问题是求平面欧几里德最小生成树的第n - k小边。
平面欧几里德最小生成树是经典问题，可以做到O(nlogn)。具体做法是先对平面点进行三角剖分，时间复杂度是O(nlogn)，三角剖分的边就是可能的在最小生成树的边。因为是平面图，所以有O(n)条边，在其上应用 Kruscal 算法即可。
#include <cstdio>
#include <cmath>
#include <cstring>
#include <cstdlib>
#include <algorithm>
using namespace std;
#define TRUE 1
#define FALSE 0
#define Vector(p1, p2, u, v) (u = p2->x - p1->x, v = p2->y - p1->y)
#define Cross_product_2v(u1, v1, u2, v2) (u1 * v2 - v1 * u2)
#define Cross_product_3p(p1, p2, p3) ((p2->x - p1->x) * (p3->y - p1->y) - (p2->y - p1->y) * (p3->x - p1->x))
#define Dot_product_2v(u1, v1, u2, v2) (u1 * u2 + v1 * v2)
#define Org(e) ((e)->org)
#define Dest(e) ((e)->dest)
#define Onext(e) ((e)->onext)
#define Oprev(e) ((e)->oprev)
#define Dnext(e) ((e)->dnext)
#define Dprev(e) ((e)->dprev)
#define Other_point(e, p) ((e)->org == p ? (e)->dest : (e)->org)
#define Next(e, p) ((e)->org == p ? (e)->onext : (e)->dnext)
#define Prev(e, p) ((e)->org == p ? (e)->oprev : (e)->dprev)
#define Visited(p) ((p)->f)
#define Identical_refs(e1, e2) (e1 == e2)
typedef enum 
{right, left} side;
typedef double Real;
typedef int boolean;
typedef struct point point;
typedef struct edge edge;
struct point
{
Real x, y;
edge *entry_pt;
};
struct edge
{
point *org, *dest;
edge *onext, *oprev, *dnext, *dprev;
};
#define MAXV 1000001
#define MAXE 3000001
struct EDGE
{
int u, v;
double w;
};
EDGE E[MAXE], mst[MAXE];
int n, en, k;
double dist[MAXV];
int p[MAXV], d[MAXV];
void UFinit(int m)
{
int i;
for(i = 0; i < m; i++)
{
p[i] = i, d[i] = 0;
}
}
int UFfind(int x)
{
int i = x, j, k;
while(p[i] != i) i = p[i];
j = x;
while(p[j] != i)
{
k = p[j];
p[j] = i;
j = k;
}
return i;
}
void UFunion(int x, int y)
{
int i = UFfind(x), j = UFfind(y);
if(d[i] > d[j]) p[j] = i;
else
{
p[i] = j;
if(d[i] == d[j]) d[j]++;
}
}
bool cmp(const EDGE &e1, const EDGE &e2)
{
return e1.w < e2.w;
}
void kruskal(void)
{
if(k > n) k = n;
int i, kk;
sort(E, E + en, cmp);
UFinit(n);
for(i = 0, kk = 0; i < en && kk < n - 1; i++)
if(UFfind(E[i].u) != UFfind(E[i].v))
{
UFunion(E[i].u, E[i].v);
mst[kk++] = E[i];
}
printf("%d\n", int(ceil(mst[n - k - 1].w) + 0.00000001));
}
point *p_array;
static edge *e_array;
static edge **free_list_e;
static int n_free_e;
void alloc_memory(int n)
{
edge *e;
int i;
p_array = (point *)calloc(n, sizeof(point));
n_free_e = 3 * n;
e_array = e = (edge *)calloc(n_free_e, sizeof(edge));
free_list_e = (edge **)calloc(n_free_e, sizeof(edge *));
for(i = 0; i < n_free_e; i++, e++)
free_list_e[i] = e;
}
void free_memory(void)
{
free(p_array);
free(e_array);
free(free_list_e);
}
edge *get_edge(void)
{
if(n_free_e == 0)
printf("Out of memory for edges\n");
return (free_list_e[--n_free_e]);
