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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。
. r# f$ j# C# V- F5 c4 g' M" W1 c) w# Lclc,clear; L% L1 R0 ]. d1 o: h1 k. c \2 Q& p
a = zeros(9,9);: U- |$ x3 O9 U
a(1,[2:4]) = [20,20,100];$ e; o1 R: a7 m1 ^* l) ~
a(2,[5 6 8]) = [30,10,40];* ^$ G( `: @* E6 M, p
a(3,[7 8]) = [10,50];+ m* _/ C$ Z6 k$ y
a(4,[5:8]) = [20,10,40,5];7 {: g: G4 P& n# a( k# ?; }
a([5:8],9) = [20,20,60,20];
) d$ [1 l ~# b; v {a = sparse(a);
4 s6 \$ }9 n2 |% t5 b- a[b,c] = graphmaxflow(a,1,9)( v2 P/ L2 C8 P0 \: {" c; f- P
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: ![]()
+ U5 _7 d/ N& X% zclc,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)' \2 i( j. m6 b
最大流量为11 自定义Matlab代码: , W9 O: Q1 K8 ?2 y7 v: n, Q0 n; ]
2 D& l2 L. o0 f) _, C; ~
最小费用求解
2 t! F" Y# x o- Y3 R) g8 M9 e- `5 o- E& ?: J; u! g
Lingo:
9 R# C* _$ u2 W( M% B$ I) @+ i: D. x# ~& n% i: l
model:$ k9 J9 ]" Z! c9 Q
sets:0 x3 i* B q6 l1 ^$ V! g
nodes/s,1,2,3,t/:d;
' w3 |! @1 ^% ?arcs(nodes,nodes)/s 1,s 2,1 3,1 t,2 1,2 3,3 t/:b,c,f;
! h* w+ c8 x# f2 f: Lendsets
: E! i( Q# h! q* C Rdata:4 K; [- ~# S2 `6 z3 g3 r' T3 m# l
b = 4 1 6 1 2 3 2;
2 j0 i; m6 B. H' ^; j, N( e! ] u- kc = 10 8 2 7 5 10 4;$ ]4 K" ]0 E1 A3 U
d = 11 0 0 0 -11;. u( q* `+ x* q! f2 c
enddata
8 w5 @$ F- I9 x# jn = @size(nodes);) k3 @! ]" o% K) V, v/ ?
min = @sum(arcs:b*f);) B- D5 p* V& F! q1 I1 ?. U" k. a( _! D
@for(nodes(i) sum(arcs(i,j):f(i,j))-@sum(arcs(j,i):f(j,i)) = d(i));+ L3 ^$ C, u8 ]
@for(arcs bnd(0,f,c));
7 v1 T4 r( ]+ G3 r& H/ r) r6 a0 `( Nend& N4 ^, p6 D& g# S# ~
1# f/ f# u6 f# F: M( D5 k
2) c i) [# q: U2 }# u" z1 i6 u
3- V0 c0 N5 O: j z$ q0 N. y8 `
4: O/ u% V( M9 e7 t* `* [* ?# Z
5
! C% Y, d- s9 R2 w67 {- @3 ]7 ] s* ~" [7 h6 X D! ^
7
# b% ^, Y) _: ^* E8 K5 D/ I8& J, o) E g& G5 _* [7 j9 h
9
0 T: M% ?/ o' j1 ~4 K& k6 Z0 e10/ {! ?9 \' A! c. A. p1 m
11! t% A5 y2 F6 h8 Z# d" ?
12
! C" { p; ~4 g, t. D* @; N13
1 Z' C h. |1 e14
) @! L* L1 G" \. X. C+ w" d% D) G15% F! _; I9 [6 n/ O
Matlab实现:3 U, I- D! I9 m. m9 d* Z
' l, a3 s z2 i% f, R7 [9 v
1 k: C! o8 D0 c% ln = 5;3 s/ A1 \0 F" M' W) n% j
%弧容量
( D6 G, k% g* a" _) Ma = zeros(5);$ f8 R0 v6 m- Z
a(1,[2 3]) = [10 8];
; M0 ^# }/ W/ R/ B6 }6 }a(2,[4,5]) = [2 7];
: p8 e- b* o: B1 i* la(3,[2 4]) = [5 10];
* t5 r6 `! P) U5 P. t& Sa(4,5) = 4;
1 y! q" f/ D- z9 j# OC = a;; A6 u( h5 ?; e
%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];8 o9 p# E0 U. |) e x3 f O" T% t
%弧上单元的费用
' f! [6 W2 F$ O3 ha(1,[2 3]) = [4 1];) G0 t, y k1 q
a(2,[4,5]) = [6 1];0 ~9 R, i# K$ ]+ w4 E; K: e
a(3,[2 4]) = [2 3];: s3 \2 k6 ?7 u W
a(4,5) = 2; {$ [- ~. P! h6 p& N( H a
b = a;% A. W. P1 A6 O4 l4 _9 k/ Z6 L7 b
