CF715B. Complete The Graph

題意：

給一張 n 個點，m 條邊的無向圖，要求設定一些邊的邊權
使得所有邊權都是正整數，最終 S 到 T 的最短路為 L

1 ≤ n ≤ 1000, 1 ≤ m ≤ 10000

假做法：

spfa求s到t最短路且滿足可變邊最少

然后把不在最短路上的可變邊標為inf，最短路上的可變邊修改成使最短路長為L

假的原因：

其他的賦值為inf只是保證了經過其他可變邊的路徑不會更短，沒有保證不經過其他可變邊只是少經過了幾條可變邊、導致比最短路長的路徑不會在最短路修改后更短

存在繞過某條可變邊的路徑p，本來不是最短路且不經過最短路上可變邊x，但你修改x之后，p會變成當前最短路，于是這個做法就掛掉了。

就是說走了非可變邊，把你修改的那條可變邊繞過去了

修正：

真做法1：

應當選擇滿足d<L的路徑中經過可變邊最少的一條

可以在最短路上加維，\(d(i,j)\)表示1到i經過j條可變邊的最短路

復雜度\(O(mn\log{mn})\)

真做法2：

隨便求一條最短路，同樣其他賦值inf，然后枚舉最短路上的可變邊，依次修改改可變邊的值，修改后再求最短路看看會不會被繞過去。最后一定會收斂出答案

無解：不經過可變邊就可以<L，經過可變邊也比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:假做法的代碼

#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';
}