M p f(p, q) = (p + q) O(1)

l k

M = E, I S = {S,..., S t } E S i = p i {,..., t} S S q S Y E q X S X Y = X Y I X S X Y = X Y I S q S q q p+q p q S q p i O q S pq p i O S 2 p q q p+q p q p+q p fp, q AM S O fp, q p + q p p+q p AM M p fp, q = p + q O 2 Ok k A M S O M = E, I p + q = k A M F S = {S,..., S t } E S i = p i {,..., t} q S i=0 i=0

S p+q p+q p O p tp ω + t p+q ω q F ω < 2.373 S = {S,..., S t } n S i = p {,..., t} q x 0, O x q 2 op+q t n q S S S x p x q 2 op+q n p q C F, χ F U n χ U p 2 F A U p F χa A F A U p B U\A q F χa A F F B = F, χ F χa A U p n p q F F, χ χa A U p l l M,..., M l k k l = 2 l 3 S 2 Ok k A M S O k n m k 2k 2 Ok nm Ok 2k 2 Ok nm n Ok 3k nm n/ n O6.75 k+ok mn 2 n n Ω fn n f

k G n m k G k Ok!nm Ok!2 k n Oc k n n c O4 k+o 3 k nm O2.69 k n n

? =? =

Σ {0, } Σ Σ L Σ N x, k L k x +k 3 3 ϕ 3 3 ϕ, k k k ϕ ϕ fk L Σ N A f : N N c

x, k Σ N A x, k L O fk x, k c A

R 3 v = 0, v 2 = 0, v 3 = 0 0, v 4 = 0 0 0 v 6 = 2 0 0 v 7 =, v 5 = v,..., v n λ v + + λ n v n = 0 λ = = λ n = 0 v, v 2, v 3 v 2, v 4 v 2, v 3, v 5 v, v 2, v 3, v 5 A B B A v B A A = {v, v 2, v 3 } B = {v 2, v 4 } B {v } 0 0 0 0, G V, E V E

S G V G, S {e, e 2, e 3 } {e 2, e 4 } {e, e 6 } {e 7 } A B A A B A = {e 2, e 3, e 6 } B = {e, e 5 } B {e 2 } M = E, I E I E I A A A I A I A, B I A < B e B \ A A {e} I I E M I B, B 2 e B e 2 B 2 B \ {e } {e 2 } M rankm

M = E, I B B 2 B < B 2 e B 2 \ B B {e} I B I 3 G = V, E M G M G EG A EG V G, A A, B EG A < B C V G, A V G, A B C A B B V G, A V G, B A A < B e B V G, A A {e} M G G E n E I = {A E A k}. I A I A I A A A, B I A < B k A {e} I e B \ A A {e} < k + E, I U k,n U k,n k k E k

U U P = {U,..., U t } U t i= U i = U U i U j i j U i E t E,..., E t t k,..., k t I E I = { A E A E i k i i {,..., t} }. M = E, I E i = 0 k i i {,... t} I A I A A A I A E i k i i {,... t} A E i A E i k A I A, B I A < B i {,..., t} A E i < B E i e B \ A E i A {e} I M = E, I E k,..., k t E v e F E I A E {v e e A} E, I A, B I A < B {v e e A} A {v e e B} B b B v b {v e e A} A {b} I E, I M e E e v e M M = E, I F M = E, I F 2 M M e = i, j 0 i j F F

U k,n GF s s > n E = {e, e 2,..., e n } U k,n e i α i GF s v i =, αi, α2 i,..., αk i e i k A E A > k {v i e i A} A E A = k {v i e i A} k k M A 2 3 n 2 2 3 2 n 2 2 k 3 k n k M A α j α i, 0 i<j n e i A,e j A GF s M A {v i e i A} GF s s > i k E i + E,..., E t M = E, I, M 2 = E 2, I 2,..., M t = E t, I t. M M t = E, I E = I = { X E X E i I i i {,..., t} }. t i= E i

