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TA的每日心情 | 开心 2021-8-11 17:59 |
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签到天数: 17 天 [LV.4]偶尔看看III 网络挑战赛参赛者 网络挑战赛参赛者 - 自我介绍
- 本人女,毕业于内蒙古科技大学,担任文职专业,毕业专业英语。
 群组: 2018美赛大象算法课程 群组: 2018美赛护航培训课程 群组: 2019年 数学中国站长建 群组: 2019年数据分析师课程 群组: 2018年大象老师国赛优 |
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最大流: ![]() 注意Matlab 中的最大流问题是必须是单源和单汇问题,因此这里需要构建虚拟的源点S和汇点G。
+ t7 g# c- c: zclc,clear
1 k+ R# F7 k2 G; Q3 Za = zeros(9,9);
4 `* _. r- N4 u/ ^a(1,[2:4]) = [20,20,100];, H; C8 P* E# d6 |6 W) C- s
a(2,[5 6 8]) = [30,10,40];
/ @( A/ ^- i9 e. e/ |1 S7 Va(3,[7 8]) = [10,50];! P J" \( y7 R, @' U) U5 N+ f
a(4,[5:8]) = [20,10,40,5];
1 }/ o' `( {; c5 _6 Ca([5:8],9) = [20,20,60,20];/ e/ c1 M3 k3 h) u. L
a = sparse(a);9 Z2 `2 A; N. V( ?+ }. R1 U# ^
[b,c] = graphmaxflow(a,1,9)
6 f5 C# y; a1 z: d' T/ h% o最大流拓展:最小费用最大流 仅仅是在求得最大流之后进行对最小费用的求解: 7 r- C0 ^2 G) V: _8 p( h4 J
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)
% O/ T' N/ z6 D3 P. p3 H最大流量为11 自定义Matlab代码: . p! p2 M% D# h. b3 c! ?
" v9 a8 d1 }3 ?9 x" v最小费用求解
3 X/ w, }( R( t4 d% V; L* j& q1 B
Lingo:" }& U* _: ` ~% t4 |5 W
% P, l e/ D# G+ N J& [* j6 J: kmodel:
" u4 S7 l5 [! @sets:
8 e: j; }: `6 y( {/ `, l0 e4 Snodes/s,1,2,3,t/:d;6 b% [; X$ l9 B/ R: @* k$ }2 I
arcs(nodes,nodes)/s 1,s 2,1 3,1 t,2 1,2 3,3 t/:b,c,f;* X$ N# L8 S* g: c6 ^7 ?' h1 X
endsets+ @" L& I5 N- I: Q8 c; S2 W" r
data:
' U& C4 I( n5 c' t* a: B. Ub = 4 1 6 1 2 3 2;
1 K0 `. l c# `* c {! E9 ^0 A0 Bc = 10 8 2 7 5 10 4;+ O6 f7 J+ [ l. L* j* m$ W$ B
d = 11 0 0 0 -11;
* F; T5 C {- t( j" z5 B/ ^6 Kenddata; @* H( |0 `. @
n = @size(nodes);
?! t6 P, t; f# ymin = @sum(arcs:b*f);
6 i5 I3 G: S1 [: v@for(nodes(i) sum(arcs(i,j):f(i,j))-@sum(arcs(j,i):f(j,i)) = d(i));
/ M9 V: J; L8 R6 _3 f4 \8 p9 F4 n. W@for(arcs bnd(0,f,c));; z' b3 ~3 e, c/ n9 U% B
end7 }. Y' Z4 o% ?! g9 u" T2 H
14 z% \+ z: I& `% F" T4 j; G( S1 ~/ N9 i
2$ R7 _/ m5 {" A6 i
3; P9 H: M- o) \; f
4
# Z7 j8 h7 \9 ~50 j# t# a: q% @) y; M
6# C. B' c, f! g* i2 S3 `
7" b+ B% [' q2 e" S, z3 V
8. q7 L7 P6 ~% m8 n0 }- g
9
. f; K; ^4 x( t, U" J% r r10& N9 m& T. H1 a* J& t
11
* S( U' t% R- l7 B9 B; u0 _129 z2 z. q; v" E
13
' U# d+ Z5 P6 x' u( Y14( a6 T2 M: F H3 ~8 k
15
) {' j1 O! Z! b: t. C6 e) R' dMatlab实现:/ A6 ?) Z% A7 y$ W+ J$ b
7 f$ \% S, f# t4 J* X7 n9 j& R
6 V2 w A, ` U; Wn = 5;# Y5 H! I/ Q4 {* ]% s5 p
%弧容量
7 z, n& A2 {; u5 I& xa = zeros(5);
% l( l3 i/ V/ e7 X' za(1,[2 3]) = [10 8];1 @5 C5 \6 r1 @
a(2,[4,5]) = [2 7];
4 d6 v2 O3 Y$ z% \; q, [a(3,[2 4]) = [5 10];
. F) n: {. F$ La(4,5) = 4; S6 b# {( t5 \! r0 X v$ r
C = a;5 w# z; B- ]+ c7 a
%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];
/ F. ?( K6 j8 g5 F# j" T& i/ T8 j%弧上单元的费用2 S( c6 K* h+ ~
a(1,[2 3]) = [4 1];
( Y) a. p$ F& v( {a(2,[4,5]) = [6 1];) o- c. v, _( L" r* n
a(3,[2 4]) = [2 3];% k9 ]& D3 X' p/ J3 Y5 ]
a(4,5) = 2;
! j, s6 F+ ]6 eb = a;9 D* C. z; k6 ^1 s- a& \
%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];# h$ ]9 D% h# l0 v$ |( U" x8 ~9 W3 K
%wf表示最大流量。wf0表示预定的流量值) l4 a- S K8 {# ], ~4 z$ {
wf = 0;: o/ |3 u V1 d
wf0 = Inf;) l( f9 X1 _5 O7 \
%取初始可行流f为零流' _: e: A4 ~! q5 F' q8 @
for i = 1:n' m4 x, {# Q+ i% X7 G
for j = 1:n7 K8 s/ u) _9 ]* U) D( C/ A4 I% d
f(i,j) = 0;% _' x8 J/ J L; e- s5 T
end8 k- @& W$ Q' h4 Z( i4 M
