CF715B. Complete The Graph

题意：

给一张 n 个点，m 条边的无向图，要求设定一些边的边权
使得所有边权都是正整数，最终 S 到 T 的最短路为 L

1 ≤ n ≤ 1000, 1 ≤ m ≤ 10000

假做法：

spfa求s到t最短路且满足可变边最少

然后把不在最短路上的可变边标为inf，最短路上的可变边修改成使最短路长为L

假的原因：

其他的赋值为inf只是保证了经过其他可变边的路径不会更短，没有保证不经过其他可变边只是少经过了几条可变边、导致比最短路长的路径不会在最短路修改后更短

存在绕过某条可变边的路径p，本来不是最短路且不经过最短路上可变边x，但你修改x之后，p会变成当前最短路，于是这个做法就挂掉了。

就是说走了非可变边，把你修改的那条可变边绕过去了

修正：

真做法1：

应当选择满足d<L的路径中经过可变边最少的一条

可以在最短路上加维，\(d(i,j)\)表示1到i经过j条可变边的最短路

复杂度\(O(mn\log{mn})\)

真做法2：

随便求一条最短路，同样其他赋值inf，然后枚举最短路上的可变边，依次修改改可变边的值，修改后再求最短路看看会不会被绕过去。最后一定会收敛出答案

无解：不经过可变边就可以<L，经过可变边也比L大

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cstring>
#include <queue>
#include <vector>
using namespace std;
typedef long long ll;
const int N = 1e5+5, M = 1e6+5;
const ll inf = 1e16;int n, m, L, s, t;
struct edge {int u, v, ne; ll w;} e[M];
struct meow {int u, v; ll w;} a[M];
int cnt=1, h[N], mark[M];
inline void ins(int u, int v, ll w) {if(w == 0) w=1, mark[cnt+1] = mark[cnt+2] = 1;e[++cnt] = (edge) {u, v, h[u], w}; h[u] = cnt;e[++cnt] = (edge) {v, u, h[v], w}; h[v] = cnt;a[cnt>>1] = (meow) {u, v, w};
}
inline void paint(int x, ll v) {a[x].w = v;e[x<<1].w = e[x<<1|1].w = v;
}ll d[N];
int inq[N], pre[N];
int q[N], head, tail;
inline void lop(int &x) {if(x==N) x = 1;}
void spfa0(int s) {memset(d, 0x3f, sizeof(d));head = tail = 1;d[s] = 0; q[tail++] = s;  inq[s] = 1;while(head != tail) {int u = q[head++]; lop(head); inq[u] = 0;for(int i=h[u]; i; i=e[i].ne) if(!mark[i]) {int v = e[i].v;if(d[v] > d[u] + e[i].w) {d[v] = d[u] + e[i].w;if(!inq[v]) q[tail++] = v, lop(tail), inq[v] = 1;}}}
}void spfa(int s) {memset(d, 0x3f, sizeof(d));memset(inq, 0, sizeof(inq));head = tail = 1;d[s] = 0; q[tail++] = s;  inq[s] = 1;while(head != tail) {int u = q[head++]; lop(head); inq[u] = 0;for(int i=h[u]; i; i=e[i].ne) {int v = e[i].v;if(d[v] > d[u] + e[i].w ) {d[v] = d[u] + e[i].w;pre[v] = i;if(!inq[v]) q[tail++] = v, lop(tail), inq[v] = 1;}}}
}
int chose[N];
vector<int> li;
int main() {//freopen("in", "r", stdin);ios::sync_with_stdio(false); cin.tie(); cout.tie();cin >> n >> m >> L >> s >> t;s++; t++;for(int i=1; i<=m; i++) {int u, v, w;cin >> u >> v >> w;u++; v++;ins(u, v, w);}spfa0(s);if(d[t] < L) {cout << "NO"; return 0;}spfa(s);if(d[t] > L) {cout << "NO"; return 0;}if(d[t] == L) {cout << "YES" << endl;for(int i=1; i <= cnt>>1; i++) cout << a[i].u-1 << ' ' << a[i].v-1 << ' ' << a[i].w << '\n';return 0;}int x = t;while(x != s) {if(mark[pre[x]]) chose[pre[x]>>1] = 1, li.push_back(pre[x]>>1);x = e[pre[x]].u;}for(int i=1; i<=cnt>>1; i++) if(mark[i<<1] && !chose[i]) paint(i, inf);for(int i=0; i<li.size(); i++) {int now = li[i];int delta = L - d[t] + 1;paint(now, delta);spfa(s);if(d[t] == L) break;}cout << "YES" << endl;for(int i=1; i <= cnt>>1; i++) cout << a[i].u-1 << ' ' << a[i].v-1 << ' ' << a[i].w << '\n';
}

