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Hungarian.cpp
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#include <bits/stdc++.h>
using namespace std;
#define N 101 //max number of vertices in one part
#define INF 100000000 //just infinity
int cost[N][N]; //cost matrix
int n, max_match; //n workers and n jobs
int lx[N], ly[N]; //labels of X and Y parts
int xy[N]; //xy[x] - vertex that is matched with x,
int yx[N]; //yx[y] - vertex that is matched with y
bool S[N], T[N]; //sets S and T in algorithm
int slack[N]; //as in the algorithm description
int slackx[N]; //slackx[y] such a vertex, that
// l(slackx[y]) + l(y) - w(slackx[y],y) = slack[y]
int prv[N]; //array for memorizing alternating paths
void init_labels()
{
memset(lx, 0, sizeof(lx));
memset(ly, 0, sizeof(ly));
for (int x = 0; x < n; x++)
for (int y = 0; y < n; y++)
lx[x] = max(lx[x], cost[x][y]);
}
void update_labels()
{
int x, y, delta = INF; //init delta as infinity
for (y = 0; y < n; y++) //calculate delta using slack
if (!T[y])
delta = min(delta, slack[y]);
for (x = 0; x < n; x++) //update X labels
if (S[x]) lx[x] -= delta;
for (y = 0; y < n; y++) //update Y labels
if (T[y]) ly[y] += delta;
for (y = 0; y < n; y++) //update slack array
if (!T[y])
slack[y] -= delta;
}
void add_to_tree(int x, int prevx)
//x - current vertex,prevx - vertex from X before x in the alternating path,
//so we add edges (prevx, xy[x]), (xy[x], x)
{
S[x] = true; //add x to S
prv[x] = prevx; //we need this when augmenting
for (int y = 0; y < n; y++) //update slacks, because we add new vertex to S
if (lx[x] + ly[y] - cost[x][y] < slack[y])
{
slack[y] = lx[x] + ly[y] - cost[x][y];
slackx[y] = x;
}
}
void augment() //main function of the algorithm
{
if (max_match == n) return; //check wether matching is already perfect
int x, y, root; //just counters and root vertex
int q[N], wr = 0, rd = 0; //q - queue for bfs, wr,rd - write and read
//pos in queue
memset(S, false, sizeof(S)); //init set S
memset(T, false, sizeof(T)); //init set T
memset(prv, -1, sizeof(prv)); //init set prev - for the alternating tree
for (x = 0; x < n; x++) //finding root of the tree
if (xy[x] == -1)
{
q[wr++] = root = x;
prv[x] = -2;
S[x] = true;
break;
}
for (y = 0; y < n; y++) //initializing slack array
{
slack[y] = lx[root] + ly[y] - cost[root][y];
slackx[y] = root;
}
//second part of augment() function
while (true) //main cycle
{
while (rd < wr) //building tree with bfs cycle
{
x = q[rd++]; //current vertex from X part
for (y = 0; y < n; y++) //iterate through all edges in equality graph
if (cost[x][y] == lx[x] + ly[y] && !T[y])
{
if (yx[y] == -1) break; //an exposed vertex in Y found, so
//augmenting path exists!
T[y] = true; //else just add y to T,
q[wr++] = yx[y]; //add vertex yx[y], which is matched
//with y, to the queue
add_to_tree(yx[y], x); //add edges (x,y) and (y,yx[y]) to the tree
}
if (y < n) break; //augmenting path found!
}
if (y < n) break; //augmenting path found!
update_labels(); //augmenting path not found, so improve labeling
wr = rd = 0;
for (y = 0; y < n; y++)
//in this cycle we add edges that were added to the equality graph as a
//result of improving the labeling, we add edge (slackx[y], y) to the tree if
//and only if !T[y] && slack[y] == 0, also with this edge we add another one
//(y, yx[y]) or augment the matching, if y was exposed
if (!T[y] && slack[y] == 0)
{
if (yx[y] == -1) //exposed vertex in Y found - augmenting path exists!
{
x = slackx[y];
break;
}
else
{
T[y] = true; //else just add y to T,
if (!S[yx[y]])
{
q[wr++] = yx[y]; //add vertex yx[y], which is matched with
//y, to the queue
add_to_tree(yx[y], slackx[y]); //and add edges (x,y) and (y,
//yx[y]) to the tree
}
}
}
if (y < n) break; //augmenting path found!
}
if (y < n) //we found augmenting path!
{
max_match++; //increment matching
//in this cycle we inverse edges along augmenting path
for (int cx = x, cy = y, ty; cx != -2; cx = prv[cx], cy = ty)
{
ty = xy[cx];
yx[cy] = cx;
xy[cx] = cy;
}
augment(); //recall function, go to step 1 of the algorithm
}
}//end of augment() function
int hungarian()
{
int ret = 0; //weight of the optimal matching
max_match = 0; //number of vertices in current matching
memset(xy, -1, sizeof(xy));
memset(yx, -1, sizeof(yx));
init_labels(); //step 0
augment(); //steps 1-3
for (int x = 0; x < n; x++) //forming answer there
ret += cost[x][xy[x]];
return ret;
}
int main () {
int tc;
cin >> tc;
while (tc--)
{
cin >> n;
for (int i=0; i<n; i++)
for (int j=0; j<n; j++)
scanf("%d",&cost[i][j]);
long long ans = hungarian();
cout << ans << endl;
}
return 0;
}