56 !#"$% '&)(*,+-/.0214365#7398:;
-=<?>A@?B+C-DCEGFIHJ!CKL#MN=OQP.,RCS T#UVWXCY G = (V, E) O[Z?\ (i, j) ∈ E>^]`_ c !bac d6e;S?%bA/gf.6h#e>^ij s L`k[l>M6md=>i#j#n>A#?opq %#N>C#3# (2.2) >Ahrs#tuOvdCw>,abxy!ez={.RS^M :#|#{ X c x

6ijij ij}(i,j)∈E~X

X

xij −{j|(i,j)∈E}

xji ={j|(j,i)∈E} n − 1,
 −1,i=s(6.4)∀i ∈ V \ {s}0 ≤ xij ≤ n, ∀(i, j) ∈ E.

 .#3
 6.1. WX?Y G !#9i=j s L`k)Z?i=j i ∈ V \ {s} n>bAP#/M%&^>OvSM?V&v/&ij s L'k i n>^P?bv-$=3A+!\ (s, i) o G !
3/C>]_o c = +∞ Oe/M,%N/O.C#e 6.1 6boA/S^TU=Vsi b ¡¢£u¤,¥¦§©¨«ª=¬ o^­®M% ¯!# (6.4) >(*+?-6°±uo²uv³=md"RSCv/a,x´V

6.1?µ ¶#·¸¹º . S »3>&¼6=½,o6¿¾À¼?L?Á?ÂuvSCv`Ãx :¹º 6.1. i#j s Lk k n/>#PM==3Ai#j k n/>?>ÄÅ,^P??/M%V¹º 6.2. = P = (s = i , i , . . . , i = k) Pi?j k n#> 66-,kÆ#3ÇZ ` = 2, 3, . . . , h−!6È,v9d P = (i , . . . , i ) ij i n>A/./0%V11k`

1

`2h`

¹º 6.3. É#Ê >^i/j i ∈ V n/>#ËÌuo d(i) .CÍ/MOCRA3CbA P P?i=j k n>#=.0/%6Î q >AÏÐ=ÑCÒ }~ #3kd(j) = d(i) + cij ,∀(i, j) ∈ Pk .

(6.5)



57qpQsspqkrk

=½ 6.1 O½ 6.2.

6.2:> >o;cd6½ 6.1 ovS6vÃ?x´V/N#3Aij k n/>Ab^oÍvAd=/S
M6P3=N!9  Q P `S)d=S^M6V&9vC36.2

X

cij < 0(6.6)

P 1,Î?-$==3 Q o8`%- ¾ 3i=j p L k  k nL x6PË#Ì9¾-1SAM6VA.3 (6.6) P 1 dC<`x´.vbÃCxÀL, Q o!M%#",O!i#j k n>^ËÌ ¾«-,%9>C.39i=j k n>Auoe q %#NO ?.bR#S TUVw>½ 6.2 & $%=&! M,NOQP.bRCSM6V 6.2 > '> !?"#?d#3Aij k n/>A#?oOv9)3 (Ò P = (s, . . . , p, . . . , q)  q n/>?#!?-,cgd=P = (s, . . . , p, . . . , q, . . . , k)
&>,O)?euvSCv`Ãx V?M,%O3(i,j)∈Qqk

X

cij <Xcijo*ÎM9i=j s Lk q n=>?# P = (s, . . . , r, . . . , q) PCvSM6V#v6ÎPuc)d3 P >)+-,
1Æ! P o8%ij k n>bA P = (s, . . . , r, . . . , q, . . . , k) (i,j)∈P(i,j)∈Pq

q0kX

cij <XcijO-ucd3 P P#/.=0%#NO !.-/vS^M6V
=½ 6.3 > (6.5) PÏÐ }~ .0%NO=½ 6.2 L`k109!#2L6`SM6V#ÑÒobv/SvÃx :S3 P = (s = i , . . . , i = k) OgM9=3 d(i ) = 0 -=>6.
(i,j)∈Pk0(i,j)∈Pkkk

1h1d(k) = d(ih ) = (d(ih ) − d(ih−1 )) + (d(ih−1 ) − d(ih−2 )) + · · · + (d(i2 ) − d(i1 ))58qkPkspr

Plt

6.3:l

½ 6.4.

