复习计划-强连通分量

于 2020-09-15 20:17:27 发布
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分类专栏： 复习计划 联通图
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本文链接：https://blog.youkuaiyun.com/Zenith_Habitant/article/details/108608496
版权
  • 强连通分量
  • P1262 SCC缩点+BFS+贪心
//#define LOCAL
#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define mem(a, b) memset(a,b,sizeof(a))
#define sz(a) (int)a.size()
#define INF 0x3f3f3f3f
#define DNF 0x7f
#define DBG printf("this is a input\n")
#define fi first
#define se second
#define mk(a, b) make_pair(a,b)
#define pb push_back
#define LF putchar('\n')
#define SP putchar(' ')
#define p_queue priority_queue
#define CLOSE ios::sync_with_stdio(0); cin.tie(0)

template<typename T>
void read(T &x) {x = 0;char ch = getchar();ll f = 1;while(!isdigit(ch)){if(ch == '-')f *= -1;ch = getchar();}while(isdigit(ch)){x = x * 10 + ch - 48; ch = getchar();}x *= f;}
template<typename T, typename... Args>
void read(T &first, Args& ... args) {read(first);read(args...);}
template<typename T>
void write(T arg) {T x = arg;if(x < 0) {putchar('-'); x =- x;}if(x > 9) {write(x / 10);}putchar(x % 10 + '0');}
template<typename T, typename ... Ts>
void write(T arg, Ts ... args) {write(arg);if(sizeof...(args) != 0) {putchar(' ');write(args ...);}}
using namespace std;

ll gcd(ll a, ll b) {
    return b == 0 ? a : gcd(b, a % b);
}

ll lcm(ll a, ll b) {
    return a / gcd(a, b) * b;
}
const int N = 2e5 + 5;
int n, m;
int val[N], nval[N];
int head[N], cnt;
struct node {
    int t, next;
}edge[N << 1];
void add (int f, int t)
{
    edge[cnt].t = t;
    edge[cnt].next = head[f];
    head[f] = cnt++;
}
int times , tot ;
int dfn[N] , low[N] , vis[N] , st[N] , scc[N] , type = 0, ed[N];
void tarjan (int u)
{
    dfn[u] = low[u] = ++ times;
    st[++ tot] = u;
    vis[u] = 1;
    for (int i = head[u] ; i != -1 ; i = edge[i].next)
    {
        int v = edge[i].t;
        if (!dfn[v])
        {
            tarjan(v);
            low[u] = min (low[u], low[v]);
        }
        else if (vis[v])
            low[u] = min (low[u] , dfn[v]);
    }
    if (low[u] == dfn[u])
    {
        type ++;
        do {
           scc[st[tot]] = type;
           vis[st[tot]] = 0;
           tot --;
        }while (u != st[tot + 1]);
    }
    return ;

}
int in[N];
vector <int> eg[N];
queue <int> q;
void bfs (int u)
{
    q.push(u);
    ed[u] = 1;
    while (!q.empty())
    {
        int no = q.front();
        q.pop();
        for (int i = head[no] ; i != -1 ; i = edge[i].next)
        {
            int v = edge[i].t;
            if (!ed[v])
            {
                ed[v] = 1;
                q.push(v);
            }
        }
    }
}
int main (void)
{
    read (n, m);
    mem(head,-1);
    for (int i = 1; i <= n ; i ++) {
        nval[i] = val[i] = INF;
    }
    for (int i = 1 ; i <= m ; i ++)
    {
        int x , v;
        read (x, v);
        val[x] = min(val[x], v);
    }
    read (m);
    for (int i = 1 ; i <= m ; i ++)
    {
        int u , v;
        read (u, v);
        add (u, v);
    }
    for (int i = 1 ; i <= n ; i ++)
    {
        if (val[i] != INF && !ed[i])
             bfs (i);
    }

    for (int i = 1 ; i <= n ;  i++) {
        if (!dfn[i])
            tarjan(i);
    }
    for (int u = 1 ; u <= n ; u ++)
    {
        for (int j = head[u] ; j != -1 ; j = edge[j].next)
        {
            int v = edge[j].t;
            if (scc[u] != scc[v])
                in[scc[v]] ++;
        }
    }
    ll sum = 0;
    for (int i = 1 ; i <= n ; i ++)
        nval[scc[i]] = min (val[i], nval[scc[i]]);
    for (int i = 1 ; i <= type ; i ++)
        if (!in[i])
            sum += nval[i];

    int id = 0;
    for (int i = 1 ;i <= n ; i ++)
        if (!ed[i])
        {
            id = i;
            break;
        }
    if (id)
    {
        cout << "NO" << endl;
        cout << id << endl;
    }
    else
    {
        cout << "YES" << endl;
        cout << sum << endl;
    }

