Dominator Tree Certification and Independent Spanning Trees

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1 Domintor Tr Crtiition n Inpnnt Spnnin Tr Louk Gorii 1 Rort E. Trjn 2 rxiv: v3 [.DS] 7 Mr 2013 Otor 29, 2018 Atrt How o on vriy tht th output o omplit prorm i orrt? On n ormlly prov tht th prorm i orrt, ut thi my yon th powr o xitin mtho. Altrntivly on n hk tht th output prou or prtiulr input tii th ir input-output rltion, y runnin hkr on th input-output pir. Thn on only n to prov th orrtn o th hkr. But or om prolm vn uh hkr my too omplit to ormlly vriy. Thr i thir ltrntiv: umnt th oriinl prorm to prou not only n output ut lo orrtn rtiit, with th proprty tht vry impl prorm (who orrtn i y to prov) n u th rtiit to vriy tht th input-output pir tii th ir input-output rltion. W onir th ollowin importnt intn o thi nrl qution: How o on vriy tht th omintor tr o low rph i orrt? Exitin t lorithm or inin omintor r omplit, n vn vriyin th orrtn o omintor tr in th n o itionl inormtion m omplit. W in orrtn rtiit or omintor tr, how how to u it to ily vriy th orrtn o th tr, n how how to umnt t omintor-inin lorithm o tht thy prou orrtn rtiit. W lo rlt th omintor rtiit prolm to th prolm o inin inpnnt pnnin tr in low rph, n w vlop lorithm to in uh tr. All our lorithm run in linr tim. Prviou lorithm pply jut to th pil o only trivil omintor, n thy tk t lt qurti tim. Thi work i rwrittn n xpn omintion o two onrn ppr, Domintor Tr Vriition n Vrtx-Dijoint Pth, Pro. 16th ACM-SIAM Sympoium on Dirt Alorithm, pp , 2005, n Domintor, Dirt Bipolr Orr, n Inpnnt Spnnin Tr, Pro. 39th Intrntionl Colloquium on Automt, Lnu n Prormmin, pp , Dprtmnt o Computr Sin, Univrity o Ionnin, Gr. E-mil: louk@.uoi.r. 2 Dprtmnt o Computr Sin, Printon Univrity, 35 Oln Strt, Printon, NJ, 08540, n Hwltt-Pkr Lortori. E-mil: rt@.printon.u. Rrh t Printon Univrity prtilly upport y NSF rnt CCF n CCF Rrh whil viitin Stnor Univrity prtilly upport y n AFOSR MURI rnt. Th inormtion ontin hrin o not nrily rlt th opinion or poliy o th rl ovrnmnt n no oiil normnt houl inrr. 1

2 G D 1 Introution Fiur 1: A low rph n it omintor tr. A vxin prolm whn runnin omputr prorm i to vriy tht th output prou y th prorm i orrt. On option i to ormlly prov th orrtn o th prorm, ut i th prorm implmnt ophitit lorithm, thi my t t hllnin n t wort yon th powr o urrnt prorm vriition thniqu. A on option i to implmnt hkr [6]: n lorithm tht, ivn n input-output pir, vrii tht it tii th ir input-output rltion. For om prolm vn thi my hr. Thr i thir option: umnt th oriinl lorithm to prou not only th ir output ut lo orrtn rtiit, n implmnt rtiir: n lorithm tht, ivn n input-output pir n rtiit, u th rtiit to vriy tht th pir tii th ir input-output rltion. Th umnt lorithm houl iint th oriinl (to within ontnt tor), n th rtiir houl t lt iint th oriinl lorithm, muh implr, n y to prov orrt. W r th ollowin importnt intn o thi prolm: vriy th orrtn o th omintor tr o low rph. A low rph i irt rph G with int trt vrtx uh tht vry vrtx i rhl rom. A vrtx v omint vrtx w i v i on vry pth rom to w; i v w, v i trit omintor o w. Th omintor rltion in omintor tr D uh tht v omint w i n only i v i n ntor o w in D. (S Fiur 1.) Th omintor tr i ntrl tool in prorm optimiztion n o nrtion [13], n it h pplition in othr ivr r inluin ontrint prormmin [44], iruit ttin [3], thortil ioloy [1], mmory proilin [40], onntivity n pth-trmintion prolm [18, 19, 35], n th nlyi o iuion ntwork [25]. Lnur n Trjn [38] v two nr-linr-tim lorithm or omputin D tht run t in prti [24] n hv n u in mny o th pplition. Th implr o th run in O(mlo (m/n+1) n) tim on n n-vrtx, m-r low rph with n > 1. Th othr run in O(mα m/n (n)) tim. Hr α i untionl invr o Akrmnn untion in ollow: α (n) = min{k > 0 A(k, ) > n}, whr A i in rurivly y A(0,j) = j+1; A(k,0) = A(k 1,1) i k > 0; A(k,j) = A(k 1,A(k,j 1)) i k > 0, j > 0. Thi i liht vrint o Trjn oriinl inition [53]. Suquntly, mor-omplit ut truly linr-tim lorithm to omput D wr iovr [2, 7, 8, 21]. 2

3 Howoonknowthtthtrprouyonothtutomplitlorithm i in t th omintor tr? Thi nturl qution w k o th on uthor y Stv Wk in Rrhr in prorm vriition hv k th m qution [58]. Thi ppr i our nwr. Forml vriition o ny o th t lorithm h to our knowl nvr n on. Int, only muh implr ut muh lowr lorithm h n ormlly vrii [58]. It m hr vn to uil t hkr o th omintor tr. W tk th thir pproh to vriition: umnt th lorithm tht omput th omintor tr to prou in ition orrtn rtiit, uh tht it i impl to vriy th orrtn o th tr with th hlp o th rtiit. Hr wht impl mn i not riorou: in linr tim i nry n uiint or ontrutin th tr n hn or vriyin it, runnin tim nnot th mur o impliity. Our ol i linr-tim rtiir tht voi th thnil omplition o th t omintor-inin lorithm n i impl poil, impl nouh tht on oul ily ormlly prov th orrtn o th rtiir. (W lv thi tk or tho kill in prorm vriition.) Our rtiit i prorr o th vrti o D with rtin proprty, whih w ll low-hih. Sin low-hih orr i prorr, it n m in th rprnttion o D, or it n rprnt prtly. Vriyin tht n orr i low-hih i ntirly trihtorwr n n on ily in linr tim. In ition to proviin rtiir, w vlop linr-tim lorithm to omput low-hih orr ivn G n D. By in on o th lorithm to n lorithm to omput D, on otin rtiyin lorithm [41] to omput D. To otin our rult, w vlop nw thory out omintor n rlt onpt, thory tht h itionl pplition. W otin our inition o low-hih orr y nrlizin notion o Plin [43] n Chriyn n Ri [10], whih thy u in lorithm or ontrutin pir o ijoint pnnin tr. Suppo th omintor tr D i lt; tht i, h vrtx v h only on trit omintor,. Thn only n v r ommon to ll pth rom to v. By Mnr Thorm [42], thr r two pth rom to v ontinin no ommon vrtx othr thn n v. Whitty [57] prov tht uh pth n rliz or ll v y pir o tr: thr r pnnin tr B n R o G, root t, uh tht or ny v th pth rom to v in B n R hr only n v. W ll uh tr ijoint. Whitty tully prov omthin tronr: thr r two ijoint pnnin tr B n R root t uh tht, or ny itint vrti v n w, ithr th pth in B rom to v n th pth in R rom to w hr only, or th pth in R rom to v n th pth in B rom to w hr only. W ll uh tr tronly ijoint. Two tr n ijoint without in tronly ijoint, th xmpl in Fiur 2 how. Plhn[43] n inpnntly Chriyn n Ri[10] v implr proo o Whitty rult uin wht Chriyn n Ri ll irt t-numrin n intrmiry. Givn irt rph with n vrti n two itint vrti n t, irt t-numrin i numrin o th vrti rom 1 to n uh tht i numr 1, t i numr n, n vry othr vrtx v h n ntrin r rom mllr vrtx n n ntrin r rom lrr vrtx. Th proo o Whitty, o Plhn, n o Chriyn n Ri iv polynomil-tim ontrution o irt t-numrin n o tronly ijoint pnnin tr, ut thir ontrution m to rquir Ω(nm) tim in th wort. Huk [28] ltr v n O(nm)- tim lorithm to in two ijoint pnnin tr. Non o th proo or lorithm i 3

