4 a6 N, K1 a" Z/ Y9 j( B
clc,cleara = zeros(5);a(1,[2 3]) = [10 8;a(2,[4,5]) = [2 7;a(3,[2 4]) = [5 10;a(4,5) = 4;a = sparse(a);[b c] = graphmaxflow(a,1,5) * a' t5 O& w. L# v3 d, W/ B
最大流量为11
自定义Matlab代码：
F* l0 P( F, n( B: u: l, E3 J; q; V 最小费用求解 ! q; d$ u8 m8 S 8 l8 e* Z: R9 t6 mLingo:" ?8 l, h3 i% S# Z
% s5 G/ T/ c/ l. I
model:9 v( w( g" H |- d
sets:8 M. X3 `( h7 D( V( y, R! r' ~8 Y
nodes/s,1,2,3,t/:d;) J4 {4 L+ ]( a$ E) s
arcs(nodes,nodes)/s 1,s 2,1 3,1 t,2 1,2 3,3 t/:b,c,f; - V2 @% d# q* v$ |endsets' M: [, I9 L" I% |4 M% O& r+ G
data: # }. d; t$ y! J5 cb = 4 1 6 1 2 3 2;, ]# c) P4 D5 z9 w$ P
c = 10 8 2 7 5 10 4; & w' U8 [. W" ~5 o/ |. h7 v' ~% \d = 11 0 0 0 -11; 8 A& E1 g4 @2 ~enddata ! L. Q9 U1 U% k1 v, q* A' E+ l$ in = @size(nodes);. `' t) t c" z7 Q
min = @sum(arcs:b*f);1 `7 D! }6 ]" ~$ s$ ]7 U9 T
@for(nodes(i)sum(arcs(i,j):f(i,j))-@sum(arcs(j,i):f(j,i)) = d(i));8 k' O( T; D# n
@for(arcsbnd(0,f,c)); 9 \# ]8 p$ w9 K k4 Cend ( P. T8 \6 H5 o5 u N/ O1, z; q {6 T% ?' O& A( \1 M Q/ Y
28 w* ~9 q$ G5 J" Y
3. F. o9 u, k2 z" F. m; R5 \3 c
4 4 Y' B& n) e" `0 H( H* Q) N5 0 B" o o' C+ @1 t! F( u$ y! @6; S' b+ [8 a( I' x; a8 M/ k
7 . \4 D2 Y8 B4 i) V8 . v3 Q2 ^ k1 z" v; @: [/ S9 " l! ]' L/ L9 z3 {# ^& a2 T10 ) \! n7 l: i% q- P( @7 B11. F6 u2 q# s: ]4 `1 M& {% T
12 $ |$ g8 n1 N7 H' r6 ^/ x3 m133 s9 f4 O9 t5 y7 B; o
146 E4 \# d a; ~7 d
15, t5 ] q9 l. [/ z; r# B
Matlab实现：" Q, z" V$ q# V8 W
4 M D( H$ q/ }9 Z3 L" t" t 9 k, {8 y4 _: m* d, ~4 Sn = 5; C N+ S7 ^" ]; Q8 W; [%弧容量+ w# l; p/ ]9 k& I3 m- _5 A, s
a = zeros(5);6 @& [8 B9 g$ k4 ~% J
a(1,[2 3]) = [10 8];7 Q) P3 v- Y+ E( @! I3 B
a(2,[4,5]) = [2 7];. d& i* L- _( R$ a2 @3 M4 p
a(3,[2 4]) = [5 10];4 @* L; w; I& @, v) W
a(4,5) = 4; 7 X$ K7 Y/ o0 S4 Q% D6 m6 _- f' `- RC = a; 5 U, @. m" d8 @/ i6 d%C = [0 15 16 0 0;0 0 0 13 14;0 11 0 17 0;0 0 0 0 8;0 0 0 0 0]; & H! m% Y, v( \6 X) ~) b%弧上单元的费用 / u8 T3 s7 |+ o- ta(1,[2 3]) = [4 1]; , a9 F. F) y" _a(2,[4,5]) = [6 1]; 8 C b+ i0 h3 Y- y xa(3,[2 4]) = [2 3];( o7 D& z n+ V9 |, W
a(4,5) = 2; $ G8 V3 Z. N- [* ]/ M% Nb = a; ) L$ E# v7 O+ `/ |%b = [0 4 1 0 0;0 0 0 6 1;0 2 0 3 0;0 0 0 0 2;0 0 0 0 0]; ( a+ j9 a) Q8 b$ Y%wf表示最大流量。wf0表示预定的流量值 3 t$ w" [* B( p9 T4 @wf = 0; ; E" W: o/ o" `( Awf0 = Inf;& M& h" p* U0 e a
%取初始可行流f为零流+ a% I; R* F2 [7 f* h% k! R
for i = 1:n ' S$ U! }$ x9 y; x) j) w3 { for j = 1:n - g) V1 U$ I/ X& ` Y7 k( f f(i,j) = 0; 1 K1 n2 j' W9 j end " a% b! t8 t# w, E! Tend ! y# p' k0 C4 k5 n8 G3 D. H- ~while (1)3 p# n8 z# I2 t. J
%构造有向赋权图) {& a5 s( [6 G; x+ Q" g1 b
for i = 1:n- d6 @, h; j- i( _7 Z: Z4 u
for j = 1:n: C2 e8 ?0 z6 u/ f
if j~=i! {/ d+ a; y/ a
a(i,j) = Inf;) ~( Z3 O/ J! i2 \
end, ^2 _* Z9 F, T' j1 [" ^
end [# z8 k; T k: ]( U5 F
end , l8 g* y% \/ D" v l8 K, I" C for i = 1:n' ~- k# K1 ]- I7 }* Q
for j = 1:n % Z2 b2 B1 ]( t if C(i,j) > 0 && f(i,j) == 00 h. a* F9 ^1 N5 o1 ^+ Y7 k
a(i,j) = b(i,j); : J' u% J% n2 V% i) I c" x elseif C(i,j) > 0 && f(i,j) == C(i,j) 0 T( {6 w. [9 i$ q a(j,i) = -b(i,j);1 t8 S6 j3 G( R' Q% Y! m4 a+ R
elseif C(i,j) > 0 9 v% \% ]- P( f) f a(i,j) = b(i,j); % g8 o5 h& L1 J. U. J c$ l a(j,i) = -b(i,j); 1 {. O5 ?& A& r( b1 @3 Y end ) e$ \% x" _0 s9 ]( Y ~% ` end9 m9 Y. e, P5 y
end5 G7 m3 c0 d/ b9 G3 A- ?
