Er -Orrin Ulrik Brn Dprmn o Compur & Inormion Sin, Univriy o Konnz ulrik.rn@uni-konnz. Ar. Givn ionn rp G =(V,E) wi{, } E, n -orrin i n orrin v 1,...,v n o V u = v 1, = v n,n vry or vrx o ir-numr n lowr-numr nior. Prviou linr-im -orrin lorim r on prproin p in wi p-ir r i u o ompu lowpoin. T ul orrin i rmin only in on p ovr rp. W prn nw, inrmnl lorim o no rquir lowpoin inormion n, rouou inl p-ir rvrl, minin n -orrin o ionn omponn o {, } in rvr urp. 1 Inrouion T -orrin o vri in n unir rp i unmnl ool or mny rp lorim,.. in plnriy in, rp rwin, or m rouin. I i loly rl o or imporn onp u ionniviy, r ompoiion or ipolr orinion. T ir linr-im lorim or -orrin vri o ionn rp i u o Evn n Trjn [2, 3]. Er [1] prn lily implr lorim, wi i urr implii y Trjn [7]. All lorim, owvr, prpro rp uin p-ir r, nilly o ompu lowpoin wi in urn rmin n (implii) opn r ompoiion. A on rvrl i rquir o ompu ul -orrin. W prn nw lorim voi ompuion o lowpoin n u rquir only inl p ovr rp. I ppr o mor inuiiv, xpliily ompu n opn r ompoiion n ipolr orinion on ly, n i i rou in ppliion o non-ionn rp. Mo noly, i n opp r ny rvrl n will rurn n -orrin o ionn omponn oninin {, } in w n rvr o rp unil n. T lorim n u uiliz in lzy vluion, or inn wn only orrin o w vri i rquir, n on impliily rprn rp r oly o rvr mor n on. T ppr i orniz ollow. In S. 2 w rll i iniion n orrponn wn ionniviy, -orrin n rl onp. Sion 3 i ri rviw o p-ir r n lowpoin ompuion. T nw lorim i vlop in S. 4 n iu in S. 5. R. Mörin n R. Rmn (E.): ESA 2002, LNCS 2461, pp. 247 256, 2002. Sprinr-Vrl Brlin Hilr 2002
248 Ulrik Brn 2 Prliminri W onir only unir n impl rp G =(V,E). A (impl) p P =(v 0, 1,v 1,..., k,v k )ing i n lrnin qun o vri V (P )= {v 0,...,v k } V n E(P )={ 1,..., k } E u {v i 1,v i } = i, 1 i k, nv i = v j impli i = j or {i, j} = {0,k}. Tln o P i k. A p i ll lo i v 0 = v k,nopn orwi. ArpG i onn i vry pir o vri i link y p, n i i ionn i i rmin onn r ny vrx i rmov rom G. W r inr in orrin vri o ionn rp in wy urn orwr n kwr onnn. Drminin u n orrin i n nil prproin p in mny ppliion inluin plnriy in, rouin, n rp rwin. Diniion 1 (-orrin [5]). L G =(V,E) ionn rp n V.Anorrin = v 1,v 2,...,v n = o vri o G i ll n -orrin, i or ll vri v j, 1 <j<n,rxi1 i<j<k n u {v i,v j }, {v j,v k } E. Lmm 1 ([5]). A rp G =(V,E) i ionn i n only i, or {, } E, i n -orrin. Svrl linr-im lorim or ompuin -orrin o ionn rp r vill [2, 1, 7]. All o r on priion o rp ino orin p. An orinion in irion o in o. An orinion (lo ll ipolr orinion) o rp G i n orinion u rulin ir rp i yli n n r only our n ink, rpivly. T ollowin lmm i olklor. Lmm 2. A rp G =(V,E) n -orinion i n only i i n -orrin. T n rnorm ino or in linr im. Proo. An -orrin i oin rom n -orinion y opoloil orrin, n n -orinion i oin rom n -orrin y orinin rom lowr-numr o ir-numr vri. A qun D =(P 0,...,P r ) o (opn) p inuin rp G i =(V i,e i ) wi V i = i j=0 V (P j)ne i = i j=0 E(P j), 0 i r, illn(opn) r ompoiion, ie(p 0 ),...,E(P r ) i priion o E n or P i = (v 0, 1,v 1,..., k,v k ), 1 i r, wv{v 0,v k } V i 1 n {v 1,...,v k 1 } V i 1 =. An r ompoiion r wi {, } E, ip 0 =(, {, },). Lmm 3 ([8]). A rp G =(V,E) i ionn i n only i, or {, } E, i n opn r ompoiion rin wi {, }.
