                                                            1

          
                       -᪠
    㤠⢥  பᬨ᪮ ਡ஥
     _________________________________________________________
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
                                
                                
                       
                 믮 ୮ ࠡ
        "   "
     
     
     
     
     
     
     
     
     
                                       ⠢⥫:   ન ..
                                                    ࡠ祢 ..
     
                                                                 
    㤥⮢    "᫨⥫ "
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
                          .-
                              1995

                                                            2

                                
                                
     騥 ⮤᪨  㪠 饭  ࠢ
 ஥஢   ᫨⥫  ⥩   樥
ᮮ饭     ⮢,     :   ஥஢   
।   ᫨⥫쭮          롮   ய᪭
ᯮᮡ⥩.
     ᭮ 襭    ।⢮ 롮
᪨ ⥬  ।   ਪ ணࠬ.
襭   ᬮ७  祡 ᮡ /1/. ணࠬ
ᯥ祭     믮  ୮  ࠡ  "⨬
  ।   ᫨⥫   ⥩"   ࠧࠡ⠭
㤥⠬ 㯯  4810 ᮢ  ਥ   ஬
  IBM .  ࠦ  ᢮ ୮.
                                   

       1.      
                

     稥 ⢠  ਥ 䥪⨢  㭪樮஢
᫨⥫ ⥩  ᫮  ਠ  ࠪ
   ஥஢.   饭    ஥஢
।   ⥫  䥪⨢,    
⨬樨,     ⠫   ⥫   ॢ      ࠧ
࠭祭.   ᨬ  ᭮ ⥫ 䥪⨢
ࠧ ᫥騥  ਠ ⠭    ଠ
஥஢ ᫨⥫ ⥩:
     -  ⥧        ६  -  ।  ⠪
         ᫨⥫쭮   ,      ᯥ稢
        প  ᮮ饭    ⮨  
       .
     - ⥧      ⮨ -   ।
       প  । ᮮ饭, ⥫  
            ꥬ  ଠ樨   ⮢,  
       ࠡ⪥   . ஥஢    ⠪  ,
        㤥   ⮨.
     - ⥧       -    饩
       ⮨    ।  প   । ᮮ饭.
       ஥஢ ,  ᨬ .
      ᪨  ᫨⥫  ⥩  饣  짮
 ਥ   ᨭ⥧   ⮨,  
᫨⥫ ⥩ ᯥ樠쭮 祭 - ᨭ⥧  
६  .
      ।  ,    ᭮ ᫨⥫
⥩,    業஢ ( । ன) 
 ଠ樥  ,  ᪨ ( ன ॢ)
 ᯥ祭 㯠 ⮢  .
     ⮤ ஥஢      । .
       ᪨  ⥩  ᯮ  
⮤ ஥஢  ⨬ ॢ쥢,     ६  
஥஢ 業஢  ⥩    ।  ⮯
ᯮ  ਡ   ᪨  .   ⢮
砥  १⮢    ⮬  ।  ⥬,  ᪮쪮
㤠 ⨪ ᯮ  ⬥.
      ࠬ  ⮤᪨ 㪠  㤥 ᬠਢ
 ஥஢  । (業஢) ᫨⥫
⥩ 饣  짮,    ⢥     ᨭ⥧
     ᯮ  ⮨.     ⢥   