M M t I A I A I A A A, B I A < B A < B j {,..., t} A E j < B E j A E j I j B E j I j e B E j \ A E j A {e} E j I j A {e} E i I i i j A {e} I U E,k, U E2,k 2,..., U Et,k t E = t E i i= A i {,..., t} A E i I i A E i k i F M i M i = E, I i i {,..., t} M 0 0 0 0 M 2 0 0 M = 0 0 0 M t M M t M,..., M t F M M t t t M = E, I M = E, I A E M A M A t M I = 0 t M A I A A A I A t A I A A t A I A, B I A < B A, B I A < B t e B \ A A {e} I A {e} t A < t A {e} I M M GF s t t M M F px,..., x n F[x,..., x n ] d S F α,... α n S S pα,..., α n = 0 d S.

s xt x 2 M r M r n R t r s GF s M RM t M t n RM t t M M X E X = t M 0 r t M X B 0 = RM 0. X M M 0 B 0 X M X M M 0 M 0 0 [ RM 0 0] > 0. R tr detrm 0 t [ RM 0 = 0] t s x. x RM t M M = E, I,..., M t = E t, I t E = t i= E i I = { I I t I i I i M M t = E, I, i {,..., t} } E, I E,..., E t I A I A I A A A, B I A < B A = t i= A i B = t i= B i A i, B i I i i {,..., t} A,..., A t B,..., B t A < B j {,..., t} A j < B j e B j \ A j A j {e} I j e / I i i j A {e} I E,..., E t M = E, I E f : E E I = {fi I I } M = E, I f = I A I B A A I A I fa = A e A f e A B = {e A fe B} B = fb B A B I B I

A, B I A < B A fa = A B fb = B A = A B = B A, B I A < B e B \ A A {e } I A fe I fe B \ A M = E, I E,..., E t E,..., E t M i = E i, I I i = I i E = t i= E i M = E, I I = {I I k I i I i} M f : E E E = e M t i= E i fe = e e E M = E, I M 2 = E, I 2 E = {a, b, c, d} I = {, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {b, d}, {c, d} } I 2 = {, {a}, {b}, {c}, {d}, {a, b}, {a, d}, {b, c}, {c, d} } M = E, I I 2 I I 2 = {, {a}, {b}, {c}, {d}, {a, b}, {c, d} } {c} I I 2 {a, b} I I 2 {a, c} / I I 2 {b, c} / I I 2

S = {S, S 2, S t } E S p p S i = p i {,..., t} E S p E S S q S Y E q X S X Y = X S X Y = S {2, 4}, {, 5}, {, 6}, {, 7}, {3, 6}, {3, 8}, {4, 8} {4, 7}. S = { {2, 4}, {, 5} } S S Y [8] Y = {2, 4} Y = {, 5} Y = S = { {2, 4}, {, 5}, {3, 6} } 2 S S = { {2, 4}, {, 5}, {3, 6}, {, 7}, {3, 8}, {4, 8} } 3 M = E, I S p E S S q q S Y E q X S X Y = X Y I X S X Y = X Y I S q S S q rep S M p + q

M = E, I S p E Ŝ q rep S S q rep Ŝ S q rep S Y E q X S X Y = X Y I Ŝ q rep S X Ŝ X Y = X Y I S q rep Ŝ X S X Y X Y I S q rep S M = E, I S p E S = S S k S i q rep S i i k k i= S i q rep S Y E q X S X Y = X Y I S = S S k i X S i X S i X Y = X Y I k i= S i q rep S k i= S i S S 2 S S 2 = {S S 2 S S S 2 S 2 S S 2 = } M = E, I k S, S 2 E p p 2 S k p rep S S 2 k p2 rep S 2 S S 2 k p p2 rep S S 2 Y E q = k p p 2 X S S 2 X Y = X Y I S S S 2 S 2 S S 2 = X S S 2 = S k p rep S S S S S 2 Y = S S 2 Y I S 2 k p 2 rep S 2 S 2 S 2 S S 2 Y = S S 2 Y I S S 2 S S 2 S S 2 Y I S S 2 k p p2 rep S S 2 q p q p+q p ω 2.373

