Constructing Incremental Sequences in Graphs
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Constructing Incremental Sequences in Graphs Ralf Klasing, Christian Laforest, Joseph Peters and Nicolas Thibault. N° 5648 August 2005. ISSN 0249-6399. ISRN INRIA/RR--5648--FR+ENG. Thème COM. apport de recherche.
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(62) r¨2wt rNmnpu¢_#w¦ u¢tq'{p`D_%`DtRn}w¦ lo`;«Rs^`'tq'` ¶ ¤ Ú t·hk`;qÍnpu¢rxt ¤ ` ^{prN`,np[HwnQn}[^`%^{prN¥^¦¢`'_*r¨q'rxtlonp{}sqÍn}utwt7rNmnpu¢_#w¦ u¢tq'{p`D_%`DtRn}w¦Hlo`;«Ns`'tq'`q'wtt^rnÊ¥2`Qlprx¦¢x`D%w^{prEÅmu¢_#wnp`D¦j uÎn}[ÇwtÇw^{prEÅmu¢_#wnpu¢rxtb{}wnpu¢r,¦`;lplXnp[wxt Ú t¹hk`;qÍnpu¢rxt `%m`'N`'¦¢rx7wxt7rxmn}u_#wx¦2rx¦¢jkt^rx_¬uvw¦©=n}u_¬`bwx¦Nrx{}uÎn}[^_Шçrx{6np[^` Çs^t^¦¢`Dl}l {}`'¦vwn}`D^P{}rx¥^=¦¢`'N_ÐPr¨ ¶ *t^ut^wtªu¢t¤q{}`'_¬`'tRnyw¦ lp`D«Rs^`Dtq`rx¨Xx{}rxs^l\ unp[7lp_#w¦¢¦¡½y¼y¼y½ºH¿=°Í¸4¼'¸»¿=¸4½´ ¶ É ` n}[^`'tslp`np[^uvl#w¦¢xrN{punp[^_ n}r£m`'N`'¦¢rx w¹2rx¦¢jRtrx_¬u¢wx¦Î©8npu¢_¬`x©Áwx^^{}rEÅmu_#wn}urNtÈw¦¢xrN{punp[_ ¨çrN{¬np[^` {prN¥^¦`D_ rx¨ÛqrxtHlJn}{psqnpu¢t^ wxt#rxmn}u_#w¦HutHq{}`'_¬`'tRn}wx¦Hlp`D«Rs^`Dtq`\¨çrN{w,N{}wx^[ ¤ wxt¬ ¡`6lp[^r Èn}[wn¡rxs^{ wxtw¦¢jmlouvl¡r¨np[^`wx¦Nrx{}uÎn}[^_&uvl¡npu¢x[Rn ¶ (* \
(63) ¹ -Æ ¡& ,+' ¯ b $¬ ¡ Ú tÑnp[^uvl.lo`;qÍnpu¢rxt ¤ `·m`'{}uN`7_#wnyqy[^utÈs^^2`'{.wtH ¦r ¡`'{¥HrNs^t^lrxtÑnp[^`¯qrRlJnrx¨ wtÑrNmnpu¢_#w¦ u¢tq'{p`D_%`DtRn}w¦Ûlp`D«Rs^`Dtq` ¶ p ±°½ ½"° !$#½'¸Î5® k¿Ä½ ®°}À} ³ G = (V, E, w) #X¸»¿ ÔkØ Ô cost(N , . . . , N ) ≤ D(V ) ± ° À »   w(e) ≥ 1 e ∈ E# opt 1. opt n. ùçú ÜÛù»ð.
(64) ±º^´Í¿=°Í²¼'¿8¸»ºR®º2¼'°p½'¾¬½'º¿ÄÀ½'²½'º2¼y½Í´¸»º\°pÀy³ k´. . ØÛØ `'n G = (V, E, w) ¥2`bw `DuN[Rnp`D.N{}wx^[ª unp[ w(e) ≥ 1 ¨çrN{6w¦¢¦ e ∈ E ¶Ï rN{Q`'N`'{}j i ¤ ¦`'n ¥2`¬wÇx{}rxsªr¨lpu)D` r¨_¬u¢t^u¢_ s^_Ìmuvw_¬`n}`'{ `n ¥2`np[`b¦vw{}x`;lJnQu¢tRnp`Dx`'{ 1 ≤ i ≤ n¤ lpsqy[np[wn D(NN) ≤ pD(V ) ¶ hku¢tq'` Gi= (V, E, w) u¢l w% ¡`'u¢x[Rnp`;¶ N{}wxi^[ unp[ w(e) ≥ 1 ¨çrx{wx¦¦ `[wEN` e ∈ E¤ p 1 ≤ D(N ) ≤ · · · ≤ D(N ) ≤ D(V ) < D(N ) ≤ · · · ≤ D(N ). ;µ `n M , M , . . . , M ¥2`wtkjÇu¢tq{}`'_¬`DtNnyw¦lp`D«Rs^`'tHq`,losHqy[n}[wn M = N ¶ Z\[ksl ¤ p 1 ≤ D(M ) ≤ · · · ≤ D(M ) = D(N ) ≤ D(V ). xµ 6l¡np[^`^u¢wx_%`'np`D{ r¨ G = (V, E, w) u¢l D(V ) ¤ `,[HwEx` D(M ) ≤ · · · ≤ D(M ) ≤ D(V ). Nµ o n ¡j ;µbwt xµ ¡`rN¥mn}wxut wtÈ¥Rj ;µbwt Rµ ¡`rN¥mn}wxut )¤ n o max p Ú n ¨çrN≤¦¦¢rp lD(V np[wn ∗ i0. ∗ i. 0. ∗ 2. 