%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]; H3 N+ P( _5 B$ }/ _1 }: y+ l! {
%wf表示最大流量。wf0表示预定的流量值. a+ N0 ?1 A5 A( f+ Z
wf = 0;
+ s' W0 N1 d3 L1 J1 kwf0 = Inf; |% y4 O% a$ n5 ]) S# c. y, u" U. F) J
%取初始可行流f为零流9 u9 u/ x+ h& M, `+ P
for i = 1:n
) U' M# }" h+ d; O for j = 1:n* N0 Q$ V( @: [2 K
f(i,j) = 0;& `% ?1 B* g% c# k {' f
end
4 E+ j, e3 Z9 l3 ?8 d9 Y$ Wend0 b0 a* [- _8 T& z1 J8 t
while (1)* N' q: L6 P- L5 m
%构造有向赋权图
2 ]7 i* m1 _. d! h3 E# V for i = 1:n8 V2 j2 M& A$ w0 ?; R4 M7 \9 a
for j = 1:n4 H! k5 {9 i2 X
if j~=i
7 E- }: i4 v9 h. { a(i,j) = Inf;
% i2 }! [" d6 ~& j* R' B7 d end
T/ D. L# C1 z+ Z a4 J end: K* J1 ~9 f8 H b/ ?. ]
end
, }' B; m" {) a for i = 1:n
, p$ _% y- r1 p R; q' J3 w for j = 1:n
1 o) }$ x7 M0 x if C(i,j) > 0 && f(i,j) == 0
$ j9 | I& p+ J, y- m a(i,j) = b(i,j);9 T+ W6 D+ [+ i4 e, s7 I
elseif C(i,j) > 0 && f(i,j) == C(i,j)
2 N0 r" S" u0 Z- F8 I9 Y8 p' F a(j,i) = -b(i,j);$ R3 [- a3 X: Z8 ?" Q; A, G
elseif C(i,j) > 0
3 `" W! c* k& A' _6 d a(i,j) = b(i,j);2 i9 r4 i# [: V3 e# N6 l! e
a(j,i) = -b(i,j);" |" v! p$ R" B5 r2 k
end
M; Y" `" I$ H' z( m end
; m* q: _( e* ~; J* y end$ Z/ Y& L( U' M* ^! S) T
%使用ford算法求最短路,赋初值4 z' w5 X$ A) H& ?/ }
for i = 2:n7 L; E# Q, n) Z( B }# |: e4 [7 V
p(i) = Inf;
9 @2 y5 z% M. Q; L" |% ? s(i) = i;
, _8 o2 I* q2 M" n2 p end9 h# X0 L4 b8 l. x2 Q0 l# j
%求有向赋权图vs到vt的最短路,赋初值: Y/ F& A* l" g k
for k = 1:n% f- O' N# d) z0 G* N7 @) M
pd = 1;
7 _7 u( L! i* ]7 J for i = 2:n/ Z9 B( d- i; g6 K q
for j = 1:n
1 k* C, G- x. g$ p) ]: M8 c3 Q b if p(i) > p(j) + a(j,i)
4 U0 `% D; _& d% y* l# b p(i) = p(j) + a(j,i);1 H! b2 A E, u! x' t
s(i) = j;8 S# f, z+ j5 s+ `8 O
pd = 0;! w. U4 I/ l* ^) b# `8 {
end( ?8 J0 `) {9 [
end* i( d7 Y/ R2 X* e; |
end
, _7 g' w* I" i, I6 I %求最短路的Ford算法结束
( t: S! D4 m8 K% W% X if pd
6 X/ a1 W# x9 \6 ]- ]% d0 ]) S break;
% \/ e" }" S2 |4 O$ C end- b0 a+ F' t& {5 A1 D3 W" Q4 [
end
& x& n/ W0 n4 i %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& O3 C* j+ C% l" O0 g( k' ~
if p(n) == Inf) l' I2 ~+ o3 c& [
break;
9 H2 q) v( k/ G. x& l end
" J1 q4 k* k& I) |/ U, {) q %进入调整过程,dvt表示调整量+ b8 o( k3 Z* q' s! j5 e: O
dvt = Inf;
! \7 c+ l7 G% X$ e3 A4 U: `6 a dvtt = Inf;
" t% X3 M/ q# A; f4 S6 ]" S: C0 G t = n;* [! X( |$ c7 F7 f& i
while(1)2 z5 X+ p* A4 g. M* l
if a(s(t),t) > 0
/ }( s( j% F! A %前向弧调整量
0 H4 e- X' ]# j9 L2 u dvtt = C(s(t),t)-f(s(t),t);
5 v7 c0 H" [* B %后向弧调整量. r' ^7 C: g4 f% \2 l9 K7 ?8 m
elseif a(s(t),t) < 0
1 I9 S' X: d8 u" d G dvtt = f(t,s(t));1 T# |7 f; [" x$ t# h* G- t' q
end
, l% n) u, c/ `& B q* j if dvt > dvtt
! Y! \) r* r" {) a dvt = dvtt;
2 J2 M6 n" n8 j end. j# }" t% a8 q8 x
%当t的标号为vs时,终止计算调整量
, R# q' v& G( U. f3 d/ D if s(t) == 1, e% f# @! c+ \1 T- C ?
