           ᪠  誮쭨  ଠ⨪.
                 ⥡, 28  - 1 ५, 1994.
                              .

                             1. 窨.
      N  祪.       묨 ୨ ᠭ 
㭨  - ᫮  1  N.     窥 묨
୨ ᠭ   楫 ᫮,  饥  ஬⪥  1 
N (묨 묨 "묨" ᫠   ᪮-
 祪).
     ਬ, N=5, 5 祪 祭 ᫥騬 ࠧ:
                      Ŀ
     "୮" ᫮    1 2 3 4 5
                      Ĵ
     "᭮" ᫮   3 3 2 4 2
                      
     室    N 祪 ᨬ쭮 ᫮ -
祪  ⠪ ࠧ,  ⮡ ⢠ ""  "" ᥫ  
ᮢ.
      ਬ    窨  "묨" ࠬ 2,  3, 4
(⢮   ஢,      ॡ    ,      -
{2,3,4}).
     :
          <N=> N, N<=50
          <""  1, "" -> "᭮"_᫮_1
                              ......
          <""  N, "" -> "᭮"_᫮_N
     :
          < ࠭ ⢥ ⮢> ⢮_⮢ S
          <""  ࠭ 祪> a1, ..., aS

                              2. .
        N 㧮,  ஭㬥஢  1  N.   ⭮,
:
     1)  ࢮ  N- 㧮 m(1)  m(N) ᮮ⢥⢥;
     2)   i-  㧠 m(i)  । 䬥᪮  
ᥤ 㧮, 㢥祭  d:
                      m(i)=d+(m(i-1)+m(i+1))/2;
       m(1), m(N), N, d  j  m(j).
     ࠭祭: N<200.
     室,  ࠬ 㫨஢ ,  ।ᬮ ஢
४⭮  ணࠬ.

                          3.  .
      N  s.
      ࠫ쭮 ᫮ N  K>s  (K  ). ⠢
,  ।騩,   s  㤠, ⮡ ⠢訥 
⠢  襥 ᫮.  뤠  浪 ⠭  㤠-
 .  ஭㬥஢ ᫥ ࠢ, 㬥 稭 
.
     ਬ. N=2435, s=1. ᫨ 㤠   (4), 稬 ᫮
235.    襥 ᫮ (⠫: 435,245,243).
     ᫮ N ।⠢ ᫥騬 ࠧ:
      ⢮  ப  p,    ப   ᫠
ai  bi,  ai<=60000, bi ⮨    祬 60 . ai 㪠뢠,
᪮쪮 ࠧ  ᫥ 砥 㯯  bi. p<30.
     N=(a1) b1 ... (ap) bp.
     ਬ, ᫨
      p=3
      a1=3  b1=123
      a1=1  b1=0
      a1=2  b1=09
      N=12312312300909
.
           ᪠  誮쭨  ଠ⨪.
                 ⥡, 28  - 1 ५, 1994.
                             ன .

               1.  (᪠ ).
      㯯  ⮨  N 祫.      
, ᫨      ⠢,    .
 ⮨  ⮬, ⮡  ⮩ 㯯 ।  (-
  ⠬ ).   ⮬ ࠧ蠥  ⮫쪮    
",      ⮣ 祫?". (।,  
⢥ ࠢ,      ⢥   ⠢  
).
      쭮 室 ᫮ ᮢ,  ஥    -
,    .
     室 :   C[i,j], ⠪  C[i,j]=1, ᫨ i 
j,  C[i,j]=0 .

                            2. ᯨᠭ.
    㫥  ६  ६  㯠  N  ࠡ.
  ஭㬥஢  1  N.    ࠡ i ࠭ ⭮
᫥饥:
   1) ६ 믮 ࠡ ti, ⭮ ⪠,
   2)  i   ⪨  ࠡ i   砫  
믮.
   ६  믮 ⮫쪮  ࠡ,    ᫨  
㯠   믮 ன ࠡ,    த 믮
,   .
    ࠧ,  㬬 ,   㤥 㯫, -
 ᫥騬 ࠧ:
    ࠢ 㬬 i*(६ 砫 믮 ࠡ i)  ᥬ i.
    ⠪ 冷 믮 ࠡ,  ⮡     -
.
   :
   <N=> N, N<100,
   < 1: ६ 믮 t1   c1> t1, c1
                               ....
   < N: ६ 믮 tN   cN> tN, cN
   뢮:
   < 㬬 => 
   <冷 믮 ࠡ:>  ࠡ  浪 믮

   ਬ: N=3
   t1=4    c1=1
   t2=1    c2=1
   t3=10   c3=3
    㬬 =14
   冷 믮 ࠡ: 2,3,1.