}
  • P1407
  • 强连通分量+建图
//#define LOCAL
#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define mem(a, b) memset(a,b,sizeof(a))
#define sz(a) (int)a.size()
#define INF 0x3f3f3f3f
#define DNF 0x7f
#define DBG printf("this is a input\n")
#define fi first
#define se second
#define mk(a, b) make_pair(a,b)
#define pb push_back
#define LF putchar('\n')
#define SP putchar(' ')
#define p_queue priority_queue
#define CLOSE ios::sync_with_stdio(0); cin.tie(0)

template<typename T>
void read(T &x) {x = 0;char ch = getchar();ll f = 1;while(!isdigit(ch)){if(ch == '-')f *= -1;ch = getchar();}while(isdigit(ch)){x = x * 10 + ch - 48; ch = getchar();}x *= f;}
template<typename T, typename... Args>
void read(T &first, Args& ... args) {read(first);read(args...);}
template<typename T>
void write(T arg) {T x = arg;if(x < 0) {putchar('-'); x =- x;}if(x > 9) {write(x / 10);}putchar(x % 10 + '0');}
template<typename T, typename ... Ts>
void write(T arg, Ts ... args) {write(arg);if(sizeof...(args) != 0) {putchar(' ');write(args ...);}}
using namespace std;

ll gcd(ll a, ll b) {
    return b == 0 ? a : gcd(b, a % b);
}

ll lcm(ll a, ll b) {
    return a / gcd(a, b) * b;
}
const int N = 2e5 + 5;
int n, m;
int head[N], cnt;
struct node {
    int t, next;
}edge[N << 1];
void add (int f, int t)
{
    edge[cnt].t = t;
    edge[cnt].next = head[f];
    head[f] = cnt++;
}
string a , b;
map <string, int> ma;
int times , tot ;
int dfn[N] , low[N] , vis[N] , st[N] , scc[N] , type = 0;
int buck[N];
void tarjan (int u)
{
    dfn[u] = low[u] = ++ times;
    st[++ tot] = u;
    vis[u] = 1;
    for (int i = head[u] ; i != -1 ; i = edge[i].next)
    {
        int v = edge[i].t;
        if (!dfn[v])
        {
            tarjan(v);
            low[u] = min (low[u], low[v]);
        }
        else if (vis[v])
            low[u] = min (low[u] , dfn[v]);
    }
    if (low[u] == dfn[u])
    {
        type ++;
        do {
            scc[st[tot]] = type;
            vis[st[tot]] = 0;
            tot --;
        }while (u != st[tot + 1]);
    }
    return ;

}
int main (void)
{
    read (n);
    mem(head,-1);
    for (int i = 1 ; i <= n ; i ++)
    {
        cin >> a >> b;
        if (!ma[a])
            ma[a] = ++tot;
        if (!ma[b])
            ma[b] = ++tot;
        add (ma[a], ma[b]);
        //cout << ma[a] << ' ' << ma[b] << endl;
    }
    read (m);
    for (int i = 1 ; i <= m ; i ++)
    {
        cin >> a >> b;
        add (ma[b] , ma[a]);
        //cout << ma[b] << ' ' << ma[a] << endl;
    }
    for (int i = 1 ; i <= 2*n ; i += 2)
    {
        if (!dfn[i])
            tarjan (i);
    }
    for (int i = 1 ; i <= 2*n ; i += 2)
        buck[scc[i]] ++;
    for (int i = 1 ; i <= 2*n ; i += 2)
    {
        if (buck[scc[i]] == 1)
            cout << "Safe" << endl;
        else
            cout << "Unsafe" << endl;