end2 z6 Z# ^( l3 ]0 D. D
while (1)! e( }9 G2 J! v
%构造有向赋权图) @) l) z4 X: p. z
for i = 1:n- U, ]% g4 i- d/ ~, C+ ^/ [. _, `& y2 Q
for j = 1:n
6 l9 [# m2 w1 B$ o9 Q if j~=i# G+ s- u; B9 q
a(i,j) = Inf; t) K% c% a3 v$ X* u+ l$ `9 E8 t
end; m9 E+ w: m$ P0 B3 P2 ~
end
) F3 R' @8 m) c. j# X3 e end- q$ L3 d% L: J
for i = 1:n) {& M( U; u% K: k0 p' H% o
for j = 1:n
% n- B: _' F: r9 b if C(i,j) > 0 && f(i,j) == 0
. a B' u" f) C9 u4 K a(i,j) = b(i,j);
( X8 L/ t: ?, r, k+ K* L elseif C(i,j) > 0 && f(i,j) == C(i,j)4 k: ^- z& s4 o* [2 u, b4 V9 V
a(j,i) = -b(i,j);
3 M. n( r4 g0 z2 C2 { elseif C(i,j) > 0" j- e; T$ R. ^& C5 H4 @
a(i,j) = b(i,j);+ f$ M* F# @7 y+ R7 R5 j+ \
a(j,i) = -b(i,j);
# G) I0 E, G' F. a6 d end
/ G' d2 `2 ]+ U# Y+ ^" z: v end
* r2 A* e; h4 i& b end
1 c1 W6 o. }- Q4 u/ S- I; k %使用ford算法求最短路,赋初值
( a- n' i; `# r9 y, U4 @7 o# S: d for i = 2:n
( G) X# ?9 a2 K' f/ S3 [4 ` p(i) = Inf;
4 S: e% C4 k$ i' d s(i) = i;* }7 {1 A* S; _
end% W% m- S& Q9 D9 O% `7 D
%求有向赋权图vs到vt的最短路,赋初值
( v e/ B9 ^& h; c3 f for k = 1:n7 q+ I1 u9 @# H6 I" M
pd = 1;/ s/ x% D1 X5 _( F- D# u
for i = 2:n! V" B1 g4 O& T* F3 K+ U/ _
for j = 1:n" E6 v" x3 t4 z `3 {9 n* m. k
if p(i) > p(j) + a(j,i)
/ y* U: |$ @) S) ]+ r/ U+ j3 P p(i) = p(j) + a(j,i);- d9 M$ J$ H) }3 n
s(i) = j;
! ?# i0 F5 \& j0 l& O. e pd = 0;& ^; X8 S3 ?9 G% M4 n
end
# W5 I' i+ R* f8 i$ S& l end( a. s6 ?$ Q6 J. r: @8 W
end1 o0 l+ B) s4 n( _+ t
%求最短路的Ford算法结束3 V. `6 }+ h; J
if pd1 H' U/ |- C8 W7 p+ q
break;
7 K4 o- ]/ e: p$ X8 Q6 o; } end
' B$ i: r% d7 y" J, M end! l, q1 m8 U @' W; b1 o% B
%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, k$ I1 n3 f* M& J* ^7 d
if p(n) == Inf
( J( u" F% p9 H0 z break; H, M. W" e. O0 q. e. o/ b/ p
end
+ t& A: A" a4 j6 W+ }. c' U6 r %进入调整过程,dvt表示调整量, l( h( A5 ]8 h' f
dvt = Inf;8 `; O/ H/ I% y4 o- S' c
dvtt = Inf;
" b" g h+ \6 h2 W t = n;
- U9 m3 x2 i1 T+ w/ l$ W' V. u' h while(1)! {, \* H& ~' q3 n, m0 K
if a(s(t),t) > 0) [% t w, a' c0 @
%前向弧调整量
% X; n4 E* s# w% \ dvtt = C(s(t),t)-f(s(t),t);( U+ I8 h5 M! P8 x% b7 O: {
%后向弧调整量( {, l- ?1 |# {5 R. l! p
elseif a(s(t),t) < 0
) O6 s( d" z* |) J( k2 ? dvtt = f(t,s(t));% s K4 t4 V/ ?' l( O7 W( ^
end
3 H; [- _) q% r/ p: u3 g2 O if dvt > dvtt l6 ]+ c2 P/ g, Q4 z" Q4 l; i+ x
dvt = dvtt;) g" g) R; w( @ G
end
! l* U4 Z9 R1 L& k8 b5 [' J %当t的标号为vs时,终止计算调整量
! S+ N3 n/ `0 I" V+ x0 }# }+ z if s(t) == 1, g' j$ v: {; R8 U% j7 j+ z. Q
break;& b% j/ ~+ j8 b" ~3 \
end5 u7 x& [, R8 ?: J% p1 A
%继续调整前一段弧上的流f4 ^; o, U6 {; s
t = s(t);1 A0 K( y6 w/ n, R+ D% K1 q5 U
end, f' j* B8 ?+ q
pd = 0;
~" p6 |2 O# ~; v; L1 p! ~ %如果最大流量大于或等于预定的流量值
1 `( O J9 ]' W' s if wf + dvt >= wf0
: r. c, ^4 g4 ] dvt = wf0 - wf;
$ H$ m+ Q/ w- z pd = 1;7 A) B4 Q2 f; q4 I; k$ u! j
end% g: ]$ x y+ b% _
t = n;9 p7 w2 D, }- r# Y' R2 _" O
%调整过程
5 ?9 q; ^9 D/ C* v; x J while(1)7 \* l* Z4 F* x% b$ t
if a(s(t),t) > 0
# [7 e; |! R# |( p2 @; | %前向弧调整
' p+ o& c* i4 j3 S$ t1 l% A$ e f(s(t),t) = f(s(t),t) + dvt;
: j" A6 R1 L( Q6 ?$ s2 z& ? elseif a(s(t),t) < 0+ ?! f4 ~' G4 c( ]2 |: N+ [
%后向弧调整6 u$ J5 M9 X. D* @% }$ `& ?: N
f(t,s(t)) = f(t,s(t)) - dvt;; Z' s" U0 \# B5 P" X. i
end ; L/ [2 ~+ _# T2 Z5 ~) ~
%当t的标号为vs时,终止调整过程
y: m) n3 s3 t if s(t) == 1 d7 b# r. |0 w+ r- G+ J2 E
break;
1 b+ h3 r4 V# a+ ?3 t end. s# j8 v. H# X, l/ x0 S% ?