%使用ford算法求最短路，赋初值+ s& {7 y+ X5 T4 z2 x. g# X0 u" z3 {! |
for i = 2:n" V) m, w$ N5 D
p(i) = Inf; 8 o! t# Y) {5 r: X s(i) = i; . w B& x8 R& [ v6 k end" P6 ? w- \6 d, V* Q% W
%求有向赋权图vs到vt的最短路,赋初值 * c3 n2 j4 S4 W for k = 1:n : e2 u$ |; X8 M6 [% D pd = 1; ) g' X; C/ n0 P' d% L. \ for i = 2:n6 Q( @ }: e9 T" w
for j = 1:n ! s# ~0 x. H9 T1 u* w3 K if p(i) > p(j) + a(j,i)) p$ ?; `1 t1 B7 Q
p(i) = p(j) + a(j,i);" q1 U: \7 g( `( O& [
s(i) = j; ) r2 Z+ @, O4 z; r; g pd = 0; ( ?6 o( q$ m- |% \1 K7 f) Y end5 _) h" V$ A/ [6 r& d6 W& G
end3 F' c5 ` j* n
end # J5 G* p7 W* r+ n, Q- ?3 U %求最短路的Ford算法结束' y% V0 M" U6 X/ v b ~ }
if pd5 ?3 \. [: L8 V/ F$ ~8 w
break; * ?9 s+ w1 I0 w$ j9 @9 e- k0 L% ?, ^, X" } end % _5 B* X3 M+ A$ u2 X$ S& q end ! t4 t t0 f4 c( q& F2 a %不存在vs到vt的最短路，算法终止，注意在求最小费用最大流时构造有向赋权图中不会含有负权回路，不会出现k = n 9 A# v* M$ u6 U: F2 X( n& B if p(n) == Inf / T3 N; N+ O3 v) V2 R+ ^ break; " O1 E# \$ v) [6 g end) U9 e" q: e2 h( ^
%进入调整过程,dvt表示调整量. R h0 B3 Z9 W9 z, f
dvt = Inf; 6 w) y5 y5 \# C6 J dvtt = Inf;% D+ n- q$ O! h |
t = n;: u7 @+ d9 l* S0 D$ {+ |/ s$ K
while(1) * K( [! m" m# q, X$ H+ V0 h J if a(s(t),t) > 0& l0 x8 j4 s, i3 ^: t, T* O9 L
%前向弧调整量 - r; Y7 n9 }3 Q: } L& A) n( i$ U9 a dvtt = C(s(t),t)-f(s(t),t);9 W Z- ~5 Q7 V
%后向弧调整量; q9 r, e' }* \$ q# S, t3 o
elseif a(s(t),t) < 0" t7 N+ v8 c& x( l4 [6 L
dvtt = f(t,s(t));+ z# O2 m9 A2 T8 N4 V$ h, P) {
end0 f( F' N& x( T% u9 r% v- x
if dvt > dvtt7 ?0 b" b9 q6 ~; L# c4 q; Q
dvt = dvtt; 9 `% U( H. t0 l/ ? end1 n- N6 Q- I% G( t3 P' u/ A0 E2 {
%当t的标号为vs时，终止计算调整量 ; j' v" M+ z$ o4 }: `' V9 t if s(t) == 1 # B$ b$ X i! Q0 C break; 4 @, S# _' v( {& m end5 H8 b) D3 g; d7 B: l
%继续调整前一段弧上的流f - k* L6 z; g) l H I' ^4 \ t = s(t); 1 X' m3 j9 s; ] end " P/ [. w# b' ~. s0 G: |$ S pd = 0;2 _3 l7 c5 X4 R: `! g" T+ S
%如果最大流量大于或等于预定的流量值, n( w$ ^7 v* s6 N# X
if wf + dvt >= wf0 0 z& I' ?4 p# q! ]) X, ?# Z dvt = wf0 - wf; ( d: q4 M {7 V pd = 1;$ u8 m6 D7 ^2 a6 E1 J& M- J+ A
end u/ w6 p$ T [ t = n; 9 [, X- \0 P& q7 P) r %调整过程 ! b4 L& i7 R& \ while(1) . f0 i- j5 O' D x) M1 D' a$ [ if a(s(t),t) > 0 2 z' G o9 C8 N3 U %前向弧调整/ R/ Q+ j' G! ?- _! t& q: U
f(s(t),t) = f(s(t),t) + dvt; * L8 ~; X' {" M |' E. R elseif a(s(t),t) < 0 ' B- U0 X$ @* ]' n7 @7 B %后向弧调整) F) q9 y! W8 h5 B: `5 \& d
f(t,s(t)) = f(t,s(t)) - dvt;) q' d1 g" x. p2 N; _9 n
end * _- s( t) m* T, X' ~
%当t的标号为vs时，终止调整过程 7 o3 \6 x/ |3 F7 W. ] \0 I" ^ if s(t) == 1% E) p+ n# B6 D2 U
break; ! s8 }" }' I- @3 |% ]( B end4 m- [( D& {1 a/ B" V; ~
t = s(t);9 E2 j, D/ k8 Y: u. `9 b& K
end5 C, f; I* a; ~; ~: C
%如果最大流量达到预定的流量值+ v* c- h6 h- h. B. P
if pd" c! w9 ] [4 e% A, _( Y
break; + B) `; @9 h% j; h4 { m0 o) ? end1 ]) B- a0 u5 i* l6 U% w
%计算最大流量; F/ I6 C. R4 Q% [1 C, W5 Q
wf = 0;5 N4 N$ K- R6 G* V
for j = 1:n4 Z+ T% e: l3 a% q* |9 b, M; d0 K
wf = wf + f(1,j); 2 Q9 }3 |# n% s0 g% h3 b7 t end B+ V( S% v. j8 k% `
