function [Matching,Cost] = Hungarian(Perf)
% [MATCHING,COST] = Hungarian_New(WEIGHTS)
% 匈牙利算法
% 给定一个n x n矩阵的边权矩阵，使用匈牙利算法求解最小边权值和问题，类似最小树问题
% 如果矩阵中出现inf，则表示没有边与之相连
% 输出：
% Matching 为一个 n x n的矩阵，只有0 和 1
% COST 为对应Matching处为1所在位置的元素和
% 初始化变量
Matching = zeros(size(Perf));
% 移除Inf，加速算法执行效率
% 针对每一列找inf
num_y = sum(~isinf(Perf),1);
% 针对每一行找inf
num_x = sum(~isinf(Perf),2);
% Find the columns(vertices) and rows(vertices) that are isolated
x_con = find(num_x~=0);
y_con = find(num_y~=0);
% 缩减矩阵
P_size = max(length(x_con),length(y_con));
P_cond = zeros(P_size);
P_cond(1:length(x_con),1:length(y_con)) = Perf(x_con,y_con);
if isempty(P_cond)
Cost = 0;
return
end
% 计算边权矩阵
Edge = P_cond;
Edge(P_cond~=Inf) = 0;
% 修正权效矩阵
cnum = min_line_cover(Edge);
% 对未标记的行和已标记的列画纵、横线P
Pmax = max(max(P_cond(P_cond~=Inf)));
P_size = length(P_cond)+cnum;
P_cond = ones(P_size)*Pmax;
P_cond(1:length(x_con),1:length(y_con)) = Perf(x_con,y_con);
%*************************************************
% 主要求解步骤
%*************************************************
exit_flag = 1;
stepnum = 1;
while exit_flag
switch stepnum
case 1
[P_cond,stepnum] = step1(P_cond);
case 2
[r_cov,c_cov,M,stepnum] = step2(P_cond);
case 3
[c_cov,stepnum] = step3(M,P_size);
case 4
[M,r_cov,c_cov,Z_r,Z_c,stepnum] = step4(P_cond,r_cov,c_cov,M);
case 5
[M,r_cov,c_cov,stepnum] = step5(M,Z_r,Z_c,r_cov,c_cov);
case 6
[P_cond,stepnum] = step6(P_cond,r_cov,c_cov);
case 7
exit_flag = 0;
end
end
% 移除所有的虚拟节点
Matching(x_con,y_con) = M(1:length(x_con),1:length(y_con));
Cost = sum(sum(Perf(Matching==1)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% STEP 1:找每行最小元素
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [P_cond,stepnum] = step1(P_cond)
P_size = length(P_cond);
% Loop throught each row
for ii = 1:P_size
rmin = min(P_cond(ii,:));
P_cond(ii,:) = P_cond(ii,:)-rmin;
end
stepnum = 2;
%**************************************************************************
% STEP 2: 寻找P_cond中的0元素
%**************************************************************************
function [r_cov,c_cov,M,stepnum] = step2(P_cond)
% 定义变量
P_size = length(P_cond);
r_cov = zeros(P_size,1); % 记录 元素为0 所在的行
c_cov = zeros(P_size,1); % 记录 元素为0 所在的列
M = zeros(P_size); % A mask that shows if a position is starred or primed
for ii = 1:P_size
for jj = 1:P_size
if P_cond(ii,jj) == 0 && r_cov(ii) == 0 && c_cov(jj) == 0
M(ii,jj) = 1;
r_cov(ii) = 1;
c_cov(jj) = 1;
end
end
end
% 重新初始化操作
r_cov = zeros(P_size,1); % A vector that shows if a row is covered
c_cov = zeros(P_size,1); % A vector that shows if a column is covered
stepnum = 3;
%**************************************************************************
% STEP 3: 给定每一类一个0
%**************************************************************************
function [c_cov,stepnum] = step3(M,P_size)
c_cov = sum(M,1);
if sum(c_cov) == P_size
stepnum = 7;
else
stepnum = 4;
end
%**************************************************************************
% STEP 4: 寻找没有被覆盖为0的元素，如果在该行没有0元素，则返回到第5步Step 5.