4 x y x y x y h i h i h i G B R Fiur 2: A rph with two ijoint ut not tronly ijoint pnnin tr. Grph G h lt omintor tr. Spnnin tr B n R r ijoint, ut th pir,h violt th onition or tron ijointn. pilly impl. Not tht i D i lt, ontrutin it i trivil, ut D provi no hlp in ontrutin two ijoint or tronly ijoint tr. Tht i, ontrutin two uh tr i irnt prolm thn inin D. Furthrmor, vriyin tht two pnnin tr r ijoint or tronly ijoint i not y. W xtn th onpt to ritrry low rph. Lt G low rph with vrtx t V, r t A, n trt vrtx. Lt T tr root t with vrtx t V (not nrily pnnin tr o G), with t(v) th prnt o v in T. A prorr o T i totl orr o it vrti otinl y oin pth-irt trvrl o T n orrin th vrti y irt viit. A prorr i o T i low-hih on G i, or ll v, (t(v),v) A or thr r two r (u,v) A, (w,v) A uh tht u i l thn v, v i l thn w, n w i not nnt o v. Two pnnin tr B n R root t r inpnnt i or ll v, th pth rom to v in B n R hr only th omintor o v; B n R r tronly inpnnt i or vry pir o vrti v n w, ithr th pth in B rom to v n th pth in R rom to w hr only th ommon omintor o v n w, or th pth in R rom to v n th pth in B rom to w hr only th ommon omintor o v n w. (S Fiur 3.) Th thr inition xtn th notion o t-numrin n ijoint n tronly ijoint pnnin tr, rptivly, to low rph with non-lt omintor tr. W prov tht vry low rph h pir o tronly inpnnt pnnin tr n it omintor tr h low-hih orr. W vlop linr-tim lorithm rlt to th onpt, nrlizin n iniintly improvin th prviou rult. W prnt vry impl rtiir tht, ivn low rph n trwithvrtxorr,vriithtthorrilow-hihnthtthtrithomintor tr o th low rph. W prnt n lorithm tht, ivn low rph n it omintor tr with low-hih orr, ontrut pir o tronly inpnnt pnnin tr. Th tr hv th itionl proprty tht thy r r-ijoint xpt or ri. (A ri i n r (u,v) uh tht vry pth rom to v ontin (u,v).) Tht th tr r tronly 4

5 [1] [8] [2] [1] [5] [4] [2] [6] [7] [8] [7] [6] [3] [3] [4] [5] G D [1] [1] [8] [2] [8] [2] [5] [4] [5] [4] [7] [3] [7] [3] [6] [6] B R Fiur 3: Th low rph o Fiur 1, it omintor tr with vrti numr in low-hih orr (numr in rkt), n two tronly inpnnt pnnin tr B n R. inpnnt ollow immitly rom th t tht th orr i low-hih. W prnt thr lorithm to ontrut low-hih orr, ivn low rph n it omintor tr. Th irt ppli only to ruil low rph (in low), l tht inlu yli low rph. Th on n thir pply to ritrry low rph, ut oth rquir itionl input: th on rquir loop-ntin inormtion (in in Stion 5), th thir rquir inormtion out mi-omintor (in in Stion 6). Th thir lorithm u th mi-omintor inormtion to ontrut pir o inpnnt pnnin tr, rom whih it thn ontrut low-hih orrin. Althouh oth tp r impl, th orrtn proo o th irt tp i urpriinly omplit. Mny prorm ontrol low rph r ruil. For th, th irt low-hih orr lorithm n u to rtiy th omintor tr. For ritrry low rph, ithr th on or thir lorithm n u. Mny pplition o omintor n loop inormtion wll; or th, th on lorithm i pproprit, in th loop inormtion mut omput nywy. Th t lorithm or inin omintor o o y omputin mi-omintor, mkin it y to xtn th lorithm to prou th inormtion n y th thir lorithm. I th low rph i not ruil, th pplition o not n loop inormtion, n omintor r omput y impl lorithm tht o not omput mi-omintor, uh th itrtiv lorithm o Coopr t l. [11], th on lorithmn u i loopromput, whih n onin nr-linr tim [54] uin 5

6 ijoint t union [53] or truly linr tim [7] uin itionl thniqu. Thu, pnin on th itution, t lt on o our thr lorithm n u to rtiy th omintor tr. Our ppr i rwrittn n xpn omintion o two onrn ppr [22, 23]. It ontin ix tion in ition to thi introution. Stion 2 introu om trminoloy n vlop om proprti o omintor tr. It lo prnt our lorithm to ontrut pir o tronly inpnnt pnnin tr ivn low-hih orr n to vriy th orrtn o omintor tr with low-hih orr. Stion 3 introu th riv rph, whih in t llow u to ru th prolm o ontrutin low-hih orr to th o lt omintor tr. Stion 4 prnt our lorithm or inin low-hih orr on ruil low rph. A prt o th implmnttion o thi lorithm, w vlop implii linr-tim lorithm or pil o th ynmi lit orr prolm [5, 14]. Thi lorithm my itl hv itionl pplition. Stion 5 n 6 ontin our lorithm to in low-hih orr on n ritrry low rph. Th lorithm in Stion 5 u loop inormtion, n th lorithm in Stion 6 u mi-omintor inormtion. Stion 7 iu othr pplition n opn prolm. 2 Domintor, Low-Hih Orr, n Stronly Inpnnt Spnnin Tr 2.1 Trminoloy Lt T root tr. W not y t(v) th prnt o vrtx v in T; t(v) = null i v i th root o T. I v i n ntor o w, T[v,w] i th pth in T rom v to w. I v i propr ntor o w, T(v,w] i th pth to w rom th hil o v tht i n ntor o w. Tr T i lt i it root i th prnt o vry othr vrtx. A prorr o T i totl orr o th vrti o T uh tht, or vry vrtx v, th nnt o v r orr onutivly, with v irt. Equivlntly, or h non-root vrtx v, t(v) i l thn v, n i x i nnt o v ut y i not, thn y i l thn v or rtr thn x. Th poil prorr o T r xtly tho tht n otin y totlly orrin th hilrn o h vrtx, oin pth-irt trvrl o T, n orrin th vrti in th orr thy r irt viit y th trvrl. Throuhout th rt o thi ppr, G i low rph with vrtx t V, r t A, trt vrtx, n no r ntrin : r ntrin n lt without tin ny o th onpt w tuy. W not y n th numr o vrti n y m th numr o r. To impliy oun w um n > 1. Sin m n 1, thi impli m = Ω(n). W not th omintor tr o G y D: i v, th prnt (v) o v in D i th immit omintor o v, th uniqu trit omintor o v omint y ll trit omintor o v. (Som uthor u iom(v) to not th immit omintor o v.) 2.2 Th prnt n ilin proprti Lt T root tr who vrtx t i ut o V. Tr T h th prnt proprty i or ll (v,w) A, t(w) i n ntor o v in T. Sin h no ntrin r, ut vry othr 6

7 vrtx h t lt on ntrin r (ll vrti r rhl rom ), th prnt proprty impli tht T i root t n h vrtx t xtly V. Tr T h th ilin proprty i v o not omint w or ll ilin v n w. Th prnt n ilin proprti r nry n uiint or tr to th omintor tr. Thorm 2.1. Tr D h th prnt n ilin proprti. Proo. Tr D h th ilin proprty y inition. Trjn [55, Lmm 7] prov tht D h th prnt proprty. For ompltn, w inlu proo hr. Suppo D violt th prnt proprty. Thn thr i n r (v,w) uh tht (w) (th immit omintor o w) i not n ntor o v in D; tht i, (w) o not omint v. But thn thr i pth rom to w tht voi (w), onitin o pth rom to v voiin (w) ollow y r (v, w). Thu (w) o not omint w, ontrition. Lmm 2.2. Suppo T h th prnt proprty. Lt u n v vrti uh tht v n u i not nnt o t(v) in T. Thn ny pth rom u to v ontin t(v). Proo. Conir ny pth rom u to v. Lt (x,y) th irt r on thi pth uh tht y i nnt o t(v) in T. By th prnt proprty, t(y) i n ntor o x in T. But in x i not nnt o t(v) in T, it mut th tht y = t(v). Tht i, th pth ontin t(v). Corollry 2.3. Suppo T h th prnt proprty. I v i n ntor o w in T, thn v omint w. Proo. Suppo v. By Lmm 2.2, ny pth rom to v ontin t(v), o t(v) omint v. Th orollry ollow y th trnitivity o ominn. Corollry 2.4. Suppo T h th prnt proprty n v. I x i vrtx on impl pth P rom t(v) to v, thn x i nnt o t(v) ut not propr nnt o v. Proo. I x wr non-nnt o t(v), th prt o P rom x to v woul ontin t(v) y Lmm 2.2, o P woul not impl. I x wr propr nnt o v, th prt o P rom t(v) to x woul ontin v y Lmm 2.2, o P woul not impl. Thorm 2.5. A tr T h th prnt n ilin proprti i n only i T = D. Proo. Thorm 2.1 iv th i hl o th thorm. Suppo T h th prnt proprty. By Corollry 2.3, i v i n ntor o w in T, thn v omint w. Suppo v omint w ut v i not n ntor o w. Sin v w, w o not omint v. By Corollry 2.3, w i not n ntor o v in T, o v n w r unrlt. Lt u th nrt ommon ntor o v n w in T, n lt x n y th hilrn o u tht r ntor o v n w. By Corollry 2.3, x omint v n y omint w. By th trnitivity o th omintor rltion, x omint w. Sin oth x n y omint w, on mut omint th othr, whih violt th ilin proprty. Thi iv th only i hl o th thorm. By Thorm 2.5, to vriy tht tr T i th omintor tr, it ui to how (1) T i root tr, (2) T h th prnt proprty, n (3) T h th ilin proprty. It i trihtorwr to vriy (1) in O(n) tim, umin tht T i ivn y it prnt untion: 7