end6 X2 H/ \8 J' d: S/ ]5 `- m+ Y3 r
%计算最小费用2 ~2 R. ?0 o2 |1 @; ]- L' \
zwf = 0;1 n2 ?3 I2 Q; M a# B5 }( P7 n
for i = 1:n 4 Y" i3 b+ r, e: @2 S for j = 1:n4 t+ |! I- U, p: f. R. @( D7 z' s
zwf = zwf + b(i,j)*f(i,j); + O% O: C' A7 \* W, Z U4 s$ @/ u. Q end 3 k/ P! O$ `- {3 Cend: J! f% Q; X' v; ]: l$ l# A
%最小费用最大流- Q2 x& G6 V+ F& d8 k# o# s
f : T0 k% @( x4 M- t, W%最小费用最大流量! s/ |3 R {! f
wf 8 p1 h9 D3 P, U6 J i$ m" X( y4 b%显示最小费用& q2 V- y3 g0 `# @( S( k5 }: K& u
zwf- ~- W. x& k6 c! F# ?4 c5 m
1 7 b s0 w9 {" X4 r0 H, H29 R3 L+ s( k# K
3 6 G, o# T3 C- n8 m9 D, q6 e7 H# k4; i* ~1 u3 m2 \0 Q' z
58 o/ r8 J$ }6 {. u" j6 m( i
6: {% |1 }9 e* p& I9 i i
7' }4 C z: W" l9 @3 x0 \
88 b+ v3 Y6 m% a( B3 F
9 " O% d* q+ \$ R3 z* J10 I' x1 ~$ e( e. s, h* y
11 . M2 L- e, I3 W M2 r. ^8 |12 1 ]) f$ j* u+ I13 8 x3 \0 c$ D9 X: C+ d. e2 s8 A14 ; D h# k; }1 Q# h6 I15( {6 A' J! i, C9 b
161 N7 ^4 {3 l! p
17# a7 T9 g& q- `( y8 r+ M
18 : m* @! @' r" m5 Z2 v) |19& ] x1 |) x. p
20" V- O6 V" r! I2 W' L
21 ( `8 |' h7 t: C+ f22 . w, ~/ j s5 i( k. Z$ l t233 v9 W4 j+ O0 K8 o) i' \8 L
240 T: M9 \& M, N7 `6 N
25- t" K! u% L$ x& l# z; s
26 7 J1 }; C% j! E2 \' f' M; V27 & H& I/ W% P3 G" M6 h4 ?4 M0 X28 : t* n3 C: l% U2 M3 F$ J$ Q29 & ~& l' e- m9 X) v% ?6 t30$ R4 l- F5 J# k& \5 [
31 " t. I# O& X1 b3 l" r* p$ C7 ^32 0 Y. M0 T6 s) E# Y33% k) S7 K; ?+ V+ U
34 ! K1 |$ n# W5 h0 \, H! m2 j$ @35 Y" ?- |4 k: `% f# ?9 l' Z$ ^; @" i36" ~ Q2 g, Q% t6 j/ K9 B' g
37 g( n& y0 j* h% f" y+ m+ L! ]
38 9 @! k- |7 J/ d5 x, W; o) t39 , L6 P8 X" @+ S y0 c; P40/ n9 ^/ b# M3 Z/ ]1 U
414 {0 ^" y4 e7 c/ r
42 " k/ h' Y# \- c! c' P' K43, P8 R; E* V( _3 B3 U
44 1 l; V& w2 `1 m3 j! {# I8 K451 @$ M. Z4 K7 i* t& k. ^3 Q
46% ?- [4 C1 s' m9 b. V
47$ Y* R8 W4 R; x; I* D8 n! u" }
48 7 s* x- h8 q' ~/ ]495 M* _5 Q) [: K! P) n$ M3 x
50 , \2 G' I2 C3 G4 M: n51 - w1 T" a: q x( N* t- x' n) K4 j522 C0 J+ b4 A2 H4 k" ]$ k
53! U) p5 M0 Y& A. m9 \2 r# M5 l$ ^
54) F$ W* G# Q: d. D
55! l h, Q* q' r6 K* z6 J" {
56+ {8 u0 h& d1 Q* j/ f7 R
57$ ]6 R6 O/ U, @ Y6 t" o u5 b) @
58: u' c5 z2 b! Q2 f. {5 @# |, ]2 } o
59 ! A- Y6 j. e, U$ V$ P7 G2 o60, v Y& @$ G8 _+ y
61: \" R2 P+ U5 @$ b# G# `
62 4 R; Z* W+ @' j$ k63% k& j) Z1 ]2 f b; b
646 W9 I6 K" F$ a6 b3 O9 q& s
65! V) c9 A: E j a8 D) X
66 4 A* ~+ K G R" i1 `( r; R0 R67 - Z' m6 Q) m+ s0 R! ]3 m68 " @6 c: x9 K- z8 ]( T693 Q" q3 w0 ` j; [/ [( U4 X
70; S: X* o. C$ k9 h
71 " M/ u( O& ?5 W# F8 F5 H72$ K3 ?. Y7 W. P; H' t( I* Z3 ]) |
73. G3 { r; z; V& C' G
74! R2 i) j# k" w: n! L
75 ; o- {5 ~2 l% L3 z# y' O767 c( p5 ^. `9 Z# I- j2 o6 `
77! M, B/ r; u J- U) @
78 + O$ m3 ?# j, a9 |79 - P5 q7 r3 Z$ b1 j P5 l80 / s2 l5 p5 Z1 E; e1 `9 h0 h81 ( X3 B4 _1 d8 _82 $ b- M. D$ H9 g$ u( r836 {; z7 l$ B+ l$ o
847 b9 \* t& @1 f! n5 H; K/ P/ k' X