}
void free_edge(edge *e)
{
free_list_e[n_free_e++] = e;
}
void splice(edge *a, edge *b, point *v)
{
edge *next;
if(Org(a) == v)
{
next = Onext(a);
Onext(a) = b;
}
else
{
next = Dnext(a);
Dnext(a) = b;
}
if(Org(next) == v)
Oprev(next) = b;
else
Dprev(next) = b;
if(Org(b) == v)
{
Onext(b) = next;
Oprev(b) = a;
}
else
{
Dnext(b) = next;
Dprev(b) = a;
}
}
edge *make_edge(point *u, point *v)
{
edge *e;
e = get_edge();
e->onext = e->oprev = e->dnext = e->dprev = e;
e->org = u;
e->dest = v;
if(u->entry_pt == NULL)
u->entry_pt = e;
if(v->entry_pt == NULL)
v->entry_pt = e;
return e;
}
edge *join(edge *a, point *u, edge *b, point *v, side s)
{
edge *e;
e = make_edge(u, v);
if(s == left)
{
if (Org(a) == u)
splice(Oprev(a), e, u);
else
splice(Dprev(a), e, u);
splice(b, e, v);
}
else
{
splice(a, e, u);
if(Org(b) == v)
splice(Oprev(b), e, v);
else
splice(Dprev(b), e, v);
}
return e;
}
void delete_edge(edge *e)
{
point *u, *v;
u = Org(e);
v = Dest(e);
if(u->entry_pt == e)
u->entry_pt = e->onext;
if(v->entry_pt == e)
v->entry_pt = e->dnext;
if(Org(e->onext) == u)
e->onext->oprev = e->oprev;
else
e->onext->dprev = e->oprev;
if(Org(e->oprev) == u)
e->oprev->onext = e->onext;
else
e->oprev->dnext = e->onext;
if(Org(e->dnext) == v)
e->dnext->oprev = e->dprev;
else
e->dnext->dprev = e->dprev;
if(Org(e->dprev) == v)
e->dprev->onext = e->dnext;
else
e->dprev->dnext = e->dnext;
free_edge(e);
}
double px[MAXV], py[MAXV];
void read_points()
{
int i = 0;
while (1)
{
int t1, t2;
scanf("%d", &t1);
px[i] = t1;
if (t1 == -1)
break;
scanf("%d", &t2);
py[i] = t2;
i++;
}
n = i;
}
static void print_edges(int n)
{
edge *e_start, *e;
point *u, *v;
int i;
for(i = 0; i < n; i++)
{
u = &p_array[i];
e_start = e = u->entry_pt;
do
{
v = Other_point(e, u);
if(u < v)
{
E[en].u = u - p_array, E[en].v = v - p_array;
E[en++].w = hypot(u->x - v->x, u->y - v->y);
}
e = Next(e, u);
}while(!Identical_refs(e, e_start));
}
}
void merge_sort(point *p[], point *p_temp[], int l, int r)
{
int i, j, k, m;
if(r - l > 0)
{
m = (r + l) / 2;
merge_sort(p, p_temp, l, m);
merge_sort(p, p_temp, m + 1, r);
for(i = m + 1; i > l; i--)
p_temp[i - 1] = p[i - 1];
for(j = m; j < r; j++)
p_temp[r + m - j] = p[j + 1];
for(k = l; k <= r; k++)
if(p_temp[i]->x < p_temp[j]->x)
{
p[k] = p_temp[i];
i = i + 1;
}
else if(p_temp[i]->x == p_temp[j]->x && p_temp[i]->y < p_temp[j]->y)
{
p[k] = p_temp[i];
i = i + 1;
}
else
{
p[k] = p_temp[j];
j = j - 1;
}
}
}
static void lower_tangent(edge *r_cw_l, point *s, edge *l_ccw_r, point *u, edge **l_lower, point **org_l_lower, edge **r_lower, point **org_r_lower)
{
edge *l, *r;
point *o_l, *o_r, *d_l, *d_r;
boolean finished;
l = r_cw_l;
r = l_ccw_r;