                                                            3

ᮥ  । । ᫨⥫ ⥩
ᯮ .  樨 ᫨⥫쭮   ⮬
ᮯ⮢ 設,     ।  - 㣨 .
᫨    ᫨⥫쭮      㯫묨  
㯫묨,           ᮮ⢥騩            㤥
ਭ஢.  ᫨       ᨬ  ,  
ᮮ⢥騩   㤥 ਥ஢  䮬  䮬.
   騬 ꥪ⮬, 祬 ਥ஢ ,
⠪   ᫥  ८ࠧ      ।⢮  
  ॡ   ࠧࠢ . 室  ⮣ 
⢥  ᫨⥫쭮   㤥 ᯮ짮 
 뢠   "".
       ஥஢   ।   ᫨⥫쭮   
६  ॡ    吝    ⮩  .
祬,  砥       ᨫ쭮 易
(    設       設),
⠪   ᫠ 吝 (    設  
     ,    ⭮  -  ),    ।
吝        ᯮ짮    ।
砩襣  ﭨ.   ⢨⥫쭮,   ᫨      १
믮 ⬠  ﭨ   設    ன
뤥 設  ,   㤥 吝,   ⨢
砥 - .
     ᬮਬ          ⬮   ।
砩  ﭨ   -     .     
। 砩訥  ﭨ   設 襭 1)
   ⠫ 設.
     ਬ砭 1.   뢠  襭, ᫨   
㣥 ਯᠭ  (ﭨ).
          訬    浪  ᫠  権
(᫮     ࠢ)    冷  ()    㤮
⠢ O(3n2),  n - ᫮ 設  2).
     ਬ砭 2.   ""      砥  ந室  
᪮ ᫮ order - 冷.
     稬 १ G = (V,E) ਥ஢ 吝 ,
 V = {v1,v2,...vn} - ⢮ 設 .
    E   V x V         - ⢮  .
        L0 =  ||L0ij|| - (n x n)- ⥫ ᮢ
  G. 祬, L0ij -  㣨 (vi, vj).
     ᫨ (vi, vj)   E,     L0ij = +    ,    +    - ᮡ-
 ᫮     묨 ᢮⢠   ७ ⥬
⢥ ᥫ.
                  (n x n)-     L = ||Lij||
宯࠭ :
     <i, k, j> = min { Lij, Lik + Lkj }.
 㤮⢠  ⬠  㤥 , 
ॡ  砩訥   設 v1  G.
樠:  S = { 2, 3,...,n }, Lij := L0ij   i,
j.
      1.   = 2  ३  蠣 2.
      2.  i*  ⠪,   L1i* = min L1i.
                                         j  S
      3. S:= S\i* .
      4.  j  S 믮 L1i := <1, i*, j> .
      5.  ᢮ =+1  , ᫨   < n, ३  蠣 2. 
            ⨢ 砥  砩 ⥩ .
      ⨥   ,  ஥ ࠥ   ஫
 ஥஢   ᫨⥫  ⥩  묨

                                                            4

 ࠭祭  ६ । ଠ樨.         
 뢠  ᨬ쭠  砩 ⮢  
(ᨬ   ᥬ ࠬ 設).
      ""     吝 䮬  ॢ.
    ⥬  ᢮⢮,    ன 設
  ⢥   楯.  ,  뢠  
ॠ   ᫨⥫ ⥩,  ⮣ 筮.
   ⥫ ୮   ᫨⥫ ⥩
 ᫮  ᥪ ᨬ  ⥩   ࠬ
㧫   3),         ⥫      뢠   
஥஢  ᫨⥫쭮 .
     ਬ砭 3.   ଠ쭮  ஭    楯    
()   ⥮ਨ 䮢 ⫨.  뢠 楯,
᫨   㣨 ⫨. ᫨, ஬ ⮣, ⫨  設,
  뢠 ⮩ 楯.
        ᥪ 楯:
     - 設  ᥪ騥 楯,     楯 
       ᮤন  設;
     - ᥪ騥   㣠 楯,     楯 
       ᮤন  .
      VW  - ࠧ騬  ⢮ ⢮ A  ,
饥 ⥬  ᢮⢮,   猪   楯    V    W
ᮤন   A.
     筮 VW  - ⤥騬  ⢮   ⢮ S
設    (  ᮤঠ饥 V   W), 饥 ⥬ ᢮⢮,
   楯  V  W 室 १ 設  S.
         ⥮६,    易    
.
             1. ᨬ쭮        ᫮         設
ᥪ     楯,  ᮥ      ࠧ
ᬥ 設  V  W  G, ࠢ 쭮  設
 VW-⤥饬 ⢥.
             2. ᨬ쭮  ᫮   ᥪ  
㣠   楯, ᮥ   ࠧ  設 V    W
 G, ࠢ 쭮   VW-ࠧ饣 ⢠.
       ࠧ१   ⠪  ࠧ饥   ⢮,  
ᮡ⢥ ⢮ ண   ࠧ騬.
     ६ 2     ⢥ত,    ᪠ ᫠
ᥪ   㣠 楯  ⭠  ।
魮 쭮 ࠧ१.
     ࠪ᪨ ᯮᮡ  । 쭮  ࠧ१  
⥮६ ठ-ᮭ.      㫨஢    ⨥
, ஥      裡    ᬮ७  ࠭ᯮ
⥩.
          ।            D=(N,E),
ᬠਢ    㭪樥  b, ਯ뢠饩   㣥
(ei, ej)  E  ஥   ⥫쭮  ᫮   bij,   뢠
ய᪭ ᯮᮡ.
     設  ( ⢮ N) ਭ 뢠 㧫.
        㣥 (ei, ej)   D   ⥪ ⮪ xij,
祬
     0 < xij < bij ,   i,j.
     