ps:假做法的代码

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cstring>
#include <queue>
using namespace std;
typedef long long ll;
const int N = 1e5+5, M = 1e6+5;
const ll inf = 1e16;int n, m, L, s, t;
struct edge {int u, v, ne; ll w;} e[M];
struct meow {int u, v; ll w;} a[M];
int cnt=1, h[N], mark[M];
inline void ins(int u, int v, ll w) {if(w == 0) w=1, mark[cnt+1] = mark[cnt+2] = 1;e[++cnt] = (edge) {u, v, h[u], w}; h[u] = cnt;e[++cnt] = (edge) {v, u, h[v], w}; h[v] = cnt;a[cnt>>1] = (meow) {u, v, w};
}ll d[N];
int inq[N], cou[N], pre[N];
int q[N], head, tail;
inline void lop(int &x) {if(x==N) x = 1;}
void spfa0(int s) {memset(d, 0x3f, sizeof(d));head = tail = 1;d[s] = 0; q[tail++] = s;  inq[s] = 1;while(head != tail) {int u = q[head++]; lop(head); inq[u] = 0;for(int i=h[u]; i; i=e[i].ne) if(!mark[i]) {int v = e[i].v;if(d[v] > d[u] + e[i].w) {d[v] = d[u] + e[i].w;if(!inq[v]) q[tail++] = v, lop(tail), inq[v] = 1;}}}
}
namespace test {int pre[N];
void spfa1(int s) {memset(d, 0x3f, sizeof(d));head = tail = 1;d[s] = 0; q[tail++] = s;  inq[s] = 1;while(head != tail) {int u = q[head++]; lop(head); inq[u] = 0;for(int i=h[u]; i; i=e[i].ne) {int v = e[i].v;if(d[v] > d[u] + e[i].w) {d[v] = d[u] + e[i].w;pre[v] = i;if(!inq[v]) q[tail++] = v, lop(tail), inq[v] = 1;}}}
}
}
void spfa(int s) {memset(d, 0x3f, sizeof(d));memset(cou, 0x3f, sizeof(cou));memset(inq, 0, sizeof(inq));head = tail = 1;d[s] = 0; cou[s] = 0; q[tail++] = s;  inq[s] = 1;while(head != tail) {int u = q[head++]; lop(head); inq[u] = 0;for(int i=h[u]; i; i=e[i].ne) {int v = e[i].v;if(d[v] > d[u] + e[i].w || (d[v] == d[u]+e[i].w && cou[v] > cou[u] + mark[i])) {d[v] = d[u] + e[i].w;cou[v] = cou[u] + mark[i];pre[v] = i;if(!inq[v]) q[tail++] = v, lop(tail), inq[v] = 1;}}}
}
int chose[N];
int main() {//freopen("in", "r", stdin);ios::sync_with_stdio(false); cin.tie(); cout.tie();cin >> n >> m >> L >> s >> t;s++; t++;for(int i=1; i<=m; i++) {int u, v, w;cin >> u >> v >> w;u++; v++;ins(u, v, w);}spfa0(s);if(d[t] < L) {cout << "NO"; return 0;}spfa(s);if(d[t] > L) {cout << "NO"; return 0;}if(d[t] == L) {cout << "YES" << endl;for(int i=1; i <= cnt>>1; i++) cout << a[i].u-1 << ' ' << a[i].v-1 << ' ' << a[i].w << '\n';return 0;}int x = t;int flag = 0;while(x != s) {if(mark[pre[x]]) chose[pre[x]>>1] = 1, flag = pre[x] >> 1;x = e[pre[x]].u;}if(!flag) {cout << "NO"; return 0;}for(int i=1; i <= cnt>>1; i++) if(mark[i<<1] && !chose[i]) a[i].w = inf;for(int i=1; i<=cnt; i++) if(mark[i] && !chose[i>>1]) e[i].w = inf;int delta = L - d[t] + 1;a[flag].w = delta; e[flag<<1].w = e[flag<<1|1].w = delta;test::spfa1(s);if(d[t] != L) {cout << d[t] << "nooooo\n";int x = t;while(x != s) {if(test::pre[x] != pre[x]) {cout << "wrong\n";cout << mark[pre[x]] << "   " << mark[test::pre[x]] << '\n';}x = e[pre[x]].u;}}cout << "YES" << endl;for(int i=1; i <= cnt>>1; i++) cout << a[i].u-1 << ' ' << a[i].v-1 << ' ' << a[i].w << '\n';
}