1. 2. ∗ i0. ∗ i0 +1. n. ∗ i0. i0. 2. ∗ i0. i0. i0 +1. n. D(Mi ) D(Ni∗ ). 2≤i≤i0. D(Mi ) D(Ni∗ ). maxi0 +1≤i≤n. ∗ n. ≤ √D(V ) = D(V ). D(V ).. cost(N1opt , . . . , Nnopt ) ≤ cost(M1 , . . . , Mn ) ≤. p. . D(V ).. Z\[^`t^`'Åkn¡n}[^`'rN{p`D_ ^{prkuvm`Dl¡w%¦r ¡`'{¥HrNs^tnp[wn _¬wn}qy[^`;lnp[^`s^^2`'{\¥2rxs^tHrx¨ Z\[^`Drx{}`'_& Z rxN`n}[^`'{ ¤ Z\[^`'rN{p`D_#lwtªbxu¢x`,w%npu¢x[Rn¥2rxs^tH.rxtnp[^`, ¡rx{ylon qDwxlp`,qrNlonr¨Xwt.rNmnpu¢_#w¦Ûu¢tq{}`© ¶ _¬`DtNnyw¦Ûlp`D«Rs^`Dtq`e¨çrx{\n}[^`q¦vwxl}l\r¨ x{yw^[Hl unp[ªwx¦¦Û`;mx` ¡`'u¢x[Rn}l\wn¦`;wxlon ¶ p ±°¸»º ʺH¸»¿Á½' !ª¾#Àº !$#½¸Î5® k¿Ä½! .®°pÀy³ k´ #X¸»¿" À» ÔkØ Ô & cost(N , . . . , N ) ≥ D(V ) !½ D®N ½ #½¸Î5® k¿=´bÀ¿Â½yÀE´Í¿ # ØÛØ `'n ¥H`np[` ¡`'u¢x[Rnp`;ËN{}wx^[£ut uNs^{}` [^`'{}` u¢l¬wt wx{p¥uÎn}{}wx{pjÇq' rxtlon}GwxtNn ¶ =Z\(V[^`m, Euvw_¬, w`n}`'){rx¨ G uvl D(V ) = K ¶ÊÏ rx{`DÏ x`D{pj i ¤ 1¤ ≤ i ≤ 5K¤ ¦¢`>n N1 ¥H`w N{prNs^r¨ lp*u )'` i rx¨ _¬u¢t^u¢_ s^_ mu¢wx_¬`np`D{ )¶ `n M , M , . . . , M ¥H`bwtkjÇu¢tq{}`'_¬`'tRnyw¦Ûlp`D«Rs^`'tHq`e¨çrx{ Ú ¨ M 6= {a, b} ¤ n}[^`'t cQnp[`'{} u¢lp` ¤ M = {a, b} ¤ wt.¨çrN{6wx¦¦ M losHqy[ ≥ = K¶ G ¶ n}[wn M ⊂ M ¤ Z\[ksl ¤ cost(M , . . . , M ) ≥ K = pD(V ). Z\[^`¬^{}rkr¨ = = K¶ uvl\`DwNlou¢¦¢j#x`'t`'{yw¦¢u )D`D#n}r#wtkjqrN_¬^¦`'np`x{yw[ unp[ªw¦¢¦`DmN` `DuN[Nnyl K `Å^q`Dmnwbnp{}u¢wxt^x¦¢`6 unp[ `;mx`6 `DuN[Rn}l K wtHÇw Hwu¢{rx¨Ûx`'{pnpuvq`;lÊn}[wn\uvlmuvl4|Jrxu¢tNn¡¨ç{}rx_ np[^`6np{}uvwt^N¦`ewtÇq'rxt^t`DqÍn}`D¥Rj#wxt `;mx` unp[. ¡`'u¢x[Rn, ¶ ,++
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(78) ±º^´Í¿=°Í²¼'¿8¸»ºR®º2¼'°p½'¾¬½'º¿ÄÀ½'²½'º2¼y½Í´¸»º\°pÀy³ k´ . q'¦uv«Rs^`¬ut arxtHlouvm`'{n}[^`#u¢tq{}`'_¬`'tRnyw¦Xlo`;«Rs^`'tq'` u¢t rx¥mnywu¢t^`D¹¥kjwx^^ut^ n}[^`N`'{pnpuvqG`Dl¶ r¨ G u¢tªn}[^`rx{ym`D{ v , v , . . . , v ¶ Ql GN uvlQ,w.tª. .s, tRN ¡`'u¢x[Rn}`DGx{yw^[.rx¨X^u¢wx_%`'np`D{6wn _¬rRlJn 2 ¤ wxtkj%lps^¥lp`n S ⊆ V uÎn}[ |S| ≥ 2 uvllpsqy[#n}[wn D (S) = 1 rx{ D (S) = 2 ¶ Ú n¨çrx¦¢¦¢r lXnp[Hwn ¨çrN{w¦¢¦ 2 ≤ i ≤ i ¤ ) = D (N ) = 1 NEµµ DD (N ¨çrN{w¦¢¦ i + 1 ≤ i ≤ n ¶ (N ) = D (N ) = 2 Ú t¹w{pnpuvqs¦¢wx{ ¤ np[^`%u¢tq{}`'_¬`'tRnyw¦Xlo`;«Rs^`'tq'` N , . . . , N l}wnpuvl *`Dl D (N ) = D (N ) ¨çrN{ewx¦¦ 2≤i≤n ¶ `'tq'` ¤ o n opt 1 0. 