85 4 U S2 h8 E( O1 J( S, p2 S867 c# A: J: O5 W- J3 W$ Y6 [
87 7 X1 g3 {8 y" U R4 c88 , I _5 }# R' Z8 R, n( ^" I89 # B8 x( y& X4 q% m3 ?90% j) A& q4 B" Q" l* _( D# D
91 0 [) V' a: P/ W92# f$ H/ v1 q# A* Q9 i ~
93 6 u8 X9 j/ o2 K2 `7 @/ |94 & q& ^/ k8 I2 e( E! i95 % k" O% i$ X+ U% O) G96 % ?' }0 i+ C) f971 v, L5 C% j! A
98 : @! Z9 q9 j; |- ~5 W9 Q99: D$ V; A. ~( I8 C0 V. _5 G' d
100 + J/ i; G% r" R7 `& o5 P101 % L# v/ |6 ~+ _) V7 F# G6 ]102 ! b: I5 s# W$ @103 ' n0 J2 t: v$ t7 O* f; S$ w0 U% v# O104+ r' Q3 m6 \% S; O! |) h! c
105 ' z9 b) K) N4 h5 u( q. @/ b9 j106 % G% d; L2 E# A7 L: `& Q107 3 m3 j( `- ~) R/ A+ B1085 z& f2 P4 g3 i$ b0 s7 k
109$ p; k# X3 D& i5 g0 F
110 6 u$ h5 V; J4 N' ?% {5 w111 h6 k5 b8 G3 {7 ^112 7 q0 _2 A7 f+ q1 a/ H$ R1131 b4 T7 Q* X: z
114 - L8 ^8 F/ \, z- f115 4 z% O1 [ G* X0 t* E& n' b116$ A! x5 z3 r% Y! b
117- w( w: f% Y& @; o
118 " Z ^+ r3 d3 f' V119 ( T7 {' ~: v3 _1 M1201 }1 ~& P0 U L* `5 P, C* v
121 6 i9 H& X4 n2 F. k+ G& H9 V122 ; a$ x6 @7 A* ^+ U" x4 i1236 D) W2 |8 _6 [6 \& d" C
1245 y3 l. d2 ]. S5 K
125 & B7 M" W9 G6 [0 A( u6 d1 F z1263 X# e' T# ~1 c5 h3 [
127( z% d% I8 K. k5 M
128 : T. r/ D( Q) f$ H1 r+ l/ F. v129* I7 v1 O5 c' d; O
1305 R& c$ a0 B4 @$ r
131 3 a9 }3 L& W! W132- j: l3 |7 I, m: ~. |& @- p
133 $ L0 k1 Z* D7 {: [* y8 G2 B# F1345 `7 C/ T7 z' ]
135( ? v6 z, b( A/ T d0 C, w% Z
136$ |5 S+ l& ~# t+ Q
137 % u! r) T, }1 s5 d0 I( G138 ; k4 @ k5 ^4 q) N4 Y139 - ]) Q. f& M5 d140. Z2 l8 e! J, }. d7 P. J- `) ?
匹配问题： ! L J" y# j& O- z/ b( \1 d4 w3 N - d3 J& F9 j+ ?5 A0 b最大匹配： 0 t; L5 O) G; I2 I8 m8 R" g& ?# E8 l/ ?/ N
1 b3 l( v3 o5 e; d代码实现： , t& t7 X9 }8 j# {" z9 Z2 D0 k 1 |7 E( i. r" x8 ~m = 5; * ~1 |" |' z9 Q: o+ L1 j9 nn = 5; # i5 p9 c5 v' ^A = [0 1 1 0 0;1 1 0 1 1;0 1 1 0 0;0 1 1 0 0;0 0 0 1 1]; # j2 S. \ Y8 OM(m,n) = 0; 2 d6 d) `' ] Ufor i = 1:m , b( a) Z7 H) h& `, M, z+ N0 V for j = 1:n 7 g* e1 j3 J- p5 a %求初始匹配M7 A7 p+ [) T- k6 r" N W. a
if A(i,j) : G% j6 s" x# }1 Q4 v
M(i,j) = 1; ! ~* i# n8 q* ]! T6 b5 w break;6 B% {/ [6 `! F/ r0 I5 F+ r
end* i" M' n5 e( S7 K& T, I
end ! i3 f8 D4 e$ R; w6 m) W& N %仅含一条边的初始匹配M8 b! b4 }0 x- Q- X: e6 A
if M(i,j) . q9 l) V# b. W break; / N8 c7 B l) w6 v* q) {( p9 I end ' D0 G) |0 Y' h( T1 y' t7 }end3 w) H/ L. e% F/ H3 o
; {6 ^; j) {1 `9 Q) l; }# Awhile (1)8 U! D& \, w4 Y% m& c
%记录X中点的标号和标记 % b4 X& x8 L" y/ u for i = 1:m ) m5 Z" K% z0 u6 I( ^; o7 B x(i) = 0;/ T$ A9 M7 |3 |7 T: | y
end 5 B- A9 t1 }0 R0 I5 b' Q %记录Y中点的标号和标记 2 Q' ]! O# O; I* V; U0 x8 [ for i = 1:n " L, J! f& o6 i# v( }7 B7 c0 N* ^ y(i) = 0; $ x9 z9 B" ?6 M/ S, u& z end# e6 S! c5 h6 N$ B' ?. h- B8 _
%寻找X中M的所有非饱和点8 }) ^' e( x8 ]1 Q1 h+ l r- v+ Z# ]
for i = 1:m " J* I2 N- g5 B6 p5 E# J2 N v. W pd = 1;- Y2 m0 @) G2 }9 L" X7 t
for j = 1:n 6 r4 K& u' r* a* c+ p if M(i,j) & `, B. {6 r# L' {* a; I pd = 0;" `* h3 m, ?8 u* p+ g+ O5 ?