        xij -    xji = 0,    j =\= Ns  Nt,    (1)
     i        i
      㧫  Ns   Nt 뢠 ᠬ  筨  ⮪
ᮮ⢥⢥. ⮪ ⥪   १ 筨  ⥪ 

                                                            5

 १  ⮪.  稭  ⮪  १  寮    
ᠬ Ns  Nt ࠢ:
     
        xsj =   xjt .                                  (2)
     j        j
     ய᪭ ᯮᮡ  ࠧ१    㬬  ய᪭
ᯮᮡ⥩ ਭ   .      ⥫
१,  뢠   ⥮६      ᨬ쭮   ⮪   
쭮 ࠧ१.
             3  (ठ-ᮭ). ᫨   
楯,    Ns   Nt,  ᨬ ⮪  Ns  Nt ࠢ
쭮 ய᪭  ᯮᮡ ࠧ१, ࠧ 㧫 Ns
 Nt.
     ᫨ ⥯    ய᪭  ᯮᮡ    㣨
ࠢ ,   ய᪭  ᯮᮡ 쭮  ࠧ१
 ᫮ ᨬ  㣠 楯.
      襭   ।  ᫠ 設  ᨬ
楯 筮   㧥  (஬ 筨  ⮪) 
ன 㧫  ᮥ饩  㣮 ⠪ ࠧ,   
.1.
     
     --------->*--------->  ====>    ------->*---------->*----->
     
                           . 1
     ய᪭  ᯮᮡ   祭     (   묨
ய᪭묨 ᯮᮡﬨ )  ᫮ ᨬ 楯.
     ᬮਬ ⥯  ठ-ᮭ, ।祭
 宦  ᨬ쭮 ⮪    쭮  ࠧ१  
.
      稭  ந쭮 ⮪  筨 Ns 
⮪ Nt.  ⮨  ⥬᪮ ᪥  
⥩   Ns    Nt,  㢥稢  ⮪  (  ⠭
⮪),       ᮮ⢥饬  㢥祭  ⮪  (
⮪).
      1.   ⠭  ⮪.   蠣 1  㧥
室     ﭨ:
     - " 祭",
     - "祭   ᬮ७",
     - "祭  ᬮ७".
     砫   㧫   祭. ⪠  ந쭮  㧫
ᥣ ⮨     ⥩. ࢠ   -   㧫  Ni,
 㪠뢠,     ᫠ ⮪   Ni   Nj. 
 ⪨  - ᫮, 㪠뢠饥 ᨬ 稭 ⮪,
  ᫠쨧 筨 Ns  Nj,   ࠭祭
 ய᪭  ᯮᮡ . ० ᥣ 筨 Ns 砥
 (S+,  E(s)=   ). ࢠ   ⮩ ⪨ 砥, 
 ᫠  ⮪  㧫 Ns   㧥; ᨬ     砥,
 稭 ⮪  ࠭祭 ᢥ.  㧥 Ns "祭
  ᬮ७".   ⠫ 㧫 " 祭".
     롥६   祭    ᬮ७ 㧥 Nj. 
   (i+, E(j))   (i-, E(j)).
      㧫 㤥 뢠 ᥤ, ᫨  ᮥ 㣮.
   㧫, ᥤ   Nj,  뤥   㧫 Nk,   
祭    xjk < bjk .
     ਯ襬  㧫 Nk  (j+, E(k)),
       E(k)=min { E(j), bjk - xjk }.

