G(V,E), , .
( ):
G(V,E) n . :
- G ;
- G n-1 ;
- G n-1 ;
- G , ;
- G ;
- G , .
, ( , x1) 1, x1 ( ) .
v u, $ u v. u v. , , .
. d(v) . h(v) . . , 2.
. () , .
( ).
:
- ( ) .
- .
1( ): (, , ..) n "" , , , -, "" ( ), , -, , .
, n , (SST - shortest spanning tree, MST - minimal spanning tree) , , "" .
: ( ) .
. (xi,xj) cij. , .
, .
v G=(V,E) G'ÍG , v, G':
|
|
d(v,G')=min(v,w)ÎE(G),wÎG'Weight(v,w).
- SST '=<, G>;
- |E(SST ')|=n-1, SST=SST '; ;
- I G, SST ', SST' (I={u | uÎV(G), uÏSST', (u,v)ÎE(G), vÎSST'}), wÎI, SST' : d(w,SST')=minvÎI d(v,SST');
- (w,u), d(w,SST'), SST ';
- 2.
2( ): A B , , ( , , , , , ).
:
( - "" ), , () - , () (, ) . () , A B.
: . ( ).
( .)
: y p G=(V,E) . p y s t (s≠t).
:
p vi p - : d(s)=0, d(vi)= ¥ vi≠s;
, s, , s - ;
s (c:=s);
.
:
- uj, c, , :
d(uj):=min{d(uj), d(c)+Weight(c,uj)} (*)
(c,uj) - , c uj, Weight(c,uj) - ;
- ¥, ; (" ");
- ( , , (1)!) x , , (c',x), , d(x) = d(c')+Weight(c',x), c'= c' - , (c'=, (1) uj=x (*)), (c:=x);
- c=t, d(t), : s t, (3) ( , ); (" ");
- (1).