    }

}
  • [JSOI2010]连通数
  • 反图+SCC+Topo+状态枚举
#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define mem(a, b) memset(a,b,sizeof(a))
#define sz(a) (int)a.size()
#define INF 0x3f3f3f3f
#define DNF 0x7f
#define DBG printf("this is a input\n")
#define fi first
#define se second
#define mk(a, b) make_pair(a,b)
#define pb push_back
#define LF putchar('\n')
#define SP putchar(' ')
#define p_queue priority_queue
#define CLOSE ios::sync_with_stdio(0); cin.tie(0)

template<typename T>
void read(T &x) {x = 0;char ch = getchar();ll f = 1;while(!isdigit(ch)){if(ch == '-')f *= -1;ch = getchar();}while(isdigit(ch)){x = x * 10 + ch - 48; ch = getchar();}x *= f;}
template<typename T, typename... Args>
void read(T &first, Args& ... args) {read(first);read(args...);}
template<typename T>
void write(T arg) {T x = arg;if(x < 0) {putchar('-'); x =- x;}if(x > 9) {write(x / 10);}putchar(x % 10 + '0');}
template<typename T, typename ... Ts>
void write(T arg, Ts ... args) {write(arg);if(sizeof...(args) != 0) {putchar(' ');write(args ...);}}
using namespace std;

ll gcd(ll a, ll b) {
    return b == 0 ? a : gcd(b, a % b);
}

ll lcm(ll a, ll b) {
    return a / gcd(a, b) * b;
}
const int N = 3005;
map <pair <int,int> , int > ma;
int n, m;
int head[N], cnt;
bitset <N> bs[N];
struct node {
    int t, next;
}edge[N*N];
void add (int f, int t)
{
    edge[cnt].t = t;
    edge[cnt].next = head[f];
    head[f] = cnt++;

}
char ch[N];
int times , tot , in[N];
int dfn[N] , low[N] , vis[N] , st[N] , scc[N] , type = 0;
void tarjan (int u)
{
    dfn[u] = low[u] = ++ times;
    st[++ tot] = u;
    vis[u] = 1;
    for (int i = head[u] ; i != -1 ; i = edge[i].next)
    {
        int v = edge[i].t;
        if (!dfn[v])
        {
            tarjan(v);
            low[u] = min (low[u], low[v]);
        }
        else if (vis[v])
            low[u] = min (low[u] , dfn[v]);
    }
    if (low[u] == dfn[u])
    {
        type ++;
        do {
            scc[st[tot]] = type;
            vis[st[tot]] = 0;
            tot --;
        }while (u != st[tot + 1]);
        bs[type][type] = 1;
    }
    return ;

}
vector <int> eg[N];
int pp[N];
void topo()
{
    queue <int> q;
    for (int i = 1 ; i <= type ; i ++) {
        if (!in[i])
            q.push(i);
    }
    while (!q.empty())
    {
        int no = q.front();
        q.pop();
        for (auto v : eg[no])
        {
            bs[v] |= bs[no];
            in[v] --;
            if (!in[v]) {
                q.push(v);
            }
        }
    }
}
int main (void)
{
    scanf("%d",&n);
    mem(head,-1);
    for (int i = 1 ; i <= n ; i ++)
    {
        scanf("%s", ch);
        for (int j = 0 ; j < n ; j ++)
            if (ch[j] == '1') {
                add(j + 1, i);
            }
    }
    for (int i = 1 ; i <= n ; i ++)
        if (!dfn[i])
            tarjan(i);
    for (int u = 1 ; u <= n ; u ++)
    {
        for (int j = head[u] ; j != -1 ; j = edge[j].next)
        {
            int v = edge[j].t;
            if (scc[u] != scc[v])
                eg[scc[u]].pb(scc[v]), in[scc[v]] ++;
        }
    }
    for (int i = 1 ; i <= n ; i ++)
        pp[scc[i]] ++;