t = s(t);
$ {+ m. A: }$ [; P4 S+ Q end
" D/ R# b' C* v; Z: |* P+ Y7 K& x %如果最大流量达到预定的流量值
! m# S4 b4 G* E3 L( W: P if pd8 c! r& [$ x$ ^, `6 `9 e. r* B
break;0 a! S$ F& \# v: G, I4 n; m( `" a
end
0 G" \9 c% a) A %计算最大流量
$ F0 ]' c! E0 H2 S2 E7 A0 e2 \ wf = 0;# q# G4 o4 q8 ^/ k5 N8 ~% x
for j = 1:n
# X/ Z5 L" d$ n, |% w9 d wf = wf + f(1,j);3 b" i5 _( p( G r( `
end* R# T8 a, ?# l3 y% V% S
end
% k$ K% e1 x* C# ^( K%计算最小费用
2 I) T" x+ L% }5 Xzwf = 0;8 [( e5 O# T* J, |
for i = 1:n
# ?5 T+ T1 ~/ A% ~% ^$ P; P/ H for j = 1:n
; r8 G# T$ o9 ^' }$ @ zwf = zwf + b(i,j)*f(i,j);/ h5 r% A- Y. b+ i. y; Y
end
; r. I& U, x: Z5 V) B0 F1 ?end
/ Y0 t9 D- x: C3 W2 |9 _3 Z%最小费用最大流9 h2 k/ q& x2 e: v
f# J% Y8 c0 \7 f! g( E+ I
%最小费用最大流量
% g9 X2 b. {9 _wf4 g6 j3 P+ Y1 ^3 L' q3 u% h
%显示最小费用
2 e/ y5 s6 J, V; n4 Rzwf
, J) R8 k- O9 c4 M, Y1
# T' M0 g/ w$ x2 b2
( t% _4 g0 X! l34 p9 h/ ] k9 f0 O) G
4& m) K* f7 M+ F5 g! x% q
5
# R. Z5 J" t3 `0 ?5 o% a4 k6
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1 P8 P& N S8 |' c$ {8
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101 G# S3 K; O8 t! e
11: g' o! L( A) V) J- e
12
1 [. b* C& p! O1 q7 N5 K0 a13+ `7 u( Y$ |9 C7 {
14
6 L( J' @3 H2 X; |7 L( [# M158 z5 e/ I6 c, b" o
16
! u, l. W. K7 v& }8 ~$ R17* W. {, u% e- f! }
18
* }8 D& u3 t( W( a. R19
6 s3 L5 g' Z6 Z" v J, |7 y; ^ p203 m. f& f# A# U1 A) |6 K; `; ]
21, H' D: ]5 Y9 }2 g8 A
22
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24' |* p- {, v5 Y. ~
25
1 x; g4 o3 }0 K268 c2 Y7 u" V; B* n5 p! ~
275 @, }4 }" a3 n7 d9 [
285 O4 a% a+ n1 r9 Z
297 D% x8 @' ]/ A. a! I% C/ P4 c/ M8 t
30
) I$ l- o6 h# O- T& f! H31
6 s8 B | D5 C2 a- p9 _32) Z+ X: O6 @; S" r' D# K0 F5 @
33' K! J) Y6 C4 ?3 e+ D
347 q& M! w% f$ x7 z$ w3 v
35
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37' D( q. f; n# c' g! L& a# L