% 否则，直到覆盖该元素为0。
%**************************************************************************
function [M,r_cov,c_cov,Z_r,Z_c,stepnum] = step4(P_cond,r_cov,c_cov,M)
P_size = length(P_cond);
zflag = 1;
while zflag
% 寻找第一个未被覆盖的0元素
row = 0; col = 0; exit_flag = 1;
ii = 1; jj = 1;
while exit_flag
if P_cond(ii,jj) == 0 && r_cov(ii) == 0 && c_cov(jj) == 0
row = ii;
col = jj;
exit_flag = 0;
end
jj = jj + 1;
if jj > P_size; jj = 1; ii = ii+1; end
if ii > P_size; exit_flag = 0; end
end
% 如果没有被覆盖为0元素时，则转到step 6
if row == 0
stepnum = 6;
zflag = 0;
Z_r = 0;
Z_c = 0;
else
% Prime the uncovered zero
M(row,col) = 2;
% 在该行，有一个起始0元素，覆盖该行，用一个含有0元素的列去修正该列
% Cover the row and uncover the column containing the zero
if sum(find(M(row,:)==1)) ~= 0
r_cov(row) = 1;
zcol = find(M(row,:)==1);
c_cov(zcol) = 0;
else
stepnum = 5;
zflag = 0;
Z_r = row;
Z_c = col;
end
end
end
%**************************************************************************
% STEP 5: Z0 为已寻到的0元素in Step 4.Z1为Z0列的其它0元素，Z2 为主要的0元素，去掉所有
% 0的序列，保留主要的0元素， 返回到 Step 3.
%**************************************************************************
function [M,r_cov,c_cov,stepnum] = step5(M,Z_r,Z_c,r_cov,c_cov)
zflag = 1;
ii = 1;
while zflag
% 寻找列所在的起始0元素
rindex = find(M(:,Z_c(ii))==1);
if rindex > 0
% 保存起始0
ii = ii+1;
% 保存起始0所在的行
Z_r(ii,1) = rindex;
% The column of the starred zero is the same as the column of the
% primed zero
Z_c(ii,1) = Z_c(ii-1);
else
zflag = 0;
end
% Continue if there is a starred zero in the column of the primed zero
if zflag == 1;
% Find the column of the primed zero in the last starred zeros row
cindex = find(M(Z_r(ii),:)==2);
ii = ii+1;
Z_r(ii,1) = Z_r(ii-1);
Z_c(ii,1) = cindex;
end
end
% UNSTAR all the starred zeros in the path and STAR all primed zeros
for ii = 1:length(Z_r)
if M(Z_r(ii),Z_c(ii)) == 1
M(Z_r(ii),Z_c(ii)) = 0;
else
M(Z_r(ii),Z_c(ii)) = 1;
end
end
% 清除
r_cov = r_cov.*0;
c_cov = c_cov.*0;
% Remove all the primes
M(M==2) = 0;
stepnum = 3;
% *************************************************************************
% STEP 6: 在每一行叠加一个很小的值，在每一列减去一个很小的这个值
% 返回到step 4
%**************************************************************************
function [P_cond,stepnum] = step6(P_cond,r_cov,c_cov)
a = find(r_cov == 0);
b = find(c_cov == 0);
minval = min(min(P_cond(a,b)));
P_cond(find(r_cov == 1),:) = P_cond(find(r_cov == 1),:) + minval;
P_cond(:,find(c_cov == 0)) = P_cond(:,find(c_cov == 0)) - minval;
stepnum = 4;
function cnum = min_line_cover(Edge)
% Step 2
[r_cov,c_cov,M,stepnum] = step2(Edge);
% Step 3
[c_cov,stepnum] = step3(M,length(Edge));
% Step 4
[M,r_cov,c_cov,Z_r,Z_c,stepnum] = step4(Edge,r_cov,c_cov,M);
% Calculate the deficiency
cnum = length(Edge)-sum(r_cov)-sum(c_cov);
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