end% I; t5 h. P& u: I8 e
end- n2 v% u9 K! S1 l3 }
if pd 2 q1 [2 A! }" B0 q. \; c- j x(i) = -n-1;+ ]# y# e L" O$ S, R- c9 e
end & f4 n! O% _* S; T; e3 k! b# `/ L4 ~ end6 ]+ [ W) U; I0 |4 F( S4 Z7 G
pd = 0;5 c% J# l9 @- J6 D9 M/ W
while(1)+ `9 t1 g2 p8 q& ]
xi = 0; ' \7 r3 q( Y l6 Y) W- J' w U for i = 1:m M) _) G) i% f6 V" C7 P if x(i) < 0; z t3 p& g w7 d5 V. e
xi = i; 8 `) e K7 ^, f8 i$ |% [ break;' k. C3 e: L G
end + U! w6 @9 } _# F, r2 y E, ~ end 4 i: ?* L" w% z. l& x if(xi == 0) , L, S: N; _3 W: \$ T; w8 o% u' V pd = 1;. g5 h: Z( h( T% y
break;: P" N# r; e N% G5 ~
end + z7 i) v: f0 v8 r! {" j6 N- F$ g x(xi) = x(xi)*(-1);9 m3 P t6 C Z% T: \9 g- U
k = 1;3 J) q' w7 J4 u' L" P/ m$ r
for j = 1:n4 u* ~. E/ H3 e v$ p
if A(xi,j)&&y(j)==0 ' A" G7 L& U7 d; S3 z* T6 F8 m2 [* J y(j) = xi;1 B4 b& b8 M; h S t- m7 {1 `
yy(k) = j;4 q: ^6 k% O0 |
k = k + 1;3 f6 d, g5 E# |. z
end 9 k2 t) K0 A& ~$ O I1 N% c end* Q; F1 I/ X! `4 O% g+ x
if k > 1. h( ? i5 [1 ?
k = k - 1;: x) @1 `, B* L" M0 f# ]
for j = 1:k4 K5 A5 ?; x6 t8 A; p
pdd = 1;4 r y3 K3 b* q0 b. q7 _! Z
for i = 1:m$ ?+ t6 Q' s: @) }% v& g
if M(i,yy(j))6 L s; R/ r: z# p
x(i) = -yy(j);# u5 W) i7 O7 b- _/ G5 p
pdd = 0; ( r* Q( I: U$ c break;5 M x' R8 i: ]0 L
end( {# g) `9 l; Z7 J; e
end $ N* b1 O% ^: L: U( J if pdd 9 X8 N1 h- v; U6 a0 } break; 0 Y6 U5 b! J3 T end; C' x, h. F/ l. g
end ( N4 @1 T4 k1 t$ S+ a7 F if pdd , l8 X& X4 H4 l- Y* b
k = 1;7 D+ ]- |, L2 M5 ^ t6 n7 C7 E, O! F
j = yy(j); 2 F. G3 n% u [! r V/ z while(1) 0 N6 Y% ?" T) ~$ O8 m6 M# K/ {" [ P(k,2) = j;3 X- S! \6 J5 i G; y
P(k,1) = y(j);; I9 ~' ^! y$ E5 m, ~) S
j = abs(x(y(j))); % n/ `! O$ l( f' m- x1 m if j == n+1# \8 E* z3 L' v8 ?
break;+ {" ?( @/ c% G" R1 B
end7 _$ f' E; B/ f1 s9 e6 E
k = k+1;: T B5 {+ w6 _" R
end8 v" s. p# d1 d8 Q. l
for i = 1:k $ P E$ k" S9 r" V* B if M(P(i,1),P(i,2)) 6 H- x7 ]4 c6 l7 k0 k M(P(i,1),P(i,2)) = 0; l% @6 X v$ e2 Q$ u% H* u6 J else8 ?% ?: f* [+ e3 m' Z
M(P(i,1),P(i,2)) = 1; ( {! M. o4 K% |# y end) ?6 F7 ], k- F* p5 ~
end 6 G) `1 S, ]8 z0 @8 W+ Q* G; J break; : {4 k5 i% x) ?! x3 j5 N end$ p1 y, O- |' B$ _; J6 e
end/ z9 \, i2 Z) A- \
end' F7 D9 _8 C1 x! X; S4 X0 o
if pd# z( E/ m- p, \! l& m
break; 9 W( U1 G$ R5 `! o# n end8 d; w0 `! L- h$ i( S
end! i: o& f6 k4 d
1, F( Z8 n- E* l1 r0 I
2 % t9 w- N& m5 v& i2 l9 w n38 U) Q6 j, n& _2 R w- L
4 4 M; S" J( {1 H( F7 ?9 m5 ( H" B$ ]8 J8 l2 r' ^9 Y6' D8 X" p5 E6 T# H' B
7 % ~* Q' y) P9 o" Q" Y% y( m" E" `8* [+ J* ^/ f' ]4 \8 N
98 f7 W# A: G& k4 \# c/ j' _
10 o6 ]9 q9 H2 U) n
11 4 v8 \" |, D/ B; {12 1 I; o. ~7 u0 A2 a9 l8 W13 $ O s' w2 j9 x# w4 l3 J14 0 ^, t, i t& G4 l1 N7 i$ A15; _& k0 e. C" i
16- H* m7 B8 b6 J6 D! V% \% r, j& s5 W
17 % g5 ^ z' k- Q( S- f: K18 {/ w. d1 J$ S: J8 ]
19 $ g$ K$ U9 E9 `! E20 6 i0 A7 k( w) e# R$ o21 9 q5 Y# A/ f( l' k S22 - \3 i& ~) L3 L C0 z9 p/ r- H" ~) ^23 1 _2 r4 c1 D# B# P* f" `6 m" d, y9 Q24 0 W& S& M/ u1 O Z$ @* }8 H' N5 F25 3 A( A1 @" M# A( j26 # k9 L8 J+ d6 D+ ~5 N3 U; a27( {, k; F6 W% @& _$ P2 } C
28) Z& t1 @# K, j4 ~) t
29 & R( g1 t/ p- Q30 . `; O# s0 S6 R9 b( t5 l# M7 h31+ }! _3 s6 z$ l) B
32 5 [; E) Z- i* \3 R a6 n, s4 D338 }. g R8 X8 [9 h2 G
34 0 H$ L$ I% F1 D# J }2 q6 }35/ H) e% h5 d5 ?6 a0 d; h
36- f- |7 B$ G; P& p/ Z
37$ V' v: j( U. D H: ^0 [ M u
38' C# l5 }# R; p% W( ~( k3 [" q
39 5 n/ s! X9 z0 E' k40. D3 C. s/ ^' w5 v0 l* O
41 & g) q% @ D8 P3 h" k42- Z" e" v+ V" o+ _( n! p
43* A9 Q& e* V4 F- a4 |5 e! N
44 9 Z& q% s$ F- ?452 Q( C! L4 d# k X# V/ u+ Z' a
46 3 I) M% J) i+ P) w0 a+ \& f47, P$ u5 v2 b# [7 L' Q q4 ?