Er -Orrin 249 No, ivn n opn r ompoiion P 0,P 1,...,P r rin wi {, }, i i riorwr o onru n -orinion. Simply orin P 0 rom o, np i =(u,...,w), 1 i r, romu o w (rom w o u) iu li or (r) w in pril orrin inu y P 0,...,P i 1. Sin orinion o n r onorm o orr o i npoin, no yl r inrou, n n r only our n ink. 3 Dp-Fir Sr n Bionniviy Srin rom roo vrx, p-ir r (DFS) o n unir rp G =(V,E) rvr ll o rp, wr nx i on o inin o mo rnly vii vrx n unrvr. An {v, w} rvr rom v o w i ll r, no y v w, iw i nounr or ir im wn rvrin {v, w}, n i i ll k, no y v w, orwi. For onvnin w u v w or v w o no rpiv wll r i u n wn v n w, or p (v, {v, w},w). W no y v w (poily mpy) p o r rvr in orrponin irion. No r orm pnnin DFS r T = T (G) roo, i.. v or ll v V n v w impli w v. Forv V l T (v) ur inu y ll w V wi v w. W will u rp in Fi. 1 our runnin xmpl. DFS i i o mny iin rp lorim [6], wi on mk u o ollowin noion. T lowpoin o vrx u V i vrx w lo o in T (G) wiw = u or u v w. I no u p xi, u i i own lowpoin. Lowpoin r imporn moly or ollowin ron. 1 9 7 4 2 3 5 12 6 11 8 10 14 13 15 Fi. 1. Runnin xmpl wi numr in DFS rvrl orr, r pi oli, n k (rrwn rom [7])
250 Ulrik Brn Lmm 4 ([6]). A rp G =(V,E) i ionn i n only i, in DFS r T (G), only roo i i own lowpoin n r i mo on r u i lowpoin o (in i, i roo). Prviou linr-im -orrin lorim ir onru DFS r n imulnouly ompu lowpoin or ll vri. In on rvrl o rp y u i inormion o rmin n -orrin. A nw ionniviy lorim y Gow [4] rquir only on p ovr rp n in priulr o no rquir lowpoin ri quion wr w n -orr vri o ionn rp in imilr mnnr. 4 An Er Alorim W prn nw linr-im lorim or -orrin vri o ionn rp. I i r in n i minin, urin p-ir r, n orrin o mximum rvr urp or wi u orrin i poil wiou ponil n o moiiy i lr on. I i inrou vi r prliminry p ini ow lorim lo ompu on ly n opn r ompoion n n -orinion. Puo-o or ompl lorim i ivn n o i ion. 4.1 Opn Er Dompoiion L G =(V,E) ionn rp wi {, } E, nlt DFS r o G wi roo n. W in n opn r ompoiion D(T )= (P 0,...,P r ) uin lol inormion only. In priulr, w o no mk u o lowpoin vlu. L P 0 =(, {, },) n um w v in P 0,...,P i, i 0. I r i k l i no in E i, w in P i+1 ollow. L P 0 : P 1 : P 2 : P 3 : P 4 : P 5 : P 6 : (rivil r) Fi. 2. An r ompoiion D(T ) oin rom DFS r T
Er -Orrin 251 v, w, x V u w, x V i, v w E i, w x, nx v ( Fi. 2). Uin l vrx u on r p rom x o v wi u V i (ponilly v il), w P i+1 = u v w. Sinw x u, P i+1 i opn. I i ll r o v w, nrivil i u = v. Torm 1. D(T ) i n opn r ompoiion rin wi {, }. Proo. Clrly, D(T )=(P 0,...,P r ) i qun o -ijoin opn p. I rmin o ow y ovr nir rp, i.. V r = V n E r = E. Fir um r i n unovr vrx n oo u V r u u minimum ln. L w lowpoin o u. SinG i ionn n u,, Lmm 4 impli r xi x, v V wi w x u v w n x u. I ollow w, x V r y minimliy o u, o ompoiion i inompl, in v w ii ll oniion or nor r. Sin ll vri r ovr, ll r r ovr y onruion. Finlly, n unovr k ii ll oniion o in nor (rivil) r. 4.2 -Orinion W nx rin ov iniion o n opn r ompoiion o oin n -orinion. W y k v w, n likwi i r, pn on uniqurw x or wi x v. ErinD(T ) r orin in ir qunil orr: P 0 i orin rom o, wrp i,0<i r, iorin orin o r i pn on. S Fi. 3 or n xmpl. T ollowin lmm ow orinin k n ir r prlll o r y pn on nily prop ino ur. I {v, w} E i, 0 i r, i orin rom v o w l v i w,0 i r. P 1 = pn on P 2 = pn on P 3 = pn on P 4 = pn on P 5 = pn on P 6 = pn on Fi. 3. Orinion o r in D(T )