0. 0. 0 1. 0. 0 0. opt i opt i. 0 0. 0 2. 0 n0. ∗ i ∗ i. opt n0. 0. 0. 0. 0. 0. 0. opt 1. 0. opt n0. opt i. 0. 0. ∗ i. 0. D (Mi ) max2≤i≤n0 D 0 (N ∗ ) i n 0 opt o = max 0 D (Ni ) 2≤i≤n max2≤i≤n0 D0 (N ∗ ). D0 (Mi ) D0 (Ni∗ ). .. Ú t G ¤ wxtkjÇlos^¥Hlo`'n S ⊆ V uÎn}[ n|S| ≥ 2ouvl¡lpsqy[np[wn D (S) = 1 rx{ D (S) = 2 ¤ lpr n}[^`6rNt^¦¢j¬nJ r 2rNl}lpu¥^¦¢` wx¦s^`;ler¨ max w{}` 1 wt 2 ¶ Z\[^u¢le_¬`Dwxtl6np[wneu¨ M , . . . , M u¢lwxt u¢tq'{p`D_%`DtRn}w¦ lo`;«Rs^`'tq'`bq'rxtlonp{}sqÍn}`Dªu¢t72rx¦¢jkt^nrx_¬uvw¦npu¢_%o `b¥kj.wxtw¦¢xrN{punp[_Ð unp[·w^^{}rEÅmu_#wnpu¢rxt {ywn}urªlonp{}u¢qnp¦¢j¹¦¢`Dl}ln}[wt 2 ¤ np[`'t max wtH D (M ) = D (N ) ¨çrx{ wx¦¦ i ¤ = 1¤ Z\[ksl ¥kjqy[rRrRlou¢t^%np[`¦¢wx{pN`Dlon utRnp`Dx`D{ losqy[.n}[wn rxt^` 2≤i≤n¶ `D¦j M) =µ D¤ wx(N t¹np)[^`D={p`'1¨çrx¤ {}`%w qDwt¯qrNtlJn}{psHqÍn,ut¯¤ 2rx¦¢jkt^rx_¬uvw¦
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(81) b Ë $¬ Ú tÈnp[^uvl%lo`;qÍnpu¢rxt ¤ `m`'N`'¦¢rxÈwxtËrx^npu¢_¬wx¦2rx¦¢jkt^rx_¬uvw¦©=n}u_¬`w¦¢xrx{}unp[^_ npr *t wtÈu¢tq'{p`D_%`DtRn}w¦ lp`D«Rs^`Dtq`ªrx¨x{}rxs^Hl# uÎn}[Ñlo_#wx¦¦¬½}¼y¼y½'º¿8°Í¸4¼¸»¿8¸4½Í´ `ªn}[^`'tÑ{prN`.np[wnrxs^{w¦¢xrN{punp[^_ u¢lw © wx^^{}rEÅmu_#wn}urNt%wx¦Nrx{}uÎn}[^_¨çrN{Ênp[`^{}rx¥^¦¢`'_ r¨ *¶tmÉu¢t^ wxt¬rxmn}u_#w¦HutHq{}`'_¬`'tRn}wx¦lp`D«Rs^`'tHq`\¨çrx{np[^4` ^u¢wx_%`'np`D{ ¶ Ô 4 Ø R $ ½G`;q'q'`'tRnp{}u¢q'uÎnJj ± ÑÀÑ®x°p±²;³ M ⊆ V #X¸»¿ &{prkrn r ∈ M ¸v´ E(M, r) = ®°p±²;³ #X¸»"¿ ¸v´bÀN{prNs^.r¨Êlo*u )'` r¨ max{d | = ¼yiÀ »1Â!½ ≤ i¿4´ ≤wNlpnlp rmquvwnp`;Ç{}rkrn %´Í²¼ ª"¿ i À¿ _¬u¢t^u¢_ s^_Ð(u,`Dq'r)q'`':tRunp{}uv∈quMnJj¹}.¸ ¿ ^½'°p½%½ /N¸v´ÍM¿4´À ⊆½'V°Í¿Á½ / r |M . » ¸ ∈M 6ºÇrNmnpu¢_#w¦utq'{p`D_¬`'tRn}wx¦ lo`;«Rs^`'tq'`e¨çrx{ E(M , r ) = min{E(M, r) : M ⊆ V, |M | = i, r ∈ M }. n}[^`#`DqDq`DtNn}{puvqunJj¸v´ÇÀºÈ¸»º¼°p½'¾#½ºH¿ÁÀÂX´'½ '²½'º2¼y½± ®°}±²;³H´ M = {r }, M , . . . , M = V #X¸»¿ ±°bÀ» i 1 ≤ i ≤ n ´y²!¼ ¿ ^À¿ |M | = i i. 0. 0. 0. 0. 0. 2≤i≤n0. D (Mi ) D 0 (Ni∗ ). 1. 2≤i≤n0. 0. D 0 (Mi ) D 0 (Ni∗ ). 0. 0. 0. ∗ i. ∗ i. ∗ i. ∗ i. ∗ i. opt 1. max. 2≤i≤n. ÜÜ ê76/ 3.-98. E(Miopt , ropt ) E(Mi∗ , ri∗ ). ). i0. ∗ i. opt i. (. 0. ∗ i ∗ i0. 0. i0. G. 0. i. i0. 0. n0. = min. . max. 2≤i≤n. . E(Mi0 , r0 ) E(Mi∗ , ri∗ ). . opt. opt 2. M10 ⊂ · · · ⊂ Mn0 = V, : |Mi0 | = i, M10 = {r0 }. opt n. . ..