P 1 0#S^M6V,&9v& (6.5) P*Î_[==3Z#\ (i, j) ∈ P !È,v9d d(j) − d(i) = c O- 13ij

kd(k) = cih−1 ih + cih−2 ih−1 + · · · + ci1 i2 =P  ky'S^M6V

X

cij(i,j)∈Pk

Cµ ¶· . ½ 6.1 !,a1 3  3ZCij k n/>A P !?#P uS/dC-#,&^>O?eM,%N/O!/v=S^M6V_d/3 P o\,> 2O =3kK

T =[Pk ⊆ Eo ev=SCv`Ãx V\  T CZ?## P oÃx »! /M,%NO).  P oAÄÅ O)
evdu& Ê O uk-N/O! Ê W=X?Y G >  o¼ ¾%#NOQP=.bR#S^M6V
! T P  /.-=bO Mb%O 3CZ?#!#P-;S-=N=OgL'k3 6.3 !&/M
axQ! >##!ba6cd  Q P?° '_['%A6»=.MVi=j k, ` n>o? O M3/½ 6.2 !
P = (s, . . . , p, . . . , q, . . . , r, . . . , k), P = (s, . . . , p, . . . , t, . . . , r, . . . , `)a1´3/> (Ò P = (s, . . . , p, . . . , q, . . . , r) O P = (s, . . . , p, . . . , t, . . . , r) >9]_`OC&!
!v¾ -21)SM6Vv6Î^Pbcyd3 P o P O=#Î P = (s, . . . , p, . . . , t, . . . , r, . . . , k)d(r)&QO-¿1 3 T = (T \ P ) ∪ P OgM,%NO).  Q "! #'_Q'SAM6Vk∈Vkkk`0rrk0k0r0k¹º 6.4. MCm=d/>ij k n>!a?c^d=W=XY G > uo° .R%V

59q
λq

P
λ

1-1sp- λspspk

k
k

1- λ

λ-1

1- λ

P’rr

6.4:

½ 6.5.

µ¸#¹ . Z?i=j k n># P o c d= (6.4) > x oÍuvd ,SCv`Ãx V## P 9ÄÅ-=>6.3CZ?\ (i, j) ∈ E o  vLC8 1 STU/VN.=3
>uOR 1, (i, j) ∈ P
δ(P ) =
 0, (i, j) 6∈ P >uOR3O[e q /=3 x ∗kkkkk∗x∗ij =X

δ(Pk )(6.7)! a6c duky%N=OQP-,#L¿1 SAM6V!3AZCij k n/>A# P o#,%g*,!9#=Mb%>6.v`Ãx´L  (6.4) 9|=J69 (2.1) >Ahrs/CtC./0%9Î q 3 !# }~ O $}~ vALT'ky/d#";k»?3 x P  boO%#N=OC&bvdSM6V9!! - boOk-P?/M%OM//3 6.4 > ! =Ma,x!0/%#"#Ðj k $%=j s L`k  Ä&>Axo ,>!'uo^8vd(=$)%C».M6VbN>+*,>!,3 ' >babxQ!i=j k n=.-[¼>9^ P = (s, . . . , p, . . . , q, . . . , k), P = (s, . . . , p, . . . , r, . . . , k) P?v3=^ 6! λ, 1 − λÄ&#P?C`%#N=O!?-1SM6VN>bOR3ij s L`k k n  Ä&>?uo 7;%9>?!ÏÐ(> 0)-=Jk∈Vkij∗
ij

0.#M6VvL,v93

c(λ) = λX(i,j)∈Pc(λ) ≥ min


 X


(i,j)∈PX

cij + (1 − λ)cij ,cij(i,j)∈P 0

X(i,j)∈P 0cij


,0 < ∀λ < 1,

60

P 1 ¼,NOL k 3-Q¼>9^ P , P >2x0^ =!$  Ä&o^uv9du&/J=#uv
S TU ; #>9 />,#Pij k n>A##=.#M6V,N>#L k ,Lu%9>C : /3 uo ,»#3 6.4 !9P??=M=3^i#j s L`k[Zi#j k n=>#|J>b/3¼S1# P P##/Mb% ; vd3=N!  Ä &o^/MNO.  k['%# !`& (6.4) >9 !
O-b%3,N/Og.#M6V¹º 6.5.  (6.4) !9#P#/M9/=3  ;b ¦ Og-,%  x P?M%V0k∗ Ng=L'k[­®=Mb%t X (Dijkstra’s algorithm) 3uA9/ P

6.2

cij ≥ 0,∀(i, j) ∈ E(6.8)