252 Ulrik Brn Lmm 5. For ll 0 i r, ov orinion o P 0,...,P i yil n -orinion o G i,n i i pril orr iyin: I w x E i n w i x (x i w), n w i v (v i w) or ll v T (x) V i. Proo. T proo i y inuion ovr qun D(T ). T invrin lrly ol or P 0. Aum i ol or om i<rn l P i+1 r o v w. L w x E i r v w pn on n um i i orin rom w o x ( or i ymmri). T l vrx u V i on x v ii w i u. All vri o P i+1 xp w r in T (x), n in P i+1 i orin lik w x, invrin i minin. Corollry 1. T ov orinion o D(T ) yil n -orinion o G. 4.3 -Orrin W inlly ow ow o minin inrmnlly n orrin o V i urin onruion o D(T ). Srin wi rivil -orrin o P 0,lP i = u v w, 0<i r, rov w. IP i i orin rom u o w (w o u), inr qun o innr vri V (P i ) \{u, w} o P i in orr ivn y orinion o P i immily r (or) u. Lmm 6. For ll 0 i r, orrin o V i i linr xnion o i. Proo. T proo i in y inuion ovr qun D(T ). T invrin lrly ol or P 0. Aum i ol or om i < r n P i+1 = u v w pn on w x. Iw i x ( or i ymmri), innr vri o P i+1 r inr immily or u. By Lmm 5 n our invrin, u i r w, o orrin i lo linr xnion o i+1. Corollry 2. T ov orrin yil n -orrin o G. No inrin innr vri o n r P = u v w nx o i inion w rr n i oriin u my rul in n vri o nor r in in wron rliv orr urin lr o lorim. An xmpl o i kin i own in Fi. 4. 4.4 Alorim n Implmnion Dil I rmin o ow ow ov p n rri ou in linr im. W our lorim on p-ir r, u unlik prviou lorim, DFS i no u or prproin u rr mpl o lorim. T lorim i ivn in Al. 1, wr o or DFS-r mnmn i implii. E im DFS rvr k v w, w k wr r w x i pn on i lry orin (no x i urrn il o w on DFS p). I w x i orin, r o v w i orin in m irion n inr ino orrin. For r in nwly
Er -Orrin 253 inrion inion,,,,,,,,,, inrion oriin,,,,,,,,,,,,,,, Fi. 4. Exmpl owin i i imporn o inr n r nx o i oriin in r rr n inion o i inin k Alorim 1: Er -orrin Inpu: rpg =(V,E), {, } E Oupu: li L o vri in ionn omponn o {, } (in -orr) pro r(r w! x) in or v w pnin on w x o rmin u L on x v lo o v; P o u v w; i w x orin rom w o x (rp. x o w) n orin P rom w o u (rp. u o w); inr innr vri o P ino L ri or (rp. r) u; or r w x o P o pro r(w x ); lr pnni on w x; n (vrx v) in or nior w o v o i v w n (w); i v w n l x urrn il o w; mk v w pn on w x; i x L n pro r(w x); n in iniiliz L wi ; (); n
254 Ulrik Brn,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Fi. 5. Inrmi p o lorim
