Proving Equivalences

Time Limit: 4000/2000 MS (Java/Others) Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 6006 Accepted Submission(s):
2051

Problem Description

Consider the following exercise, found in a generic
linear algebra textbook.

Let A be an n × n matrix. Prove that the
following statements are equivalent:

1. A is invertible.
2. Ax = b has
exactly one solution for every n × 1 matrix b.
3. Ax = b is consistent for
every n × 1 matrix b.
4. Ax = 0 has only the trivial solution x = 0.

The typical way to solve such an exercise is to show a series of
implications. For instance, one can proceed by showing that (a) implies (b),
that (b) implies (c), that (c) implies (d), and finally that (d) implies (a).
These four implications show that the four statements are
equivalent.

Another way would be to show that (a) is equivalent to (b)
(by proving that (a) implies (b) and that (b) implies (a)), that (b) is
equivalent to (c), and that (c) is equivalent to (d). However, this way requires
proving six implications, which is clearly a lot more work than just proving
four implications!

I have been given some similar tasks, and have already
started proving some implications. Now I wonder, how many more implications do I
have to prove? Can you help me determine this?

Input

On the first line one positive number: the number of
testcases, at most 100. After that per testcase:

* One line containing
two integers n (1 ≤ n ≤ 20000) and m (0 ≤ m ≤ 50000): the number of statements
and the number of implications that have already been proved.
* m lines with
two integers s1 and s2 (1 ≤ s1, s2 ≤ n and s1 ≠ s2) each, indicating that it has
been proved that statement s1 implies statement s2.

Output

Per testcase:

* One line with the minimum number
of additional implications that need to be proved in order to prove that all
statements are equivalent.

Sample Input

2
4 0
3 2
1 2
1 3

Sample Output

4
2

Source

NWERC
2008

和上题一样

PS：再次N写错

#include <iostream>

#include <cstdio>

#include <cstring>

#include <algorithm>

#include <cmath>

using namespace std;

const int N=2e4+,M=5e4+;

typedef long long ll;

inline int read(){

    char c=getchar();int x=,f=;

    while(c<''||c>''){if(c=='-')f=-;c=getchar();}

    while(c>=''&&c<=''){x=x*+c-'';c=getchar();}

    return x*f;

}

int n,m,u,v;

struct edge{

    int v,ne;

}e[M];

int h[N],cnt=;

inline void ins(int u,int v){

    cnt++;

    e[cnt].v=v;e[cnt].ne=h[u];h[u]=cnt;

}

int dfn[N],low[N],belong[N],dfc,scc;

int st[N],top=;

void dfs(int u){

    dfn[u]=low[u]=++dfc;

    st[++top]=u;

    for(int i=h[u];i;i=e[i].ne){

        int v=e[i].v;

        if(!dfn[v]){

            dfs(v);

            low[u]=min(low[u],low[v]);

        }else if(!belong[v])

            low[u]=min(low[u],dfn[v]);

    }

    if(low[u]==dfn[u]){

        scc++;

        while(true){

            int x=st[top--];

            belong[x]=scc;

            if(x==u) break;

        }

    }

}

void findSCC(){

    memset(dfn,,sizeof(dfn));

    memset(belong,,sizeof(belong));

    memset(low,,sizeof(low));

    dfc=scc=top=;

    for(int i=;i<=n;i++) if(!dfn[i]) dfs(i);

}

int outd[N],ind[N];

void point(){

    memset(ind,,sizeof(ind));

    memset(outd,,sizeof(outd));

    for(int u=;u<=n;u++)

        for(int i=h[u];i;i=e[i].ne){

            int v=e[i].v;

            if(belong[u]!=belong[v]) outd[belong[u]]++,ind[belong[v]]++;

        }

}

int T;

int main(){

    T=read();

    while(T--){

        n=read();m=read();

        cnt=;

        memset(h,,sizeof(h));

        for(int i=;i<=m;i++){u=read();v=read();ins(u,v);}

        findSCC();

        point();

        int cnt1=,cnt2=;

        for(int i=;i<=scc;i++){

            if(ind[i]==) cnt1++;

            if(outd[i]==) cnt2++;

        }

        if(scc==) printf("0\n");

        else printf("%d\n",max(cnt1,cnt2));

    }

}

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