                                                            6

      㧫  Nk ⥯  "祭     ᬮ७".  ᫥
⮣ ᥬ  ᥤ   Nj 㧫  Nk,    祭   
 xkj > 0, ਯ뢠  (j-, E(k)),
       E(k)=min { E(j), xkj }.
      㧫  Nk ⥯  ⠪ "祭      ᬮ७".
   㧫, ᥤ   Nj,   ⪨.   㧥  Nj
⠥ 祭   ᬮ७,          
ᬠਢ  ⮬ 蠣.
      ,    ᥤ  Nj 㧫 祭,
 ⠫      祭 (  ᥤ  Nj 㧫
    祭).  ⠪  㧥 Nj ⠪ ⠥
祭   ᬮ७.   "+"    "-"    ࢮ  
⮪ 㪠뢠,    ⮪  蠣 2.
     த  ਯ뢠   ⪨  㧫,     
ᥤ   祭    ᬮ७  㧫    ,
   㧥 Nt   祭,   㤥 
   㧥  ⮪ Nt  祭. ᫨ Nt 
     祭,             Ns    Nt,
㢥稢饣  ⮪,   ,   ᫥⥫쭮,   祭   ⮪
ᨬ. ᫨   Nt  祭,    蠣 2   ,
㢥稢騩 ⮪.
      2.   ⮪.  ।,   ⮪  Nt  
 (k+, E(t)).   xkt  (xkt + E(t)). ᫨  
   (k-, E(t)),   xtk   (xtk - E(t)). ⥬
    砥 室  㧫 Nk. , ᫨ 㧥 Nk
   (j+,  E(k)),    xjk      (xjk+E(k)),  
३  㧫 Nj. ᫨ 㧥 Nk   (j-, E(k)),  xkj
   (xkj - E(k))  ३  㧫 Nj. ᫥ ⮣ ६
  ⪨ 㧫   ३  蠣 1.
     ᬮਬ  ⥯   ।⢥        ⬮
⨬樨   । ᫨⥫쭮  .  
 ᯮ   ⢥  ⮨,  ࠭祭
।⠢ 㪠  ᫠  ᥪ    㣠  楯
 ࠬ  㧫 .  ਬ    㯫묨
 । .
      稭  ࠡ   樨    ⥬
᫥⥫쭮 ⠥        ॡ    ॡ஬,
롨      稪 砩  ᥫ.  ᫨    ⮬
ॡ     吝      ॡ୮  ᨬ  ⥩
㤮⢮,   ॡ 㤠.  ᫨    ।
᫮ ⠪  ⮪ 该,   祭   
 室  ᫥饬 ⠯ ⨬樨  .
 ⠯ ⮨  ᫥⥫쭮 ਬ ⠪ 뢠 -
࠭ଠ権 ⥩.
      -࠭ଠ権   ⮨    ᫥饬.  롨ࠥ  
ॡ (vi, vj)   (vk, vl)  ⠪,   ॡ (vi, vl)  (vk, vj)
  室 . ᫨ ⮨ ᫥  ॡ
㤮⢮ ࠢ  wil+wkj < wij+wkl,    ॡ (vi,
vj)   (vk, vl)       (vi,vl)  (vk, vj)
ᮮ⢥⢥.   (-࠭ଠ ) 믮,
᫨ 祭   易  㤮⢮ ॡ  
ॡ୮ ᨬ  楯.  稢,   
    ਬ    -࠭ଠ.  ࠧ,
祭   쭮 ⨬쭮  ⭮襭 
-࠭ଠ.
     ᫥ 宦  쭮  ⨬쭮    
    ७.    楫ᮮࠧ    ⮬  砥,  ᫨
ᯮ 㣠  ॠ (  㣮 १  ॠ樨)

                                                            7

᫥⥫쭮 砩  ᥫ.     
    饬  砥  㤥  ⫨    ०    
१ 㤥     쭮 ⨬쭠  .
⥫   롮   ।   쭮   ⨬   
믮   ⮨.