break;* Z* Q; v6 P, ^4 x. j- ~+ O
end
2 q& h5 U* x$ M. u( q( ] %继续调整前一段弧上的流f
/ i3 G+ r) ]2 J/ E t = s(t);
6 [2 m4 ?* a7 G+ ~, T end
- S8 Y0 d# m4 m. B pd = 0;. J* \! Y4 x6 P8 g6 _$ P" @! n
%如果最大流量大于或等于预定的流量值
% B7 g( J( s. n v& N if wf + dvt >= wf0
- c$ M* Q9 L$ ^7 z! K dvt = wf0 - wf;
3 Y) g6 V) w0 S/ s: ^" E pd = 1;
: z' P6 o) J, n9 B3 a' t* Y- U end
) }) i5 |1 h6 g Q t = n;$ b. v r: G7 m+ t4 Z) H% o
%调整过程0 P& w8 o4 G! w# e
while(1)
0 x; h$ e3 o6 N: D n$ L if a(s(t),t) > 0
/ x1 J3 I" x( C %前向弧调整6 @; ]& s) x1 x
f(s(t),t) = f(s(t),t) + dvt; - x! Y; `. O$ B3 N
elseif a(s(t),t) < 08 ^; W, d/ ?* ?# s
%后向弧调整
; W- A- M4 P; p) g7 c f(t,s(t)) = f(t,s(t)) - dvt;
. `1 w$ q3 q4 C& Y* T end # V, l2 |, T! O, b1 @* @
%当t的标号为vs时,终止调整过程" K- z9 i% a! E j# I
if s(t) == 1
3 B! p# I3 u# K1 Z7 X break;& q9 ?* t, e! ?$ W1 P2 G
end
/ c3 X, q: m" ^& e, R4 C- P5 O t = s(t);2 B% q1 S. e; o' i2 g
end- b7 ?( r k! l8 `" X( D
%如果最大流量达到预定的流量值5 I8 [) Z; t7 m# E4 x2 V! m
if pd
9 W, e7 d. [7 \4 u break;
% M% [6 y& z! A end
/ q1 u9 M' H" w) t* I %计算最大流量
9 @: p! B4 Y$ A, \: [ wf = 0;+ q/ ?: r- o/ E I# N
for j = 1:n
6 S! d7 M2 ~8 \) D wf = wf + f(1,j);
( ?# y* s. \! J) p5 H end' h$ z+ K. `% C
end$ V: ?% X/ t; s3 |8 E, j. p3 w
%计算最小费用
1 U I0 O, E% x, H) ]zwf = 0;/ a) n ~: M& m+ Q
for i = 1:n" N4 Y' T; P3 }
for j = 1:n
2 M1 ^' r3 C3 B% ] zwf = zwf + b(i,j)*f(i,j);: f" Z0 r# D, a: ^2 j
end
) X* k4 C' ]' ]# X+ c6 M2 pend3 V. q' v. l1 y. |. R; _) S
%最小费用最大流0 f: ]3 c: ?# n
f
4 K5 P" G) i# Q# v; N: L& C, E$ t: `%最小费用最大流量
6 [- x. g7 H* B: Mwf
& o$ E) u2 F/ a%显示最小费用; S4 l) u4 j( t: W
zwf
4 Q, s3 s0 T1 q8 Z; I5 T1
# e! H6 q5 ~4 I' D6 k4 a r4 X) `- ~/ u3 Q2& p1 T8 v& T1 h: v% Z4 {, ]) s
3) L1 M* k: P/ v1 s0 N
44 x- `1 b. o0 D0 Z A3 G
5
( H/ K: G. k7 M0 P# J( u6
$ w2 C- T5 a% n X79 {3 m" M* G, V( P/ Q- B7 @
8# H/ \7 H" ?3 s1 f
9
3 y0 C% p Q6 b' E' d10: k6 S+ Q" C$ T- |2 H3 p
11
3 Y ~7 U; F# `, d& o12
) r) ^! n6 b2 ]0 `- |1 M n135 x2 M+ `, r q: {
14
+ B/ \$ i) j! e7 \/ ~6 \7 l15
7 u# X) C2 [0 q8 @7 x" [) q16) S" y* G- m# X8 \
17
6 R: j8 P9 j& S: f& ^18. O) `* N5 q5 Z" s) d( G
19
' T* Q* C2 L7 d0 B20
, g+ b x' Z1 M; c: d& v5 l( v213 J1 s: l5 C& d4 L" J! w
22
% u7 I7 K$ |# Q% x3 [23& m% B- @0 m3 K% L& B# L* d6 d
24# m. j% w6 O2 V% }$ A' [
25) _ }# [, o) A+ q
26
$ }$ D0 w4 L, ?6 W, Y, O27! L/ d; o" r2 L3 j" E5 c. J
28. x: t& U9 b3 l6 a4 ]2 V! Z
29 b' Y& q# e. c# V# Y" w! o9 S e
30
+ _0 d) N, g+ b7 c" M31+ d, v- ~$ n: }$ }0 L p( r' I0 F8 Y
323 ^- ?5 A; ?- ^; I8 A
33, w- J3 b& m7 s) W F
348 c0 N" r5 R" s0 i: \
35
/ O* v" Z! o3 k! Z3 M9 F% X2 l6 u7 y36
" S7 a, x- C1 P h0 s37
' i( p5 C4 `3 W8 k" W- i38
4 C* T# U w# m O39
9 B8 i5 B& r9 D* P }, N40
0 R# _6 C5 F6 w1 }% z# F( s41
( U$ I& } l8 u; u) c' [1 }+ c' G42
9 W& p4 t) y& J# |& m E432 f$ c1 P) q3 o) Z8 w
44
' E6 A5 w3 V) j! ^459 \; I* V# Q; r$ {) U* U
46
* o9 K/ W* N# S6 l47' h3 S9 G# H2 w' |( Z1 S( A
486 {! y. T$ O, r7 b1 F
49
" D$ t; ^' m% L/ g% T50. A+ C% o2 ^* X3 m# @& @, x
51$ e7 D% o, u8 Q7 p7 {8 b