48 + D9 Y( f0 F; d: C49 ( t% u$ B) z1 ?/ X# d$ m; e6 \501 Q! y$ U4 D/ j; F! B( @
51 ; p8 Q% m4 ?( t* ~! W' y7 ^52 . n' z. @, _7 o' q( N53 5 ^8 z1 `7 s+ i1 @9 Y+ A7 y$ o4 a541 S/ ?+ e( x- j+ u
555 Z6 p, Q8 K& k3 n5 N& O+ U. U" G
56, E3 Q: q* @$ ^$ x
57 ; L' ^6 L% ]' t H58% F, N* F$ h! L8 R# ]9 G
592 n! _# X# l9 k3 A" \! J
60 ( V9 Y6 Z3 K9 h2 c! U6 G% U61 2 Z' s J# v! K( B623 r! M4 K8 J q
639 m0 ^* L: `1 | ^- z$ L+ _. F
64 5 W1 o: m1 w. e" X. w( l& N- I65 9 H4 K3 a9 V) E66 # P$ s1 ^+ S& Z" z67 + l3 M% M l3 j S) Q# T% F3 b68- r p8 F3 q5 _. f& A# i, v* Z
69 * `. }9 R0 d$ e! |( M' {4 U+ e* L( O70 $ l% E% g! F/ Z. q7 w7 d71 : K% a% J5 z5 P9 d3 u0 M72# W' z' S- U$ k
731 A# J1 V% ~! Q; X2 E1 M
74 4 N. Z/ ^* e) C1 W% P75 - j# q W4 {4 l" S- K& V9 ]76 - U( R* F7 b: t4 p77# g0 ~0 V! [0 q4 S1 D" Q! V
78& h: z; d6 y) N! q) {; w
795 `# ?# ]: ]( b
80$ Y$ \$ S9 b; r& z' D
811 `4 K+ P9 J% i9 _+ q# P. P
82. |( I- K8 i) p; g$ j/ y) F; t
83( T7 e9 H2 L, j) N d. h3 g2 F
84 4 C/ W4 Q2 q* k {# G5 d- N0 }3 W85 . M( N p, v! I: c86 5 A b/ J5 U8 |" o& w87 % ?- w$ Y: T5 y; h, K8 x889 v, o1 b3 |% x+ E- h- b
89 / J- r+ u7 S; a+ u; ?# C% I905 r1 b' a2 L& p' b
91 ( A7 u0 A) F. p1 z( P92 ) u* o' Z; B5 O* z: b( T* s93 7 j" G% Z! H7 A, d6 z. I) U- N2 d/ N941 O L. U' H- A8 g2 D
95 * ~9 Q* E) [( J& p* j96+ s) R' Y2 j1 T, i0 ~) Y: z
97 2 l: p! n+ n9 h98! _6 |6 |8 n! N" Z
99/ b6 x& `5 b8 A: C* s
1009 w# u3 v+ C3 X5 g8 V4 @
101 % Q) ]/ h+ J3 U; D102 . U, W3 R% t& R2 T4 O; e" E" v& P3 k103' x* l7 L2 d" } V
最佳匹配 - X0 I/ ?; e! o. H# u3 L' ^ " _$ y1 O! L X& ?# d- f代码实现： : D. l; y6 g( ^6 @6 I6 `, C) h/ G/ U! B6 @+ [
n = 4;8 j7 s0 [8 Z6 x( `, V5 u0 k8 V; G
A = [4 5 5 1;2 2 4 6;4 2 3 3;5 0 2 1];2 x0 T' D' B1 Q5 W8 p- h( z
M(n,n) = 0;. F2 a) v$ G8 |- i' [
for i = 1:n 9 W( _: k( a; z9 ^( t L(i,1) = 0; 9 j5 r5 s: B `2 {' w. `# L5 k7 v L(i,2) = 0;3 P" v! u8 y2 w2 v7 x) ?3 `
end 2 f0 l4 A+ {4 ~1 t' M6 {%初始化可行点标记L & d0 _% B2 a2 x4 ?7 G5 Yfor i = 1:n" S# Y! \5 [9 Q
for j = 1:n6 u( U- o. g1 v) g
if L(i,1) < A(i,j) , b* e8 f9 ~# K2 \4 Y2 V L(i,1) = A(i,j);4 @. Y' s% w- _
end 2 w+ x, X: q9 ]# m9 a end ) r0 ~& n" j# |; R3 Dend ; b# \) a; P/ U P% c% [' y%生成子图Gl 2 ]; U, p$ G* f2 e2 r s/ m1 z2 }5 ]for i = 1:n" E! ^0 J5 q T1 g
for j = 1:n& f% X! X& M, o! V, G C
if L(i,1) + L(j,2) == A(i,j)' r/ a7 T7 A, A$ p
Gl(i,j) = 1; . i, v4 D6 w* @2 _% C. o! O A# ` else 1 T! y7 Q" d& E: u" a7 w Gl(i,j) = 0; - U" ^, ~ }6 {, P% u2 T end, r1 W* l9 l: I
end5 E& M' Q9 u v4 F) J. H
end* [. s% N* m2 {
%获得仅含Gl的一条边的初始匹配M) m( [% V5 z6 \4 C E; v! y* @ I
ii = 0;1 L2 e ~$ x7 f! b- o9 e' \
jj = 0; 0 b3 M6 t6 {, S) Q1 A; hfor i = 1:n/ l$ H' T, g( x8 ~8 T& I
for j = 1:n4 ?- r; U" o/ T- D
if Gl(i,j)- {% w3 S5 q' V) {# M7 ~# ~4 \
ii = i; % t$ R' p% [0 |6 E9 n jj = j; 9 l. }/ i& u4 m" T1 ` break; 8 l( J9 L) R) s# k8 c( ~* ]/ l end6 o9 \) ^2 y3 b; @
end1 I; b3 u$ p$ _7 M2 Z' p% m