38
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40
% o. j( F* F8 j. @5 k9 o& y414 F* I" t/ ` u9 o7 x5 ^# e+ w
42
y% X! Y4 R& l3 ]/ {/ {* u43+ ^. r! l, C# i
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45
6 X% k" U% |/ k46$ N/ S& w1 }: U% X; ~ h
47
3 b. v! {: M, o; y' e$ S- q* y2 @48& h" ~4 @3 p! [% M) l- G! |) N2 y
49
' \( B. @% {9 L( |50
+ K) J6 H: r% e" K516 I2 R5 h2 v8 U* ^
52
# |: k7 u, L4 G* m53
9 O5 ~4 s5 s) L. [54, g; f/ i) F: ?
55
# V! J, e( b: {+ R; b56
/ e* ], a# t6 `! a9 a7 g57
5 u* l2 b6 S+ [% a! J6 {. @58
' v* v, V& `+ z% m0 C m1 ], J: `, A594 [; A6 O: e- D* o1 o4 A# K4 G1 {
60# x9 ]2 ]- U7 [" w5 {3 L
61' v. n0 P j Z5 n) ^
62
4 h5 T1 F- a& I! B4 A% F, _630 ?4 ?* J( B! f! r- k) q
64" L& G1 C$ N' V/ A1 w
65
2 D* p& S/ _ G66 v# J; K& q2 _8 i5 r) c! g
67
# ^: G6 O+ O4 _( y/ m68
' W+ X1 b% [5 H) P G69
$ U5 w" x6 h! T: g0 r' V70
' Z! K9 N. L" }! B# | b718 O, d4 }. t5 |9 e" L# E) F
72- l4 b% ]* t0 ]
73$ z, V/ I0 g0 B5 b/ _7 y5 V( l& N
74$ c7 M4 Y% _2 s1 q) v2 s
756 c3 `& ]& b& b I
764 W4 ]1 h; T3 u9 r' e
778 q2 n" q2 G+ W { O" E9 K
78
) E$ a7 n6 J& e' J* J79
3 ]5 O& |/ [- |80, ^: L9 J" `7 ^( \
81
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83
& w4 V1 O: _& W! ^/ Q84
) Z" X$ \* g! i7 V4 i9 s1 ~85
& x& @3 k3 C+ O: s+ o. K86
1 w9 Z0 B5 C! w% P2 q- Y87
: }# b! O k" ~) {88: q) n% X% R7 \3 M6 X
89
2 J, g G% I; s7 t- h90
! L _0 L* c7 s0 f+ Z91 b6 t) M9 O9 g* r/ n5 H! s( U+ b
921 d: r# A0 d( E, w
93
1 g& k, g# u7 o) w$ @94
o5 I' ?/ E* M) A9 x( F2 \5 I& D953 U# d' T# [8 M$ D9 e' B. f' Q$ {; r
96
$ t+ @2 y# l" v/ r/ \( A2 S: ~& P. e978 p$ i( Q+ @6 |# d% G
983 v8 n/ e. p( [6 }' ]$ w4 d
994 z! f2 k1 @/ R: {* d/ D: Q+ C
100 H( z) `6 z! O. T5 G3 p' `
101
2 T6 L y' C# u( `" h) P/ |102
" O' ?4 T/ H+ }" C$ D103# N* \6 N2 M' Z B( r2 ]
104# }2 [: U& Q _( T
1058 |, u5 P3 y1 y k
106' V/ U, j1 p; J& D7 \& e
107
9 J7 w, j5 w) E+ H% f1086 R1 K$ O: J" E/ C0 ^: y/ c
1094 N6 P- Y; r( r$ N4 ~
110" u+ i; I/ |8 {! ^+ l6 l
111- Z" V( \( W( a5 s3 T! N6 x
112
P3 |( M# ]: F* B; U1 U3 ~) t% g: v113
8 X. x1 h' i: P0 h4 i# f) ]) U9 f0 g114
3 s6 ?% Y) M# [' G; ^5 o$ w115+ K4 R* \' D& a* e# d
1167 W: f" i3 L! \ d
117- b. {6 B9 n, C0 {0 u% A! y
118
; D/ w: e4 s9 G119
5 q/ m2 L9 e- D7 T) a4 a3 M/ K120( l; @ r2 j" b: y( W
121
# U9 G, H# G v. ~# Z122
! B9 s( w' h$ S123
" d8 k: `- w, h1248 r' [" F E) v2 w- W" b
125/ W1 S- Q7 V8 H% s0 @
126% Q6 P3 K& R( m b3 c3 t* `
127. a: n% F! U* T; [! f! v' }
128
) ]6 o! g+ s! Q0 f# K+ P129! e) v6 i3 }% s5 X9 \
130
' y4 g, E6 K& n. {% I7 e1319 R; d- M" _ i. s
132( y6 T# I( W: m) m! w" ^+ ~4 [
1335 u+ L, T+ t* p- V" ?
134
- M9 r9 R* c) \$ L& ~135- L' J. z7 U" i* A7 e8 h& y
136( x C: G- w' K' b& ?2 `- y, N
1372 d- \! w# A2 r0 r1 f8 c
138
- g, R. D9 J( u% e1 t3 u139* s6 D& A v, R2 t5 E. X
140
" h9 D: O% w- V, a, r4 c1 J匹配问题:
8 s9 `, q, P9 h" l- ~- Y3 b/ @. D4 J: |3 t. B- z4 H
最大匹配:
1 G t5 D4 w# ?% p + _% d+ V+ n) h6 M: F6 A ~
- g# V, T) x" S6 g/ _
代码实现:
" V1 c$ H ]4 q7 Y% V# A2 y# l2 s, l. f+ M4 f
m = 5;
7 N s$ u( \2 B9 a- s% vn = 5;5 p0 M8 [8 X& Q4 R- M4 k5 i9 m
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];) ?7 [: n* q1 c
M(m,n) = 0;
4 k% N! y+ p0 cfor i = 1:m1 V$ ^1 ~" t# A
for j = 1:n! x& j' m6 Z( k7 E
%求初始匹配M
: l5 R" B# l, U if A(i,j) - R1 D& ?$ y. z' N0 ^5 @
M(i,j) = 1;
5 |+ \( y- I, j/ g break;# u8 ] V5 x0 K) j" ]# ^9 H
end$ Z5 A; O$ |/ H' q
end
4 D: f4 D `; M8 G: C+ R %仅含一条边的初始匹配M
0 v. O6 p U2 | if M(i,j)
% N/ F: J$ W3 _% I% M break; C; y! U8 p5 A9 G* S, H, E! D
end3 O% X" x, d, ^4 z. T8 Y, j! X2 ]
end1 Y, j: L# u* ~& m/ N* {
