6 、 图论—NP 搜索

6.1 最大团

//最大团
//返回最大团大小和一个方案,传入图的大小 n 和邻接阵 mat
//mat[i][j]为布尔量
#define MAXN 60
void clique(int n, int* u, int mat[][MAXN], int size, int& max, int& bb, int* res, int* rr, int* c) {
int i, j, vn, v[MAXN];
if (n) {
if (size + c[u[0]] <= max) return;
for (i = 0; i < n + size - max && i < n; ++ i) {
for (j = i + 1, vn = 0; j < n; ++ j)
if (mat[u[i]][u[j]])
v[vn ++] = u[j];
rr[size] = u[i];
clique(vn, v, mat, size + 1, max, bb, res, rr, c);
if (bb) return;
}
} else if (size > max) {
max = size;
for (i = 0; i < size; ++ i)
res[i] = rr[i];
bb = 1;
}
}
int maxclique(int n, int mat[][MAXN], int *ret) {
int max = 0, bb, c[MAXN], i, j;
int vn, v[MAXN], rr[MAXN];
for (c[i = n - 1] = 0; i >= 0; -- i) {
for (vn = 0, j = i + 1; j < n; ++ j)
if (mat[i][j])
v[vn ++] = j;
bb = 0;
rr[0] = i;
clique(vn, v, mat, 1, max, bb, ret, rr, c);
c[i] = max;
}
return max;
}

6.2 最大团(n<64)(faster)

/**
* WishingBone's ACM/ICPC Routine Library
*
* maximum clique solver
*/
#include <vector>
using std::vector;
// clique solver calculates both size and consitution of maximum clique
// uses bit operation to accelerate searching
// graph size limit is 63, the graph should be undirected
// can optimize to calculate on each component, and sort on vertex degrees
// can be used to solve maximum independent set
class clique {
public:
static const long long ONE = 1;
static const long long MASK = (1 << 21) - 1;
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char* bits;
int n, size, cmax[63];
long long mask[63], cons;
// initiate lookup table
clique() {
bits = new char[1 << 21];
bits[0] = 0;
for (int i = 1; i < 1 << 21; ++i) bits[i] = bits[i >> 1] + (i & 1);
}
~clique() {
delete bits;
}
// search routine
bool search(int step, int size, long long more, long long con);
// solve maximum clique and return size
int sizeClique(vector<vector<int> >& mat);
// solve maximum clique and return constitution
vector<int> consClique(vector<vector<int> >& mat);
};
// search routine
// step is node id, size is current solution, more is available mask, cons is
constitution mask
bool clique::search(int step, int size, long long more, long long cons) {
if (step >= n) {
// a new solution reached
this->size = size;
this->cons = cons;
return true;
}
long long now = ONE << step;
if ((now & more) > 0) {
long long next = more & mask[step];
if (size + bits[next & MASK] + bits[(next >> 21) & MASK] + bits[next >>
42] >= this->size
&& size + cmax[step] > this->size) {
// the current node is in the clique
if (search(step + 1, size + 1, next, cons | now)) return true;
}
}
long long next = more & ~now;
if (size + bits[next & MASK] + bits[(next >> 21) & MASK] + bits[next >> 42]
> this->size) {
// the current node is not in the clique
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if (search(step + 1, size, next, cons)) return true;
}
return false;
}
// solve maximum clique and return size
int clique::sizeClique(vector<vector<int> >& mat) {
n = mat.size();
// generate mask vectors
for (int i = 0; i < n; ++i) {
mask[i] = 0;
for (int j = 0; j < n; ++j) if (mat[i][j] > 0) mask[i] |= ONE << j;
}
size = 0;
for (int i = n - 1; i >= 0; --i) {
search(i + 1, 1, mask[i], ONE << i);
cmax[i] = size;
}
return size;
}
// solve maximum clique and return constitution
// calls sizeClique and restore cons
vector<int> clique::consClique(vector<vector<int> >& mat) {
sizeClique(mat);
vector<int> ret;
for (int i = 0; i < n; ++i) if ((cons & (ONE << i)) > 0) ret.push_back(i);
return ret;
}