5.1

(V(G),E(G),M(E,V))V(G)()E(G)M(E,V)= (V(G),E(G))G=(V,E)

5.1- vivja(i,j)=10 vivjvjvi viVja(i,j)=10 ViVjVjVi

5.1 Viejb(i,j)=1 Viejb(i,j)=-1 b(i,j)=0 Viejb(i,j)=1 b(i,j)=0 nm.

ejViejViejViejVi

ABCDe1e2e3e4e5e6e7

abcde1e2e3e4e5e6

V={a, b, c, d}E={e1,e2,e3,e4,e5,e6}|V|n |E|m.

e1=AD,e1ADADDA , e1=(a, b), abe1ab/()

D(A)=5,d(A) =1,d(A) =4, d(A) +d(A) =d(A)=5

=2 deg(v)=2|E|=2m ()0 e=(u,v)uv1 2 deg(v)=2|E| =2m()

=3+2+3+2=10,=5 10=2(2*5) =3+3+3+3=12 =6 12=2(2*6)

Vo(odd)Ve(even), edeg(v)+ odeg(v)= deg(v)=2m Ve edeg(v)2k odeg(v)=2m-2k=2(m-k) Vo2n1-1,2n2-1,,2nt-1t 2(n1+n2+nt)-1-1--1=2n'-t 2n-t=2(m-k) t=2(m-k)-2n'=2(m-k-n')

:A(5) B(3)2:B(3),D(3)2

()=()11111 =

()=4+1+1+2=8()=1+2+3+2=8=

nKn=n(n-1)/2 Kn:nn-1(n-1)nn(n-1), n(n-1)=2=2mn(n-1)=2mm=n(n-1)/2 m=C(n,2)c(n,2)=n(n-1)/2(n-1)n(n-1)/2

=n(n-1)/2

:Gn ,, 0n-1. 0,1n-1n2 0,0,.

32233333

237 2323ii 77723=23*7=161161=2m .

5.2

5.2

5.2 () , WarShall

a(1,3)=013,0a(1,5)=115,1

12n-1AA2A3An-1

5.3Euler

5.3.1 .a-b-da-b-aa-b-d-c-a5.3.2 5.3.3

5.3.1 G.5.3.2G2 deg(a)=5deg(b)=3deg(c)=3deg(d)=3 deg(a)=deg(d)=3 deg(1)=2 deg(2)=4 deg(3)=4 deg (4)=2 deg(5)=2

5.4 185920

5.4 this5.4.1 5.4.2 5.4.1 GnGn-1G5.4.2 GnGnG

5.4 n=524Hn=625Hn=62=4

5.4 5.4.3 G=VSW(G-S)|S|S={1,5}3W(G-S)=3|S|=2W(G-S)|S|H

5.4 1W(G-S)=1|S|2W(G-S)=1|S|3W(G-S)=1|S|4W(G-S)=1|S|5W(G-S)=1|S|6W(G-S)4

5.4 TSP(Traveling Salesman Problem) 1-2-3-4-5-14+16+11+13+10=54 1-2-4-3-5-14+3+11+4+10=31 NP(Non Polynomial)

5.5 5.5.1

5.5 5.5.2 1-3-4-111-2-4-122-4-3-231-3-2-14 5.5.1 =-+2d=4m=6n=4 d=m-n+2

5.5 5.5.3 3()35.5.2 3d=2mm=3n-6d=2n-4

5.5 3(34). ,,2=2m 3d3d3d=2m d=m-n+23(m-n+2)=2mm=3n-6m=3d/2d=m-n+2d=3d/2-n+2d=2n-4

5.5 5.5.3 m3n-6d2n-4 m=C(5,2)=10,n=5,3n-6=9,m3n-6 m=C(6,2)=15,n=6,3n-6=12, m3n-6

5.5 4.

5.5 Powell(1) (2)11(3)(2)

5.5 6(1)B(4)D(4)A(3)C(3)E(3)F(3) (2)BBAA(3)DDFF CCEE

5.6 5.6.1

5.6 5.6.2 5.6.1 1

5.6 5.6.3 5.20(a)5.20(b),5.20(c)

5.6 -Kruskal(1)e1i=1(2)i=n-1(3)(3) e1,e2,...,eiGe1,e2,...,eiei+1{e1,e2,...,eiei+1}ei+1(4)i=i+15.20(a)(1)2(2,5)(3,4)i=2(2)3(2,4)(4,5)(2,5),(2,4) i=3(3)4(1,2)i=4

5.6 -Prim(1)V0U(2)UV-U()(v,r)T,vU,r V-U.(3)U=U+{r}(4)|U|

5.6 - ,5.21(a)A(b)A-C-B-E-FA-D A0 10 11 . ADC1AB2AE3AF4

5.6 - 222 5.21(b) 5.21(b)5.22

5.6 - 5.6.4 Ttv1,v2,,vtw1,w2,,wtTl(vi)vi w(i)l(vi)HuffmanHuffman(1)(2)W1W2(3)W1+W2W1W2(4)1(1)(5)(5)Huffman

5.6 - 2,3,5,7,11,13,17,19,23,29,31,37,411000101110100101111

5.6 - 010 10101111010110101010101000100010110111110111101110011000

5.6 - 010 1000101110100101111 100()0101() 110()100()1011()110(0110)523

5.7 5.2412Dijkstra(1)D(1)=01iD(i)=W(1,i)D(i)=S={1}(2)S=SDiS=S+{i}(3)(3)Sijd(i)+w(i,j)

5.7 (1)D(1)=0D(2)=7D(3)=1S={1}S={2,3,4,5,6}(2)i=3S={1,3}S={2,4,5,6}(3)d(3)+w(3,2)=6

5.7 2113

5.7 ,5.26

5.8 5.245.25 14

5.8 Edmond-Karp1(1)2.27(7,0)70

5.8 Edmond-Karp1

5.8 Edmond-Karp1

5.8 Edmond-Karp1

5.8 1=n0

5.8 1=n0

5.8 1=n0

5.8 1=n0

5.8 =n0