M = E, I p + q = k A M F S = {S,..., S t } p E q S S p+q p O p+q p tp ω + t p+q q ω F q p+q p RA CA A = α i,j I RA J CA A[I, J] = α i,j i I, j J A I J I [n] Ī = [n] \ I I = i i I n n A J CA = [n] A = I [n], I = J I+ J A[I, J] A[Ī, J]. A M M rankm rankm t k p S = S t > k p e E xe F k A M S i S s i = s i [I] I [k] p s i [I] = A M [I, S i ] s i F p k k p t HS = s,..., s t s i W rankh S H S W s,..., s t W = rankh S k p S = {S α s α W} S = l W = S l k p S = {S i i l} W = { s,..., s l } S q rep S S S S / S Y E Y q S Y = S Y I S S S Y = S Y I Y = q S Y = S Y = p+q = k S Y I A M A M [RA M, S Y ] 0 s = s[i] I [k] p s[i] = A M[I, S] y = y[l] L [k] q

y[l] = A M [L, Y ] γ s, y = I [k] p k p = k k p = k q I+ J s[i] y[ī]. γ s, y = A M [RA M, S Y ] 0. s W s = l i= λ i s i λ i F 0 γ s, y = I I+ J s[i] y[ī] = I+ l J λ i s i [I] y[ī] I i= l = λ i I+ J s i [I]y[Ī] = i= I l λ i A M [RA M, S i Y ] i= γ s, y 0 S S S S = {S i S i S λ i A M [RA M, S i Y ] 0}. l λ i A M [RA M, S Y ] 0 S i= A M [RA M, S Y ] 0. S Y S Y I S q rep S Y = q < q M k = p + q Y Y q S Y = S Y I S S S Y = S Y I Y \ Y S Y = S Y I S q rep S S q rep S H S S i s i A M [I, S i ] I [k] p p p Op ω p+q H S O tp ω W k p t HS O t p+q ω p W p+q S q rep S O p tp ω + t p+q ω q p

rankm = p + q rankm > p + q p + q M M = E, I rankm > p + q A M F S = {S,..., S t } p E q S S p+q p F O p+q p tp ω + t p+q q ω q F A B F F A F B F = U n n p q n p q C F, χ F U χ U p 2 F A U p F χa A F A U p B U\A q χa A B C F A U p χa n p q F U n A χa C = F, χ n p q F τn, p, q τ Q n, p, q χa A U p C ζn, p, q n, p, q

x 0, n p q ζn, p, q 2 O p+q p+q x p x q τ I n, p, q 2 O n, p, q 2 O p+q p+q x q τ Q n, p, q 2 O p + q O n p+q p+q x p x q p + q O n p+q p+q x q p + q O n n p + q O n U n S p U q x 0, O x p x q 2op+q n n + S x q 2op+q n, q S S S x p x 2 op+q n q S x p x 2 op+q n S = S q S > x p x 2 op+q n C = F, χ q n p q S F S = S S S = F A S χa F χa A S S A S S F x p x q 2op+q n, A S F S C = F, χ x p x p+q O 2op+q n n A S q χa x q 2 op+q p + q O n χa x 2 op+q p + q O n q O x p x q 2op+q n n + S x q 2op+q n S q rep S A S B U B = q A B = A S A / S.

A S A B = C = F, χ n p q χa A B A / S F F A A S F χa A F F B = A F A B = A S n p q F χ F, χ n p q x 0, n p q ζn, p, q = O p + q + n x p x q 2 τ I n, p, q = O n ζn,p,q x p x n Op+q q n, p, q = O x q p + q + n τ Q n, p, q = O x p x n O q n p q C = F, χ t = x p x p + q + n F = {F q,..., F t } U n F i U U F i x F i U F A U p χa = {F F A F } C = F, χ C n p q A U p B U\A q i {,..., t} A F i F i B = = x p x q. F F i A F i F i B = x p x q t e p+q+ n = n p+q+. n p n p q A U p B U\A q χa F A χa A