                                                            8

             2. -  N1

             "⨬  ।
                     ᫨⥫ ⥩"

                                   

      ࠡ:  祭 ⬮ ⨬樨    
ਬ.
     
     2.1. ⮤᪨ 㪠  ⨬樨 
          । ᫨⥫ ⥩

     ணࠬ,  ॠ騥    ⨬樨    IBM  PC,
뢠 ARTNET.
     ⮨ ७  ।   ᨬ  
殮 ।  ணࠬ  ᮮ⢥⢨  䠬
    ﭨ    1977.  ⮨  ७㥬  㯫
 ⮭쭮  ⠢ ($ /./ ):
     -  25  - 3.3;
     - ᫥騥 150  - 1.65.
       㤮⢠   ࠡ  ணࠬ     ⥬     
᪠.
     
     2.2. ਬ ᯮ짮 ⬠ ठ-ᮭ

       㭪⮬   믮  ୮  ࠡ  
। ᫠  ॡ୮-ᨬ 楯   ன  設
   ⬠ ठ-ᮭ.
     ᬮਬ ਬ  ⮣ ⬠   , ࠦ
 .3.
     
              +----+   1,1   +----+    3,0     +----+
              | S  |-------->| 1  |----------->| T  |
              +----+         +----+            +----+
                |              |1,1             /|\
                |              |                 |
                |             \|/                |
                |     2,1    +----+     2,2      |
                +----------->| 2  |--------------+
                             +----+
     
                           .3
     
        ਥ஢ 㣥 ⠢  ᮮ⢥⢨
 ᫠.  ࢮ   - ய᪭ ᯮᮡ 㣨,  ஥
-室 ⮪    㣥  (  ⢥  室  ⮪  
ᯮ짮   ᫠, 㤮⢮騥  ᫮ (1)  (2). 
⭮,    㫥묨).
      1.  ਯ襬 㧫  NS    (S+,  @    ),    @  -
᪮筮.  NS   ᥤ 㧫.  N1  
, ⠪   bs1-xs1=0     㣮 ⮪  X1s>0.  N2
ਯ襬  (S+,1), ᪮ bs2-xs2=1  
     E(2) = min (E(S), bs2-xs2) = min ( @, 1) = 1.
      㧥  NS 祭  ᬮ७.  㧥 N2 祭  
ᬮ७.   N2     祭  ᥤ  㧫,  
 N1   NT.  NT       祭, 
㧥 N1 砥  (2-,1), ᪮ x12=1 > 0  

                                                            9

     E(1)=min(E(2), x12)=1.
      㧥  N2 祭  ᬮ७,  㧥 N1 祭  
ᬮ७.  N1  ⮫쪮  ᥤ 祭 㧥,
  NT.  NT    (1+,1), ᪮
     E(t) = min (E(1), b1t-x1t) = 1.
     
      ⮩ ⠭ ⮪ ࠦ  .4.
     
                              (2-,1)           (1+,1)
              +----+   1,1   +----+    3,0     +----+
       (S+, @)| S  |-------->| 1  |----------->| T  |
              +----+         +----+            +----+
                |              |1,1             /|\
                |              |                 |
                |             \|/                |
                |     2,1    +----+     2,2      |
                +----------->| 2  |--------------+
                             +----+
                             (S+,1)
     
                             .4
     
        㧥 NT 뢠 祭,  室  蠣
2.
      2.  ᪮   ⪠  㧥   NT     (1+,1),   
㢥稢 x1t  1.  १ 砥 x'1t=x1t+1=1.
     室   㧫 N1,  饬   (2-,1),  㬥蠥
x12  1. 砥 x'12=x12-1=0.
     室  㧫 N2, 饬  (S+,1),  ਡ塞 1
 xs2. 砥 x'2s=x2s+1=2.
     ⥫ १  樨  ⮪ ࠦ
 .5.

                                                            10

     
              +----+   1,1   +----+    3,1     +----+
              | S  |-------->| 1  |----------->| T  |
              +----+         +----+            +----+
                |              |1,0             /|\
                |              |                 |
                |             \|/                |
                |     2,2    +----+     2,2      |
                +----------->| 2  |--------------+
                             +----+
     
                           .5
     

      3. ਯ뢠  (S+, @ ).