52
" |9 [1 `4 {2 l9 @0 n4 _) A53
/ n' Y( J9 s7 u& |2 x4 d54 Z) l. F* `8 W. y) p7 j
55
7 ], }+ I2 u9 N$ l; J8 `: N56
5 Y B7 B9 }4 W6 m3 r* h57
2 [! Z+ F9 j0 Y! u) v( C" V58
3 F7 b: R: B: b" Z59; q% N+ d" ^$ A9 J% [, Z
60
9 n2 e8 V: G0 k1 @( U3 d+ c61" Z6 R9 A; c5 J% e" Q% L9 b! {
62
" o8 ^+ T6 h+ Q& }63
# c: E1 n2 ?2 G a5 ~# b3 x) `64
- t/ O) r/ X4 B2 ~7 F* g65
$ v; W1 E3 r; F" T1 `667 m8 @+ Q" k* Z4 k8 n1 v$ X
67
3 ?* H* [ h0 l9 t68, v% o v* @ E7 R; T& ]6 y
69
* ^* C* j3 [' ^; B" q! `706 |6 d; H" M0 T% C: T, \( M% ]
71
J' {6 l! c. `3 B- z# X72 X5 R& l2 x4 G' X1 R
738 ?7 X+ y( M2 x0 ?, X% n9 c
740 V4 R* X" J+ Y- `
751 }1 N; O! R( M0 r& ]4 l% P% y
76
u- w; M2 F% B6 H+ A9 c77
( W0 ~ K+ k: D$ o4 @7 n: K9 E# Q78
) \ ^ j2 O& n: s3 x. \# _: J. n" k79
& Z( g! ]( O& k80: A9 f" h' W K4 c8 m
81, G& D; G+ O; o% H
82
, R; g3 H# d3 C# o1 a83% M% `2 c; T* f% K
84- T- f" n+ R7 m) Z9 g! v
85' ~0 R- }6 J7 A/ _" O3 L
86
4 j4 t( a% d2 P1 S+ ~- b* a87) {- w' a+ F, z/ M8 r
88: t6 W1 l: H, v! r B2 C
89! C4 T0 O+ k' W/ V x
90% s# P8 y% Y/ x. v
91
8 O& e# {7 {. V& P923 ?% g, i+ o" t s
937 y z5 G3 @1 c3 W: T( X0 E
94
: [- _8 C, j* S95- i A7 ^% [( A/ [
96- O3 D, B6 |* U, F% W1 j2 U
97, |# [5 s$ o4 h/ Q$ h# t
98
/ d* D' c6 z* y2 w99
8 Z) i% w9 t8 @4 w# u100# W( b' _8 R2 U1 _) i0 f
101
0 u, A. k3 b4 q102/ c7 Y5 x& K- \ v
103
" N3 D3 c5 l3 G' R6 Y% u104( |4 {/ B* [7 r4 {* I; F& |: p
105
4 Q* b# r5 J- Y, e! s: y. x106
$ p* V1 y$ M! S* k4 x107; t9 {; g8 W2 |/ ]& c5 E
1084 q+ N4 g" b# G/ G$ C5 F; u$ N
109
3 }% I" ], L2 k( T1102 ~+ ]+ b3 a: b. s
111) R- d0 @7 C- x
112
6 S# C% R4 G' V5 a* X' I" l" f113! R; o1 L0 I A4 h: T0 h: q
114' g: ^& U6 Z6 K. l: f8 _ J: @
115
& G/ f) v1 O+ W+ ^116
# i9 T- N( ?& ^* _' W1175 r s( ~4 x$ c) k& F
1186 f. A6 ^" e* L
119
% y: u, l( z7 a. }6 J9 \% n120
" F3 T( W! j7 w1219 G7 R+ g& F( @
122* q* N1 d; ~: z: g( H2 P V( f
123
' w. @0 |" k- h! v, k124
9 A# t& s+ I$ y- @4 S, V125
1 @5 ~7 @+ q. j7 A$ Q6 Q1264 j. t& A- A0 X/ H
1275 Q& V+ S! S7 P3 }% f) k
128
; q% e, ]6 W+ `: j129
2 o, V2 z S% f7 A5 C8 I' X* r130
" k2 s" b; n" `4 Z2 r7 R131
! y# @7 a7 E8 n( W$ z ~, r% H1324 H' B/ _: c* M
133
2 Y- K% j! Y: X. F1349 J' o* d2 o0 c) X( W, @) n
1350 \9 u1 ]) p3 r' `
136 e0 o k! w# Z* `
137& C1 o2 W, e1 a6 M8 M
138, ]/ C/ @1 @2 U* A. e
139. w* g& W5 J' C. H. T+ N
140
- v5 `2 U( C. J& j匹配问题:
5 c! |; L6 w8 O5 i! P+ i% C X6 N O2 c- t
最大匹配:' F4 h0 R N* N* s, F
" r" f- I- G6 P: u* [
3 B' {, V" d: [/ W- |' h代码实现:4 M9 R: u$ \ J8 N: |! Z7 c, ?5 ^
, n3 _& A8 G- x R
m = 5;5 W8 `/ P- @+ Z
n = 5;
2 f# x+ G8 I) XA = [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];& R; B& _6 W% F) C
M(m,n) = 0;! a3 p! b& Q+ Q
for i = 1:m; d! \5 Q, N; U, Y! F
for j = 1:n
/ l& \' Y+ @# y: u %求初始匹配M* N& G5 [' ]; p W/ z9 m5 _$ }2 L4 ?
if A(i,j) 6 O6 h3 G* n8 T
M(i,j) = 1;
0 k' S5 v9 b* h$ D$ W; i; n break;3 Z0 n. x( x9 [. Y! j) g
end2 N& {+ n- G: v* H9 @
end- ]: p& e6 a5 K. O; N
%仅含一条边的初始匹配M( x1 y% X3 v* W/ n
if M(i,j)4 L8 }$ V* K5 m2 N2 K/ ?
