Q - Marriage Match IV
Do not sincere non-interference。 Like that show, now starvae also
take part in a show, but it take place between city A and B. Starvae
is in city A and girls are in city B. Every time starvae can get to
city B and make a data with a girl he likes. But there are two
problems with it, one is starvae must get to B within least time, it’s
said that he must take a shortest path. Other is no road can be taken
more than once. While the city starvae passed away can been taken more
than once.So, under a good RP, starvae may have many chances to get to city B.
But he don’t know how many chances at most he can make a data with the
girl he likes . Could you help starvae? Input The first line is an
integer T indicating the case number.(1<=T<=65) For each case,there
are two integer n and m in the first line ( 2<=n<=1000, 0<=m<=100000 )
,n is the number of the city and m is the number of the roads.Then follows m line ,each line have three integers
a,b,c,(1<=a,b<=n,0<c<=1000)it means there is a road from a to b and
it’s distance is c, while there may have no road from b to a. There
may have a road from a to a,but you can ignore it. If there are two
roads from a to b, they are different.At last is a line with two integer A and B(1<=A,B<=N,A!=B), means the
number of city A and city B. There may be some blank line between
each case. Output Output a line with a integer, means the chances
starvae can get at most. Sample Input
3
7 8
1 2 1
1 3 1
2 4 1
3 4 1
4 5 1
4 6 1
5 7 1
6 7 1
1 7
6 7
1 2 1
2 3 1
1 3 3
3 4 1
3 5 1
4 6 1
5 6 1
1 6
2 2
1 2 1
1 2 2
1 2
Sample Output
2
1
1
- 題意：一個人 從 城市 A 到 B 的最短路徑有幾條，
這裏特別需要注意：每條路經只能走一次，走過之後就不能再走了，而且只能走最短的路徑 - 思路：把不是最組成短路徑（
這裏 最短路可能有多條）點邊剔除掉，把剩餘的邊重新建圖，邊權設置爲1，跑一遍最大流。
那麼我們我現在要解決的問題是怎麼判斷某一條邊 是組成最短路徑的邊呢？
我們先做一些假設：
- 假設要判斷的邊是 （u ，v），其長度是 w（u，v），假設圖的 源點爲 s 、匯點爲 e。
- 正向跑最短路 的到的從 s 到其他點的最短距離存放在 dis1[ ] 數組中，
dis[ u ] 爲s到u的最短距離； - 逆向跑最短路（但是帶到權值還是 正向的權值） 的到的從 e 到其他點的最短距離存放在 dis2[ ] 數組中，dis[ v ] 爲
u到e (注意這個方向是u到v)的最短距離
- 最後我們只要在遍歷所給的每一條邊時：
如果dis1[ u ] + w(u, v) + dis2[ v ] = dis1[ e ] 成立。那麼我們就可判斷這條邊就是組成最短的路徑的邊。
最後把這些 邊新建圖跑最大流，就能得出 路徑方案數了。 - 其實剩下的我們還要考慮一下：爲什麼最大流跑出來的就是我們所要的 答案？？？？？？？？？
題解（Spfa + ISAP）
#include<iostream>
#include<queue>
#include<cstring>
#include<cstdio>
#include<algorithm>
using namespace std;
#define INF 0x3f3f3f3f
const int maxn = 10005;
const int maxm = 200005;
struct Edge
{
int v,w,next;
} edge1[maxm], edge2[maxm], edge[maxm];
int n,m,s,e;
int head1[maxn], head2[maxn], head[maxn];
int dis1[maxn], dis2[maxn];
int use[maxn];
int k1,k2,k;
void Add(int u, int v, int w, int head[], int & k, Edge edge[])
{
edge[++ k] = (Edge){ v, w, head[u]}; head[u] = k;