                          3. אַ㣮쭨.
    ணࠬ, ⢥  ,    N  -
㣮쭨 i ࠧ஢ (ai,bi), i=1,...,N, ᫮  让 -
㣮쭨  ࠧ (a,b).    㣮 讣 אַ㣮쭨
 न (0,0),  ஭ ࠫ  न; 쪨
אַ㣮쭨  稢,   ⠪, ⮡  ஭ ᫥ -
 ⠢ ࠫ  न.  ⠢  אַ-
㣮쭨 i   ᯮ짮  .  אַ㣮쭨 i  -
४뢠.
   ࠭祭: N<8, a,b,ai,bi - 楫 ᫠.
   ⢥ 뤠   '' - '',   砥 '' 室 뤠
   אַ㣮쭨 i,  ࠧ饣 ,  न  
  ࠢ 孥 㣫    i.
.
         ᪠  誮쭨  ଠ⨪.
               ⥡, 28  - 1 ५, 1994.
                      . 襭.

                           1. 窨.
      㬥 祪   ᨢ F.   i-
窥 ᠭ  :  ""    i,  ""  -  F[i].
 C[i] - 稪,  뢠騩,  ᪮쪮 ࠧ ᫮ i -
 । "".
     ᫨ - ᫠ i  । "" (..  C[i]=0), 
    ᪮ ⢥.  ⥫쭮,   ᬠ-
   ()   ᮮ⢥騬 "" ᫮
i ( "" F[i]).  塞 i   ⢠  ""  ᥫ  
㬥蠥 ᮮ⢥騩 稪 [F[i]]  1. ᫨ ᫮ C[F[i]]
⠫ ࠢ 0,  ᫠ C[F[i]]  । "",    
ᬠਢ    "" ஬ C[F[i]],  ..

const N1=10;
type mas=array[1..N1] of integer;
var F: mas;
    N,i: integer;

     procedure mapping(F: mas);
     var C: mas;
         j: integer;
         Stack: array [0..N1] of integer;
     begin
        Stack[0]:=0;
        for j:=1 to N do C[j]:=0;
        for j:=1 to N do C[F[j]]:=C[F[j]]+1;
        for j:=1 to N do
           if C[j]=0
           then begin Stack[0]:=Stack[0]+1;
                      Stack[Stack[0]]:=j
                end;
        while  Stack[0]<>0 do { while Stack is not empty }
           begin
             j:=Stack[Stack[0]]; { remove i from the top of Stack }
             Stack[0]:=Stack[0]-1;
             C[F[j]]:=C[F[j]]-1;  { decrease counter }
             if C[F[j]]=0 then begin Stack[0]:=Stack[0]+1;
                                     Stack[Stack[0]]:=F[j]
                               end;
           end;
        for i:=1 to N do
           if C[i]<>0 then writeln(i,' in set');
     end;

begin
 write('N=');  readln(N);
 for i:=1 to N do
  begin
    write('"p" p ',i,', "p" -');
    readln(F[i]);
  end;
 mapping(F);
end.
( ਠ ୠ "Communications of the ACM")

                       2.  (.祭).
      ࠬ 㫨஢   ४⭮    ணࠬ
 ,     㧠 m(i)    ⥫
᫮.   m(1)  m(n).  ⢨⥫  㧮  拉
m(i). 㤥  ⥪騥 塞  m'(i).  ந-
쭮 m'(2) -  ண 㧠.  m'(2)-m(2)=s, m'(1)=m(1) 
                     m'(i)=d+(m'(i-1)+m'(i+1))/2,
, ⠪ 
                      m(i)=d+(m(i-1)+m(i+1))/2,

                     m(i+1)=-2*d+2*m(i)-m(i-1),
                    m'(i+1)=-2*d+2*m'(i)-m'(i-1),

        m'(3)-m(3)=[-2*d+2*m'(2)-m'(1)]-[-2*d+2*m(2)-m(1)]=2s
        m'(4)-m(4)=[-2*d+2*m'(3)-m'(2)]-[-2*d+2*m(3)-m(2)]=3s
                                .....
                          m'(n)-m(n)=(n-1)s.
    室 ࠧ m'(n)-m(n)=(n-1)s. 塞 s.

