模型建立
一般题目中会明确告诉你要求第二短路或者次短路
思路
主要思路一共有两种,一种为记录路径然后删边,另一种是开一个dis2记录次短路.
暴力删边
好像没什么可讲的
以洛谷P1491 集合位置为例
直接上代码吧..
#include <bits/stdc++.h>
#define ll long long
#define N 100010
#define M 210
using namespace std;
int n, m, add_edge; bool bian[M][M];
double px[M], py[M], dis[M];
int head[M * M * 2], from[M], vis[M];
struct CCC {
int next, to; double dis;
}edge[M * M * 2];
struct node {
int x; double dis;
bool operator < (const node &b) const {
return dis > b.dis;
}
};
int read() {
int s = 0, f = 0; char ch = getchar();
while (!isdigit(ch)) f |= (ch == '-'), ch = getchar();
while (isdigit(ch)) s = s * 10 + (ch ^ 48), ch = getchar();
return f ? -s : s;
}
double distant(double x, double y, double a, double b) {
return sqrt((x - a) * (x - a) + (y - b) * (y - b));
}
void add(int from, int to, double dis) {
edge[++add_edge].next = head[from];
edge[add_edge].to = to;
edge[add_edge].dis = dis;
head[from] = add_edge;
}
void dijkstra(int sy) {
for (int i = 1; i <= n; i++) dis[i] = 444444444444.00;
memset(vis, 0, sizeof vis);
priority_queue<node> q;
q.push((node){1, 0}), dis[1] = 0;
while (!q.empty()) {
node fr = q.top(); q.pop();
int x = fr.x;
if (vis[x]) continue;
vis[x] = 1;
for (int i = head[x]; i; i = edge[i].next) {
int to = edge[i].to;
if (bian[x][to] || bian[to][x]) continue;
if (!vis[to] && dis[to] > dis[x] + edge[i].dis) {
dis[to] = dis[x] + edge[i].dis;
if (sy) from[to] = x;
q.push((node){to, dis[to]});
}
}
}
}
int main() {
n = read(), m = read();
for (int i = 1; i <= n; i++)
px[i] = read(), py[i] = read();
for (int i = 1, x, y; i <= m; i++) {
x = read(), y = read();
double d = distant(px[x], py[x], px[y], py[y]);
add(x, y, d), add(y, x, d);
}
dijkstra(1);
double an = dis[n];
int sy = n; double ans = 444444444444.00;
while (sy != 1) {
bian[from[sy]][sy] = 1, bian[sy][from[sy]] = 1;
dijkstra(0);
if (dis[n] > an)
ans = min(ans, dis[n]);
bian[from[sy]][sy] = 0, bian[sy][from[sy]] = 0;
sy = from[sy];
}
if (ans == 444444444444.00) puts("-1");
else printf("%.2lf
", ans);
}
the other
以洛谷P2865 [USACO06NOV]Roadblocks G为例
这个题以为有重边,然后暴力删边操作之后只有90分就很难受.
在用(dijkstra)时候,就直接更新次短路,用(dis[])代表最短路,用(dis2[])代表次短路,
然后在松弛的时候同时更新两个数组,要判断三个条件
((u)是当前考虑的点,(v)是与(u)有边相连的点,(d(u,v))表示从(u)到(v)的边长)
(1.)如果(dis[v]>dis[u]+d(u,v)),则更新(dis[v])
(2.)如果(dis[v]<dis[u]+d(u,v))(不能取等,否则(dis2[v])和(dis[v])可能相等)
且(dis2[v]>dis[u]+d(u,v)),则更新(dis2[v])
(3.)如果(dis2[v]>dis2[u]+d(u,v)),则更新(dis2[v])(显然,如果2成立,
更新后(dis2[v]=dis[u]+d(u,v)<dis2[u]+d(u,v)),即(3)一定不成立)
如果上述三个条件中有任意一个成立,则将v入队。
code
#include <bits/stdc++.h>
#define ll long long
#define N 100010
#define M 5010
using namespace std;
int n, m;
int head[N << 1], add_edge, nx, ny, ndis;
int dis[M], dis2[M], vis[M], from[M], bian[M][M];
struct CCC {
int next, to, dis;
}edge[N << 1];
struct node {
int x, dis;
bool operator < (const node &b) const {
return dis > b.dis;
}
};
int read() {
int s = 0, f = 0; char ch = getchar();
while (!isdigit(ch)) f |= (ch == '-'), ch = getchar();
while (isdigit(ch)) s = s * 10 + (ch ^ 48), ch = getchar();
return f ? -s : s;
}
void add(int from, int to, int dis) {
edge[++add_edge].next = head[from];
edge[add_edge].to = to;
edge[add_edge].dis = dis;
head[from] = add_edge;
}
void dijkstra(int sy) {
memset(dis, 0x3f, sizeof dis);
memset(dis2, 0x3f, sizeof dis2);
priority_queue<node> q;
q.push((node){1, 0}); dis[1] = 0;
while (!q.empty()) {
node fr = q.top(); q.pop();
int x = fr.x;
for (int i = head[x]; i; i = edge[i].next) {
int to = edge[i].to;
if (dis[to] >= dis[x] + edge[i].dis) {
dis[to] = dis[x] + edge[i].dis;
q.push((node){to, dis[to]});
} else if (dis2[to] > dis[x] + edge[i].dis) {
dis2[to] = dis[x] + edge[i].dis;
q.push((node){to, dis[to]});
}
if (dis2[to] > dis2[x] + edge[i].dis)
dis2[to] = dis2[x] + edge[i].dis;
}
}
}
int main() {
n = read(), m = read();
for (int i = 1, x, y, d; i <= m; i++) {
x = read(), y = read(), d = read();
add(x, y, d), add(y, x, d);
if (!bian[x][y]) bian[x][y] = d, bian[y][x] = d;
else bian[x][y] = min(bian[x][y], d), bian[y][x] = bian[x][y];
}
dijkstra(1);
cout << dis2[n] << "
";
}