(82) . ÂÀ´Í¸»ºR® . #. # À ±°p½´Í¿ #. ½'¿Ä½°y´ # $ m ¸ }À²m ¿. Ô 4Ø ½'¿ r ∈ V Àº ǽ'¿ S }½b¿ ^½´'½'²½'º2¼y½ ¼y±ºH¿ÁÀ¸»º¸»ºk®¿ ^½ À ²½´ {d (r, u) : u ∈ V } ´'±°Í¿Á½! ¸»º¸»º¼°p½yÀ´y¸»ºk®#±° N½° º±¿Ä½6¿" À¿ |S| ≤ n = |V | # ±º´y¸ N½°e¿ ^½Ê³^À°Í¿=¸»¿8¸4±º FG1(r), . . . , Fn(r) ± ´Í²¼7 ¿" À¿ ¸v´,¿ ^½ ÀÂβ½¸»º ®x°p±²;³ M ⊆ V ¸v´bÀ ¥{p`;Vwxkn}[m© H{}lon\lps^F¥lpj`(r)n ¨ç{}=rx_Ò{u{prk:rdn Gr(r,∈u)M ¸ ¸»¿´'jÀth¿8¸v´ ½´0' S}, 1 ≤ j ≤ n. |M | = 1 ¿" ½º M = {r} # |M | ≥ 2 ¿" ½º7¿ ^ ½'°p½%½ N/ ¸v´Í¿=´,À k ≥ 2 ´Í²¼ª ¿" À¿. *. . . ∀j, 1 ≤ j ≤ k − 1, Fj (r) ∩ M = Fj (r). . Fk (r) ∩ M 6= ∅. . . . #. Z\[`ª¨çrx¦¢¦r u¢t^£w¦¢xrN{punp[_ ¨çrx{ ¡`;lJn ( q'q`DtRnp{}u¢q'uÎnJj^µ&*tH^lw¯N{prNs^"r¨,lou*)'` i r¨_¬utu_bs^_ `;q'q'`'tRnp{}u¢q'uÎnJj¬¨çrN{wtkj i ¤ 1 ≤BEi ≤ n ¶ , Ø BE # ±°½yÀN¼ r ∈ V ¼y±º^´Í¿=°Í²¼'¿À '°p½yÀ ¿
(83) °y´Í¿
(84) ´y² ´'½¿ M (r) ⊆ V y°p±¾&°p±;±¿ r #X¸»¿ |M (r)| = i# T # ^±;±E´'½ r À º %¸»¿=´6ÀE´y´'±D¼'¸4À¿Ä!½ ®x°p±²;³ M (r ) ´Í²¼ ¿ ^À¿ E(M (r ), r ) = min{E(M (r), r) : ∀l > k Fl (r) ∩ M = ∅ i. i. i. r ∈ V}. #. i. i. i. i. i. i. i. i. q'rxrtn}lo`np{}n}s[qwnpn
(85) `D¨çbrx{Xu¢t#wx¦H¦ rN¦jkt^rN_%¤ n}uvw[^¦k`\n}u_¬w{p` npusnplpu¢rxu¢t^t Qu |-mlJn}{}w lw¦¢xrx{}uwnp[^t_np¶ [^Z\` [kwxsl}l lp¤rkq'u¢wnp`;,xqD{}wrxt¬s^¥2`QqrNtlJn}qD{psHwtbqÍnp¥H`;` r∈V. F1 (r), . . . , Fn (r). Mi (r). u¢t.HrN¦jkt^rN_¬u¢wx¦Hn}u_¬` ¶ Z\[`#¨çrx¦¢¦r u¢t^ª¦¢`'_¬_#w7lp[^r lnp[Hwn%Q¦Nrx{}uÎn}[^_ qrNtlJn}{psHqÍn}l wªx{}rxs^¯rx¨ lpu*)'` r¨ _¬utu_bs^_ `;q'q'`'tRnp{}u¢q'uÎnJj ¶ Z\[`6uvm`Dw r¨n}[^`6{prkr¨Ûuvlnpr¬lo[^r n}BE[wn¡¨çrx{\wNuN`'tÇ{}rkrn r ∈ V ¤ n}[^`exi {}rxs^Çrx¨ lou*)'` i rx¨_¬u¢t^u_bs^_ `Dq'q'`'tRnp{}uvqunJjÇwxl}lormquvwn}`Dnpr r u¢lw%¥^{}`DwNknp[^© *{ylon\lps^¥lp`n ¨ç{}rx_&{}rkrn r ¶ Qlwx¦Nrx{}uÎn}[^_ qy[`D!q -ml¡`Dwxqy[{}rRrxn r ∈ V ¤ unt^`Dq'`Dl}lpwx{pu¢¦¢j *t^l¡n}[^`,{pu¢x[Rnlps^¥lp`n ¶ BE
(86) Ô Ë 6 ®R±°Í¸»¿ k¾ BE ¼y±º´Í¿=°Í²¼'¿4´#À®x°p±²;³Ñ± %´Í¸ ;½ i ± Ǿb¸»ºH¸»¾b²m¾ ½}¼y¼y½'º¿8°Í¸4¼¸»¿ ! ±°À º ! i Mi (ri ). i. i. 1≤i≤n. i. #. ØÛØ `n 1 ≤ i ≤ n ¶PÏ rx{wx¦¦ r ∈ V ¤ ¦`'n M (r) ⊆ V ¥2`¹wxtRj x{}rxs^r¨lpu)D` i unp[ wt¦¢`n M (r) ⊆ V ¥H`w¬¥{p`;wxkn}[m© *H{}lonlos^¥Hlo`'n ¨ç{prN_&{prkrxn r r¨Xlou*)'` i ¶ Z\[ksl ¤ ¨çrx{6wtkj r ∈ M (r) ¡ ` [wEN` E(M (r), r) = max {d (u, r) : u ∈ M (r)} ≤ max {d (v, r) : v ∈ M (r)} = r∈V¤ `Dtq` ¤ E(M (r), r) ¶ min {E(M (r), r) : r ∈ V } ≤ min {E(M (r), r) : r ∈ V } . Rµ `n M (r ) ¥H`·w£x{}rxsrx¨ lo*u )'` i q'rxtlonp{}sqÍn}`D"¥kj"¦¢xrx{}unp[^_ BE ¤ 1 ≤ i ≤ n ¤ wxt ¦¢`n H ¥ `¹w£x{}rxs^"rx¨lpu )D` i uÎn}[ _%u¢t^u¢_ s^_`;q'q`DtRnp{}u¢q'uÎnJjwt wNlplprmquvwnp`;È{}rkrn r ∈ M ¶ ¡j M 0 i. 0 i. 00 i 00 i. 0 i. G. 00 i. i. ∗ i. i. 00 i. 0 i. G. 0 i. i. ∗ i. ∗ i. ùçú ÜÛù»ð.