if(ii) ( ]* C6 v# H0 c, C6 W break; ( D6 Z9 G+ C& L) h end$ k% b# w7 q; S0 G3 n- F
end1 w2 e4 n! N0 t) X' v ^; b
M(ii,jj) = 1; ~: z2 k9 X2 M4 ^- L: K* @
for i = 1:n8 s( E+ W7 g( h
S(i) = 0; 0 p S3 Q |! w' a2 a0 \5 M M% K T(i) = 0;3 v( @6 ?4 f- T3 m2 K- j% I
NIS(i) = 0;$ Y3 ^ j" a/ r. [$ q7 r
end# r5 T1 y0 g( ? S4 O7 N) m
$ `7 o# z) n/ Y& g, b8 h/ h, f/ l - ]& ?2 l, n" \2 Jwhile (1)# f+ N9 D" d7 _1 z/ a& M
for i = 1:n 6 H8 T4 z9 X/ } k = 1; " Q8 t/ K# C4 a" r& B. ~3 L for j = 1:n & W$ D5 W! T* S, o$ R) a+ s( B' A) R if M(i,j); N# C/ s0 a- m. x: ^, K
k = 0; , J2 O( M7 c# A( o! l5 T6 e$ v break;; I, ?3 B# s" r' V
end: p. u& Y2 @) D! O1 W( D7 b4 n; p
end# C y$ l. M2 v g( O! ?; S H
if k 4 E! H* O8 |* ?5 H break;/ ~0 B7 J, h. N- F
end 7 L/ R8 k# j- |5 n& a6 X end * r$ j. |$ d) s8 ]6 X, g%获得最佳匹配M,算法终止 % o! ^; l! W. L2 r' ?2 s. r+ U! W if k == 0 / g, v7 J9 r* l* e8 }) g% } break; w9 o7 W0 S: V# D. M. a end( v1 c4 Z1 p4 F+ A
& u s, i1 Y3 Z; I1 h& |
, m7 G2 h$ B& R/ F( L%S = {xi} 4 \( h1 E# ~5 AS(1) = i;! u1 v0 o) \/ i" b( C$ D6 g
jss = 1;. t3 S( Q* P5 \) Q: L$ ]% Y8 \8 d
jst = 0; 6 Z: j" m! p0 @4 G; d$ Iwhile(1) 4 |1 ^% E% @ H jsn = 0;; }' D3 {3 g* }$ ^, c S
%选择NL的值 9 c; r3 f4 P6 z) ~ for i = 1:jss 0 w) i- E0 j9 n* ~0 P' _ for j = 1:n& [! ~- R' b* l
if Gl(S(i),j) + T6 L: l, n. e( ]$ [( p jsn = jsn + 1; # u) E' J, p; F8 t- \8 h NIS(jsn) = j;9 F {6 ]' V! G, r3 h
for k = 1:jsn-1 7 T' W1 R% D% ~* ~, U2 p6 I z if NIS(k) == j ( ^. T% S! h! ]3 B' U! W jsn = jsn - 1;5 ^# A; m G2 _0 Q2 b! M
end) u% F( d, {; H- ?7 \
end 3 E. L4 w# b, Y' q end& F3 ]* B. B2 B0 ]; n
end * v6 e/ j S0 I, v end 2 |5 G& l+ _, V$ K! L %判断NL(S) = T ? ! _/ N: @" o! D; ^+ d% Z2 s if jsn == jst & o* P( [: b7 L' v pd = 1;* F$ U* q2 a7 z& F/ K1 D3 T' y% O; N
for j = 1:jsn0 Y5 E' ?. E( E; O2 h
if NIS(j) ~= T(j)) ~" I8 u: G8 Z: W) m
pd = 0; : p% x! |2 k$ E4 c b break; * z/ ]( d" p" j" _ end! V: w) |: L+ t
end: C' O( g8 W& r6 u g
end 5 G: ^& N6 i) v0 K %如果NL(S) = T 计算al的值 / v1 b8 p4 _1 ^& X if (jsn == jst) && pd # z- J; B6 x4 l' G al = Inf;2 @) z; R& E6 K: S& `
for i = 1:jss5 }. y' O! L! f) ^
for j = 1:n # k7 b' p5 m. M9 q: y" B' P pd = 1;! D0 x1 t. F6 _) W' Z
for k = 1:jst , k8 m8 {' L! \) J; X3 D9 h5 d0 N if T(k) == j + C8 h1 [* U1 ~, w2 Q* w pd = 0;7 v* @. ]. U# m7 O
break; 0 d: |& }, m, k" q1 G$ y9 o end1 F% i+ e% E v6 {& ^- b
end ) E- B* s+ S& _- w if pd && (al > L(S(i),1) + L(j,2) - A(S(i),j)) 0 c5 E3 |8 M, {; S* Z1 I( c1 | al = L(S(i),1) + L(j,2) - A(S(i),j);( s" o. p8 e0 P, s4 a% s- G% S/ b
end1 _6 }+ w# y5 ~
end ' m2 L V, B- X) D end& B m1 j$ f P
%调整可行点标记 # c- R# E9 @7 [1 `! I* {1 q5 P4 t+ z for i = 1:jss D6 `; t2 E- z5 B
L(S(i),1) = L(S(i),1) - al; $ B4 y; e- Y$ Z7 j( k end( ^! z( l2 F* ?0 @/ v0 u
%调整可行点标记 ' h7 P. V3 U: T8 c4 X% ` for j = 1:jst& m# Z M# y* T, G- {- R' A& O, h% k. ~' Z
L(T(j),2) = L(T(j),2) + al; 6 `( h" \0 d4 e/ r& w6 a end& J* r) i: B1 v9 M4 W+ j