break;+ Q% \( |8 h. X& D
end7 f+ Y# ~% V, ^) m/ _$ }
end
! e4 s6 `" X7 p" {2 l
( O' l; c3 x# xwhile (1)9 m( G/ J6 m: B0 r- P9 k% w
%记录X中点的标号和标记$ k1 V& Z: Y) t! |! l
for i = 1:m, ]6 p, ]3 e0 t& F
x(i) = 0;/ ?7 N, k1 t z
end
( ?$ n0 H5 A( b! _$ B# Y %记录Y中点的标号和标记
% U L- I$ o1 g' c: M2 D3 f for i = 1:n
9 z5 b9 P7 h+ S. M7 k. S y(i) = 0;
' u7 V' [7 X$ g6 T) H5 C% E4 a! A end- s/ l4 i R5 [; X9 M6 @; }
%寻找X中M的所有非饱和点
& T' `3 E# ]- W for i = 1:m
9 k, I) a" X) d) N/ l pd = 1;. o" j3 A) ]) e
for j = 1:n
' C$ T7 B! m+ g4 d if M(i,j)
( N. _2 {- A8 n9 j pd = 0;' l* I- D I: i& c: N7 Q( K
end i( {0 {* H- I4 e7 I% w) R
end
7 T% l, N3 T3 S2 A1 A1 Y if pd
" S5 ~2 i- \7 V: g# H x(i) = -n-1;* }) n: g% V8 d
end
7 i4 T0 Y% Q3 W- @ end# J3 \( a" A' O; p* O
pd = 0;
$ ], F& k& }% \3 @8 X while(1)% `/ T: {# `9 ? _- q' }9 G! X
xi = 0;5 L+ D4 _$ M: s5 O
for i = 1:m
' c" W! g& H! m if x(i) < 0& f# u% _' R+ V8 r* T
xi = i;7 g, b' y" r9 w8 y3 i) k( o% W
break;! W- z8 O7 G; o M0 _
end, |6 n! v; Z {% o# l
end
3 f+ b% f2 ]& W4 o5 R/ }1 t if(xi == 0)6 y) a, R9 p( X0 V% X) @3 R
pd = 1; e' m9 c6 U! m6 j. \ c; @0 H
break;
7 D: g( z0 |7 P S! O end7 Q3 w, G2 s( a6 l$ O, H5 w& N
x(xi) = x(xi)*(-1);
: g! m& ?2 A/ n/ N k = 1;
; b* r2 O' X7 R- P for j = 1:n
6 V" r Z$ T6 O1 s if A(xi,j)&&y(j)==0
* j; L. B4 {/ t/ Y y(j) = xi;5 \6 Q7 [ }" E1 N: R3 L
yy(k) = j;
; [# {- L. R7 j$ `, @7 s! j k = k + 1;
" H# L: O" l$ q( d7 V$ o2 ^ end
! P, e0 f( M8 b- a n( o- f end1 H. W" P8 b m6 P9 x8 m! a2 z5 r- x
if k > 1
3 e$ w, p$ p" \* R k = k - 1;
: y" A2 c" S& F$ k for j = 1:k i3 [- }! J6 |* [, E
pdd = 1;
1 ?5 [3 q( Y/ C' n for i = 1:m' v/ s1 J, `2 \1 q
if M(i,yy(j))/ m4 v b0 Z! ]7 a0 H
x(i) = -yy(j);3 r$ R# J, n, {* z/ ~
pdd = 0;
2 o1 g# \& J, N/ `$ n9 T break;
4 D2 v! o- `8 k& H5 L end
3 e9 M6 N3 W+ N3 |8 E0 d end
8 h) o% Z4 O) H8 [ `6 X; r5 q0 F( ]' [ if pdd. \2 [8 I5 L! G Y% W7 j
break;6 L1 I# I( k# t* G9 H8 _7 _
end% c5 S; l. O& `% {4 A$ P# m
end
) w# D2 I# B7 K" D if pdd
3 j2 ^% D! H) K' C/ j k = 1;3 X m4 ^: h1 i( F$ Q* u
j = yy(j);
/ Z& c, _7 m! K8 `$ f while(1). |6 R3 z/ N& I* q- N. [
P(k,2) = j; T: Z3 K/ B1 E1 E
P(k,1) = y(j); M0 @! o+ P% z9 l$ j$ e* q4 {
j = abs(x(y(j)));
_& N# m& U) p# t if j == n+14 j! W S* f3 y* |, z
break;: ^( G+ i) a" a3 @! ]4 `
end; r7 y% V" [' N n* D
k = k+1;" f! q" v1 \4 X$ q
end6 f) V( l7 K G; {9 m, b
for i = 1:k$ c( d) N q$ y
if M(P(i,1),P(i,2)): i* O' s# S0 Y" m
M(P(i,1),P(i,2)) = 0;: H$ b- n( f0 l) v5 X
else1 ?2 J( v( N- s
M(P(i,1),P(i,2)) = 1;* S9 j$ n; D3 k/ A6 T
end/ {! l( E# _9 [6 ]: ]$ d
end
& z/ ?% ?7 P9 D5 } break;4 b0 d0 b& t" K0 b
end4 I" d6 w2 {( c3 ?% Q" K/ W; j4 |% j
end
' j4 }0 G) M7 q2 p- ?; h0 s9 _" t Z end
8 W; `) S( B; G: M if pd- U2 z7 E, Y% U; m( Q& K! B6 P
break;1 a6 g6 ` X' {! N/ ^1 v
end
# o$ ~( ?- w# F$ {5 A end" ?. K2 u. b4 C( |' f- p
1/ y# P9 @4 q: ?& c
29 R: O4 m% ?9 m& ^- k
3
( ~# c7 Q, n7 ~6 n6 E$ {4
# c' o9 W% r2 l! A0 J# J5
6 z7 L: a) Z. p& n0 O2 l6 C: s5 F1 k4 I$ u4 m& y$ y1 |- W
7 L; H% T' K+ q; R
8
5 f- Q- z" m5 A# x3 n, k+ B% x/ U/ M9
% m- L: B/ \. ~101 O+ U$ y& `: Z3 q% N/ S
11
" W3 J( c6 E8 ~ t* s12
" x3 {7 T! f6 u% w" }13, O. h% M Y6 V' {; T7 r6 I
14% F' c$ n6 M& i7 k2 x
15