(87) ±º^´Í¿=°Í²¼'¿8¸»ºR®º2¼'°p½'¾¬½'º¿ÄÀ½'²½'º2¼y½Í´¸»º\°pÀy³ k´ n}[^`m` *t^unpu¢rxtËrx¨¦¢xrx{}unp[^_ BE ¤ ¡`Ç[wEx` E(M (r ), r ) = min{E(M (r), r) : r ∈ V } wt ¥kj¹np[^`m` *tuÎn}urNtËr¨ M ¤ ¡`#[wEN` E(M , r ) = min{E(M (r), r) : r ∈ V } ¶ Z\[ksl ¤ ¥kj Rµ ¤ Ql M u¢l.w x{}rxs^Gr¨blpu)D` i unp[ np[`¯lp_#w¦¢¦`;lJn.`DqDq`DtNn}{puvqunJj ¤ E(M (r ), r ) ≤ E(M , r ) ¶ E(M , r ) = E(M (r ), r ) ¶ Z\[`¡t`Åknw¦¢xrN{punp[^_ ( ¨çrN{ Ú tq{}`'_¬`'tRnyw¦ ¡`Dlon q'q'`'tRnp{}u¢q'uÎnJj^µ
(88) q'rxtlonp{}sqÍnylXwt%rxmn}u_#wx¦^utHq{}`'_¬`'tm© nyw¦Ûlp`D«Rs^`Dtq`r¨
(89) x{}rxsl¨çrx {\np[`,`Dq'q'`'tRnp{}uvqunJj ¶ ( , Ø & ( # H¿Ä±À° °Í¿ ½yÀx#X!¼ ¸» "¿ r ∈ V ' M (r) = {r} # ±° ½yÀx!¼ i 1 ≤ i ≤ n ' À ±º^´Í¿=°Í²¼'¿¡À °}½}À ¿
(90) °y´y¿´Í ² Í´'½'¿ M (r) y°}±¾ °p±D±¿ r #X¸»¿ |M (r)| = i # ±¾³²m¿Á½ "¿ ½°}À¿8¸4± # T ^. ; ± E ± ' ´ ½ À º . » ¸ 4 ¿ ´ E À y ´ ' ´ D ± ' ¼ 4 ¸ À Ä ¿ ½ ! #. ' ´ !½ '²½º¼}½ M (r ), . . . , M (r ) ´Í²!¼ .¿ ^À¿ # r ∈V i. i i ∗ ∗ i i. i. ∗ i ∗ ∗ i i. i. i. i. ∗ i. ∗ i. i. 00 i. i. ∗ i. 0 i. i. . . . 1. i. i. E(Mi (r),r) E(Mi∗ ,ri∗ ). 0. 1. max. . E(Mi (r0 ), r0 ) E(Mi∗ , ri∗ ). . = min. . 0. max. . 0. n. E(Mi (r), r) E(Mi∗ , ri∗ ). . :r∈V. . .. rn}`#np[wn¨çrx{bw¦¢¦ n}[^`wxl}lprkq'u¢wnp`;¹lp`D«Rs^`Dtq` qDwt¯¥2`qrNtlonp{}sqÍn}`D·ut 2 rx¦¢jkt^rx_¬uvw¦n}u_¬`,srlpu∈t V Q¤ u |-mlJn}{}w lwx¦Nrx{}uÎn}[^_&wtHn}[Mwn ¨ç(r), rN{Qw.¦¢¦. .r, M∈ V(r)¤ wtHwx¦¦ i ¤ 2 ≤ i ≤ n ¤ n}[^` {ywn}ur q'wt¯¥2`ÇqrN_%smnp`;u¢t¯HrN¦jkt^rN_%uvw¦
(91) n}u_¬`#¥kjsHlou¢t^ª¦¢xrN{punp[^_ BE npr7qrN_¬^smnp` Z\[ksl ¤ M (r ), . . . , M (r ) q'wt¥2`qrxtHlJn}{psqnp`;u¢t2rx¦¢jkt^rx_¬uvw¦n}u_¬` ¶ E(M , r ) ¶