8 w jut n to hk tht th prnt untion i in or ll vrti othr thn n tht thr r no yl. It i lo y to vriy (2). Thi tk O(1) tim pr r, or totl o O(m) tim, ivn n O(1)-tim tt o th ntor-nnt rltion. Thr r vrl impl O(1)-tim tt o thi rltion [51]. Th mot onvnint on or u i to numr th vrti rom 1 to n in ny prorr o T, n to omput th numr o nnt o h vrtx v, whih w not y iz(v). I vrti r intii y numr, v i n ntor o w i n only i v w < v+iz(v). I T i ivn y it prnt untion, w n numr th vrti n omput thir iz y irt uilin lit o hilrn or h vrtx n thn oin pth-irt trvrl, ll o whih tk O(n) tim. Th hrt tp in vriition i to how (3). W hll prov tht tr with th prnt proprty h th ilin proprty i n only i it h low-hih orr. W prov uiiny in thi tion n nity in Stion 4 or ruil rph, n in Stion 5 or ritrry rph. Unlik th prnt proprty, whih i y to tt, it i not o y to tt or th xitn o low-hih orr, lthouh it i y to tt i ivn orr i low-hih. Thu w pl th urn o ontrutin uh n orr on th lorithm tht omput th omintor tr, not on th vriition lorithm: th orr rtii th orrtn o th tr. 2.3 Contrution o Two Stronly Inpnnt Spnnin Tr To prov tht tr with th prnt proprty n low-hih orr h th ilin proprty, n u it i intrtin in it own riht, w how how to ontrut pir o tronly inpnnt pnnin tr, ivn tr T with th prnt proprty n low-hih orr. Alorithm 1: Contrution o Two Stronly Inpnnt Spnnin Tr B n R Lt T tr with th prnt proprty n low-hih orr. For h vrtx v, pply xtly on o th ollowin to hoo r ((v),v) in B n (r(v),v) in R (i C 1 n 2 oth pply, pply ithr on): C 1: Thr r two r (u,v) n (w,v) uh tht u < v < w in low-hih orr n w i not nnt o v in T. Choo two uh r, n t (v) = u n r(v) = w. C 2: (t(v),v) i n r n thr i nothr r (u,v) uh tht u < v in low-hih orr. Choo uh n r, n t (v) = u n r(v) = t(v). C 3: (t(v),v) i th only r ntrin v rom non-nnt o v. St (v) = r(v) = t(v). Alorithm1ppli to thrphinfiur1withth low-hihorr infiur3prou th tr hown in Fiur 3. Lmm 2.6. Lt T tr with th prnt proprty n low-hih orr. For ny vrtx 8

9 v, t lt on o C 1, 2, n 3 ppli. Thu Alorithm 1 in (v) n r(v) or ll v. Proo. Lt v vrtx to whih C 1 o not pply. Th inition o low-hih orr impli tht (t(v),v) A. I (t(v),v) i th only r ntrin v rom non-nnt o v, thn C 3 ppli. Othrwi, thr i n r (u,v) uh tht u t(v) n u i not nnt o v. Sin C 1 o not pply, u < v in low-hih orr, o C 2 ppli. Lmm 2.7. For ny vrtx v, thr i pth in R rom t(v) to v ontinin only t(v) n vrti no l thn v in low-hih orr. Proo. Suppo th lmm i l. Lt v th lrt vrtx in low-hih orr or whih it il. Thn r(v) t(v), o r(v) > v n r(v) i not nnt o v. Lt x th ilin o v tht i n ntor o r(v). Sin r(v) > v n th orr i prorr, ll nnt o x r rtr thn v. Th hoi o v impli tht th lmm hol or x n ll it nnt. It ollow tht thr i pth in R rom t(x) to x to r(v) tht ontin only t(x) n vrti no l thn x. Ain (r(v),v) to thi pth iv pth rom t(v) = t(x) to v tiyin th lmm, ontrition. Thorm 2.8. B n R r tronly inpnnt pnnin tr root t. Proo. Firt w prov tht B n R r pnnin tr root t, thn tht thy r inpnnt, n inlly tht thy r tronly inpnnt. Sin h vrtx v h n ntrin r ((v),v) with (v) < v, B i pnnin tr root t. By Lmm 2.7, vry vrtx i rhl rom in R, o R i lo pnnin tr root t. Suppo B n R r not inpnnt. Lt v th minimum vrtx uh tht B[,v] n R[,v] hr vrtx othr thn omintor o v. By Corollry 2.3, t(v) omint v, o B[,v] n R[,v] oth ontin t(v). By th hoi o v, B[,t(v)] n R[,t(v)] hr only omintor o t(v). By Corollry 2.4, no vrti on B[,t(v)] or R[,t(v)] r propr nnt o t(v), ut ll vrti on B[t(v),v] n R[t(v),v] r nnt o t(v), o B[t(v),v] n R[t(v),v] mut hr vrtx othr thn t(v) n v. But B[t(v),v] ontin only v n vrti l thn v, n y Lmm 2.7, R[t(v),v] ontin only t(v) n vrti no l thn v, o B[t(v),v] n R[t(v),v] hr only t(v) n v, ontrition. Thu B n R r inpnnt. Suppo tht B n R r not tronly inpnnt; lt v n w vrti uh tht v < w n B[,v] n R[,w] hr vrtx othr thn ommon omintor o v n w. Lt u th nrt ommon ntor o v n w in T, whih omint oth. Sin B n R r inpnnt, B[,u] n R[,u] hr only omintor o u, whih r ommon omintor o v n w. By Corollry 2.4, B[,u] n R[,u] ontin no propr nnt o u, ut ll vrti on B[u,v] n R[u,w] r nnt o u, o B[,u] n R[u,w] ontin only u in ommon, o R[,u] n B[u,v]. I ollow tht B[u,v] n R[u,w] hr vrtx othr thn u. Thi impli tht v n w r unrlt. Lt x n y th hilrn o u tht r ntor o v n w, rptivly. All nnt o x r l thn ll nnt o y. Pth B[u,v] ontin only vrti l thn y, n R[u,w] ontin only u n vrti no l thn y, o B[u,v] n R[u,w] hr only u, ontrition. Thu B n R r tronly inpnnt. 9

10 Rmrk: For two itint vrti v n w, th low-hih orr tll u whih pir o pth, B[,v] n R[,w], or R[,v] n B[,w], hr only th ommon omintor v n w: th ormr i v < w, th lttr i v > w. Alorithm 1 run in O(m) tim, ivn n O(1)-tim tt o th ntor-nnt rltion: or h vrtx v, w xmin th ntrin r until inin two tht llow C 1 or 2 to ppli, or runnin out o ntrin r, llowin C 3 to ppli: v mut hv t lt on ntrin r rom non-nnt; i thr i xtly on uh r, it mut (t(v), v). To tt th ntor-nnt rltion, w u th O(1)-tim tt ri ov, with th low-hih orr rvin th prorr. C 3 ppli only whn (t(v),v)i ri (ll pth rom to v ontin (t(v),v)), o B n R r r-ijoint xpt or th ri. Trjn [54] prviouly v nr-linr-tim lorithm, uquntly improv to linr tim [7], to ontrut pir o pnnin tr tht r r-ijoint xpt or th ri. Hi ontrution n not prou inpnnt pnnin tr, howvr. I w r willin to llow B n R to hr non-ri, thn w n impliy Alorithm 1 y ominin C 2 n 3 into on : C 2 : (t(v),v) i n r. St (v) = r(v) = t(v). Thorm 2.9. I T h th prnt proprty n h low-hih orr, thn T h th ilin proprty, n hn T = D. Proo. Apply Alorithm 1 to ontrut two tronly inpnnt pnnin tr B n R root t. Lt v n w ilin with ommon prnt u in T. Aum without lo o nrlity tht v < w. Th pth B[,v] voi w, o w o not omint v. Th pth B[,u] ollow y R[u,w] ontin no vrti rtr thn u n l thn w n hn voi v, o v o not omint w. Thu T h th ilin proprty. By Thorm 2.5 T = D. A qution ri y Alorithm 1 i whthr vry pir o tronly inpnnt pnnin tr i th rult o pplyin Alorithm 1 to om low-hih orr o D. Th nwr i no. Conir th rph G n it pnnin tr B n R hown in Fiur 4. Th omintor tr D o G i lt (v impli (v) = ), n B n R r tronly inpnnt. Dltin r in D rom G n rvrin r in R rult in th yl Γ hown in Fiur 4. I B n R wr ontrutil y Alorithm 1 rom om low-hih orr o D, Γ woul yli. 2.4 Vriition o Domintor Tr with Low-Hih Orr Now w turn to th prolm o vriyin tht vrtx orr i low-hih. Lt tr T pii y it prnt untion, with vrtx orr ivn numrin rom 1 to n. W n vriy tht th orrin i low-hih ollow. Contrut lit o th hilrn o h vrtx in inrin orr, y innin with mpty lit n in h vrtx v in inrin orr to th k o th lit or t(v). Do pth-irt trvrl o T to vriy tht th orrponin prorr i th m th ivn orr. Durin th trvrl, omput th iz o h vrtx. Atr th trvrl, tt tht th orr i low-hih y xminin th r ntrin h vrtx n vriyin th xitn o th on or two r n to mk 10