B F n p+q+ n p n p n p q +... n = q np+q+ p!q! n p q +... n n p+q+ np+q n p+q+ = n. C = F, χ A U p i {,..., t} A F i = x p χa t x p χa t x p χa E [ χa ] = t x p = p + q + n. x q χa 6E[ χa ] 2 6E[ χa ] n p+q+ A χa 6E [ χa ] n q+ n. C = F, χ n p q 2 n > 0 2 n 0 F 4 U F F t n p q 2 n ζn,p,q O t n Op+q = O x p x n Op+q q F F A F F χa F k U V n r {f,..., f t } f i : U V i {,..., t} k k S U S = k i f i S U n k {f,..., f t } f i : U [k 2 ] i {,..., t} t = Ok O n Ok O n n

n p q F, χ τ I n, p, q ζn, p, q n, p, q τ Q n, p, q n p q ζ n, p, q ζ p + q 2, p, q p + q O n τ I τ n, p, q = O I p + q 2, p, q + ζ p + q 2, p, q p + q O n n n, p, q p + q 2, p, q p + q O n τ Q n, p, q = O τ Q p + q 2, p, q + p + q 2, p, q p + q O n. U n p q p+q {f,..., f t } f i : U [ p + q 2] i {,..., t} p + q 2 p q F, χ F [ p + q 2] S U f i S = {f i s s S} T [ p + q 2] f i T = {s U fs T } Z U f i Z = {f i S S Z} W [ p + q 2] f i W = {f i T T W} n p q F = f i F i {,...,t} A U p χa = i {,...,t} f i A = A f i χ f i A. C = F, χ n p q F χ f i A f i A F A F F χa A U p B U\A q χa A B {f,..., f t } p + q i {,..., t} f i A B F, χ p + q 2 p q χ f i A f i A χ f i A A B f i f i B f A f i i χ f i A χa χa A B O p+q O n n p+q O τ I p + q 2, p, q F ζ p + q 2, p, q O ζ p + q 2, p, q p + q O n n

F F τ I n, p, q F F p + q O n ζ n, p, q n, p, q χa i t f i A = A χ f i A p + q 2, p, q p + q O n, χa i {,..., t} f i A O p+q O n i f i A χ f i A O τ Q p + q 2, p, q f i χ f i A O χ f i A p+q O = O p+q 2, p, q p+q O χa O p+q O n χa O τ Q p + q 2, p, q + p + q 2, p, q p + q O n τ Q n, p, q n p q p q [n] U P = {U,..., U t } [n] i {,..., t} x y z x U i z U i y U i Pt n [n] t t n + t Pt n =. t Z p s,t t p, p 2,..., p t Z t p i = p i= 0 p i s i {,..., t} s t Zs,t p p+t t Z p s,t P p t [p] t p t p, p 2,..., p t p i p q s = 2 p + q t = p+q s n p q F, χ τ I n, p, q ζn, p, q n, p, q τ Q n, p, q n p q ζ n, p, q P n t p,...,p t Z p i= s,t t ζn, p i, s p i t

τ I n, p, q = O ˆp s,p s ˆp q n, p, q n, p, q = P n t Z p s,t τ I n, ˆp, s ˆp + ζ n, p, q n O p,...,p t Z p s,t i= t n, p i, s p i τ Q n, p, q = O n, p, q n O + Pt n Zs,t p t τ Q n, ˆp, s ˆp. ˆp s,p s ˆp q s = 2 p + q t = p+q s U = [n] n ˆp s ˆp Fˆp, χˆp ˆp 0 ˆp s ˆp p s ˆp q U U A B A U = {A U A A} A B = {A B A A B B}. n p q F, χ F = F p U F p2 U 2 F pt U t A U p U,...,U t P n t p,...,p t Z p s,t i s p i q χa = [ χp A U U χp2 A U 2 U 2 {U,...,U t } P n t p,...,p t Z p s,t U i U i A =p i,s p i q χ p A U U ]. F, χ n p q A U p B U\A q {U,..., U t } U i {,..., t} A B U i p+q t = s i {,..., t} pi = A U i q i = B U i = s p i p i p q i q F pi, χ pi n p i q i i i χ pi A U i A U i B U i F i χ pi A U i A U i F i F i B U i = F = t i= F i U i A F B F = χa F χa χa A B O τ I n, ˆp, s ˆp ˆp s,p s ˆp q n ˆp s ˆp ˆp s, p s ˆp q F n O τ I n, p, q