(92) Ô Ë & 6 ®R±°Í¸»¿ k¾ ( º ´,Àº·±}³¿=¸»¾#À¸»º2¼'°p½'¾¬½'º¿ÄÀ´'!½ '²½º¼}½ ±°¿ ^½b½y¼}¼y½'º¿8°Í¸4¼¸»¿ ! # ØÛØ `n ¥H`%np[`%u¢tq'{p`D_%`DtRn}w¦Xlp`D«Rs^`Dtq`¬qrNtlJn}{psHqÍnp`; ¥kj ( ¤ ¦¢`n M M =(r{r) = },{rM}, M, . (r. . ,),M. . . , M¥2`\wx(rtbrN)mnpu¢_#w¦mu¢tq{}`'_¬`DtNnyw¦mlp`D«Rs^`'tHq`¡¨çrx{ np[` `Dq'q'`'tm© n}{puvqunJj ¤ wt ¦`'n M ¥2` wQN{prNs^ rx¨2lpu )D` i r¨_¬u¢t^u_bs^_`;q'q'`'tRnp{}u¢q'uÎnJjwxt wNlplprmquvwn}`D{}rkrn r ∈ M ¤ ¦¢xrN{punp[_ ( q'rxtlonp{}sqn}lwt¹utq'{p`D_¬`'tRn}wx¦
(93) lp`D«Rs^`Dtq`¬lon}w{pnpu¢t^ uÎn}[¹`;wxqy[72rNl}lpu¥^¦¢` 1 ≤ i ≤ n¶ {}rkrn ¤ utHq¦¢smu¢t^,np[`Qlo`;«Rs^`'tq'` M (r ), . . . , M (r ) lon}wx{on}ute unp[ M (r ) = {r } ¶ dªrx{}`© rN`'{ ¤ ¥kj,np[^`^` *t^unpu¢rxt¬rx¨Q¦Nrx{}uÎn}[^_ ( ¤ np[^` N{prNs^l M (r ), . . . , M (r ) wx{p`\¥^{}`DwNknp[^© *{ylon lps^¥lp`nyl¨ç{}rx_Ì{}rkrn Z\[ksl ¡` [HwEx` (r ), r ) ≤ E(M , r ) (1 ≤ i ≤ n) ¤ wt n o ¡j%np[`em` *t^unpu¢rxtr¨Q¦Î© ¡`QrN¥mn}wxut max r n ¶ ¤ o ≤E(M max Nrx{}uÎn}[^_ ( lp`'`np[^`ªlp`DqrNt£wx{on¬r¨Qnp[^`.w¦¢xrN{punp[^_ǵ ¤ wtËnp[^`¨4wxqnb. n}[wn Mn , . . . , M o u¢l wxtrNmnpu¢_#w¦Ûu¢tq'{p`D_%`DtRn}w¦Ûlp`D«Rs^`Dtq`e¨çrN{ np[^`,`;q'q'`'tRnp{}u¢q'uÎnJj ¤ `rN¥mn}wxut max = n o 2≤i≤n. 2≤i≤n 1. E(Mi (r),r) E(Mi∗ ,ri∗ ) ∗ ∗ i i. n. i. 1. 0. 0. n. . . opt 1. 1. 0. 0. opt. ∗ i. opt 2. 2. 0. ∗ i. . opt. 1. . opt. 2≤i≤n. n. i. opt. . 0. n. opt n. opt. E(Mi (r ),r E(Mi∗ ,ri∗ ). ). opt. opt. 2≤i≤n. 1. 1 opt. opt. E(Miopt ,r opt ) E(Mi∗ ,ri∗ ). opt i. n. opt. max2≤i≤n. .. opt. opt. opt. 2≤i≤n. E(Miopt ,r opt ) E(Mi∗ ,ri∗ ). ∗ i. opt opt n 1 E(Mi (r0 ),r0 ) E(Mi∗ ,ri∗ ). . ÉrNm` npu¢_#lp[^wr¦ u¢tnpq[{}w`'n6_¬`DtN¦¢xnyrNw{p¦Ûulpnp`D[^«R_ s^`D t(q `euv¨çlrx{\w#n}x[^©Á`wx^m^uvw{}_¬rEÅm`un}_#`'{wn}urNtwx¦Nrx{}uÎn}[^_ ¨çrx{Qnp[^` ^{prN¥^¦¢`'_Òrx¨*tmu¢t^wxt ¶ . ÜÜ ê76/ 3.-98.
