14.Graph2

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1 (1 ) 14 (2018/01/17) 1. ) ( 3

2 u TF TGo] rf TG i T D gk sgk M gkf M pj T FS [ K PAS tl ) : 2 F ( G M TG mn TG Gy

3 [ ] (1) swve x G N h G U n v, : = r U p v c, v, h, =, o 2 / p, v c ( o c c ), ( g /: ) o v h g /: UE ig /: ) ovg D Vt c o / p

4 [ ] (2) c p v m ) p 1 :.. + m ) y p p m m. 2 - nu n r l p,, -,.,. p pi m s c r : s c r t) E m c

5 A B

6 [ ] (1) = ( () ).(1, V c E A,, E = c E c c

7 [ ] (2) ( ) c c

8 [ ] (1) (Dijkstr) : V = {1, 2,...,n} E (i, j) E C(i, j) ( (i, j) C(i, j) = ) : 1 j(= 2,...,n) (j)... (1) S := {1}; or j := 2 to n o (j) :=C(1,j); (2) n 1 (S = V ) (2-) S (u) u (2-) S := S { u }; (2-c) u (u, v) E (v) min( (v), (u) +C(u, v)); (v u )

9 [ ] (2) 3= 5 ) ) ) 0.) ) ) ) 0. ) ( ) ) (, ) 0. ) (

10 o S S, gcj c O D 2 S D S O S n i - D j c S S M O S v l c > E S l co i C > i S S is i - D S ( 2 S s i O S r S its 2 s > i - u ) S n ) SO S )

11 : (1) - - l A [ l j l = = -1 = ] tnor m syl l S[ j l h n, m syf l j ) A k ) + ) ) ) Cxl ) h i l S F kj A h l j ] i j Pk l j - SA j - 0 ) ] k

12 : (2) S P S ) ( ( S A

13 : (3) c C v x c 1 1 h j m u ( } } } P ) :1 A } mrl h j v w S c{ V ) :1 A = u }s m ) A i, 1 2 ), 4 2 ), 2 3 ), 3 4 ), 4 3 ), 3 5 ), 5 4 ) V SV C cv C V m u c WP

14 (Brth First Srch) (Dpth First Srch) )= F! G B s c (!, ( Q g h v = Dq(Q) or u v S B u = v +1 Dijkstr = F Enq(Q, u) v E

15 (Brth First Srch) (Dpth First Srch) ) ) D (, ) D s DFS (G, u) ) g h (3/) (2/9) (1/10) (11/1) s g h S F G (7/8) (4/5) (12/13) (14/15)

16 (Dirct Acyclic Grph)DAG i ) * + ) * + ) * Tc i : S r T T n r T ) ( p g p

17 DFS

18 G F < < jd A < < j< < <S i A < G <DAG< G D :

19 : Minimumcost Spnning Tr (1 1 2 ) 1.. l h S SG T T M : : : : = M= T.. M i c T S g S

20 MST OV P 2 ( ) K

21 MST Prim (1) (Prim) : V = {v i, 1 i n} (v i,v j ) C(v i,v j )( (v i,v j ) C(v i,v j )= ) : T... (1) T := ; ( ) U := {v 1 }; V U w ( V U) U (u, w) C min (w) min (w) = u U : C min (w) :=C(v 1,w), min (w) :=v 1 or w = v 2,...,v n V U. (2) U = V ( n 1 )... O(n 2 ) () U V U U V U (u, v), u U, v V U ( C min (v))...o(n) () U := U {v}; T := T { (u, v)}; C min (v) := ; (c) V U w ( V U) C(v, w) <C min (w) C min (w) :=C(v, w) min (w) :=v;... O(n)

22 MST Prim (2)? : U 1 5 (5) (5) c () (3) () (4) (2) = c (3) () 4 (2) = 1 5 c (3) 4 2 = 1 5 c (3) 4 2 = 1 5 c = c 3 : C min () min () 1 O(n 2 ) 4 2

23 MST Kruskl (1) (Kruskl) : V E : T... (1) T := ; ( ) S := E ( ) ; C V ( ) : C := {{v } v V }) (2) C 2 C 2 () S (v 1,v 2 ) () C v 1 v 2 c(v 1 ) c(v 2 ) (c) c(v 1 ) c(v 2 ) C c(v 1 ) c(v 2 ) C (C := C { c(v 1 ),c(v 2 ) } { c(v 1 ) c(v 2 ) };) T (v 1,v 2 ) (T := T { (u, v)};)

24 MST Kruskl (2)? c 4 2 = c 4 2 = c 4 2 = c 4 2 = c 4 2

25 MST Kruskl (3) ( : Vmlk g v V V K o s V K lc V E V V i ) 2 V nv O o u V - C SVu V OVu V 2 g u C V O u VrK r ) 2 c c

26 2 2 ) n n (2 n c g rl i t i n Gc p o rli g : G u g G ( )

27 ) ( (, g g g V V G g h i r r p p E p r h i 2 r : r t

28 Hopcrot-Krp P M M ( (1,), (2,), (3,), (4,c)) c 2. 5? c-4-2 or 5-1 c-4 ( ) c 3. ( ) c

14.Graph2

14.Graph2 アルゴリズム論第 (1クラス) 第 14 回 (2018/01/17) 情報学専攻庄野逸 (shouno@uc.c.jp) ( 3 号館 313 号室 ) 本のお題 [ 復習 ] グラフとはグラフの基本概念と語グラフの実現法グラフをいた問題最短経路問題 Dijkstr アルゴリズム,APSP 問題と Floy アルゴリズム有向グラフ (DAG) とトポロジカルソート最全域 (Minimum