; Z/ u" d }7 J0 ]9 h9 {( _while (1)
, E+ |& c" N( O' ]5 t# E0 t* X %记录X中点的标号和标记
?, I, f% |- \" j for i = 1:m
& r7 A/ b$ {) Q( N9 o x(i) = 0;
* }. Q' Z/ F' v6 q* O, R2 l end0 S4 Z( i) @' P$ f' c" `
%记录Y中点的标号和标记
* G; s. d) c* X) O for i = 1:n. \( V4 B1 F: n# z4 @: j
y(i) = 0;6 w( H8 J4 T& L: S N9 n* p
end
; e: ?/ H7 A2 j %寻找X中M的所有非饱和点( m; H, i4 t" b, y& L& `( ~+ I
for i = 1:m
- A+ ]5 q. x* P, N7 Y) f pd = 1;
; N0 G7 c. m# D for j = 1:n
V3 M2 A$ d0 ?# E# f, F) ? if M(i,j)* y2 T7 M2 l5 C) g N
pd = 0;
# x0 k7 Z. |( }" Y- x end2 J4 R8 [3 i& f* W, x
end
1 d9 \- E% m, |4 x. C if pd5 p* d- U' \. O
x(i) = -n-1;2 s# z4 R3 M& E- M
end
. V6 i% i0 Z3 R3 O end/ ^$ P; _ B0 K3 ]2 g+ h- P5 }
pd = 0;
O/ _: v8 k% }% j/ B$ u while(1)
% Z9 l- t8 `3 ? xi = 0;3 q# x; N6 ~0 z6 B M
for i = 1:m
* \3 ?% \' n7 M if x(i) < 0* ?, k" `+ K/ a) z
xi = i;! `# b/ I' R2 l4 X$ A4 t# g
break;* y$ n- D- }, K- d% t2 ^* J* r
end
4 G2 Z/ V# F1 L/ u! R. }- |5 s end
8 ]" A: S) n% _& a if(xi == 0)& Q+ u2 \, F! x, X* Y
pd = 1;
5 n7 K; M4 n3 F# ^$ D7 w& | break;6 m9 J) _' g0 y1 D
end- Q7 f1 M% g6 m! G5 e
x(xi) = x(xi)*(-1);
, U8 R$ R, h: r6 I! N; E k = 1;. }0 j$ T8 U; m3 A
for j = 1:n, @& E# j" U8 M) H( L1 q3 y
if A(xi,j)&&y(j)==0
! X# w$ v9 |% L3 [ j( n y(j) = xi;
: d/ U+ ?/ g' `: n& a/ E9 k- ~" v, s0 b6 J yy(k) = j;
2 F) W# Z0 L; o. C7 m k = k + 1;' k. K! H4 C* u- ]
end8 N5 O3 n" {$ V. \! {, t
end
2 A. ]7 b4 r& N/ P# Q {( Q, X if k > 15 | |: w5 g2 F0 e
k = k - 1;
/ q r# `- w) U for j = 1:k; k' {5 H6 v7 Q) ~8 V1 q8 k
pdd = 1;
( [' \& z( z2 A- Y) W for i = 1:m
, I ]" p T- d4 Q' u( N+ k& G) r if M(i,yy(j))! o& B S z" B" G1 L" x
x(i) = -yy(j);/ D3 L1 w& f! J% y) s
pdd = 0;! D( q1 f/ m) c4 z
break;
" W& i% J) U3 \ m* H3 }( Y) H' f end
7 O2 l1 o" ~; N end
c9 v0 i, W; c5 v s if pdd8 c# ]- l9 m' U4 s I" v' }- x
break;
+ w6 ~, }% ~( Z$ ?! r. x. r& e end
) X# t$ L) e' x! F% P end8 c+ z. D! A ~
if pdd ; o3 A2 T) i; k, ]4 ?9 A
k = 1;" s+ Z0 c% a6 [& v8 h8 ?! W4 u% w
j = yy(j);2 p- M- Y$ M% d" I9 N) w0 Z# p8 _- E
while(1), G5 X6 u; T' H3 ^
P(k,2) = j;2 s# w# {* A1 A+ z! d+ F. _' o
P(k,1) = y(j);2 x: }9 m% w2 @2 p8 i
j = abs(x(y(j)));
* Y; [0 ?, ~% p- Q2 _ y if j == n+1- }* b% Z6 B0 M/ j$ A
break;
% m. e* y9 k: B end) A; s) e& g3 ?* j( f' d+ O- F, o
k = k+1;$ T( w# V" s+ R! R' h
end3 Q, a2 n% S2 d' f
for i = 1:k, r/ t7 q7 @. Y; S) Y3 T4 m/ U
if M(P(i,1),P(i,2))
* J0 Q* T0 V7 ^; W# r. }3 ?/ M M(P(i,1),P(i,2)) = 0;
2 y6 c- w. W1 y( n4 t else5 t4 J* s) m! s- F1 A
M(P(i,1),P(i,2)) = 1;
. `( C: o% l0 x2 h; s1 _/ g end$ B, o; X2 q& \+ V% o+ ]1 L
end
9 {: c' T5 x! T9 g* J break;
t b: q$ i% ]7 I) v end2 y; J: O" [8 S$ r5 B% U
end
5 }0 n$ K6 {, K end
! V) @, F3 ^8 _ g- M if pd
# i# R0 r$ V) ^# c6 D/ N X break;
0 m! b) U n* E2 s end2 v4 W$ K2 b& G- y" B* |) T/ p: ^
end
) d' |4 V# q+ r3 }& l9 h1
& K, @, o* Y4 F- V. j2
* ] v9 W, m) X1 G& e0 D1 n3. r4 c$ Q3 ~5 I$ q
4
7 i8 h/ X1 B5 c+ z* t" w4 E0 V5
8 K! V, k" q( p1 S6 n6
9 F3 S5 `/ P/ W8 i, U" v; [7
! k( ]4 H: q N86 B; @% a0 w. x3 j% P7 C
9
# Z: b7 Y. r1 ?+ Z* x10
0 y/ f" S: ~& z' h11' P1 B B- G* R
12
; f9 n2 I$ ~6 y& N0 S138 \! Q7 I! O% [9 ]4 S5 C. Y2 D
14
" Y3 u/ G6 L0 L! ?15
$ |2 M/ n) w; C& p" c* F1 h7 ~165 ~( A" Q7 j1 U. }& t
17+ z; g- P: v( i
18% }0 m9 i4 x; i5 w; L, e- m" b3 S$ B
19
& y1 B9 X2 e- z% A1 y6 z208 E8 {! R8 O( i
21$ ?. N L) d- g) v3 v$ |' f
225 H b6 M/ @2 ? h* X: ^4 o2 o
23
& M0 k" [- w! o/ H6 a% e24+ }0 r1 m6 U. B8 b9 A2 S' V% T
25
2 e4 w& c+ }& i/ J/ Z, T. x26+ U- e# R# ?9 Z) ^/ y- l2 d
27
9 L! S0 x X( w28
+ ]- k K) H$ x# V- |# l1 C# Z29: n0 K! Y8 x4 E3 y6 n. ?