(94) ;. ÂÀ´Í¸»ºR® . #. # À ±°p½´Í¿ #. ½'¿Ä½°y´ # $ m ¸ }À²m ¿. ÔkØ Ô ½¿. y½Q¿ ^½6¸»º¼°p½'¾#½ºH¿ÁÀÂm´'½'²½'º2¼y½¼}±º^´Í¿8°Í²¼¿Ä½! ! 6  ®N±°Í¸»¿" m¾ ¢½¿ N , . . . , N M y½b, À. º·. . ,±yM³2¿8¸»¾¬À ¸»º¼°p½'¾#½ºH¿ÁÀ´'½'²½'º2¼y½ # $ ^½'º. . 1. opt 1. n. (. Àº . opt n. cost(M1 , . . . , Mn ) ≤ 4. cost(N1opt , . . . , Nnopt ). ¦¢`n ØÛMØ ¥2Ï `Nr ¹w {x`'N{}rx`'{}s^j È1r¨e≤lpu*)'i` ≤i rxn¨6¤ _¬¦`'u¢t^n u_bNs^_ ¥2`·`;q'wËq`DtRxnp{}{}rxu¢s^q'uÎnJj£rx¨,wtHlpÈu*)'` wxl}ilprkrq'¨u¢w_¬np`;u¢t^£u¢_ {prks^rx_ n rmuvw∈_¬`'Mnp`'{ ¶ ¤ w tH` n ¥H`wxtrNmnpu¢_#w¦u¢tq{}`'_¬`DtNnyw¦Ûlp`D«Rs^`Dtq`e¨çrx{\n}[^`,`DqDq`DtNn}{puvqunJj ¶ M ,M ,...,M opt 1. ∗ i. ∗ i. opt 2. max. 2≤i≤n. ∗ i. ∗ i. opt n. . D(Mi ) D(Ni∗ ). ff. ≤. 2 max. ≤. 2 max. =. ≤. ≤ ≤. 2≤i≤n. . E(Mi , r) D(Ni∗ ). . E(Mi , r) E(Mi∗ , ri∗ ). . ff. M1 = {r}
(95) D(Mi ) ≤ 2E(Mi , r)). ! ∗ ∗ . #"%$ E(Mi , ri ) ≤ & ∗ ∗ ∗ E(Ni , ci ) ≤ D(Ni ), c∗i ∈ Ni∗ ' 2≤i≤n & opt opt ff E(Mi , r ) ()*+*, .-/$ 2 max M1opt = {ropt } ' 2≤i≤n E(Mi∗ , ri∗ ) opt opt M1 , . . . , Mn 0 . ff . ! 2 1 E(Niopt , copt ) *, 3 4 *+ 356 2 & & 7 ! 2 max 4 8 4 $ 2≤i≤n E(Mi∗ , ri∗ ) opt N1 = {copt } ' ff opt opt D(Ni ) c ∈ Niopt , 9: 2 max opt opt opt E(Ni , c ) ≤ D(Ni )) 2≤i≤n E(Mi∗ , ri∗ ) opt ff ∗ ∗ D(Ni ) D(Ni ) ≤ D(Mi ) 4 max ∗ ∗ ∗ ≤ 2E(Mi , ri )) 2≤i≤n D(Ni ). ff. . ¶ rn}Z\`\[^np`,[t^wn`'Åk n `6npq'[^w`Dtrxt^{}`'r_ÐnÊrNlp¥m[^n}rwx ut#l¡wn}t#[wwxn ^n}^[^{}` rEÅkwu¢_#^w{pn}rEuÅmrNu¢tb_#{ywwnpn}u¢urxrt¦¢`D{}l}wlXnpu¢npr¬[Hwrt¨ ¨ç¨çrxrx{{np[^¦¢uvxlXrN^{pu{}nprx[¥^_ ¦¢`'_ (¥k jbqDZ\wt^[^t^`Drxrxn{}`'¥H_ ` u¢_¬^{}rx`D ¶ ±°Ç½ E½'°"! ¿ ^½'°p½Ç½/N¸v´y¿=´¬À #½'¸Î®5 k¿Ä½ ®°pÀy³ ¯´y²¼! ¹¿ ^À¿Q¿" ½¬¸»º2¼'°p½'¾¬½'º¿ÄÀ ÔkØ Ô ´'½!'²½º¼}½ ¼y±º^´Í¿=°Í²¼'¿Ä½ ! 6 ®R±°Í¸»¿ k¾ ( ®¸½Í´ 2 4. <;. 0<<1. . . M1 , . . . , Mn. 4 cost(M1 , . . . , Mn ) = . 1+ cost(N1opt , . . . , Nnopt ). ØÛØ `n ¥2`n}[^`. `DuN[Nn}`DÈx{yw^[Ëu¢t u¢xs^{}`ª rN{¬`Dwxqy[ ¦`'n ¥2`wªN{prNs^· r¨loGu*)'` ()i r¨ _¬u¢t^u_bs^_ `DqDq`'tRn}{puvqunJj ¤ ¦`'Ï n r ∈ M¶ËÏ ¥2`un}lbwxl}ilo¤rmq'1u¢w≤np`D¹i {}≤rRrx7n ¤¤ wt¯M¦¢`n ¥2`¬wÇx{}rxs7r¨lpu)D` rx¨_¬ut^u¢_ s_gmu¢wx_¬`np`D{ u¢x`Dt ¦¢xrN{punp[^_ ( u¢¦¦Xq'rxtlonp{}sqÍn N wxtËu¢tq{}`'_¬`'tRnyw¦\lo`;«Ns`'tq'i`lon}w{pnpu¢t^7 unp[ `DwNqy[Ër¨ ¶ n}[^`N`'{pGnpuvq() `;l ¤ wtHË u¢¦¦np[^`Dt qy[^rkrRlo`np[^`¥H`;lJn u¢tq'{p`D_%`DtRn}w¦lo`;«Ns`'tq'`¨çrx{n}[^``;q'q`DtRnp{}u¢q'uÎnJjw_¬rxt^¬n}[^`Dlp` ¶ Z\[^` utq'{p`D_¬`'tRn}wx¦ lo`;«Rs^`'tq'`,{p`'nps^{}t^`; ¥kj Q¦Nrx{}uÎn}[^_ ( uvl np[`lo`;«Ns`'tq'` M (a), . . . , M (a) rN¥mn}wxut^`; ¥kj wN^mu¢t^ennp[^` N`'{pnpuvq`;l
(96) or¨ G () u¢t·np[^`#rN{}^`'{ a, b, c, d, e, f, g ¶ Ú tm`D`D ¤ np[^uvllp`D«Rs^`'tHq`¬¦`;wx^l,npr max ¤ = ∗ i. 0. ∗ i. ∗ i. ∗ i. . 0. . 1. 7. 2≤i≤7. E(Mi (a),a) E(Mi∗ ,ri∗ ). 0 2 1+. ùçú ÜÛù»ð.