11 G Γ B R Fiur 4: Two tronly inpnnt pnnin tr not ontrutl y Alorithm 1. th orr low-hih, uin th numr n iz to tt th ntor-nnt rltion in O(1) tim. Thu w otin th ollowin lorithm to vriy omintor tr with low-hih orr: Alorithm 2: Vriition o Domintor Tr D with Low-Hih Orr Stp 1: Vriy tht th ivn tr D i tully tr. Stp 2: Contrut lit o th hilrn o h vrtx in inrin orr. Do pth-irt trvrl o D to vriy tht th prorr nrt y th rh i th m th ivn orr n to omput th iz o h vrtx. Stp 3: Chk tht th r tiy th prnt proprty n tht h vrtx h th on or two ntrin r n to mk th orr low-hih, uin th numr n iz to tt th ntor-nnt rltion in O(1) tim. Stp 1 n 2 tk O(n) tim; Stp 3 tk O(m) tim. W onlu thi tion with lmm out th trutur o tronly onnt urph tht w hll n in Stion 5. A urph o G i tronly onnt i thr i pth rom ny vrtx in th urph to ny othr vrtx in th urph ontinin only vrti in th urph. Lmm Lt T tr with th prnt proprty, n lt S th t o vrti o tronly onnt urph o G. Thn S onit o t o ilin in T n poily om o thir nnt in T. 11

12 G Fiur 5: Th riv rph o th low rph o Fiur 1. Proo. Lt x th nrt ommon ntor in T o th vrti in S. I x S th lmm hol: S ontin x n poily om o it nnt in T. I x S, thr r t lt two hilrn o x in T tht hv vrti in S nnt, n ll vrti in S r nnt o uh hilrn. Any pth rom non-nnt o uh hil o x to nnt mut ontin x y th prnt proprty. Thu ny uh hil mut in S. 3 Th Driv Grph Thorm 2.9 tt tht tr with th prnt proprty n low-hih orr h th ilin proprty n hn i D. It rmin to prov th onvr, nmly tht D h low-hih orr, n to vlop t lorithm to in uh n orr. W o thi in th nxt thr tion. In thi tion w introu th riv rph [55], whih mk our tk ir y in t ruin th prolm o inin low-hih orr to th o lt omintor tr. Lt T tr with th prnt proprty. (Tr T oul D or tr lim to D.) By th inition o th prnt proprty, i (v,w) i n r, th prnt t(w) o w i n ntor o v in T. Th riv r o (v,w) i null i w i n ntor o v, (v,w) othrwi, whr v = v i v = t(w), v i th ilin o w tht i n ntor o v i v t(w). Th riv rph G i th rph with vrtx t V n r t A = {(v,w) (v,w) i th non-null riv r o om r in A}. S Fiur 5. Thi inition ir rom th oriinl [55] in tht it omit l-loop (r o th orm (v,v)), whih r irrlvnt or our purpo. Lmm 3.1. Tr T h th prnt proprty in G. Proo. Eh riv r l rom vrtx to on o it hilrn or ilin. Lmm 3.2. I P i impl pth rom to v in G, thr i impl pth rom to v in G ontinin only vrti on P. Proo. By inution on th numr o r on P. Th lmm i immit i P h no r. Suppo P h t lt on r. Lt (u,v) th lt r on P. Sin P i impl, u nnot nnt o v y Corollry 2.3. Thu (u,v) h non-null riv r (u,v). Sin 12

13 u i n ntor o u in T, u i on P y Corollry 2.3. By th inution hypothi, thr i impl pth P rom to u in G ontinin only vrti on th prt o P rom to u. Ain (u,v) to thi pth iv th ir pth P. Corollry 3.3. Grph G i low rph. Proo. By Lmm 3.2, in G ll vrti r rhl rom. Lmm 3.4. I (v,w) i th non-null riv r o n r (v,w), thr i pth in G rom v to w ontinin only w n nnt o v in T. Proo. By Corollry 2.3, v omint v. Sin G i low rph, thr i pth P rom to v. Pth P ontin v. By Corollry 2.4, th prt o P rom v to v ontin only nnt o v in T. Ain (v,w) to th n o thi pth iv pth tiyin th lmm. Corollry 3.5. I G i yli, o i G. Proo. I G ontin yl, o woul G y Lmm 3.4. Lmm 3.6. Tr T h th ilin proprty in G i n only i it h th ilin proprty in G. Tht i, T i th omintor tr o G i n only i it i th omintor tr o G. Proo. Lt v n w ilin in T. Suppo thr i impl pth P rom to w in G tht voi v. Thn th pth P rom to w in G ivn y Lmm 3.2 lo voi v. Suppo thr i impl pth P rom to w in G tht voi v. Lt u th ommon prnt o v n w. By Corollry 2.4, thr i pth in G rom to u tht ontin no propr nnt o u n hn voi v. Thu ll w n to how i tht thr i pth rom u to w in G tht voi v. Th prt o P rom u to w onit o n r rom u to hil o u, ollow y qun o r rom on hil o u to nothr, no uh hil in v. Th irt uh r i lo n r o G. Conir on o th riv r (x,y) on P rom on hil o u to nothr. By Lmm 3.4 thr i pth in G rom x to y tht ontin only y n nnt o x in T, n hn voi v. Thu w n rpl h r o th prt o P rom u to w y v-voiin pth in G. Thror thr i pth in G rom to w tht voi v. Lmm 3.7. A prorr o T i low-hih on G i n only i it i low-hih on G. Proo. Lt v. Suppo th orr i low-hih on G. By Thorm 2.9, T = D. Lt v. I (t(v),v) A, thn (t(v),v) A. I (t(v),v) A, thr r r (u,v) n (w,v) uh tht u < v < w in th ivn prorr n w i not nnt o v in T. Vrtx u i not nnt o v in it i numr l thn v. Thu (u,v) n (w,v) hv non-null riv r (u,v) n (w,v), u n w r ilin o v in T, n u < v < w in th ivn prorr. Thu th orr i low-hih on G. Suppo th orr i low-hih on G. Lt v. I (t(v),v) A, thn (t(v),v) A. I (t(v),v) A, thr r r (u,v) n (w,v) in G with non-null riv r (u,v) n (w,v) uh tht u < v < w in th ivn prorr n u n v r ilin o v in T. Sin th orr i prorr, u < v < w in th prorr. Furthrmor w i not nnt o v. Thu th orr i low-hih on G. 13