%生成子图Gl( X* e1 C/ t# ?) h
for i = 1:n8 \; Y" J; k0 i7 P+ v) ^* `( ^$ }
for j = 1:n $ G) A" d4 O% p4 S: F if L(i,1) + L(j,2) == A(i,j)- s% f' {& H2 K! o- k5 t2 T7 S) q
Gl(i,j) = 1; ) c/ O/ T0 ?3 x, r9 T else 6 g3 x* e8 i6 u Gl(i,j) = 0; ) ^+ M! f% `! _& o' L6 L4 [ end 9 M( M/ l$ {. B) C M(i,j) = 0; 4 i: @7 r2 [( W( |$ Y k = 0;5 a7 N# b/ i5 X, c/ N
end4 y; e( j& X2 i' M( Q
end : ]% h; H! X1 g' ` %获得仅含Gl的一条边的初始匹配M% Z1 |! W ?+ X9 A5 _( @% a
ii = 0; % C+ K8 r* Z% J. e- n; r" { jj = 0; # R- W% e" ]8 ~* R for i = 1:n9 P. {7 n) ^: V+ e# a
for j = 1:n8 h* W2 ]! t2 w) X
if Gl(i,j)1 y7 _' u" T% L) M9 r2 K7 o
ii = i; 0 U4 U7 D5 E$ b jj = j; / t h, m0 I p0 ?2 k break; 2 Q" K- R! O5 F- @2 I end& m4 P( v& N) w: l
end : E6 f- Y6 Q, Q if(ii)9 \: e3 U; j2 b
break; " G" O6 {# X! `& _. i end4 B+ u5 x# l7 u# M0 S
end, n$ y7 k; @0 u4 H0 b! [
M(ii,jj) = 1; 6 f: I/ x; @& {$ \& p) s break;8 D* ]' {' {! h* `* {1 e* P3 `
else & q; Z. N z, u# U; w for j = 1:jsn 7 C& M) i# r" `' s2 s; O: }0 v pd = 1; " y: @( c& t" L5 w0 ^ for k = 1:jst ! ?& o6 X1 V+ {' C+ o' y if T(k) == NIS(j)3 q4 o4 f2 a5 o1 }# N
pd =0 ' D3 } y; j7 X1 [$ v$ I break;3 w/ H+ i0 v2 r3 U( f4 A
end 4 ^$ n: h; J @# R end 8 a4 \! w; O8 p |( V0 W: h if pd 6 [) n2 i) G( k- A1 a jj = j;+ @6 R( {" A& Z8 y/ a! _" P, Z
break;3 |" t4 ]& J7 d- a2 y5 n
end: u8 \0 D7 h d8 o3 n% S" ^8 k1 N
end & F8 Q4 R/ _0 d %判断y是否是M的饱和点! Y1 v' ~5 e, I; G& N
pd = 0;) k+ Z1 a& q3 N" n' e
for i = 1:n - \4 b& i1 K' v S2 Q if M(i,NIS(jj))) z! `- ?4 x5 x8 N% L. U
pd = 1; % V! |5 v9 c" K; U3 _2 U ii = i; # u1 b6 h. J' h' u( b7 Y break;3 R7 I% u9 q: s' @% _
end( m) d0 |# s$ z
end- Q1 x1 y2 t. z. g0 j( C$ {
if pd2 W, D4 s' T, j4 H: | G
jss = jss + 1; ; z: {* d) u" V S(jss) = ii; ' S8 \; p$ ^- S# {+ I1 Y# l jst = jst + 1;' j3 _5 i' @' @+ T3 J: a
T(jst) = NIS(jj); 8 s6 x/ U4 O% K else 7 W2 R( R _! Y9 h8 l, B p4 k for k = 1:jst P( L3 M/ d- U# L" E( { M(S(k),T(k)) = 1; 7 w6 j' W$ w6 o; Q! g M(S(k+1),T(k)) = 0;4 F1 F. i4 b: ] V; Z. J" O
end& w" y8 g. Z0 a( a! S8 b) q
if jst == 0+ i$ q) R( Y( k; I
k = 0; . L* b( w0 _1 N8 u I end . r" e( F& S n: | M(S(k+1),NIS(jj)) = 1; . O7 ^" D4 U7 }0 B) h3 T) O: Y/ M break;3 p2 r2 G- h% @
end 1 Y# d8 t, ]7 O3 x- a end7 K) g; r( I+ ~% D
end . {1 j1 d3 l# ]/ h$ C1 i q end4 [- j; q; z) T
MaxZjpp = 0;2 k2 ~7 j c2 J6 F" H0 [
for i = 1:n6 F8 B# B$ p, r" b# s, e
for j = 1:n p) ]% k" ?2 A) o if M(i,j) - _, p( d8 U: k* c& ~ MaxZjpp = MaxZjpp + A(i,j);5 Q$ m5 p& j8 D4 j3 H& R
end5 j6 l& K6 h# o$ a) {
end. d; i2 d) W7 Q& \# P6 V
end " G) a* z1 m1 r& t M 1 a% R. [2 G8 B( n" [ MaxZjpp 0 ]/ V& p$ L& J0 z* S" u5 [: K* r* g: c: |. H. r; D" h4 G7 Y5 Y
( D7 [+ g' L" E+ |& L; B
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发表于 2018-10-30 11:25
sum(arcs(i,j):f(i,j))-@sum(arcs(j,i):f(j,i)) = d(i));8 k' O( T; D# n
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