(97) ±º^´Í¿=°Í²¼'¿8¸»ºR®º2¼'°p½'¾¬½'º¿ÄÀ½'²½'º2¼y½Í´¸»º\°pÀy³ k´. N. . 2 2 . 2. 4. 4. . 4. . 1+ 1+ . . Ï uNs^{}`,^ÃÊZ\[^`x{yw^[ 1+. G0 (). [u¢qy[¹u¢len}[^`¬_%u¢t^u¢_ s^_ HrRlplpu¢¥^¦`%w¦¢s^`%¨çrx{,wtkjªu¢tq{}`'_¬`DtNnyw¦Xlp`D«Rs^`'tHq`%ut Z\[^`¬rNmnpu¢_#w¦ ¢u tq'{p`D_%`DtRn}w¦ lp`D«Rs^`Dtq`¨çrx{¬n}[^`7muvw_¬`n}`'{ ¤ N , . . . , N ¤ u¢l¬rN¥mn}wxut^`;È¥kjÈGwN() ^mu¢¶ t^¯x`D{on}u¢q'`Dl¬u¢t n}[^`rN{}^`'{ e, f, g, a, b, c, d ¶ Qlpu¢t^7n}[^`lp`D«Rs^`Dtq` M (a), . . . , M (a) ¨çrx{ np[^`muvw_¬`n}`'{%^{prN¥^¦¢`'_ o o n ¦¢`DwN^l npr max n Z\[ksl ¤ `[wEx` = = 1¶ ¤ [^`'{}`DwNl max opt 1. 0. opt 7. . 1. D(Mi (a)) D(Ni∗ ). 4 2≤i≤7 o n 1+ opt o n D(N ) i (a)) /max2≤i≤7 D(Ni ∗ ) max2≤i≤7 D(M D(Ni∗ ) i. 2≤i≤7. =. 4 1+ .. 7 D(Niopt ) D(Ni∗ ). . ' Ú tn}[^u¢lQw2`'{ ¤ ¡`,[wEx`u¢tRnp{}rk^sq`;ªwbt`' _¬`Dwxlps^{}`enprqDwmn}s^{p`en}[^`u¢_%HwxqÍn6r¨ np[^`u¢tq'{p`D_%`DtRn}w¦ q'rxtlonp{ywu¢tRn rxt.np[` «Rsw¦¢uÎnJjr¨
(98) np[` lorN¦s^npu¢rxtl `[wEN`,slp`Dnp[^`%w^{prRwxqy[Çn}rlonpsmj.w#muvw_¬`n}`'{ {prN¥^¦`D_ ¤ ¥^smnÊn}[^`Qwx^^{}rNwxqy[bu¢lÊx`'t`'{yw¦^wtH%¶qDwÉ t¬¥2`slp`Dbnprlonps^j rn}[^`'{rxmn}u_¬u*)Dwn}urNt¬^{prN¥^¦¢`'_#l ¶ c6s^{Q_¬wxutªq'rx_¬^¦¢`ÅmuÎnJj{p`;los¦Înuvl\np[Hwnn}[^`^{}rx¥^¦¢`'_Ðr¨XqrNtlJn}{psHqÍnpu¢t^#wtªrNmnpu¢_#w¦Ûu¢tq'{p`D_%`DtRn}w¦ lp`D«Rs^`Dtq`%q'wxt^t^rn¥2`%lprx¦¢x`;7w^{prEÅmu¢_¬wnp`D¦jut·2rx¦¢jRtrx_¬u¢wx¦ n}u_¬`¬ unp[¹wxt¹wx^^{}rEÅku¢_#wn}urNtª{}wnpu¢r Ú tnp[^`¬^{}rmq`Dl}lrx¨m`Dx`'¦¢rxut^wÇx©Áwx^^{}rEÅku¢_#wn}urNt.wx¦Nrx{}uÎn}[^_Шçrx{ ¦¢`Dl}lQnp[wxt·#st^¦`;lpl n}[^uvl{prN¥^¦`D_ ¤ `¡^{}Prx=`D,Nnp[^P`\¶ lorN_¬`' [wn
(99) lps^{}^{puvlput^6{p`;los¦În np[wnn}[^`\{p`D¦¢wnp`;,`DqDq`'tRn}{puvqunJj,^{}rx¥^¦¢`'_ qDwtÇ¥2`lorN¦N`D¬rx^npu¢_¬wx¦¦¢j¬utHrN¦jkt^rN_¬u¢wx¦^npu¢_¬` Z\[^`wtHw¦¢jklpuvlÊr¨ÛrNs^{¡©Äw^^{}rEÅmu_#wnpu¢rxt¬w¦¢xrx{}unp[^_ uvl6npu¢x[Rn ¤ lor{p`;msqu¢t^np[^`¬Rw¥2`nJ ¡`'`Dt7n}[^`¬s^¶ ^2`'{¥HrNs^t¹r¨¡wxtªn}[^`¬¦r ¡`'{e¥HrNs^tr¨\ u¢¦¦ {}`D«Rs^u¢{}`6`DuÎn}[^`'{Qw%t^`D wx¦Nrx{}uÎn}[^_&rx{ w#lonp{}rxt^N`'{¡¦r ¡`'{\¥2rxs^tH ¶ . ÜÜ ê76/ 3.-98.
(100) E. #. ÂÀ´Í¸»ºR® . # À ±°p½´Í¿ #. ½'¿Ä½°y´ # $ m ¸ }À²m ¿. \
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