14 Alorithm 3: Contrution o Driv Ar Stp 1: Lt T tr with th prnt proprty. Numr th vrti o T rom 1 to n in prorr n omput th iz o h vrtx. Intiy vrti y numr. Stp 2: Dlt h r (v,w) uh tht v i nnt o w in T. Stp 3: For h r (v,w) uh tht v = t(w), lt (v,w) it riv r. Stp 4: Forhr(v,w)uhthtv nwrunrlt, ontruttripl(t(w),v,w). For h vrtx u > 1, ontrut tripl (t(u),u,0). Stp 5: Sort th tripl in inrin lxiorphi orr y oin thr-p rix ort. Stp 6: Pro th tripl in inrin orr. To pro tripl o th orm(t(u),u,0), t x = u, whr x i lol vril. To pro tripl o th orm (t(w),v,w), lt (x,w) th riv r o (v,w). W n in th riv r in O(m) tim with Alorithm 3. Thorm 3.8. Alorithm 3 i orrt. Proo. Stp 2 n 3 orrtly hnl th r (v,w) uh tht v n w r rlt in T. Conir n r (v,w) uh tht v n w r unrlt in T. Lt (v,w) th riv r o (v,w). Thn t(v ) = t(w) n v v, o tripl (t(v ),v,0) pr (t(w),v,w) on th tripl r ort lxiorphilly. Suppo thr i tripl (t(u),u,0) ollowin (t(v ),v,0) ut prin (t(w),v,w). Thn t(u) = t(v ) n v < u v. But vrti r numr in prorr, o u mut nnt o v, ontrition. Hn thr i no uh tripl (t(u),u,0), whih impli x = v whn (t(w),v,w) i pro, o Stp 6 orrtly omput th riv r o (v,w). 4 Ruil Flow Grph A ruil low rph [27, 52] i on in whih vry tronly onnt urph S h inl ntry vrtx v uh vry pth rom to vrtx in S ontin v. Thr r mny quivlnt hrtriztion o ruil low rph [52], n thr r lorithm to tt ruiility in nr-linr[52] n truly linr[7] tim. On notion o trutur prorm i tht it low rph i ruil. Ruiility implii mny omputtion, lthouh not th omputtion o omintor, r w n tll. A low rph i ruil i n only i it om yli whn vry r (v,w) uh tht w omint v i lt [52]. Dltion o uh r o not hn th omintor tr, in no uh r n on impl pth rom. Dltin uh r thu ru th prolm o inin low-hih orr on ruil low rph to th m prolm on n yli rph. Suh rph h topoloil orr ( totl orr uh tht i (x,y) i n r, x i orr or y) [36]. 14

15 Th ollowin lmm provi wy to in th r n to tiy th low-hih proprty on ruil rph: Lmm 4.1. Lt T tr with th prnt proprty, n lt G th orrponin riv rph. Suppo T h th ilin proprty in G (or in G ). I v, thn (t(v),v) A (n in A ) or v h in-r t lt two in G. Proo. Sin T h th ilin proprty, T = D. Suppo th lmm i l or om v. Sin v i rhl rom y impl pth, thr i n r (u,v) uh tht v o not omint u. Lt (u,v) th riv r o (u,v). Sin th lmm i l or v, u (v), n thr i no othr r (w,v) with riv r (w,v) uh tht w u. But thn u omint v, ontritin th ilin proprty. Lmm 4.1 hol or ritrry rph. For ruil rph, th onition in Lmm 4.1 i not only nry ut uiint or tr T with th prnt proprty to hv th ilin proprty: Lmm 4.2. Suppo G i ruil. Lt T tr with th prnt proprty, n lt G th orrponin riv rph. Thn T h th ilin proprty i, or ll vrti v, (t(v),v) A or v h in-r t lt two in G. Proo. Dlt ll r (x,y) uh tht y omint x. Thi o not hn th omintor, nit mk thrphyli. Suppoth lmm i l. Lt u, v pir oilin uh tht u omint v, with v minimum in om topoloil orr (n orr uh tht i (x,y) i n r, x i orr or y). Thr i pth rom to t(v) tht voi u y Corollry 2.4. Thu i (t(v),v) A, u o not omint v, ontrition. I (t(v),v) A, thr i n r (x,v) in G with riv r (x,v) uh tht x i ilin o u n v ut x u. By th hoi o v, u o not omint x, o it nnot omint v, ontrition. 4.1 Low-hih orr on ruil low rph To in low-hih orr on ruil rph, w lt r (x,y) uh tht y omint x, mkin th rph yli. W thn ontrut or h vrtx n orr lit o it hilrn in D y proin th vrti othr thn in topoloil orr n inrtin h vrtx into it t o ilin in poition trmin y th r or r who xitn i urnt y Lmm 4.1. Atr uilin th orr lit o hilrn, w otin low-hih orr y oin pth-irt trvrl o D. Thi pproh iv u Alorithm 4. Fiur 6 how how thi lorithm work. Thorm 4.3. Alorithm 4 i orrt. Proo. Atr Stp 1, G i yli n vry r h riv r. I (v,w) i n r with riv r (v,w), v i n ntor o v, o thr i pth rom v to v. It ollow tht whn vrtx v i pro in Stp 2, x i in C((v)) or vry riv r (x,v) uh tht x (v). By th onition in Lmm 4.2, ithr ((v),v) i n r or thr r riv r (u,v) n (w,v) uh tht u w. Thu th inrtion o v into C(v) will u. Stp 4 prou prorr o T. Lt v ny vrtx othr thn. I ((v),v) A, thr r 15

16 Alorithm 4: Contrution o Low-Hih Orr on Ruil Flow Grph Stp 1: Dlt vry r (v,w) uh tht w i n ntor o v in D, n in th riv r (v,w) o h rminin r (v,w) with rpt to D. Stp 2: For h vrtx v, initiliz it lit o hilrn C(v) to mpty. Apply th ollowin tp to h vrtx v in topoloil orr on G (or on G ): I ((v),v) A, inrt v nywhr in C((v)). Othrwi, in riv r (u,v) n (w,v) uh tht u w. Inrt v jut or u in C((v)) i w i or u in C((v)), jut tr u othrwi. Stp 3: Do pth-irt trvrl o D, viitin th hilrn o h vrtx v in thir orr in C(v). Numr th vrti rom 1 to n thy r viit. Th rultin orr i low-hih on G. r (u,v) n (w,v) in A with riv r (u,v) n (w,v), rptivly, uh tht v i orr twn u n w, n u n w r ilin in T. Sin th orr i prorr, v i orr twn u n w. Thu th orr i low-hih. Whn uin Alorithm 4 in omintion with th omintor rtiition lorithm (Alorithm 2), w n omit Stp 3 in Alorithm 4, in Stp 2 in Alorithm 2 o th m tr trvrl n numrin. Tht i, it ui to rprnt th low-hih orr y th orr lit o hilrn prou y Stp 2. Rmrk: I w wnt to vriy th omintor tr o ruil rph ut r not intrt in ontrutin low-hih orr, w n o th vriition uin Lmm 4.2. Givn D, w vriy tht D i tr with th prnt proprty in Stion 2, omput th riv rph in Stion 3 (vriyin th orrtn o th riv rph uin ntor-nnt tt), n vriy th onition in Lmm 4.2. Th totl tim or vriition y thi mtho i O(m). It i y to implmnt mot o Alorithm 4 to run in O(m) tim. In Stp 1 w in th r to lt uin n O(1)-tim ntor-nnt tt iu prviouly, n w in th riv r uin Alorithm 3. In Stp 2, inin topoloil orr tk O(m) tim uin ithr uiv ltion o vrti o in-r zro [36, 37] or pth-irt rh [51]. Finin th n r in Stp 2 tk O(1) tim pr r, or totl o O(m) tim. Stp 3 tk O(m) tim. Th only hr prt i ontrutin th lit o hilrn in Stp O-lin ynmi lit mintnn For ontrutin th lit o hilrn, w n t trutur tht mintin lit ujt to inrtion n orr quri: ivn x n y in th lit, whih our irt? Thi i th ynmi lit mintnn prolm. Thr r olution to thi prolm tht upport 16

17 (,) (,,) (,,,) (,,,,) (,,,,,) (,,,,,,) (,,,,,,,) Fiur 6: Computtion o low-hih orr trtin with n yli riv rph. Vrti r pro in th topoloil orr (,,,,,,,); th vrtx pro t h pplition o Stp 2 i hown ill. inrtion n orr tt in O(1) tim, ithr mortiz or wort- [5, 14]. Unortuntly, th olution r rthr omplit, pilly tho with n O(1) wort- tim oun. Fortuntly, w only n olution to pil o ynmi lit mintnn, in whih thr r no ltion n, mor importntly, th qun o oprtion i ivn o-lin in n pproprit n. For thi vrion o th prolm thr i impl olution, whih w now ri. Givn n initil lit with no itm, w wnt to prorm o-lin n intrmix qun o th ollowin thr kin o intrution, n thn numr th itm in th inl lit onutivly rom 1. tr(x,y): Rturn tru i x i tr y in th urrnt lit, l othrwi. Itm x n y mut in th urrnt lit. inrt(x): Inrt x nywhr in th lit. 17