o_l = s;
d_l = Other_point(l, s);
o_r = u;
d_r = Other_point(r, u);
finished = FALSE;
while(!finished)
if(Cross_product_3p(o_l, d_l, o_r) > 0.0)
{
l = Prev(l, d_l);
o_l = d_l;
d_l = Other_point(l, o_l);
}
else if(Cross_product_3p(o_r, d_r, o_l) < 0.0)
{
r = Next(r, d_r);
o_r = d_r;
d_r = Other_point(r, o_r);
}
else
finished = TRUE;
*l_lower = l;
*r_lower = r;
*org_l_lower = o_l;
*org_r_lower = o_r;
}
static void merge(edge *r_cw_l, point *s, edge *l_ccw_r, point *u, edge **l_tangent)
{
edge *base, *l_cand, *r_cand;
point *org_base, *dest_base;
Real u_l_c_o_b, v_l_c_o_b, u_l_c_d_b, v_l_c_d_b;
Real u_r_c_o_b, v_r_c_o_b, u_r_c_d_b, v_r_c_d_b;
Real c_p_l_cand, c_p_r_cand;
Real d_p_l_cand, d_p_r_cand;
boolean above_l_cand, above_r_cand, above_next, above_prev;
point *dest_l_cand, *dest_r_cand;
Real cot_l_cand, cot_r_cand;
edge *l_lower, *r_lower;
point *org_r_lower, *org_l_lower;
lower_tangent(r_cw_l, s, l_ccw_r, u, &l_lower, &org_l_lower, &r_lower, &org_r_lower);
base = join(l_lower, org_l_lower, r_lower, org_r_lower, right);
org_base = org_l_lower;
dest_base = org_r_lower;
*l_tangent = base;
do
{
l_cand = Next(base, org_base);
r_cand = Prev(base, dest_base);
dest_l_cand = Other_point(l_cand, org_base);
dest_r_cand = Other_point(r_cand, dest_base);
Vector(dest_l_cand, org_base, u_l_c_o_b, v_l_c_o_b);
Vector(dest_l_cand, dest_base, u_l_c_d_b, v_l_c_d_b);
Vector(dest_r_cand, org_base, u_r_c_o_b, v_r_c_o_b);
Vector(dest_r_cand, dest_base, u_r_c_d_b, v_r_c_d_b);
c_p_l_cand = Cross_product_2v(u_l_c_o_b, v_l_c_o_b, u_l_c_d_b, v_l_c_d_b);
c_p_r_cand = Cross_product_2v(u_r_c_o_b, v_r_c_o_b, u_r_c_d_b, v_r_c_d_b);
above_l_cand = c_p_l_cand > 0.0;
above_r_cand = c_p_r_cand > 0.0;
if(!above_l_cand && !above_r_cand)
break;
if(above_l_cand)
{
Real u_n_o_b, v_n_o_b, u_n_d_b, v_n_d_b;
Real c_p_next, d_p_next, cot_next;
edge *next;
point *dest_next;
d_p_l_cand = Dot_product_2v(u_l_c_o_b, v_l_c_o_b, u_l_c_d_b, v_l_c_d_b);
cot_l_cand = d_p_l_cand / c_p_l_cand;
do
{
next = Next(l_cand, org_base);
dest_next = Other_point(next, org_base);
Vector(dest_next, org_base, u_n_o_b, v_n_o_b);
Vector(dest_next, dest_base, u_n_d_b, v_n_d_b);
c_p_next = Cross_product_2v(u_n_o_b, v_n_o_b, u_n_d_b, v_n_d_b);
above_next = c_p_next > 0.0;
if(!above_next)
break;
d_p_next = Dot_product_2v(u_n_o_b, v_n_o_b, u_n_d_b, v_n_d_b);
cot_next = d_p_next / c_p_next;
if(cot_next > cot_l_cand)
break;
delete_edge(l_cand);
l_cand = next;
cot_l_cand = cot_next;
}while(TRUE);
}
if(above_r_cand)
{
Real u_p_o_b, v_p_o_b, u_p_d_b, v_p_d_b;
Real c_p_prev, d_p_prev, cot_prev;
edge *prev;
point *dest_prev;
d_p_r_cand = Dot_product_2v(u_r_c_o_b, v_r_c_o_b, u_r_c_d_b, v_r_c_d_b);