     3.   ( ਠ ᪮ ).
      s=1.
     ᬠਢ ᫮ ᫥ ࠢ.    ⠪ -
,  ன  ண  ,      -
. ᫨   ⮭,  뢠, ભ ᫥.
      d1 d2 ... dk - 筠  ᫠ N.  1<=i<
<j<=k, Ni  Nj - १ ન  N i-  j-  -
⢥⢥. 쥬   ᥫ Ni   Nj    
१ -    1-    (i-1)-,    i-  (j-1)-   j- 
(k-1)- .
      1  2 ....   i-1      i    ....   j-1     j   ... k-1
  Ni d1 d2 .... d(i-1)  d(i+1)  ....    dj  d(j+1) ... dk
  Nj d1 d2 .... d(i-1)  di      .... d(j-1) d(j+1) ... dk

       ⨩ १ ࠢ ( ),   १-
 ࠢ Ni  Nj  d(i) <> d(i+1)    j.
     ᫨ s>1,  s ࠧ ਬ ਢ  .
       s<10,    ࠭  ஢ ⮫쪮
10 ⥪ .   ⮬  㤠     ।
 ப,   .
.
                             .
                             1
1)
     20
     1)3
     2)2
     3)7
     4)5
     5)11
     6)13
     7)2
     8)1
     9)8
     10)19
     11)17
     12)15
     13)11
     14)10
     15)7
     16)1
     17)2
     18)4
     19)5
     20)17
     ⢥: s=1
            {2}
2)
     6
     1)2
     2)3
     3)4
     4)5
     5)1
     6)6
     ⢥:s=6
           {1,2,3,4,5,6}
3)
     14
     1)6
     2)5
     3)1
     4)3
     5)4
     6)2
     7)8
     8)10
     9)7
     10)9
     11)3
     12)2
     13)1
     14)9
     ⢥:s=10
           {1,2,3,4,5,6,7,8,9,10}

.

                      2

     m1        mN       N      d      j     ⢥
1)   1.0       1.0      21    1.0    15      85.0
2)  30.0     210.0      10   -5.0     8     ४
3)  40.0      58.0      10   -4.0     2     ४
4)  40.0    1370.0      20   -5.0    13     460.0
5)   1.0       3.625    22    0.9375 14     100.125
6)  50.0       8.5      84    0.25   59     383.5
7)   1.0       1.0      10    0.0     3       1.0

                  N 3
     冷 : 
                    (1) b(1)
                    a(2) b(2)
                    .........
                    a(p) b(p)
                    s
     砭:  ࠢ ᫮ ⢥ -
                  뢮   ⠭ ஢

1)
     2
     2 1234
     1 9
     5
     ⢥: 2 3 4 8 9
2)
     4
     10000 111111111111111  ( 15 )
     60000 22
     2     9
     10    00001
     3
     ⢥: 1) [150001 - 270000]
            2)270001
            3)270002

3)
     9
     50000 111111111111111  ( 15 )
     50000 222222222222222
     .....................
     50000 999999999999999
     4
     ⢥: 1) [6000001 - 6750000]
            2) [6000001 - 6750000]
            3) [6000001 - 6750000]
            4) [6000001 - 6750000]

4)
     2
     2 01
     1 02
     3
     ⢥: 2 4 6
.
                     ன . 襭.
               1.  (᪠ ).
     ᫨  訢 祫 ,     祫 B,  -
 - "",  A -  , ᫨  "",  ⮣ B - 
.   ᮬ  ᪫砥  쭥襣 ᬮ-
७   祫.  ,  ᫥ N-1 ,  ⠥ 
⢥ 祫,   ᤥ ஢, ⢨⥫쭮 
  , (     ,     -
  (N-1) ).  室  ஢ન    ᭮
᫥騬 ਬ஬:
     1  2, 2  3, 3  1.