18 inrt(x,y,tt): I tt i tru, inrt x jut tr y; othrwi, inrt x jut or y. Itm y ut not x mut in th urrnt lit; tt i Booln omintion o tru n ix numr o tr quri. Alorithm 5 iv impl n iint olution to thi prolm. Alorithm 5: O-Lin Exution o Squn o Inrt n Atr Intrution Stp 1 (o-lin): Contrut root tr with on non-root vrtx or h itm vr in th lit, n root. Th prnt o itm x i th root i thr i n intrution inrt(x), or y i thr i n intrution inrt(x,y,tt). For h no x, lt iz(x) th numr o no in it utr. Stp 2 (on-lin): Exut th intrution in qun whil mintinin n intrvl [i,j] or th root n or h itm urrntly in th lit. Hr i n j r intr uh tht i j. Th intrvl will ijoint. Initilly th root h intrvl [0, iz(root)]. Givn qury tr(x, y), nwr tru i th intrvl or x ollow tht o y, l othrwi. Givn n intrution inrt(x) uh tht th root h intrvl [i,j], rpl th intrvl or th root y [i,j iz(x)] n iv x th intrvl [j iz(x)+1,j]. (Thi orrpon to inrtin x irt in th lit.) Givn n intrution inrt(x,y,tt) uh tht y h intrvl [i,j], i tt i tru rpl th intrvl or y y [i,j iz(x)] n iv x th intrvl [j iz(x) + 1,j]; i tt i l iv x th intrvl [i,i + iz(x) 1] n rpl th intrvl or y y [i+iz(x),j]. (Th ormr orrpon to inrtin x tr y, th lttr to inrtin x or y.) Stp 3: Atr Stp 2, th root h intrvl [0,0], n h itm x h n intrvl [i,i] or om i > 0, i itint or h x. I itm x h intrvl [i,i], in numr i to x. Thi numr th itm onutivly rom 1 in lit orr. Thorm 4.4. Alorithm 5 i orrt. Proo. W n viw Stp 2 trtin with th tr ontrut in Stp 1, uttin th r (root,x) whn inrt(x) our, n uttin th r (y,x) whn inrt(x,y,tt) our. Th itm urrntly in th lit r xtly th root o th tr into whih th initil tr h n ut, xluin th initil root. Eh r ut onnt root with hil. Th intrvl o root ontin xtly mny intr th numr o vrti in it urrnt tr, plu on i it i th initil root. It ollow tht h intrvl [i,j] h i j, whih urnt tht th implmnttion i orrt. W u Alorithm 5 to implmnt Stp 2 o Alorithm 4 ollow. For h v, w ontrut qun o th oprtion tht vrti to C(v), ollow: For h ition o vrtx x to C(v) in n ritrry poition, w ontrut n oprtion inrt(x). For h ition o vrtx x or or tr nothr vrtx y pnin on whthr tt i tru, w 18

19 ontrut n oprtion inrt(x,y,tt). Eh uh tt i inl tr qury; thr r no othr quri. W o th lit oprtion uin Alorithm 5. At th n o Alorithm 5, th itm in C(v) will numr onutivly in lit orr. U o Alorithm 5 limint th n to u omplit on-lin ynmi lit orr lorithm. Alorithm 5 my hv othr pplition wll. 5 Low-Hih Orr rom Loop Ntin Inormtion In thi tion w xtn th low-hih orrin lorithm o Stion 4 to ritrry low rph. To o o w mut ovrom th irulrity u y yl. Rmlinn [46] h hown how to ru th prolm o omputin omintor on n ritrry low rph to omputin omintor on n yli rph. Hi rution run in O(mα m/n (n)) tim n n improv to run in O(m) tim: it i n xtnion o th omputtion o loop ntin ort tht w iu low. Rmlinn rution hn th rph n hn t th low-hih orr. For thi ron w o not u hi rution ut our ontrution irtly on th loop ntin ort. Th rouh i i to rptly ontrt tronly onnt urph to inl vrti until no yl xit. Thn w pply th lorithm o Stion 4 to th rultin yli rph, ut whn vrtx orrponin to ontrt urph i to pro, w xpn th urph n pro it vrti, rurivly xpnin h uh vrtx tht itl orrpon to ontrt urph. To otin qun o urph to ontrut, w u th notion o loop ntin ort. Thr r vrl wy to in uh ort [26, 39, 46, 49, 50, 54]. Th on w u w irt prnt y Trjn [54] n ltr riovr y Hvlk [26]. It i prtiulr o mor nrl inition [39, 46]. Lt F th pnnin tr nrt y pth-irt rh o G trtin rom, with (v) th prnt o vrtx v in F. Evry yl in G ontin n r rom nnt to n ntor in F [51]. Suh n r i ll k r. Numr th vrti rom 1 to n in rvr potorr with rpt to th rh, n intiy vrti y numr. (Potorr, lo known inihin orr, i th orr in whih th rh inih it vrtx viit. S [51].) An r i k r i n only i it l rom lrr to mllr vrtx [51]. Dltin th k r mk th rph yli, n mk th vrtx orr ivn y th numrin topoloil [51]. Th h h(v) o vrtx v i th mximum propr ntor u o v in F uh tht thr i pth romv to u ontinin only nnt o u in F; i thr i no uh u, h(v) = null. Th h in loop ntin ort H: h(v) i th prnt o v in H. Grph G i yli i n only i ll h r null. For h vrtx v tht i not l in H, th nnt o v in H inu non-trivil tronly onnt urph o G, ll loop. Bu H i tr, ny two loop r ithr ijoint or on ontin th othr. Fiur 7 illutrt th onpt. In nrl H pn on F. Grph G i ruil i n only i vry F in th m H. Fort H in ontrtion qun ollow. For h vrtx v in rin orr, i v i not l in H, ontrt th urph inu y v n ll it hilrn into inl vrtx v. Thi urph i th intrvl o v. Th intrvl o v i tronly onnt (jut or it i ontrt), n ll it vrti r nnt o v in F. I ll r lvin v r lt rom it intrvl, th intrvl om yli, n v i th uniqu vrtx with 19

20 h i j k l G [1] [1] [2] [2] [3] [3] [10] [4] [10] [4] i[13] [11] h[6] [5] i[13] [11] h[6] [5] j[12] l[8] [7] j[12] l[8] [7] k[9] k[9] F H Fiur 7: A rph, pth-irt pnnin tr (tr r r oli, non-tr r r h) with vrti numr in rvr potorr(in rkt), n th orrponin loop ntin ort. Thr r iv loop, with h,,, h,. 20

21 no outoin r. Thu, i on trt rom ny vrtx in th intrvl othr thn v n ollow ny pth in th intrvl, on vntully rh v without rptin vrtx. S Fiur 8. Not tht intrvl r in in th ontrt rph, whr loop r in in th oriinl rph. Mny pplition o omintor n loop ntin ort wll [7, 17, 35, 46, 54]. Suh ort n omput in O(mα m/n (n)) tim ([54], ltr riovr [45]) or y mor-omplit lorithm in O(m) tim [7]. Th lorithm rquir l mhinry thn th t lorithm or inin omintor ( [7]), ut thy r not ntirly trihtorwr. It i y to xtn th lorithm to in, or h vrtx v tht i not root o H, n outoin r rom v in th intrvl o h(v), lon with th orrponin oriinl r; in, th ort-ontrution lorithm pro y trvrin uh outoin r kwr, oin o y xminin th orrponin oriinl r. W hll um tht loop ntin ort, oit outoin r, n th orrponin oriinl r r vill; i not, thy n omput y on o th it lorithm. Lmm 5.1. Lt u vrtx, n lt (v,w) n r uh tht w i nnt o u in H. Thn v i nnt o u in H i n only i v u. Proo. Sin ll nnt o u in H r lo nnt o u in F, w > u. I v > w, thn (v,w) i k r n v i nnt o w in H n hn nnt o u in H. Suppo v < w. I v u, v i nnt o u in F n hn nnt o u in H. Suppo v < u. Thn v i not nnt o u in F, n hn nnot nnt o u in H. Lmm 5.2. Lt T tr with th prnt proprty. Lt u, n lt (v,w) n r uh tht w ut not v i nnt o u in H. Thn w i u or ilin o u in T. Proo. By th prnt proprty, i x i n ntor o y in T, thn x omint y. Thi impli x i n ntor o y in F, o x < y. Th t S o nnt o u in H inu tronly onnt urph o G. By Lmm 2.10, S onit o t o ilin n poily om o thir nnt in T. Sin v i not uh nnt, th prnt proprty impli tht w i on o th ilin. Sin u i minimum in S, u i lo on o th ilin. To urnt th xitn o r to tiy th low-hih proprty, w n n nlou o Lmm 4.1. For h vrtx u, w not y ( (u),u) th riv r o ((u),u), whih i non-null u th pth in F rom to (u) voi u. Lmm 5.3. Lt T tr with th prnt proprty. Thn T h th ilin proprty i n only i, or h u, (u) = t(u) or thr i n r (y,w) with riv r (y,w) uh tht w i nnt o u in H, y < u, n y (u). Proo. I x omint z, x mut n ntor o z in F, o x z. Thu i r (x,v) h riv r (x,v), x x. Suppo T h th ilin proprty. Lt u uh tht (u) t(u). Thn (u) i ilin o u in T. By th ilin proprty, (u) o not omint u, o thr i pth rom to u voiin (u). Lt (y,w) th irt r on thi pth with w nnt o u in H. By Lmm 5.1, y < u w, o w o not omint y, whih mn tht (y,w) h riv r (y,w). Sin y i on th pth rom to w, y (u). Thu T tii th onition in th lmm. 21