3 l/ H3 \% j' U7 l' M" @16! J; A. D# \) ?9 `
17
, ~8 E7 r# \; y0 B9 {3 L18. d5 ]0 g3 x% I6 P$ `
19
! U8 {2 x8 c* z& |20
( Z. H1 U( S5 V# N, _: \6 o0 V21
( q1 u1 j7 l0 r6 d5 a/ C! l. l222 }/ m7 k* V0 E4 o
23
( x+ z: n$ S- t4 i! i' y6 B: n24
( _! d; Z9 ?) J) ], J1 W X* w25# w: a7 J; U; I
26
- p$ U' V5 }/ e7 K$ l27) Q0 ^1 O. h/ C) B3 u1 k' F" X( m: c. Q
28
8 M0 V& u. P2 h; k' H29
_; \: t# l" T4 s' L2 M' ^! L: N, Z30) j$ M. Q# m4 k/ A
31
4 `8 O, n7 _4 l: R8 @: {32
; R$ l, F& a8 _# c333 \; ~+ R4 s% d- r
34
' E# A& s) O2 N9 M$ y35
2 k5 g4 `3 R$ u7 ~& x, t0 r36- Q: F" c8 k) P
37( B; I# o" R# T
38
0 ^% f# |8 x" m399 _. b8 y+ m$ J, e
40
* q0 A' t& F, o3 p) Z' H41
$ W9 O H! w# E9 @6 g42
' _! v) y# f0 D; ?+ s2 S6 y# I43. ]4 v6 g0 l% {' B3 m% [1 ]) e
44
3 _4 i% ~9 k5 m9 j# i( L45
( y& u5 v: {: g% P- O U: W, R f0 F46
# B' p* s6 ]9 c* V$ B47
( ]$ C/ {6 `' x9 f7 G" d5 h; J48
& T5 T+ \; `$ L( N' V. K497 Y* `8 p% M4 ~- }: E- O0 p
50
4 i% e% r @2 J" `% I4 L1 w) e! h% N- `51
- b& U7 L f* H. d- o3 V% X522 g k. \% t* _9 n5 [4 {( Z: {
53 L, r/ T" K5 m; x' A, N
546 m5 l1 t1 M8 C, q% J( ^2 G
55
" {& n$ d, A; H# }! O4 `6 l3 Z# m& N56$ @7 F: n; {8 l9 ?9 M
57
8 S$ h- V+ r! ]3 U9 Q8 m! n58
$ k1 o. [! D' s7 b59! b! n- p; _# v2 u7 `8 D
60' b0 ~+ |. E% t
61
: k" z$ A# f. ~62
+ ^; I7 N1 [% K0 Z3 M2 x63. X9 i& Q8 ^0 u4 N- q3 V* a
64
0 B9 O6 c) k. L7 m3 K- l65
+ O! H( n' P, [9 b; U66. H4 z. K5 P; z8 u; E! y
673 D% n" x q5 f# i# `, S$ D3 m1 l
68
8 F: c) h9 @- P( H* P6 V" [1 t9 i69
* }1 \' y2 _7 l* U# b4 U70# q7 D9 E+ A: j# x
71
6 D0 P6 l5 X1 R. s5 \8 Z4 d72
% R9 W/ h- Q3 T& }/ s73" A$ ]8 Q' d f
74
: `4 B$ U% h1 ]3 K75
$ P( C5 l$ ^# M$ w3 a0 t; z( E76
# } Z. G% H; N4 {9 U3 J77
8 v% E2 g9 J' U4 G, D3 Y78
* p8 B( J( ~1 ?3 B+ w/ h79
& ~! }" u9 k4 J& P5 J80
* A" r$ w5 c' |* f- ^0 V, G' e81
+ @9 {! K$ h# p5 E* `828 g# L9 ^3 e" A2 O
83
* |9 V) X' Q( S. N84- k3 U3 Z! P r: b
85
7 V0 k* v; G4 p; O0 K G86
. L K9 x; I* l' f$ n9 Q2 v, M87) r' ?0 H; I( y! k& k) B l' X
88: x9 V6 o0 l, a/ h+ j1 O& b
89
' Y% r8 S H& l2 H% l5 U905 X& [9 ?0 N4 b# u+ |) D7 F
91& k+ N$ m, j2 {" r" |1 o
92! O& T' g( }" q+ `) v/ z
930 C) }' Y( ~. R; p' P( f3 K) x# [
94
2 x" k# P4 p$ I# _95
# u5 i3 c# m# o1 s: o9 ~8 _, B/ }# y* z96
) `( h. G* M2 N# C97' V' Q4 t' r( j. c, A
98
4 T: ^" ]9 l6 E# u9 S, |' \99
( a5 f% P4 n, E" O6 h100
: j! y8 h* [3 H! F101
2 n* T! L/ |; v$ K( Q( l102
' E9 F- s& @( P7 N3 e2 I103
E3 B( K- ~9 V8 C, _! V( E最佳匹配9 f6 Y' `" `# W5 E( C2 G
: O5 ?# O- E' m; B代码实现:
2 ]" Z M4 ^+ o! j+ B- Y1 }
8 B3 S" G( f ]- I% D1 l" Fn = 4;
! L( k7 |) Y! j' E: V+ r% m' ^A = [4 5 5 1;2 2 4 6;4 2 3 3;5 0 2 1];
- l- L, E! ^* fM(n,n) = 0;# m k/ `$ x# Y* A0 T
for i = 1:n+ Z) {+ z6 O3 d+ P# A
L(i,1) = 0;4 i" T$ }: H/ Z) c) C
L(i,2) = 0;* _! ~! F# ~" J9 p# i/ i' L
end1 O; @# @& ]+ U$ L
%初始化可行点标记L
$ x6 C0 g. k5 _# t- V- Cfor i = 1:n
: r; {) {0 g6 T% [- e for j = 1:n+ o8 Y8 F" ]# L( I" i; L
if L(i,1) < A(i,j)" p& t8 R3 ~) q9 v, i
L(i,1) = A(i,j);) U# V3 ~) T/ A" o/ M
end$ H4 y& g9 c, R3 \1 a
end/ T, N7 [) U/ p4 E% O* T
end+ }3 E: V0 U2 z4 W
%生成子图Gl u: h( b) |4 O* Y5 `" M+ |
for i = 1:n ], g8 P- Q2 ?; @, y3 M5 O
for j = 1:n
]4 ~* z: j& @1 C if L(i,1) + L(j,2) == A(i,j)' a. v4 {0 h- `- H) X' F0 ?