30
5 l! m7 F; @$ }* S. |; f31
* w$ v- T; g# \+ z# f4 W32
8 j) Z% g& y. z/ n7 A33; b5 F0 P3 s/ C& ]
34: F. c" I8 V: [: }. |$ X; ^
35
7 [; [ B& ?' i36
4 \4 M" t% f6 [3 ~0 B37, P* h$ U4 m# s* T: y8 I
38
0 [4 D, @+ D' \' A8 C2 A39
2 X; @& K$ r5 B% [40
+ h6 M2 Y1 q* ]* e414 A; U$ {, d% z! ?* N" o0 O
42
/ u- q8 r7 b, D43
, T/ R5 P8 r8 t& u! P44
# R/ o* C& F% Z6 r; m+ v* _ _45' k9 c; `; o# r. n( p8 ]
461 M/ n4 u7 I8 T2 W
47
6 y b& L1 M+ e& ~ x/ ?! Z48# }1 ?$ R" o5 F, G" ~9 r9 J
49& u v) k$ u* t8 E* P; v% `( P
50
3 I# ?) x; r% |51- r9 Q. Q R$ m# s4 o+ M
52
' T: \% J. k: R/ N% r1 M3 H7 Y53: T3 `" M2 G4 T9 O5 U! I: {
54
) H% a! b1 l! N1 o5 j! ^3 g552 W+ \9 M) S% P* V' C
56; |% P5 }/ _( g% u8 M; x
578 N3 S- R, x% q3 T! W2 O$ m
58' Z: i: T9 e3 ]; m0 R# L
59
9 M5 d' l0 a3 q, U60
1 {: i) a( ~) p3 Z5 b, b61
# c- |3 M) a0 T62$ L+ q4 }( {7 j3 I. p# Z( ]9 s2 o
63' n1 J8 b d8 g- D6 k: W% M4 P
64: x5 u8 ]" K1 j. P9 C) F; { H
65
& X9 P" @3 e$ X66
5 j4 X, N Y! H T. H1 _67
; B/ H& e4 W7 E2 V% P' ]681 a* q2 a1 K1 E ^- ?
693 l+ B4 ?' r& q
703 [8 U1 }. l2 d9 h9 R7 c
71* [" P7 @8 n! z% i0 L+ w
72
9 V, S% V' c4 A2 U) d4 i1 B73
: ?/ P! W/ Y! W74% d1 D5 R( k! u
75( O* N4 t% a! c" z b3 d
76
7 y# e5 _" u- w @! `2 q5 ~) o! Q3 X77 {# z) M! s% G% Y# e
78
/ h1 U3 m2 E0 W3 U1 a791 x% j1 e* Z* j. V; J" K M2 {7 k
80
# i, k& Y' ?: |81
4 @) J, F4 e! s. G& L* _4 ^; K+ ~' b826 F+ A: Z+ g7 {$ s2 ^
83' Q+ p& w9 b9 N2 H
84
, l* {6 |8 k9 L& _- \; h852 f |2 B2 Y& L* d% [
86! f& E6 ?/ S- e( H6 Z. |
87& o" k2 `! E. k9 a1 X
88
. J; i! a- W- ~+ o' P89
7 \5 Z! ?% s7 B# L$ ~9 @90
" o0 l+ H% f+ [ P* c+ m91/ d8 p0 T0 u' C4 o! A
929 E" {. m9 |$ U* I$ l7 F5 T. S
93
/ {3 y( J2 J7 E+ ^, J ?94+ a8 ^. A/ c# h9 J& \6 L* o0 q
95
" M @8 A& k6 ?96
# J! y8 z5 _3 \- q) }97
. ?6 M6 m2 j2 l* f# E98& c1 a! i$ n! U& e/ ^/ r m# R
99
3 p& ]) U8 v9 V8 `- [# g100
8 ]$ T' q# a# L2 h( k: j$ |101. k ~4 y! B8 r% w" Y
102
. |9 c P( g4 y G+ F103
5 J9 B* v9 B% M% n* L7 f最佳匹配
1 z# O& w; f$ x. |4 K8 d4 z1 f' A/ O0 D; O3 k0 w+ J
代码实现:
9 K' D+ N4 M8 D3 i; b- Q# z g; I: [! }6 Y( [! u
n = 4;
4 L/ ?/ M% M2 ?% s h/ l6 mA = [4 5 5 1;2 2 4 6;4 2 3 3;5 0 2 1]; i. a* s; n4 ?' e; O8 t5 P, G$ I
M(n,n) = 0;
0 ~6 j9 W6 Q1 j1 W! @ Ifor i = 1:n6 |/ }6 ~: M& m: a* I0 j, Q
L(i,1) = 0;4 B) Y2 ~- u9 Z, h1 L7 P
L(i,2) = 0;
; K' M# y9 F" r% mend
8 H$ b. G3 A E& W4 ?, I$ V%初始化可行点标记L/ k t' d/ C9 ?+ A3 {
for i = 1:n
$ k% w5 v4 J. ~/ }) M for j = 1:n! J: q1 H3 C+ o1 Y* V9 W/ r
if L(i,1) < A(i,j)
) B' T# C$ L5 g: k, w6 J! E L(i,1) = A(i,j);
; \3 f& ^5 F; v: u+ P# ` end5 l* m( h( ]7 N7 \' \
end
+ U- K* ~' t; Xend
# b. _' L* U; b: e/ t7 d%生成子图Gl
0 H) k! V0 \' ]' m" e- } pfor i = 1:n7 C0 G# c/ Q5 C& r0 o7 F3 d
for j = 1:n
% W( J* t( }/ N; l if L(i,1) + L(j,2) == A(i,j)
1 n1 }. v6 Q; ]$ F& U$ d$ j4 k Gl(i,j) = 1;
2 C5 x% j( L; U9 F: K+ C4 @ else% v6 o# n- u5 W* a
Gl(i,j) = 0;
v" b/ Z0 B0 ^! D4 {( M x+ a end M; ~1 O- Y* }8 Z
end9 r0 Q9 _! b. ?2 w5 {- }
end
3 i0 ?& {& Y: V' s% w%获得仅含Gl的一条边的初始匹配M
$ L. s' h4 D: s* Q2 Qii = 0;
4 V2 g( t5 M: H& b2 fjj = 0;