22 [1] [1] [2] [2] [3] [3] [10] [4] [10] [4] i[13] [11] h[6] [5] i[13] [11] h[6] [5] j[12] l[8] [7] l[8] [7] k[9] k[9] [1] [1] [2] [2] [3] [3] [10] h[6] [4] [5] [10] h[6] [4] [5] l[8] [7] k[9] [1] [1] [2] [2] [3] [10] [4] [4] Fiur 8: Contrtion qun n intrvl (tronly onnt urph) orrponin to th loop ntin ort in Fiur 7. 22

23 Convrly, uppo T tii th onition in th lmm. Suppo thr r ilin v, u uh tht v omint u. Choo uh pir with u minimum. By Corollry 2.4 thr i pth rom to t(u) tht voi v. I (u) = t(u) thn v o not omint u. I (u) t(u), y th onition in th lmm thr i n r (y,w) with riv r (y,w) uh tht w i nnt o u in H, y < u, n y (u). By Lmm 5.2, w i u or ilin o u in T. Thu y i t(u) or ilin o u n v in T. I y = t(u), v o not omint u. I y i ilin o u, thn v o not omint y y th hoi o u, whih impli tht v o not omint u. Thu in ny v o not omint u, o th ilin proprty hol. Alorithm 6, our low-hih orrin lorithm uin loop ntin ort, i lik Alorithm 4; it uil or h vrtx n orr lit o it hilrn in th omintor tr D n thn o pth-irt trvrl o D to in low-hih orr. It inrt th vrti in inrin orr into th lit o hilrn in wy tht iv th low-hih proprty. A prt o th inrtion pro, it in, or h vrtx u, riv r ll th pivot r o u. I th pivot r o u i not ((u),u), it lo in nothr riv r ll th tt r o u. O th pivot n tt r or u, on i ( (u),u), n th othr i th riv r (y,w) o n r (y,w) tiyin Lmm 5.3. Th tt r or u i in jut or u i inrt into lit o hilrn. Th pivot r or u i in ithr jut or u i inrt or rlir, whn yl ontinin u i (impliitly) xpn. Th pivot n tt r trmin whr to inrt u in it lit o ilin. An xmpl i hown in Fiur 9. Thorm 5.4. Alorithm 6 i orrt. Proo. Conir th hoi o pivot r (x,v) or vrtx u. W lim tht (x,v) i th riv r o n oriinl r (x,v) uh tht x < u n v i nnt o u in H. Thi i immit i (x,v) i hon in Stp 2. In Stp 2, v i nnt o z in H n x < u < z, o th hoi o (x,v) th pivot r or z tii th lim. Alo in Stp 2, h oriinl r (p,q) orrponin to n r on P ntrin vrtx x i i uh tht q i nnt o x i in H ut p i not. Lmm 5.1 impli p < x i, o th hoi o (p,q) th pivot r or x i tii th lim. I (x,v) = ((u),u) in Stp 2, u will tiy th low-hih proprty. Conir th proin o vrtx u uh tht it pivot r (x,v) ((u),u). I (x,v) = ( (u),u), (u) (u),othrinr(y,w)tiyinlmm5.3,nthlorithmwilluully hoo th riv r (y,w) o uh n r th tt r or u. On o v n w i u; th othr i u or ilin o u in D y Lmm 5.2. Thu h o x n y i (u) or ilin o u in D. Furthrmor oth r l thn u, o thy hv lry n inrt into lit o hilrn whn u i out to inrt. Hn th inrtion o u i wll in. I v = w = u, th inrtion poition o u urnt tht u tii th low-hih proprty. It rmin to how tht u tii th low-hih proprty vn i on o v n w i not u. Th hoi o itionl pivot r in Stp 2 i wht urnt thi. I (u) = (u), u will tiy th low-hih proprty, in thn it o not mttr whr u i inrt into C((u)). Thu uppo (u) (u). Thn (u) i ilin o u in D. Conir th r (x,v) in Stp 2 (tr th onitionl wp in Stp 2 mk (x,v) ( (u),u)). Sin v u, v i ilin o u in D, n z i v or ilin o v in D, o z i ilin o u in D. Ar (x,v) om th pivot r o z. 23

24 Alorithm 6: Contrution o Low-Hih Orr uin Loop Ntin Fort Lt F pth-irt pnnin tr o G, with vrti numr in rvr potorr n intii y numr. Lt H th loop ntin ort in y F, with oit outoin r n orrponin oriinl r. Stp 1: Contrut th riv rph G with rpt to th omintor tr D. Stp 2: For h vrtx u, initiliz it lit o hilrn C(u) to mpty. Apply th ollowin tp to h vrtx u in inrin orr: Stp 2: Lt (x,v) th pivot r o u; i u o not yt hv pivot r, lt (x,v) = ( (u),u). Stp 2: I (x,v) = ((u),u), inrt u irt on C((u)), ompltin Stp 2. Othrwi, pro to Stp 2. Stp 2: I (x,v) ( (u),u), lt th tt r (y,w) o u ( (u),u); othrwi, lt th tt r o u th riv r (y,w) o n r (y,w) with y < u, w nnt o u in H, n y (u). Stp 2: I x = (u), inrt u irt in C((u)). Othrwi, inrt u jut or x in C((u)) i y = (u) or y pr x in C((u)), jut tr x othrwi. Swp (x,v) n (y,w) i nry o tht (y,w) = ( (u),u). I v = u, thi omplt Stp 2; othrwi, pro to Stp 2. Stp 2: Lt z th hil o u in H tht i n ntor o v. Mk (x,v) th pivot roz. FinpthP ovrti z = x 0,x 1,...,x k = uinthintrvl ou y trtin t z n ollowin outoin r in th intrvl until rhin u. For h vrtx x i othr thn z n u on P, in th oriinl r (p,q) orrponin to th r ntrin x i on P, n lt th pivot r o x i th riv r (p,q) o (p,q). Stp 3: Do pth-irt trvrl o D, viitin th hilrn o h vrtx v in thir orr in C(v). Numr th vrti rom 1 to n thy r viit. Th rultin orr i low-hih on G. 24

25 y x (,,,) h v w h w y x (,,,h,,) l v k [1] [1] [13] [2] [12] [7] [3] [6] [2] [7] h[8] l[9] k[10] [11] [12] [13] [11] k[10] h[8] l[9] i[4] j[5] [3] i[4] j[5] [6] D G Fiur 9: Computtion o low-hih orr (ini th rkt) o th low rph o Fiur 7, uin th intrvl o Fiur 8. Whn vrtx i pro, Alorithm 6 t (,) th pivot r in Stp 2 n (,) th tt r in Stp 2. In Stp 2 it in th pth P = (,h,) (hown with ol r) ini th intrvl o n t (,) th pivot r o h, whih i th oriinl r orrponin to (, h). Whn vrtx h i pro, Alorithm 6 t (,h) th tt r in Stp 2. In Stp 2 it in th pth P = (,l,k,h) (hown with ol r) ini th intrvl o h n t (,l) n (l,k) th pivot r o l n k rptivly. 25