o *uv-$uv2¾!v=STU/VvALbv3/?>@B9#9b>0k"u%$#% !&>&'.&)(?dSMV (#*'Og-,%)*+?3-,/.#j s Lk10&2!34> o657d=R 3^ij s L'k)89,!Z#ij i n>ËÌ,>:'_Î/X ¦ d(i) o/; 1y¼$,%NOg.M6V,N6>ËÌ
X6 ¦ !C#3* ?>AËÌ><:`_ybÎ=>@?/AB (permanent label) O3^>?#Ë?Ì>6:_[uÎDCEF?<AB (temporary label) P0 1 3i=j  V 6¯G,>;kybÎ P O  G>;2k[,Î  T !9ÒH'_[ S^M6V¥¦§©¨«ª¬ bS »3 P = {s}, T = V \ {s} Ovd3i=j s !-I&JuX< ¦ d(s) = 0 o<;
1 36l>ij j !?!KX/ ¦>bOCR c , (s, j) ∈ E
d(j) =
 ∞,  xÀ.-bO?R3o6; 1SMCVZL&M/.3-IJ/3!K!LL ,2kQ»#3=<?>/X/ ¦ d(j) &^3,.j s L'k1NO'63-PQ P ¸RS!TU VXW Y6Z j nL x ,>!'9.Cb>]'_(a)oÍuvSAM6V ¥¦§©¨«ª¬ #3?!KX< ¦ >ij  T >'L k['&)| _^ËÌX ¦ d(i)>^i/j i o =U. d(i) o<IJX ¦ Ov63 i o/[j OgMb%\,> :sjE(i) = {(i, j) ∈ E | j ∈ V }61

> Z=j>/X< ¦ P (a) o!/¼a,x!;v=SM6VM?m/d>Aij!&I&J;X ¦ P;kQbÎK
j=3#¼S;1 P = V O#O-ucÎbk6.MVB  ?¸^¹ .  f> ¥  gX9!c d ¥A¦§ ¨«ª¬ OvdbS=O q Î&>P w,> DIJKSTRA .0% :algorithm DIJKSTRAP := {s}; T := V \ {s};d(s) := 0; pred(s) := 0;
if (s, j) ∈ E then d(j) := csj

O pred(j) := s o{M,%else d(j) := ∞;while P 6= V do begind(i) = min{d(j) | j ∈ T }o*,ÎM6i=j i o ! ;{P := P ∪ {i}; T := T \ {i};for

Z (i, j) ∈ E(i) do

if d(j) > d(i) + cij then d(j) := d(i) + cij

O pred(j) := i o#/M,%{ij>  }X ¦ >" }

end
end;

&Z LM. pred(i) ?3^]`_ d(i) >^=.ij i >¯!"$?%gijo&%,v=Muax!#_;S^M6V#vCÎAPcgd K!C#3=N>%('o=&#O^!Z?ij i Lk<,/.j s S.=>^##,o !
) %N/OP.uRCSM6V
+* 6.6. DIJKSTRA /WXY G = (V, E) > À¼>6ij s ∈ V L kgl>Mmd>6ijn/>o!b%Vi ∈ V \ {s},.- : /#0 ,!ba#c d(1 2vSCv`Ãx V

0% LM=.3ZuX/ ¦ d(j)  (a) o)*Î=v93//PIJX ¦ 3¼bS1 j ∈ P -;k)=3

62

, .j s L'k j n9L x *#>Ab>]`_oÍuvd//%/&A>uOQ#e=Mb%V6w>XL&M/.3(b)d(i) = min{d(j) | j ∈ T }(6.9)

o*ÎM9ij i P #63 P := P ∪ {i} O _Q%gP=3=N>Aij i >=X/ ¦ d(i) P  ,/.#jL'k^>A*?>A#Ë#ÌOg-,cd=/%N/Oo#T=3 (a), (b) P *'_[d1 2 !6.
s0/%V,N),!?3 ,/.=j s L`k i n> É#Ê >b^ P >]`_)P d(i)  f2O -,%NOoT#
a#V^ P o^ij s ∈ P L`k ) cÎbOCRA3A! $?% T >ij,o k OgM9/=3 (6.9) a1 3d(k) ≥ d(i).=0%AV3Ce (6.8) a1´3i#j s L'k i S .> P !;Sg%)\,>];_MCmd  ./0%V,N9> -[¼CL k 39^ P >]`_ d(i) a1´ ¾«- 1y#-=NOQP-,L%Vw! DIJKSTRA !9,>+, b<^@#$> ,>6P#LL%9>Lbo !bvd S?v`Ãx V
/><LM=.#3i=j i > ! |T | ! */M% ,>  ;•i=j i ! #/Mb%ij j >/X/ ¦ d(j) >"! |E(i)| ! *Mb%  ,>uv  O  •P?ÏÐ=.M6VLM/ V >ij> $ ,C`%9>C.3bN)k   /./ 
1 + 2 + · · · + (n − 1) = (n − 1)n/2,

X|E(i)| ≤ mOg-1 S^M6VA,6W=X#Y>\>^*  m ?3! W=XCY.u&  m = n(n − 1) !"#?-#>9.
.MVvÎ9Pc d/3&=t X , n ! */M,%$ >(#*&%;o'?#m≤n
6Mb%N/OQP ,L 1SM6Vi∈V2(*) 6.1

2¼ ¾´,+,-t.0/to`cd=3?¼ ¾´#L'k21&3465'7 1,150 8:9 ; 7<>=?,@ACBD,E

FHG EI'J&KMLNPOQR