ζ n, p, q n, p, q χa {U,..., U t } Pt n p,..., p t Z p s,t p i = U i A p s p i q i {,..., t} i {,..., t} χ pi A U i O τ Q n, p i, s p i χa n O τ Qn, p, q O n, p, q n O + O O n, p, q n O + P n t Z p s,t n, p, q n O + P n t Z p s,t t {U,...,U t } P n t p,...,p t Z p s,t U i U i A =p i,s p i q p,...,p t Z p s,t i s p i q [ t i= [ t i= p,...,p t Z p s,t i s p i q ] τ Q n, p i, s p i ] τ Q n, p i, s p i τq n, p i, s p i O n, p, q n O + Pt n Zs,t p t τ Q n, ˆp, s ˆp ˆp s,p s ˆp q, U n x x n A U p F = {F U F = p} χa = {F F A F }. χa χa = {A} F, χ n p q F = n p n p n x n p q F n p A U p χa = x n F = {F U F = n q} A U p χa = {F F A F }.

χa χa = {U \ B} F, χ n p q F = n n q n q n x n p q F n q A U p χa F x q, x, x > n n p q ζ n, p, q = O x p x q τ I n, p, q = O n, p, q = O p+q+ x n q τ Q n, p, q = O p + q + n 2 n ζ n,p,q x p x n Op+q q x p x q n O = O2 n n O ζ 2 n, p, q = O x p x q p + q O n τi 2 n, p, q = O τi p + q 2, p, q + ζ p + q 2, p, q p + q O n n 2 2p+q2 p + q O = O x p x q p + qop+q + x p x q n n p + q Op+q = O x p x q 2 2p+q2 + n n 2 n, p, q = O x q p + q O n 2 τq 2 n, p, q = O p+q 2 + x p + q O n q s = 2 p + q t = p+q s

ζ 3 n, p, q P n t s x n s n Ot τ 3 I n, p, q = O = O n Ot Z p s,t p,...,p t Z p i= s,t n Ot p + q Ot n O p+q 2 p+q p s,p s p q p s,p s p q t ζ 2 n, p i, s p i p,...,p t Z p s,t i= t ζ 2 n, p i, s p i x p x q+s sot n Ot x p x q τi 2 n, p, s p + ζ 3 n, p, q n O s Os s 2 2 2 x p x s p + n n + ζ 3 n, p, q n O p + q O 2 p+q = O x p x q 2 4 p+q + n n + + n O p+q 2 p+q x p x q s x n Ot 3 n, p, q = 3 n, p, q P n t Z p s,t n Ot p + q Ot n O p+q 2 p+q p,...,p t Z p s,t i= x q t 2 n, p i, s p i x q+s sot n Ot

τqn, 3 p, q O O 3 n, p, q n O + P n t Z p s,t t p p,s 3 n, p, q n O + n Ot p p,s s p q s p q 2 s2 + τqn, 2 p, s p x s p n O p+q 2 p+q O x q + n Ot s O n 2 s2 + x q n O O p+q 2 p+q x q s O n ζ 4 n, p, q 2 O p+q p+q x p x p + q O n q τi 4 n, p, q O τi 3 p + q 2, p, q + ζ 3 p + q 2, p, q p + q O n n O 2 4 p+q p + q 2 p+q p+q x p x q + 2O p+q x p x q p + qo n n 4 n, p, q 3 p + q 2, p, q p + q O n p+q 2O p+q x p x q p + qo n τqn, 4 p, q O τq 3 p + q 2, p, q + 3 p + q 2, p, q p + q O n p+q 2O p+q x q p + q O n s = 2 p + q t = p+q s ζ 5 n, p, q P n t p,...,p t Z p i= s,t t ζ 4 n, p i, s p i n Ot p + q Ot s Ot 2 O n O p+q 2 p+q 2 O p+q p+q st s n Ot x p x q x p x q+s