    topo();
    ll ans = 0 ;
    for (int u = 1 ; u <= type ; u ++)
    {
        for (int v = 1 ; v <= type ; v ++)
        {
            if (bs[u][v])
                ans += pp[u] * pp[v];
        }
    }
    printf("%lld\n",ans);
}
/*
5
01100
00101
00011
00000
00000
 */
  • [APIO2009]抢掠计划
  • SCC + DP + Topo
#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define mem(a, b) memset(a,b,sizeof(a))
#define sz(a) (int)a.size()
#define INF 0x3f3f3f3f
#define DNF 0x7f
#define DBG printf("this is a input\n")
#define fi first
#define se second
#define mk(a, b) make_pair(a,b)
#define pb push_back
#define LF putchar('\n')
#define SP putchar(' ')
#define p_queue priority_queue
#define CLOSE ios::sync_with_stdio(0); cin.tie(0)

template<typename T>
void read(T &x) {x = 0;char ch = getchar();ll f = 1;while(!isdigit(ch)){if(ch == '-')f *= -1;ch = getchar();}while(isdigit(ch)){x = x * 10 + ch - 48; ch = getchar();}x *= f;}
template<typename T, typename... Args>
void read(T &first, Args& ... args) {read(first);read(args...);}
template<typename T>
void write(T arg) {T x = arg;if(x < 0) {putchar('-'); x =- x;}if(x > 9) {write(x / 10);}putchar(x % 10 + '0');}
template<typename T, typename ... Ts>
void write(T arg, Ts ... args) {write(arg);if(sizeof...(args) != 0) {putchar(' ');write(args ...);}}
using namespace std;

ll gcd(ll a, ll b) {
    return b == 0 ? a : gcd(b, a % b);
}

ll lcm(ll a, ll b) {
    return a / gcd(a, b) * b;
}
const int N = 1e6 + 5;
int n, m, x, S, T[N];
int head[N], cnt;
struct node {
    int t, next;
}edge[N];
void add (int f, int t)
{
    edge[cnt].t = t;
    edge[cnt].next = head[f];
    head[f] = cnt++;
}
char ch[N];
int times , tot , in[N];
ll val[N], nval[N];
int dfn[N] , low[N] , vis[N] , st[N] , scc[N] , type = 0;
void tarjan (int u)
{
    dfn[u] = low[u] = ++ times;
    st[++ tot] = u;
    vis[u] = 1;
    for (int i = head[u] ; i != -1 ; i = edge[i].next)
    {
        int v = edge[i].t;
        if (!dfn[v])
        {
            tarjan(v);
            low[u] = min (low[u], low[v]);
        }
        else if (vis[v])
            low[u] = min (low[u] , dfn[v]);
    }
    if (low[u] == dfn[u])
    {
        type ++;
        do {
            scc[st[tot]] = type;
            vis[st[tot]] = 0;
            tot --;
        }while (u != st[tot + 1]);
    }
    return ;

}
vector <int> eg[N];
ll dp[N];
void topo()
{
	queue <int> q;
	q.push(scc[S]);
	dp[scc[S]] = nval[scc[S]];
	while (!q.empty())
	{
		int u = q.front();
		q.pop();
		for (auto v : eg[u])
		{
			dp[v] = max(dp[v], dp[u] + nval[v]);
			in[v] --;
			if (!in[v])
				q.push(v);
		}
	}
}
int main (void)
{
	read (n, m);
	mem (head,-1);
	for (int i = 1 ; i <= m ; i ++)
	{
		int u , v;
		read (u , v);
		add (u , v);
	}
	for (int i = 1 ; i <= n ; i ++)
		read (val[i]);

	read (S, x);

	for (int i = 1 ; i <= x ; i ++)
		read (T[i]);

	tarjan(S);

	for (int u = 1 ; u <= n ; u ++)
	{
		for (int j = head[u] ; j != -1 ; j = edge[j].next)
		{
			if (!dfn[u]) continue;
			int v = edge[j].t;
			if (scc[u] != scc[v]){
				eg[scc[u]].pb(scc[v]) , in[scc[v]] ++;
			}
		}
	}

	for (int i = 1 ; i <= n ; i ++)
		nval[scc[i]] += val[i];

	for (int u = 1 ; u <= type ; u ++)
	{
		if (scc[S] != u && !in[u])
		{
			for (auto v : eg[u])
				in[v] --;
		}
	}

	ll ans = 0;
	topo();
	for (int i = 1 ; i <= x ; i ++)
		ans = max(ans, dp[scc[T[i]]]);
	write(ans), LF;
}
/*
5
01100
00101
00011
00000
00000
 */
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