26 W lim tht tr ll inrtion into C((u)), u i twn (u) n z. To prov th lim, w onir thr. I x = (u) n (x,v) i th pivot r o u, thn u will inrt irt in C((u)), in ront o (u), n ltr z will inrt irt, o u i twn (u) n z. I x = (u) n (x,v) i th tt r o u, thn ( (u),u) i th pivot r o u, o u will inrt in ront o (u) n ltr z will inrt irt, o in u i twn (u) n z. Th thir i x (u). Thn u will inrt twn (u) n x. Sin (x,v) i th pivot r o z, z will vntully inrt nxt to x, lvin u twn (u) n z in thi wll. Now onir th vrti z = x 0,x 1,...,x k = u on P. Lt S i th t o nnt o x i in H, n lt S th union o th S i. Lt (x,v) th oriinl r who riv r i (x,v). Thr r u-voiin pth rom to ll vrti in S, vi th pth in F rom to x ollow y (x,v) ollow y pth mon vrti in S. Thu u omint no vrtx in S. But (u) mut omint ll vrti in S, in othrwi thr woul pth rom to u voiin (u). Thu ll vrti in S r nnt o (u) in D. W lim tht S onit o t o ilin o u in D n poily om o thir nnt. Thi i tru o S 0, in y Lmm 2.10 S 0 onit o t o ilin in D n poily om o thir nnt, x 0 mut on o th ilin in it i minimum in S 0, n z = x 0 i ilin o u. Suppo th lim i l. Lt x i minimum uh tht om vrtx in S i i nnt o ilin o u in D ut tht ilin i not in S. By Lmm 2.10, S i onit o t o ilin in D n poily om o thir nnt. Lt (p,q) th oriinl r orrponin to th r ntrin x i on P. Thn p i in S i 1. By th hoi o x i, p i nnt in D o ilin o u tht i in S. Eithr q i nnt o in D, in whih ll vrti in S i r nnt o in D, or q i not nnt o. In th lttr, th prnt proprty impli tht q i ilin o n hn o u, n S i onit o t o ilin o n hn o u, n poily om o thir nnt. W onlu tht thr i no uh x i, vriyin th lim. Finlly w lim tht ll o th ilin o u in S r on th m i o u in C((u)) z = x 0. Thi impli tht u h th low-hih proprty, u u will twn (u) n p, whr (p,u) i th riv r o th oriinl r (p,u) orrponin to th r ntrin u on P. Suppo th lim i l or om ilin o u in S. Lt th minimum uh ilin. Lt th pivot r o th riv r (,) o oriinl r (,). Eithr (,) = (x,v), or n r in S. In th ormr, i x = (u), will inrt t th ront o C((u)) omtim tr z w inrt t th ront, o will on th m i o u z; i x i ilin o u, will inrt nxt to x omtim tr z w inrt nxt to x, o in will on th m i o u z. In th lttr, y Lmm 5.2 i ilin o n hn o u, n in i in S, i lo ilin o u. Furthrmor <, o i on th m i o u z y th hoi o. Th inrtion o nxt to put it on th m i wll. Thorm 5.5. A tr with th prnt proprty h th ilin proprty, n hn i th omintor tr, i n only i it h low-hih orr with rpt to G. Proo. Immit rom Thorm 2.9 n 5.4. Exluin th omputtion o th loop ntin ort n oit inormtion, it i trihtorwr to implmnt Alorithm 6 to run in O(m) tim. To in tt r in Stp 26

27 2, w omput, or h vrtx u, two riv r (x,v) n (y,w) with v n w nnt o u in H, x y, n x n y minimum, i two uh r xit; othrwi, (u) = (u). W n o thi y proin th vrti o H ottom-up (rom lv to root). Thi tk O(m) tim. W o th inrtion into th lit o hilrn uin th o-lin mtho vlop in Stion 4.2. W n i w wih run Alorithm 6 on th riv rph int o th oriinl rph. Thi i pplin i th loop ntin ort i not ivn ut mut omput, in it implii th trutur o th tronly onnt urph n implii Stp 2. In th riv rph, vry r l rom vrtx to hil or ilin in D, o th vrtx t o vry tronly onnt urph onit o t o ilin in D. Furthrmor vry uh urph with t lt two vrti h t lt two ntry vrti rom. To run Alorithm 6 on th riv rph, w in y omputin th riv rph G. Thn w o pth-irt rh o G to nrt pth-irt pnnin tr F n to numr th vrti in rvr prorr. Nxt w ontrut th loop ntin ort o G in y F, lon with oit in-tr n orrponin oriinl r. Finlly, w run Stp 2 n 3, inorin th itintion twn oriinl r n riv r: in G, vry r i it own riv r. Rmrk: Iwwnttovriythomintortronritrryrphutrnotintrt in ontrutin low-hih orr, w n o th vriition uin Lmm 5.3. Givn D, w vriy tht D i tr with th prnt proprty in Stion 2, omput th riv rph in Stion 3, (vriyin th orrtn o th riv rph uin ntor-nnt tt), ontrut loop ntin ort H (or ithr th oriinl or riv rph) n vriy th onition in Lmm 5.3, whih n on y proin th vrti o H rom lv to root. Th totl tim or vriition y thi mtho i O(m). Thi mtho um tht th loop ntin ort i orrt. To vriy tht it i, w vriy or h u tht ll it hilrn in H r nnt o u in F, tht th intrvl o u i tronly onnt, n tht it om yli whn u i lt. Thi lo tk O(m) tim. 6 Low-Hih Orr rom Smi-Domintor 6.1 Low-hih orr rom inpnnt pnnin tr In thi tion w vlop n ltrntiv lorithm or inin low-hih orr in n ritrry rph. Int o uin loop ntin inormtion, th lorithm u mi-omintor inormtion. Thi inormtion i omput y th t lorithm or inin omintor[2, 7, 38], mkin it y to xtn th lorithm to in not only th omintor tr, ut lowhih orr wll. Th lorithm h two tp. Th irt tp u th mi-omintor inormtion to uil two inpnnt pnnin tr. Th on tp u two inpnnt pnnin tr to in low-hih orr. Sin th two tp r inpnnt n th on tp rquir no nw i n i muh ir to prov orrt, w in with it. Lt B n R two inpnnt pnnin tr: or h vrtx v, th pth in B n R rom to v hv only th omintor o v in ommon. W not y (v) n r(v) th prnt o v in B n R, rptivly. W in y lihtly moiyin B n R: or h 27

28 vrtx v, i ((v),v) A w rpl (v) n r(v) y (v). Thn w in th riv r o th r in B n R. Lt G th urph who vrtx t i V n who r r th riv r o tho in B n R. Lt B n R th urph o G in y th riv r o B n R, rptivly. S Fiur 10. Lmm 6.1. Surph B n R r inpnnt pnnin tr in G. Proo. Any pth rom to v in B or R ontin (v), o i ((v),v) A, rplin (v) n r(v) y (v) lv B n R tr. Furthrmor uh rplmnt only limint vrti rom pth in B n R, o B n R rmin inpnnt. Similrly, or ny vrtx v, v o not omint (v) or r(v), o vry r in B n R h riv r. I v, (v) (r (v)) i on ny pth rom to (v) (r(v)), n hn on th pth rom to v in B (R). It ollow tht rplin ll th r in B n R y thir riv r prou two inpnnt pnnin tr in G. A in Alorithm 4 n 6, th tr- low-hih orrin lorithm, Alorithm 7, uil orr lit o th hilrn in D o h vrtx v. It u th r in B n R to trmin th inrtion poition. It inrt vrti in n orr trmin rurivly, unlik th itrtiv orr u in Alorithm 4 n 6 (topoloil orr in Alorithm 4, rvr potorr in Alorithm 6). Fiur 11 illutrt how thi lorithm work. Thorm 6.2. Th vrtx orr omput y Alorithm 7 i low-hih on G n hn on G. Proo. W lim tht Stp 4 mintin th invrint tht (i) B n R r inpnnt pnnin tr root t ; (ii) or vry v, ithr (v) = r (v) = (v), or (v), r (v), n (v) r ll itint; n (iii) h r in B or R i it own riv r. Lt u nonl o D ll o who hilrn in D r lv o D. Lt X th t o hilrn o u in D, nlt Y thut ox thtonit othvrti y uhtht (y) = r (y) = (y) = u. Th invrint impli tht h vrtx in Y h in-r 1 in G n h vrtx in X Y h in-r 2 in G. Sin h r in B or R i it own riv r, h r in G lvin vrtx in X ntr vrtx in X. Sin t lt on r rom u ntr X, thr mut vrtx v in X who in-r in G x it out-r in G. W lim tht v n lt in Stp 4. In, v n lt i i l in oth B n R. I not, thn v mut hv in-r 2 n out-r 1 in G. But thn v mut l in ithr B or R ; i not, it ommon hil w in B n R woul violt th invrint; B n R woul not inpnnt pnnin tr, in v i n ntor o w in oth B n R, ut v n w r ilin in D. Hn v n lt in thi lo. It ollow tht rplin (w) y (v) n ltin v prrv th invrint. Finlly, w lim tht th omput orr i low-hih. Thi i immit i G h two vrti. Suppo thi i tru i G h k 2 vrti. Lt G hv k + 1 vrti n lt v th vrtx hon or ltion. Th inrtion poition o v urnt tht v h th low-hih proprty. All vrti in G tr th ltion o v hv th low-hih proprty in th nw G y th inution hypothi, o thy hv th low-hih proprty in th ol G with th poil xption o w, on o who inomin r ir in th ol n th nw G. Suppo (w) ir; th rumnt i ymmtri i r (w) ir. By th rumnt ov, 28