      㧫  N1   N2      祭,    㧥  NT

뢠 祭.

     .
     
     
     2.3. ᠭ ୮ ⠭  ॡ
          ᭮


     ᠭ ୮  ⠭   ॡ  ᭮

ਢ  ᮮ⢥ ࠧ /2/.
     
     
     2.4.  冷 믮 ࠡ

     1.    ९⥫  ਠ    ஬  i,
ᮤঠ騩:
     - ᫮ 㧫 樨 ஥㥬  N=7+i(mod 4);
     

     -   ||A||, ᮤঠ  ॡ     ࠭

᫠ Akl  ॡ୮ ᨬ 楯  묨 ࠬ 㧫

樨

     Akl = max {|(k+i)(mod 4) - (l+i)(mod 4)|, i2 (mod 4), 1}
         ___
     k,l=1,N ;
     -  ||D||,  ᮤঠ ﭨ ( ) 
㧫 樨
     Dkl = 5[(k+l)(mod 5)] + 20(i)(mod 6) + i2(mod 10);
     -  G=(V,E)  N 設  ⢮ ॡ
     
        N-1                       N-1
     E = U (vj, v(j+1)) V  U (vj, v(j+i) (mod N)),
        j=0                       j=0
     
                                       ______

                                                            11

      vj ਭ   V   j=0, N-1.
     2.  ᯮ    ठ-ᮭ  ।  ᫮
ॡ୮ ᨬ 楯  設  G  ࠬ
     i (mod N)  (i+1)2 (mod N).
     3.   ९⥫.
     4.    室      IBM PC 
쭮  ⨬     ।  ᫨⥫쭮
.
     5.     뢮     १⠬  믮   ࠡ
(ࠢ   ⮨ 쭮  ⨬ ,   ⠪
  쭮 ⨬ ).
     6.   ९⥫.
     
     2.4. ঠ 

     1. 㫨஢ .
     2. ᠭ  室   ⨬樨  ।
᫨⥫ ⥩.
     3. ⪮ ᠭ ᯮ㥬 ⬠ ⨬樨.
     4. ᠭ   ⠫樥   蠣  ਬ  ⬠
ठ-ᮭ  宦 ᫠ ॡ୮ ᨬ 楯.
     5.   믮 (⮨ 吝 ,
  쭮 ⨬    ⮨).
     6. 뢮.
     7. .
     
     2.5. ஫ 

     1.   ਩   ࠭祭  ⨬樨 
᪮ ᫨⥫쭮  饣 짮?
     2.   ਩   ࠭祭  ⨬樨 
ᯥ樠஢ ᫨⥫쭮?
     3.   祬 ⫨稥   ⨬樨  ᪮ 
業஢ ?
     4.  ⫨  (), 楯   楯?
     5.  ⫨ ᨫ쭮 吝   ᫠ 吝?
     6.  뢠 ஬ ?
     7.  뢠 ࠧ騬 ⢮?
     8.  ⢮ 뢠 ⤥騬?
     9.  祬  設  ⥮६ ?
     10.  祬  ॡ୮ (㣮)  ⥮६ ?
     11.  । 寮᭠ ?
     12.   । ⮪  稭 ⮪  寮᭮
?
     13.  祬  ⥮६ ठ-ᮭ?
     14.    ᭮    ⠯    ᯮ㥬    ⬠
⨬樨  । ᫨⥫쭮 ?
     15.  ᭮ 蠣 ⬠ ठ-ᮭ?
     16.  祬  ࢮ 蠣 ⬠ ठ-ᮭ?
     17.  祬  ண 蠣 ⬠ ठ-ᮭ?
                                

                    ᪨ ᯨ᮪
1. ન  ..   ஥஢  ᫨⥫  ⥩:  祡
    ᮡ / .-., . 1992. 86.
2. 㤭 ..   .     ணࠬ஢: ⮤᪨
    㪠  믮  ࠡ / ., . 1983. 60
    .