s x n Ot τi 5 n, p, q O p s,p s p q τi 4 n, p, s p + ζ 5 n, p, q n O 4 s s O s 22 O 2 s x p x q + 2O x p n n+ x q p+q + n O p+q 2 p+q 2O p+q x p x q 4 s s O s 22 O 2 p+q s x p x q + n O p+q 2 p+q 2O p+q x p x q 2 2 4 s s Os p+q O x p x q + n O p+q 2 p+q 2O p+q x p x q 2 2 2 p+q4 p + q O 2 p+q O x p x q + p+q + n O p+q 2 p+q 2O p+q x p x q p+q O n O p+q 2 p+q 2O p+q x p x q 2 22 p+q4 p + q O 2 p+q p+q 2 O p+q s x n Ot 5 n, p, q 5 n, p, q = P n t Z p s,t s s p,...,p t Z p s,t i= t 4 n, p i, s p i n Ot p + q Ot 2 O st s x q+s sot n Ot n O p+q 2 p+q 2 O p+q p+q x q τqn, 5 p, q O 5 n, p, q n O + Pt n Zs,t p p s n O p+q 2 p+q 2 O p+q p+q x q s p q τqn, 4 p, s p

ζn, p, q ζ 5 p + q 2, p, q p + q O n 2 O p+q p+q x p x q p + qo n τ I n, p, q O τi 5 p + q 2, p, q + ζ 5 p + q 2, p, q p + q O n n = O 2 O p+q p+q x p x q p + qo n n n, p, q 5 p + q 2, p, q p + q O n O 2 O p+q p+q x q p + qo n τ Q n, p, q O τq 5 p + q 2, p, q + 5 p + q 2, p, q p + q O n O 2 O p+q p+q x q p + qo n U n S p U q x 0, O p + q O n n + S x q 2op+q n q S S S x p x 2 op+q n q S x p x 2 op+q n S = S q S > x p x 2 op+q n q p+q {f,..., f t } f i : U [ p + q 2] i {,..., t} t = O p + q O n O p + q O n n f i i {,..., t} S i

[p + q 2 ] p q F i, χ i S i = F i A S f i A S A S i χ ia F χ i A F A S i F S A S i S = t S i. i= S i F x p x 2 op+q p + q q A S F S F x p x q 2op+q p + q p + q O n x p x q 2op+q n. S q rep S A S B U B = q A B = A S A / S A S A B = {f,..., f t } p + q i {,..., t} f i A B F i, χ i [p + q 2 ] p q F χ i A A F F B = A / S A / S i F F A A F χ i A A F F B = A F A B = A S j S S n U U S p + q n S p + q U = U U U S q U O S p + q U S p + q U n q S S U q U X S Y U Y q X Y = Y = Y \ U Y U \ U Y Z = Y \ Y Y Z = Y X Z = q X S X Z = Y X = X Y = U n S p U q x 0, O S x q 2op+q n

q S S S x p x 2 op+q S q U n S p U q x 0, O S x q 2op+q n q S S Ŝ x p x 2 op+q q S = S S 2 q rep S,..., S m q rep S m, m S m S m 2 S i S i 2 i {,..., m } S m Sm 2 S m q S S m x p x q 2op+q S m x p x q 2op+q 2 S m S m S m x p x q 2op+q a b a b a 2 a b 2 b S m S m S m 2 x p x q 2 op+q S m S m S m S m S m 2 S m x p x q 2op+q S m x p x q 2 op+q T S m T = = m i= m i= S i = O S S O 2 i x q 2op+q n x q 2op+q n x q 2op+q n

p x = p+q U n S p U q p + q q O S 2 op+q n q q S S Ŝ p+q p 2 op+q

l l A M,..., A Ml GF s M = E, I,..., M l = E, I l k S E S k S I i i {,..., l} l = l 2 l = 2 l 3 l = 3 l 3 3 G = V, E V L V R G M L M R M G E L E R E G E G M L X E V L X M R X E