Gl(i,j) = 1;4 T' Y; r2 O& _8 l5 F" H: l6 i4 R" ?
else
8 |: g: W& E% J% m3 {9 ?7 j# w4 N' V Gl(i,j) = 0;: W$ g- \! N/ Q- K
end
3 g8 r, s( Z, e% m7 E; Y end& P$ p. [- K+ ~' y: n5 _6 m; S+ N7 Q' X
end" y; P- ~, H1 u! e
%获得仅含Gl的一条边的初始匹配M
8 i2 r3 v2 o$ f8 G9 R1 s& ^ii = 0;) p9 w. W; D! n0 T
jj = 0;
6 l0 [, R9 A0 J, Mfor i = 1:n
" U" o0 {% T! a6 I& T" c/ E for j = 1:n
1 L& L, M4 u6 T' b% Y if Gl(i,j)
3 z! `: |& m4 E" J% Z ii = i;
% D! P* Q( x. [; e/ k. z. P jj = j;) K* L+ R8 m9 H9 x# f
break;- H4 E) i9 }, T, C
end4 @) ~: ?- j, L- d1 A: U6 L+ _3 z
end
& s( N0 W$ Q2 _ if(ii)! |% V/ m- P* ~% x3 @+ g4 m
break;
5 _) w6 R8 I9 ^7 Q5 q* j6 s; [ end
1 Y; r- o# l1 m: h5 I t) Fend% A n) Q p2 E" B* Q: e9 R
M(ii,jj) = 1;( Y# Q0 c7 m. X
for i = 1:n
4 r# w& o3 P ?4 p S(i) = 0;
y/ _* l, A+ r; z1 U/ k T(i) = 0;1 e( C9 E3 V' B) g. Y
NIS(i) = 0;: c2 T9 q+ i- u* |/ g! k: ]
end) K( j$ h- w/ t( D
- p4 L0 U& R( A- A2 U6 A) u) k
) ?6 \" @9 r: `; [5 d; O% P
while (1)
& E$ Q& T) R6 _# m for i = 1:n
! A9 ^, V# J3 f" }7 K k = 1;
7 z9 w5 a$ b3 C; I+ w3 @% R for j = 1:n2 i, T1 K( t9 q$ c5 {3 I- p0 H
if M(i,j)
8 w' }. d: h/ A( J k = 0;
5 I% E3 H8 m8 [' e3 @% i break;
! G* K- K5 S: a( S& X end! h4 S& g v" o7 f2 ]
end0 k, i+ m% E8 V; N1 W
if k% ]( N! K/ D. Q1 m5 E: Z
break;
( S+ L2 p e0 F4 Z; L end% G* |6 r! H$ I2 r0 {, B% k( i( Y
end
9 W" U2 v! H( y$ S+ z7 |%获得最佳匹配M,算法终止
' `8 p% w7 ~5 s4 J. L if k == 06 S1 r& r1 D* c, v5 d
break;
, F8 q% k& t3 i: y end
6 n+ z- k' N/ g
2 b; m) p3 v; u# z2 t. U% D3 c% U$ `6 e. b+ c
%S = {xi}+ _4 Q9 F1 [9 }& W
S(1) = i;, Y+ i8 F" g) Z: `
jss = 1;1 u2 {4 Z5 H; ]8 p" ^1 }
jst = 0;2 t6 M9 } |' X+ s
while(1)
9 w @" T; ], o3 G jsn = 0;
% i" }5 D" m; u# m %选择NL的值
' X3 ]( Q1 ^! O* v; k, @# N for i = 1:jss
8 R" Y* f5 c) M! R) r+ v2 z; b% O- z for j = 1:n0 {% P) e& T" F% X i
if Gl(S(i),j)
( f" i0 C8 Q. G A6 W" Q8 X jsn = jsn + 1;
8 I7 A9 A- I' Q NIS(jsn) = j;) l1 P% M7 c0 |0 o+ ]3 ^- O
for k = 1:jsn-1
6 k1 n8 D Z6 M if NIS(k) == j6 W' y4 ^& j. t+ l
jsn = jsn - 1;, m- l, S% n I$ r, M* ~
end
- n: A- r- j1 C& x- V end
) \; P- S O0 A2 }6 M l end
+ t R7 b7 ], H- Y A9 m) j end& ^$ [0 b0 Y" e6 a0 c
end
' F4 _: k& {: Y9 M %判断NL(S) = T ?
$ }1 B d% w# ?& j% {- E; \( R* [/ v: j if jsn == jst$ G+ E$ N: U- l; q! \# G
pd = 1;6 ~- n: a6 q& c& K
for j = 1:jsn
! A" Y" K0 |9 F+ c7 ] if NIS(j) ~= T(j)# x. p! Z% J+ ]3 ?: i% A
pd = 0;
+ k2 O) ]! C9 [+ A- w, ^- n break;
4 k* C6 h2 w, o1 x2 U* s end# @$ E' ]- z/ l- u% n( ^
end8 S F0 k7 `/ K7 Z* f) {8 }
end
. T4 t5 T! C1 v" t. [9 h %如果NL(S) = T 计算al的值
+ V3 u+ r8 x. Y& } if (jsn == jst) && pd 2 k% _4 U* b1 t% D8 a/ o
al = Inf;
( H4 y. D9 R: w for i = 1:jss( N/ B/ _' A! @) l
for j = 1:n y1 M* o2 `# ~# I1 V6 u
pd = 1;
d" Q8 Q0 L$ X& M# {- l7 X* n# H& m for k = 1:jst+ a% D% F! F% ~* m$ y1 V
if T(k) == j. @, J; {$ T$ V$ Z' N, w
pd = 0;1 [+ E) E( N. R" r1 o
break;
. E% N6 h) x$ w }. y end
- a, _& j5 c" k) y& M end9 L! e1 L# i, j" z
if pd && (al > L(S(i),1) + L(j,2) - A(S(i),j))
' B4 T8 i" G/ n al = L(S(i),1) + L(j,2) - A(S(i),j);/ W, \$ F7 M. T$ ?! d: o0 Q
end
6 L S9 p) U% @2 n2 c end" m6 ?5 T6 c0 G) `2 Q
end
