or t 0, is miniml lvl non-lnr ttrn or trs. Proo. B Lmm 3, (P A 4 ) t is lvl non-lnr n Lmm 4, (P A 4 ) t is miniml. Minimlit imlis tt (P A 4 ) t os not ontin n MLNP ttrn s sur. In rtiulr, (P A 4 ) t os not ontin t rvious ttrn (P A 4 ) t 1. To s tis in Fi. 8(), osrv tt in t sur twn lvls i n j, t in is srt lvl j into two isjoint ins. Morovr, ttrn (P A 4 ) t os not mt n o t rvious ttrns (P A 4 ) α or α = 0,...,t 1sin(P A 4 ) t ontins n itionl vrt o r 3, t. Torm 2 imlis tt w n nrt n ritrr numr o irnt MLNP ttrns. Tis ivs our min rsult. t t t t -t -t t t t t +2t t +2t t () t t t () t -t -t t t t t +2t t +2t t () () Fi. 9. Dirnt ss o rmovin n (ott) rom ttrn (P A 4 ) t

12 80 A. Estrll-Blrrm, J.J. Fowlr, n S.G. Koourov Torm 3. T st o miniml lvl non-lnr ttrns or trs is ininit. 6 Conlusions n Futur Wor In tis r, w sow w two rlir ttmts to rtriz t st o lvl nonlnr trs in trms o miniml lvl non-lnr ttrns il. In ot ss, tr ws n imliit ssumtion tt t st o irnt MLNP ttrns is smll n init. Howvr, it turns out tt tr r ininitl mn irnt MLNP ttrns, n n ltotr irnt ro mit n or omlt rtriztion. Rrns 1. Brß, P., Cn, E., Dunn, C.A., Ert, A., Ertn, C., Ismilsu, D., Koourov, S.G., Luiw, A., Mitll, J.S.B.: On simultnous lnr r mins. Comuttionl Gomtr: Tor n Alitions 36(2), (2007) 2. Cimni, M., Gutwnr, C., Mutzl, P., Won, H.-M.: Lr-r uwr rossin minimiztion. In: MGo, C.C. (.) WEA LNCS, vol. 5038, Srinr, Hilr (2008) 3. Di Bttist, G., Nrlli, E.: Hirris n lnrit tor. IEEE Trnstions on Sstms, Mn, n Crntis 18(6), (1989) 4. Es, P., Fn, Q., Lin, X., Nmoi, H.: Strit-lin rwin loritms or irril rs n lustr rs. Aloritmi 44(1), 1 32 (2006) 5. Estrll-Blrrm, A.: Simultnous Emin n Lvl Plnrit. PD tsis, Drtmnt o Comutr Sin, Univrsit o Arizon (2009) 6. Estrll-Blrrm, A., Fowlr, J.J., Koourov, S.G.: Crtriztion o unll lvl lnr trs. Comuttionl Gomtr: Tor n Alitions 42(7), (2009) 7. Fowlr, J.J., Koourov, S.G.: Minimum lvl nonlnr ttrns or trs. In: Hon, S.-H., Nisizi, T., Qun, W. (s.) GD LNCS, vol. 4875, Srinr, Hilr (2008) 8. Hl, P., Kuusi, A., Lirt, S.: A rtriztion o lvl lnr rs. Disrt Mtmtis 280(1-3), (2004) 9. Ht, L.S., Pmmrju, S.V.: St n uu louts o irt li rs. II. SIAM Journl on Comutin 28(5), (1999) 10. Ht, L.S., Pmmrju, S.V., Trn, A.N.: St n uu louts o irt li rs. I. SIAM Journl on Comutin 28(4), (1999) 11. Jünr, M., L, E.K., Mutzl, P., Ontl, T.: A olrl ro to t multi-lr rossin minimiztion rolm. In: DiBttist, G. (.) GD LNCS, vol. 1353, Srinr, Hilr (1997) 12. Jünr, M., Lirt, S.: Lvl lnr min in linr tim. Journl o Gr Aloritms n Alitions 6(1), (2002) 13. Jünr, M., Lirt, S., Mutzl, P.: Lvl lnrit tstin in linr tim. In: Witsis, S.H. (.) GD LNCS, vol. 1547, Srinr, Hilr (1999) 14. Kurtowsi, C.: Sur ls rolèms s ours us n Tooloi. Funmnt Mtmti 15, (1930) 15. Suim, K., Tw, S., To, M.: Mtos or visul unrstnin o irril sstm struturs. IEEE Trnstions on Sstms, Mn, n Crntis 11(2), (1981)