- B( H/ n: I! I+ ?for i = 1:n
( x# W. P2 y4 f for j = 1:n+ e5 P! Z* k+ V
if Gl(i,j)
( a/ h( E1 N, T0 v5 _& m ii = i;
. Z4 N1 n8 Z4 r7 {! F jj = j;
1 T; H- ]5 [! Q6 d break;/ ^! i) b' [" r5 C( M
end }3 \; x0 W9 u% Y
end. ~6 {8 e) g2 G2 s( g" L2 I
if(ii)- j( l3 t4 x" @
break;! C" v: x. |' s; z
end
! _8 }6 y; k% ]1 V0 P: Send
4 o$ g0 K# h. z) L# H" P2 \# GM(ii,jj) = 1;: Z2 E- b# n' |5 k
for i = 1:n/ k# `! R# }: W% f4 \2 |* z! r
S(i) = 0;* i: X* V: x" r& b( K
T(i) = 0;
" e u( G# N/ u! H NIS(i) = 0;
; f* D9 I% h0 Yend
1 h/ M$ z' c+ o! s2 q. @
2 ^" H8 N* M4 |# @
. ?! M" N S6 ?' N( i( v; Xwhile (1)+ D- R, X; t7 M
for i = 1:n' W/ `, W" \8 c/ ~8 i0 z# k
k = 1;9 G: I1 g& i/ c% ]% z+ G' y5 F, o
for j = 1:n9 g6 v" Y! L; j5 e- S
if M(i,j)
3 v6 y& [5 r4 _; h! V* ~& D k = 0;
4 T5 ]! w3 y* Z. j, h7 o4 d$ y break;3 ~9 |) q8 _% k& r
end
# e4 k7 D. v7 X: _) k% M9 p end2 J! ]7 u) n1 O
if k
9 E: N( t/ f' V: y% w l1 c5 |6 a$ } break;3 O/ k8 a5 s7 t# h' J
end
5 z! e/ u5 w) w: q6 n1 K& _) ~ end2 q* S/ a x* F/ s9 z
%获得最佳匹配M,算法终止
& {. `# w" k) Z2 X( R if k == 0: O+ ~' V# z, W$ k' d. b
break;) N1 Y, ~8 {" g! T' ]+ _
end% l7 ~# V' v, p. R8 A
3 p" o X- J0 d% c a. f/ l& D
%S = {xi}/ h3 Z# n( Z) W1 t6 |2 V( M% X5 ]
S(1) = i;
% r* z/ t( U- V* z5 gjss = 1;$ c' }& @( [- l- S9 r3 v; c {
jst = 0;4 n- j0 d$ |1 O! D8 \
while(1)
. f4 {# X5 S$ r- Q6 l. K jsn = 0;1 z4 g) J' m t/ v9 O2 U
%选择NL的值# ]6 U9 k- g. G, z5 n$ j6 s
for i = 1:jss
9 e( J% z( s$ x& }2 }. C8 M for j = 1:n
3 g# G$ T0 Q& t6 Z& m! n& y if Gl(S(i),j)9 A. b Y, \: ^& t
jsn = jsn + 1;9 o! l# }( u# D* m7 n
NIS(jsn) = j;
8 Z }! N; }# H% V* { for k = 1:jsn-1
% X* I/ d4 U" W- I2 } if NIS(k) == j, z7 O o) A* y7 D8 q
jsn = jsn - 1;
4 {/ \6 G, J1 \, W* \ end
% S' g, M: F, w: ^" L/ J+ E end! k: x& J* r ]: s- K. |
end
/ i6 |) s# E, y2 U: r end% |. |* q3 D/ U! @& o( G4 o
end9 s% {1 W( L$ E, r
%判断NL(S) = T ?5 S. j0 d7 x6 ?9 y; W& Z8 d5 H+ ~7 }9 @
if jsn == jst
1 m, a, d N8 H9 } pd = 1;8 o# C" W* P1 R) ], @6 z0 X
for j = 1:jsn
A2 F; J- g" q. w if NIS(j) ~= T(j)
8 K& o4 {0 j2 ^/ P* a" s8 v6 U0 d pd = 0;
# F3 ~! ^7 I0 i: g break;
( A& ?, j% e/ b" h4 X: |. Y end! D* P; _- q" p4 K( j
end
/ t- C: n% O' t6 Y& y end
, y1 K5 f2 l* \! O. Y %如果NL(S) = T 计算al的值
0 q% q, t' F6 \5 D) L* \2 E if (jsn == jst) && pd
4 |3 M9 e$ Y& J3 J3 u al = Inf;
" a% K/ i$ t3 t/ t( ]0 @; m$ h for i = 1:jss
2 K/ z1 w) Z% ?( m1 T for j = 1:n% E1 ~) @+ B2 V5 [5 f- L* _1 N% P
pd = 1;
, t5 H( _& X% N2 z" R, j' E2 m for k = 1:jst6 j, |/ A' X8 P- X* T
if T(k) == j
$ s3 |. v+ @ j9 ], v1 F8 q" l, a* X pd = 0;
9 R8 L; | s- d break;
5 p% P K3 U. d( b5 C$ z: { end
; x8 }8 M& H& \, y' G) B/ L# _( t end0 V( v% H. k& b/ y c2 L7 S
if pd && (al > L(S(i),1) + L(j,2) - A(S(i),j))$ Q4 k4 G# ?; ^* ?' D) {! i
al = L(S(i),1) + L(j,2) - A(S(i),j);
* M3 o6 k. j- X7 C4 r end
. C5 ]; i, n7 E' r% a' I end P S+ K% o' v; C
end
! l" U3 F0 ^+ D* s1 q0 o; Q5 e %调整可行点标记' A. D& R8 l8 [8 Q# O, |, C4 e
for i = 1:jss
. ?7 x) D/ |" p1 l L(S(i),1) = L(S(i),1) - al;2 E/ @/ m8 y- T' H. ]7 n