cot_r_cand = d_p_r_cand / c_p_r_cand;
do
{
prev = Prev(r_cand, dest_base);
dest_prev = Other_point(prev, dest_base);
Vector(dest_prev, org_base, u_p_o_b, v_p_o_b);
Vector(dest_prev, dest_base, u_p_d_b, v_p_d_b);
c_p_prev = Cross_product_2v(u_p_o_b, v_p_o_b, u_p_d_b, v_p_d_b);
above_prev = c_p_prev > 0.0;
if(!above_prev)
break;
d_p_prev = Dot_product_2v(u_p_o_b, v_p_o_b, u_p_d_b, v_p_d_b);
cot_prev = d_p_prev / c_p_prev;
if(cot_prev > cot_r_cand)
break;
delete_edge(r_cand);
r_cand = prev;
cot_r_cand = cot_prev;
}while(TRUE);
}
dest_l_cand = Other_point(l_cand, org_base);
dest_r_cand = Other_point(r_cand, dest_base);
if(!above_l_cand || (above_l_cand && above_r_cand && cot_r_cand < cot_l_cand))
{
base = join(base, org_base, r_cand, dest_r_cand, right);
dest_base = dest_r_cand;
}
else
{
base = join(l_cand, dest_l_cand, base, dest_base, right);
org_base = dest_l_cand;
}
}while(TRUE);
}
void divide(point *p_sorted[], int l, int r, edge **l_ccw, edge **r_cw)
{
int n;
int split;
edge *l_ccw_l, *r_cw_l, *l_ccw_r, *r_cw_r, *l_tangent;
edge *a, *b, *c;
Real c_p;
n = r - l + 1;
if(n == 2)
{
*l_ccw = *r_cw = make_edge(p_sorted[l], p_sorted[r]);
}
else if(n == 3)
{
a = make_edge(p_sorted[l], p_sorted[l + 1]);
b = make_edge(p_sorted[l + 1], p_sorted[r]);
splice(a, b, p_sorted[l + 1]);
c_p = Cross_product_3p(p_sorted[l], p_sorted[l + 1], p_sorted[r]);
if(c_p > 0.0)
{
c = join(a, p_sorted[l], b, p_sorted[r], right);
*l_ccw = a;
*r_cw = b;
}
else if(c_p < 0.0)
{
c = join(a, p_sorted[l], b, p_sorted[r], left);
*l_ccw = c;
*r_cw = c;
}
else
{
*l_ccw = a;
*r_cw = b;
}
}
else if(n > 3)
{
split = (l + r) / 2;
divide(p_sorted, l, split, &l_ccw_l, &r_cw_l);
divide(p_sorted, split+1, r, &l_ccw_r, &r_cw_r);
merge(r_cw_l, p_sorted[split], l_ccw_r, p_sorted[split + 1], &l_tangent);
if(Org(l_tangent) == p_sorted[l])
l_ccw_l = l_tangent;
if(Dest(l_tangent) == p_sorted[r])
r_cw_r = l_tangent;
*l_ccw = l_ccw_l;
*r_cw = r_cw_r;
}
}
int main(void)
{
int ca;
for (scanf("%d", &ca); ca--;)
{
scanf("%d", &k);
edge *l_cw, *r_ccw;
int i;
point **p_sorted, **p_temp;
read_points();
if (k > n) k = n;
if (n == 1)
{
printf("0\n");
continue;
}
alloc_memory(n);
for (i = 0; i < n; i++)
{
p_array[i].x = px[i];
p_array[i].y = py[i];
}
for(i = 0; i < n; i++)
p_array[i].entry_pt = NULL;
p_sorted = (point **)malloc((unsigned)n * sizeof(point *));
p_temp = (point **)malloc((unsigned)n * sizeof(point *));
for(i = 0; i < n; i++)
p_sorted[i] = p_array + i;
merge_sort(p_sorted, p_temp, 0, n - 1);
free((char *)p_temp);
divide(p_sorted, 0, n - 1, &l_cw, &r_ccw);
free((char *)p_sorted);
en = 0;
print_edges(n);
kruskal();
free_memory();
}
return 0;
}