; k4 a$ C' `0 Q( E9 _2 W3 R* Z %调整可行点标记$ k4 h: X4 c* e7 G" n
for i = 1:jss
l# e9 d9 r3 _# m0 W) d8 v ~8 X2 U L(S(i),1) = L(S(i),1) - al;8 c9 I2 Q' ?6 g6 s' A' \
end
7 ~3 t0 y3 G4 u' s$ q/ W2 G/ V3 [4 W9 l %调整可行点标记$ P4 Y) k: z/ G4 Q" U i
for j = 1:jst* `( m: W* U, C- Q# a0 ~/ e8 j" C; ~
L(T(j),2) = L(T(j),2) + al;
# c [, x1 C) B1 t. [6 S! M& I/ N end7 @9 k6 t i" k: r
%生成子图Gl. \6 n& c' e2 j8 W' k2 f# G
for i = 1:n9 z/ n3 {9 O" I) L
for j = 1:n# G) W7 c- R# `) X: x6 \
if L(i,1) + L(j,2) == A(i,j)# ]) @# S3 {) K& g) Y+ T6 W# ^
Gl(i,j) = 1;
& x7 T' C+ V2 ? else
" o$ x2 I a- c# [% w0 t Gl(i,j) = 0;
; l8 g G5 l, k. k- \) i( X end* y0 _8 Q3 ~ l, E
M(i,j) = 0;8 P6 A( M5 _0 d6 n
k = 0;3 P( c- b5 P2 p \4 ^) ?# s# P' d
end
1 j/ ?* q; p- H; }$ B end
, k& X) F5 j/ C2 C) k- s5 [ %获得仅含Gl的一条边的初始匹配M$ b- R0 C9 J4 W1 D
ii = 0;$ w1 W& V* B9 b! |* j, [
jj = 0;: ?; q/ |; C5 @' p
for i = 1:n8 T$ b$ T: v) B7 B) o# k$ u
for j = 1:n7 n) B( z) O6 z
if Gl(i,j)
+ e- l: q' x* Z5 ^ ii = i; o' Z( m. B* U
jj = j;
. o; h9 w4 o5 ^, q8 G break;
3 A! ]- M& A/ g9 O; S end, y1 x, k9 C* K6 f! S
end
" M1 D% U" C% A! C2 { if(ii)
: r; c1 ~8 a/ q) x9 d7 h0 C( f break;
- Y$ W$ U2 D4 v9 c* g. ^ end
; z% |' p: M. G2 L$ ^! z, Q end9 Y8 z, x6 u& p3 U
M(ii,jj) = 1;4 q4 J7 k7 c$ v3 O, `; g( d, c3 N
break;; M& }( V% W7 P$ O
else
1 _- q$ ^1 ?$ H6 r- M- Z$ ? for j = 1:jsn
' ^, `1 s. R3 M$ K pd = 1;4 Z7 ]* |1 J1 R" G& D
for k = 1:jst
# j% Z3 ]# A+ f; T if T(k) == NIS(j)! S1 d/ W% m: ]8 z0 v* q
pd =0+ J( ]: _5 C* N) ^" p
break;4 X; ]+ O8 }8 N' M+ g
end
g$ t5 }' }: X4 u$ e0 A- y0 k$ g end
- M; [; ]7 L2 y$ O if pd" |! h& I! J6 Q/ L# F5 i- j. U
jj = j;% D* m7 r. l& c- c6 a' h
break;
6 _3 x, X, n+ e/ f9 v1 R end& W' C3 O' `. }. o, k/ k7 a1 I( }
end5 _: O: k, }8 v. b" r! D) ?6 Z. f U
%判断y是否是M的饱和点
; E) @: w( f9 H3 E( J" j+ Q4 L; F pd = 0;5 @0 P% S2 i' B* {4 {" S
for i = 1:n
9 q& T* C& i, ]3 E4 n if M(i,NIS(jj))+ z; g* ~; W" S" ]+ [. }) j# z9 k
pd = 1;5 a* s) m3 _9 g9 r) r: C
ii = i;% V4 B) q! u" d/ W8 A9 O
break;0 o3 S% d2 n# ]' \( H$ ?% X# z1 \0 y! K
end% R0 u8 F! z: ^
end# j: R; I. E# I2 G- _% ] r7 j
if pd+ p l! b9 a, F- }$ m J
jss = jss + 1;
7 S- x/ `7 L5 N0 q7 r1 { e0 S S(jss) = ii;
! n6 t; H4 b" f, Q jst = jst + 1;! J0 Z, r1 @6 Z+ \% t3 T
T(jst) = NIS(jj);4 W8 ?; f, K0 ?0 ?" D
else) @8 _8 @: t5 |+ z# e* n/ o7 F
for k = 1:jst3 Y ^4 {$ e" W u6 w# y
M(S(k),T(k)) = 1;
( P( [( X1 V7 j" m2 k9 o4 j M(S(k+1),T(k)) = 0; F2 k% a# I/ x
end8 a2 g% ~ T/ k6 Z4 j4 ?
if jst == 0; M, [( Y8 n& v1 i9 k
k = 0;8 G$ k7 ?+ P$ }8 B# s$ f
end) y! g1 ]+ B$ b* Q( P/ {7 u" W% f
M(S(k+1),NIS(jj)) = 1;
: |* d) M1 R1 \ break;
- i, k0 U7 I/ _/ m1 W' u end
m) [' A+ e4 v# C end
2 m" w, k8 X. B9 G; Z end
3 v" N- J! g$ Z4 j+ I end" |9 g9 ^, i' l m- C* h
MaxZjpp = 0;
$ [$ C6 A. {5 Z/ t" {8 A for i = 1:n) M6 n! k$ c3 }+ v
for j = 1:n' K1 U% S4 M" d& r
if M(i,j)
, d5 X" ^+ v' h, P/ P MaxZjpp = MaxZjpp + A(i,j);. q1 ~4 }8 s Y6 ]0 K2 A
end9 ]- M+ m" X% V/ C! d8 G1 [% p v
end0 Z* _* J. C9 E! j1 s9 M! z
end
8 |/ [1 r) _- n# z+ n1 Q M! M3 T# A+ E& `2 ?+ H6 g
MaxZjpp
c) }9 r! B/ D; _& f W ]4 e; ]6 Y% [% k! c' Z
3 T8 Z# ~7 M2 P* b/ j5 }
+ O% Q5 ]: ]# c6 K |
zan
|