end n8 [1 V/ w( l
%调整可行点标记
6 ^- U- n z- `3 A9 u$ x: _ for j = 1:jst
5 u: N& W |! V L(T(j),2) = L(T(j),2) + al; O# L1 C Z/ H" L y0 H1 A p1 f w
end
: n% |* K' Q. K, B %生成子图Gl: Z6 Q5 F9 |& W, h8 T7 M" j
for i = 1:n3 G3 O) T6 p/ ]' k$ k( r, C
for j = 1:n4 e6 {- B+ |. P9 J
if L(i,1) + L(j,2) == A(i,j)7 i! K" m( g) ?) l( b+ ^7 B* X
Gl(i,j) = 1;6 C0 x0 ?3 g. Y0 S; K* O) Y
else' C) `% ~& u2 z' t6 m
Gl(i,j) = 0;0 y, ?9 U1 A$ p) v. w, f
end1 I9 z* ]( v6 K% h& V
M(i,j) = 0; N+ E( A5 o8 Z9 Z6 H! n
k = 0;2 H# q, g, A" H o. a6 u" N
end- d% P ~: z& q) p. W* W. ]) b
end
3 X& k6 T8 t) B) Z4 h) ` %获得仅含Gl的一条边的初始匹配M( V0 J1 t8 }( W9 T( K& W
ii = 0;+ f0 h9 [8 d. l( Y* o( K( Z
jj = 0;
5 K0 v. ?' p, b4 @8 m7 |, N for i = 1:n
( i+ ^, u2 U2 L0 M) Y! c for j = 1:n
% w& G! p2 i+ v$ } d if Gl(i,j)
4 U9 n8 ^7 B6 A) f ii = i;6 L, i2 z# p Y% `! N. s
jj = j;4 x! I5 U( I$ ?. ^0 U3 ?& r& ?
break;8 O9 C r7 T- ?: D; Y) {
end
3 }9 H+ J5 x" I end8 ~/ I9 F, j# @; G) Z
if(ii)4 J( @ T3 M. H" f' F1 Q
break;, D# `% k) n; W! b! M! Q/ o
end
# O! L7 b" h! V% s6 Z- ], M end
" K5 t& ?" H/ D3 J0 _+ p. \3 F M(ii,jj) = 1;9 T- H% R% Y0 c
break;( @2 C, C2 e3 g4 q# S p) @- h4 T5 r
else2 Z; V3 ~! s" H6 r( w
for j = 1:jsn
* \& k4 S) e1 k6 I% h5 @4 E1 R pd = 1;* V' Y% `: p# t( [" Y- [
for k = 1:jst
) y8 m' \$ j3 z5 b% C if T(k) == NIS(j)
7 g: `" C( X9 m$ f, X' ? pd =0
7 }0 b" Z* o& L: v break;7 c& d& b9 `+ W8 R
end
: p* ]! ]$ v9 Y end
7 i/ a+ z. n- @. Y if pd4 n2 O( }! [4 b: T* J
jj = j;$ X. x' d" x8 ?* C4 v
break;
/ J$ s. M: s' x! O end
; r9 D" k2 S+ o3 p' c end7 H7 n$ N9 ~4 I9 q1 A& }$ F. A, \
%判断y是否是M的饱和点0 T7 w- n; l( ?0 m, ^# y! e, i" d3 F& c
pd = 0;
5 R. q- p1 p" j, i( w: c for i = 1:n! m3 P# X6 O6 ^. y7 g, b/ w
if M(i,NIS(jj))& M9 x+ H+ z* b; u) p& t) R
pd = 1;
; Q3 ?; X3 l* S! I$ I0 F: F ii = i;* g$ i' y5 W3 t5 m) R4 f
break;! m5 z1 L. j% d! K, L1 X
end" J# ^6 _# g1 p- ^& M# I; p
end+ s; u* q0 X2 j( I/ O1 z2 ^, s
if pd! D( a! Q' o. [2 L" u' R% v
jss = jss + 1;; I$ {- e' s- {# r" k0 G }$ @* [' n
S(jss) = ii;+ r! K0 P4 G: t' f/ ]
jst = jst + 1;
G1 Q+ [ V- z) W3 _ T(jst) = NIS(jj);
3 O$ m: W Y) ?& d1 J else
+ {7 D7 V; B) @4 c for k = 1:jst
. q4 K1 `* G, o& Q g M(S(k),T(k)) = 1;0 K3 j& V7 g; N- H
M(S(k+1),T(k)) = 0;
1 s, f+ s) k7 G+ l end0 w1 C+ h$ Z/ ^- p
if jst == 07 m7 Y# ?4 a* \/ y4 {6 Y4 i$ k
k = 0;# X# ^6 S7 f1 V) ~' b4 M! f
end
+ H# u3 b7 |8 [. Y5 } M(S(k+1),NIS(jj)) = 1;* s X8 q" V9 O- y+ V
break;
' e6 Y+ h7 A" } end& G }+ ?; Z; e% [7 k- ^
end
/ g, K0 `$ A- f3 l f* |9 `; a- F& z end
! Y6 I L/ J; Y end, b* i0 B% S5 t! v: N6 ?; `: ?* U
MaxZjpp = 0;+ P \$ N3 u. Z4 T- l: ^. y
for i = 1:n) ^/ E5 l0 k# K- T
for j = 1:n( W5 m* d8 h* y& t0 T: S
if M(i,j)7 L/ k# C5 V, K
MaxZjpp = MaxZjpp + A(i,j);
# p3 i# f6 w8 A" j. B end
/ @+ F( G$ z9 \: p end
% u* f; |. D" Y# _* e4 T+ T6 | end
' h4 O( @* X- e: o. J" k- Q( m3 p M
3 E: ~% ?+ s% ] MaxZjpp
$ k8 C: y& v7 G/ e' P: m- [3 a1 d: {$ X* i% _7 Q; }' e
/ x) @9 J; a9 q6 f* F; ]' ]+ ^8 |4 M. ^/ c: D
|
zan
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