V R X M L M R M G G X n M L M R M G X G G X M G X X n V, X T G G X M L M R T 2 2 G l 3 l k d A M GF s M = E, I rankm = kd d S E S S S = k S S I S S d O S O + k Odk GF s dk T S dk d ek d S O GF s S S S = k X I T T X S T T S S T S A S \ T B = X. X S \{A} S A B = A B = A X S \{A} X = X I. X S T dk rep S A T A B = A B I A B = A S \ {A}

T = S \ {A} {A } X = X T X S \{A} X A = B A I. T T S S T T T k ek d e d+k k d k k d+k k d k = k Okd k T k + d O T k Odk k + d O = k Odk T k l k O E O + k Okl GF s l d l A M,..., A Ml M = E, I,..., M l = E, I k l E,..., E l E i {,..., l} M i = E i, I i M i A M i M = M M l. A M M rankm = kl M d d = l x E S x = {x,..., x l } x i x E i S = {S x x E} S X E M,..., M l S x M M,..., M l, k l S S S = k S S x M S x S l O E O + k Okl x X

M,..., M l k GF s M,..., M l k M,..., M l GF s s l O E O + k Olk GF s k k 2 Ok n O D = V, A V = n A = m k k D k k D = V, A k U n,2k = E, I E = V I = {S V S 2k} u, v V P i uv = { X V u, v X, X = i uv D X } D D k u, v V X P k uv k rep P k uv D C X

C k C k C = v v 2... v r v D k r k r k r < r r 2k u = v v = v k P = v v 2... v k Q = v k+... v r Q k V P V Q = Pk uv k rep P k uv P X = V P P k uv X V Q = P P C C k X r > 2k u = v v = v k P = v... v k Q = v k+... v 2k R = v 2k+... v r Q k V P V Q = P k uv k rep P k uv P X = V P P k uv X V Q = P R P P P R C P P X V R = X V R v α P R P [v α, v k ] P v α v k { R vα = v = 2k+ R[v 2k+, v α ] R[v 2k+, v α ] R v 2k+ v α C = P [v α, v k ]QR D D C Q = k v / P [v α, v k ] P [v α, v k ] < P = P v α / R R < R k C = P [v α, v k ] + Q + R < P + Q + R = C C k C C P P C C k X

q P k uv P k uv D n m u V D U n,l = E, I E = V D I = {S V D S l} p {2,..., l} v V D \ {u} l p P uv p Puv p l 2 ol p O 2 ol m n i [p] { l l l i }. i l i P p uv D V D = {u, v,..., v n } D = V, A p n D 2,..., p v,..., v n D[i, v] P uv i l i rep Puv i D i = 2 D[2, v] = { {u, v} } uv AD i N i+ uv = w N v P i uw {v}, N v w wv AD D[i +, v] D Nuv i+ = D[i, w] {v} w N v D[i, w] w N v P uw i l i 2 ol Nuv i+ v l i 2 ol v i+ w wv AD N uv O v l i 2 ol i+ N uv rep l i+ Nuv i+ l l i+ O t n l i + t = v l i 2 ol Nuv i+ l i+ rep Puv i+ S Puv i+ Y Y = l i + S Y = S Puv i+ P = uα... α i v D S = {u, α,..., α i, v} α i N v P [u, α i ] P u α i Puα i i X = S \ {u} Puα i i Y = Y {v} X Y = Y = l i D P i uα i l i rep P i uα i

X P uα i i X Y = α i N v X {v} = X {v} = X {v} Nuv i+ S = X {v} Nuv i+ S Y = i+ i+ N uv = P uv l i+ rep Puv i+ D[i +, v] D i+ N uv P uv p P uv p p n l l l i2 O v j ol n i l i = O = O i=2 j= 2 ol n p n l v j i=2 j= 2 ol m n i [p] { l i l l i i l i } l l i. l i O8 k+ok mn 2 D D k u, v V D P V P P uv k k rep Puv k D C P P uv k k rep Puv k u, v V D u V D l = 2k p = k O8 k+ok mn n X P uv k uv P X X Q = P uv. k u,v V D X Q uv P X vu D X u v Om+n P X vu X Q k D P uv k 2k k 2 ok n 2 u, v Q n 2 2k k 2 ok n 2 4 k+ok O 8 k+ok mn n + 4 k+ok n 2 m + n 3 = O 8 k+ok mn 2.