Er -Orrin 255 orin r, w rurivly pro r o no y orin k pn on i. Ar rvrl o n w r ror l wi n -orrin o ionn omponn o {, } in rvr urp. Conqunly, lorim i rou in ppliion o non-ionn rp, n inpu rp i ionn i n only i orrin rurn onin ll vri. Torm 2. Alorim 1 ompu n -orrin o ionn omponn oninin {, } in linr im. Proo. Givn iuion ov, i i uiin o ow lorim rmin nir opn r ompoiion. E r i r o k ; i k pn on n lry orin r, r i rmin n orin. I r i no y orin k i o o pnn n pro oon r i orin. I ollow rom m rumn in proo o Torm 1 ll r in ionn omponn o {, } will vnully orin. Fiur 5 ow inrmi p o lorim wn ppli o runnin xmpl. No -orrin i irn rom oin in [7], u,.., i rvr or lowpoin- (orwi orr o n woul rvr). An iin implmnion o Al. 1 u ouly-link li L, nor ll pnin on r (inluin il) in yli li n rliz wi n rry (in li r ijoin). Sin only orinion o r i n, i i uiin o or orinion o inomin DFS vrx. In ol w n ouly-link vrx li, our vrx rry (inomin, orinion o inomin, urrn il vrx, poinr o li poiion), wo vrx k (DFS n r rvrl), n on rry (nx pnn ). T rry lo rv o ini wr n lry n rvr. 5 Diuion W v prn impl lorim o ompu n -orrin o vri o ionn rp. I rquir only inl rvrl o rp n minin, r p o DFS, mximum pril oluion or orrin, -orinion, n opn r ompoiion prolm on rvr urp. Wiou moiiion, lorim n lo u o or ionnn (imply k wr ln o rurn li qul numr o vri in rp), n i i rily n rom inuiv proo o Lmm 6 rulin -orrin v ollowin inrin propry. Corollry 3. Alorim 1 yil -orrin in wi vri o vry ur o DFS r orm n inrvl.
256 Ulrik Brn I i inrin o no orin (r) orinion vri orrpon o Trjn u o +/- ll in [7]. Sin y n inrpr orin orinion prn rr n il, y n o up, ou. Wil u o lowpoin limin n o kp rk o pnni, lowpoin r known only r rvrin nir rp. Alou Trjn lorim [7] rmin mo primoniou, w u l r lorim i mor lxil n inuiiv. Finlly, lorim n u o rmin nrlizion o ipolr orinion o non-ionn rp, nmly yli orinion wi numr o our n ink i mo on lrr n numr o ionn omponn, n wi i ipolr in omponn. Wnvr DFS krk ovr r no n orin, i ompl ionn omponn. W r n r o orin i r ny wy w wn n u o rmin, y rurivly orinin pnn r, ipolr orinion o i nir omponn. T omin orinion r yli, n or iionl ionn omponn w mo on our or on ink, pnin on ow w oo o orin ir o omponn n wr o i inin vri r u vri. Aknowlmn I woul lik o nk Roro Tmi or iniiin i rr y mkin m wr o Gow work on p- DFS, n Roro Tmi n Lu Vimr or vry lpul iuion on uj. Tnk o r nonymou rviwr or ir il ommn. Rrn [1] J. Er. -orrin vri o ionn rp. Compuin, 30(1):19 33, 1983. 247, 248 [2] S. Evn n R. E. Trjn. Compuin n -numrin. Toril Compur Sin, 2(3):339 344, 1976. 247, 248 [3] S. Evn n R. E. Trjn. Corrinum: Compuin n -numrin. Toril Compur Sin, 4(1):123, 1977. 247 [4] H. N. Gow. P- p-ir r or ron n ionn omponn. Inormion Proin Lr, 74:107 114, 2000. 250 [5] A. Lmpl, S. Evn, n I. Crum. An lorim or plnriy in o rp. In P. Ronil, ior, Proin o Inrnionl Sympoium on Tory o Grp (Rom, July 1966), p 215 232. Goron n Br, 1967. 248 [6] R. E. Trjn. Dp-ir r n linr rp lorim. SIAM Journl on Compuin, 1:146 160, 1973. 249, 250 [7] R. E. Trjn. Two rmlin p-ir r lorim. Funmn Inormi, 9:85 94, 1986. 247, 248, 249, 255, 256 [8] H. Winy. Non-prl n plnr rp. Trnion o Amrin Mmil Soiy, 34:339 362, 1932. 248