}
void Spfa(int s, int dis[], int head[], Edge edge[])
{
for(int i = 1; i <= n; i ++)
dis[i] = INF,use[i] = 0;
dis[s] = 0;
queue<int> q;
q.push(s);
int u,v,w;
while(! q.empty())
{
u = q.front();
q.pop();
use[u] = 0;
for(int i = head[u]; i != -1; i = edge[i].next)
{
v = edge[i].v;
w = edge[i].w;
if(dis[v] > dis[u] + w)
{
dis[v] = dis[u] + w;
if(! use[v])
{
q.push(v);
use[v] = 1;
}
}
}
}
}
int deep[maxn], num[maxn];
int cur[maxn], last[maxm];
void bfs(int e)
{
for(int i = 0; i <= n; i ++)
deep[i] = n, cur[i] = head[i], use[i] = 0;
deep[e] = 0;
queue<int> q;
q.push(e);
int u, v;
while(! q.empty())
{
u = q.front(); q.pop();
// use[u] = 0;
for(int i = head[u]; i != -1; i = edge[i].next)
{
v = edge[i].v;
if(edge[i^1].w && deep[v] == n) //正圖 邊存在 且 v這個節點沒有被求過
{
deep[v] = deep[u] + 1;
q.push(v);
// if(! use[v])
// {
// q.push(v);
// use[v] = 1;
// }
}
}
}
}
int Add_flow(int s, int e)
{
int ans = INF;
int now = e;
while(now != s)
{
ans = min(ans, edge[last[now]].w);
now = edge[last[now]^1].v;
}
now = e;
while(now != s)
{
edge[last[now]].w -= ans;
edge[last[now]^1].w += ans;
now = edge[last[now]^1].v;
}
return ans;
}
int isap(int s, int e)
{
int now = s; //從起點開始進行操作
bfs(e); //先找出來一條邊 被操作的增光路
for(int i = 1; i <= n; i ++) num[deep[i]] ++;
int mx_flw = 0;
while(deep[s] < n)
{
if(now == e) //如果到達匯點直接增廣，重新回到源點進行下一輪增廣
{
mx_flw += Add_flow(s, e);
now = s;
}
bool has_find = 0;
for(int i = cur[now]; i != -1; i = edge[i].next)
{
if(edge[i].w && deep[now] == deep[edge[i].v] + 1)
{
has_find = 1; //做標記已經找到一種可行路徑
cur[now] = i; //優化當前弧
now = edge[i].v;
last[edge[i].v] = i;
break;
}
}
if(! has_find)
{
int minn = n - 1;
for(int i = head[now]; i != -1; i = edge[i].next)
if(edge[i].w)
minn = min(minn, deep[edge[i].v]);
if( (-- num[deep[now]]) == 0) break; //gap 優化出現了斷層
num[deep[now] = minn + 1] ++;
cur[now] = head[now];
if(now != s)
now = edge[last[now]^1].v;
}
}
return mx_flw;
}
void init()
{
k1 = 0; k2 = 0; k = -1;
for(int i = 0; i <= n; i ++)
head1[i] = -1, head2[i] = -1, head[i] = -1;
memset(num, 0, sizeof(num));
}
int main()
{
//freopen("T.txt","r",stdin);
int t;
scanf("%d", &t);
while(t --)
{
scanf("%d %d", &n, &m);
init();
int u, v, w;
for(int i = 1; i <= m; i ++)
{
scanf("%d %d %d", &u, &v, &w);
Add(u, v, w, head1, k1, edge1);
Add(v, u, w, head2, k2, edge2);
}
scanf("%d %d", &s, &e);
Spfa(s, dis1, head1, edge1);
Spfa(e, dis2, head2, edge2);
//遍歷圖中所有的邊 去找組成所有最短了的邊都有哪些
for(int i = 1; i <= m; i ++)
{
u = edge2[i].v;
v = edge1[i].v;
w = edge1[i].w;
if(dis1[u] + w + dis2[v] == dis1[e])
{
Add(u, v, 1, head, k, edge);
Add(v, u, 0, head, k, edge);
}
}
printf("%d\n", isap(s, e));
}
reurn 0;
}