                            2. Pᯨᠭ.
       ⨬쭮 ᯨᠭ,   ஬  믮-
 ࠡ ᮢ  浪묨 ࠬ ࠡ.  ⮬ ᯨ-
 㤥 ᮮ⢥⢮ 쭮 祭 . 猪 -
  浪 믮    ࠡ 㦥    ਢ
 㬥襭 , ⮬, ⠢ ⠬ ࠡ  ࠬ k 
k+1,  稬   .
     稬 १ l(i) ६ 砫 믮 ࠡ i.
      ⮬ ।⠢     ⠪
 k-1                                          n
 SUM c(i)*l(i) + c(k)*l(k) + c(k+1)*l(k+1) + SUM  c(i)*l(i) <=
 i=1                                        i=k+2
       k-1                                            n
   <=  SUM c(i)*l(i) + c(k+1)*l'(k+1) + c(k)*l'(k) + SUM  c(i)*l(i).
       i=1                                          i=k+2

      l'(k+1)  l'(k) - ६  砫  믮  ᮮ⢥⢥
(k+1)-   k-  ࠡ ᫥ ⠭.  ⨬,   k- ࠡ 
 ᯨᠭ 믮 ᫥ ⮣,   믮 ।-
騥 k-1 ࠡ, ⮬
  l(k)=l'(k+1), l(k+1)=l(k)+t(k), l'(k)=l'(k+1)+t(k+1)=l(k)+t(k+1).
      १ 砥:
    (k)l(k) + c(k+1)(l(k)+t(k)) <= c(k+1)l(k) + c(k)(l(k)+t(k+1))
                       c(k+1)t(k) <= c(k)t(k+1)
                      t(k)/c(k) <= t(k+1)/c(k+1).
     :
     1)   ࠡ ᫨ ⭮襭 t(i)/c(i);
     2) 冷 ࠡ  ⠭ ⮣ ⭮襭.

                          3. אַ㣮쭨.
      i     ਧ⠫쭮,    ⨪쭮.
㤥      㣫.
       蠣 
     1) 㤥  ᪠    㦥  䨣 ""  묨
न⠬   㣫;
     2)      ᯮ짮 i 
             । i   ⠢  ""
            ᫨ 㤠,    諨  । ,
            
                i  ᯮ쭮
                蠣 1
              scratch:
                i  ᯮ쭮
             
          
         ᫨ ᯮ짮  i,   १. ⮯.
         ᫨  ᯮ짮   i,  襭 .
                                            scratch
.
                             .
                              2

   t        c

1) 5
   1,       0
   2,       1
   3,       1
   4,       1
   5,       1

⢥  16
   2, 3, 4, 5, 1

2)4
  1,        1
  0,        5
  1,        2
  1,        3

⢥  4
  2, 4, 3, 1

3)4
  1,        1
  100,    100
  3,        6
  4,        6

⢥ 825
  3, 4, 1, 2          3, 4, 2, 1

4)8
  999,    998
  99,      98
  98,      97
  95,      96
  94,      95
  94,      93
  2,        1
  99,       0

⢥ 569193
  5, 4, 1, 2, 3, 6, 7, 8

                               3.
#     N                i      ⢥
1)    7    200,400    100,500    
                      200,800
                      250,350
                      350,250
                      400,400
                      201,201
                      1,1000

2)    6    10,2       1,1    
                      1,1
                      1,1
                      1,1
                      1,1
                      1,1

3)    4    400,300               , 3=400,300
                      205,9
                      25,22
                      300,400
                      290,799

4)    3    500,300    200,400    
                      300,200
                      100,100
                                       Ŀ
5)    7    1200,1000  600,400                  
                      600,400          Ĵ
                      600,400                   
                      600,400                   
                      400,600          
                      400,600
                      400,600

                                        100   500
6)    6    600,500    100,400        Ŀ 100
                      100,500           Ĵ
                      200,300    400              200
                      200,300              Ĵ
                      200,300          Ĵ       200
                      100,300    100   
                                         300   300
7)    6    400,700    600,300    
                      200,500
                      100,500
                      100,500
                      200,200
                      300,200