P uv p q rep Puv p 2 p k q = 2k p x x = x x P uv p q rep Puv p 2 p k q = 2k p s p,q P p uv P p uv q rep N p uv q rep P p uv P uv p Nuv p p P uw Nuv p s p,q+ n s p,q+ s p,q s p,q+ s p,q P uv p q rep Nuv p q rep Puv p p p+q O Nuv p x q 2op+q n =O s p,q x q 2op+q n n =O x p x 2q 2op+q n n fx = x p x 2q x f x = 0 f x > 0 fx x = f x = 0 p x p x 2q + 2q x p x 2q = 0 p x + 2q x = 0 x = p p + 2q p p+2q f x > 0 fx x f x = x p x 2q p x + 2q x = fx p x + 2q x f x = fx p x 2 + 2q x 2 + f x p x + 2q x f x = fx p x 2 + 2q x 2 > 0 P uv p q rep Puv p x = p p+2q D n m u V D U n,l = E, I E = V D I = {S V D S l} p {2,..., l} v V D \ {u} l p P p uv P p uv 2l p p p 2l p l p 2 ol 2l 2p

O 2 ol m n i [p] { 2l i i i 2l i 2l 2i }. 2l 2i N j uv = P j uv x x j = j j + 2l j = j 2l j. s j,l j N j uv = P j uv s j,l j = x j j x j l j 2 ol. Pj uw s j,l j D[j, w] D j i w {v,..., v n } Nuv i+ = P uw i {v} w N v Nuv i+ s i,l i v x i i x i l i 2 ol v N i+ uv s i,l i x i+ l i+ 2 ol v n. 3 < i < p s i,l i e 2 i + s i+,l i+. s i,l i x i+ s i,l i s i+,l i+ = = x i i x i l+i x i+ i+ x i+ l+i+ 2l i l i + i+ 2l i + + e 2 i + 2l i + 2l 2i + l i+ i i 2l i+ i + + i 2l 2i l i i P uv p p n O s i,l i v j x i l+i 2 ol n =O i=2 j= 2 ol m n i [p] { 2l i i i 2l i 2l 2i } 2l 2i 2l p p 2l p l p s p,l p = x p p x p l+p 2 ol = 2 ol. p 2l 2p

P k uv k rep P k uv l = 2k p = k P k uv v V D \ {u} O 2 ok m n i [p] { 4k i i = k fk = i i 4k i 4k 2i } 4k 2i 4k i i i 4k i 4k 2i 4k 2i P k uv u, v V D O6.75 k+ok nm n P k uv u, v V D O4.5 k+ok O6.75 k+ok mn 2 k k G n m k k G G s V G G G k G k + s U n,k+ = E, I E = V G I = {S V G S k+} s, v V G P i sv = { X V G s, v X, X = i sv G X }. v V G Psv k+ Psv k+ k+ P sv 0 rep Psv k+ P sv k+ Psv k+ A Psv k+ k+ A = P sv 0 rep Psv k+ A k+ P sv A = P k+ sv

l = p = k+ { 2k + i i 2k + i } 2k+ 2i 2 ok m n i [k+] i 2k + 2i i = k+ P sv 0 rep Psv k+ v V G 5 k+ P sv k+ 0 rep Psv k+ v V G ϕ 2k+ok m 2 n = O2.69 k m n ϕ ϕ = + 5 2 P k+ sv G O2.69 k k+ m n v V G P sv G k + s G k k G k Ok 2 n Ok 2 n k k k k Ok 2 n k O2.69 k n n

t k O2.597 k n O