FUNDAMENTAL PART3 搜索与图论

搜索与图论

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1.朴素版dijkstra （适用于稠密图）

#include <iostream>
#include <cstring>
#include <algorithm>
#include <cstdio>

using namespace std;

const int N = 510;

int g[N][N], dist[N];
int n, m;
bool st[N];

int dijkstra()
{
    memset(dist, 0x3f, sizeof dist);
    dist[1] = 0;
    
    for (int i = 0; i < n - 1; i ++ )
    {
        int t = -1;
        for (int j = 1; j <= n; j ++ )
            if(!st[j] && (t == -1 || dist[t] > dist[j]))
                t = j;
            
        for (int j = 1; j <= n; j ++ )
            dist[j] = min(dist[j], dist[t] + g[t][j]);
        
        st[t] = true;
    }
    
    if(dist[n] == 0x3f3f3f3f) return -1;
    else return dist[n];
}

int main()
{
    memset(g, 0x3f, sizeof g);
    
    cin >> n >> m;
    
    while ( m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        
        g[a][b] = min(g[a][b], c);
    }
    
    printf("%d\n", dijkstra());
    
    return 0;
}

2.堆优化版的dijkstra （适用于稀疏图）

#include <cstring>
#include <iostream>
#include <algorithm>
#include <queue>

using namespace std;

typedef pair<int, int> PII;

const int N = 1e6 + 10;

int n, m;
int h[N], w[N], e[N], ne[N], idx;
int dist[N];
bool st[N];

void add(int a, int b, int c)
{
    e[idx] = b, w[idx] = c, ne[idx] = h[a], h[a] = idx ++ ;
}

int dijkstra()
{
    memset(dist, 0x3f, sizeof dist);
    dist[1] = 0;
    priority_queue<PII, vector<PII>, greater<PII>> heap;
    heap.push({0, 1});

    while (heap.size())
    {
        auto t = heap.top();
        heap.pop();

        int ver = t.second, distance = t.first;

        if (st[ver]) continue;
        st[ver] = true;

        for (int i = h[ver]; i != -1; i = ne[i])
        {
            int j = e[i];
            if (dist[j] > dist[ver] + w[i])
            {
                dist[j] = dist[ver] + w[i];
                heap.push({dist[j], j});
            }
        }
    }

    if (dist[n] == 0x3f3f3f3f) return -1;
    return dist[n];
}

int main()
{
    scanf("%d%d", &n, &m);

    memset(h, -1, sizeof h);
    while (m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        add(a, b, c);
    }

    cout << dijkstra() << endl;

    return 0;
}

3.bellman_ford

#include <iostream>
#include <cstring>
#include <algorithm>

using namespace std;

const int N = 510, M = 10010;

struct Edge
{
    int a, b, w;
}edges[M];

int dist[N], backup[N];
int n, m, k;

int bellman_ford()
{
    memset(dist, 0x3f, sizeof dist);
    dist[1] = 0;
    
    for (int i = 0; i < k; i ++ )
    {
        memcpy(backup, dist, sizeof dist);
        
        for (int j = 0; j < m; j ++ )
        {
            int a = edges[j].a, b = edges[j].b, w = edges[j].w;
            dist[b] = min(dist[b], backup[a] + w);
        }
    }
    
    if(dist[n] > 0x3f3f3f3f / 2) return -1;
    else return dist[n];
}

int main()
{
    scanf("%d%d%d", &n, &m, &k);
    
    for (int i = 0; i < m; i ++ )
    {
        int a, b, w;
        scanf("%d%d%d", &a, &b, &w);
        edges[i] = {a, b, w};
    }
    
    int t = bellman_ford();
    
    if(t == -1) puts("impossible");
    else printf("%d\n", t);
    
    return 0;
}

4.spfa

借助队列，只将dist数值修改了的结点入队

#include <iostream>
#include <cstring>
#include <algorithm>
#include <queue>

using namespace std;

const int N = 100010;

int m, n;
int h[N], w[N], e[N], ne[N], idx;
int dist[N];
bool st[N];

void add(int a, int b, int c)
{
    e[idx] = b, w[idx] = c, ne[idx] = h[a], h[a] = idx ++ ;
}

int spfa()
{
    memset(dist, 0x3f, sizeof dist);
    dist[1] = 0;
    
    queue<int> q;
    q.push(1);
    st[1] = true;
    
    while(q.size())
    {
        auto t = q.front();
        q.pop();
        
        st[t] = false;
        
        for (int i = h[t]; i != -1; i = ne[i])
        {
            int j = e[i];
            if(dist[j] > dist[t] + w[i])
            {
                dist[j] = dist[t] + w[i];
                if(!st[j])
                {
                    q.push(j);
                    st[j] = true;
                }
            }
        }
    }
    
    if(dist[n] == 0x3f3f3f3f) return -1;
    else return dist[n];
}

int main()
{
    cin >> n >> m;
    
    memset(h, -1, sizeof h);
    
    for (int i = 0; i < m; i ++ )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        add(a, b, c);
    }
    
    int t = spfa();
    
    if(t == -1) puts("impossible");
    else printf("%d\n", t);
    
    return 0;
}

spfa判断负环

#include <cstring>
#include <iostream>
#include <algorithm>
#include <queue>

using namespace std;

const int N = 2010, M = 10010;

int n, m;
int h[N], w[M], e[M], ne[M], idx;
int dist[N], cnt[N];
bool st[N];

void add(int a, int b, int c)
{
    e[idx] = b, w[idx] = c, ne[idx] = h[a], h[a] = idx ++ ;
}

bool spfa()
{
    queue<int> q;

    for (int i = 1; i <= n; i ++ )
    {
        st[i] = true;
        q.push(i);
    }

    while (q.size())
    {
        int t = q.front();
        q.pop();

        st[t] = false;

        for (int i = h[t]; i != -1; i = ne[i])
        {
            int j = e[i];
            if (dist[j] > dist[t] + w[i])
            {
                dist[j] = dist[t] + w[i];
                cnt[j] = cnt[t] + 1;

                if (cnt[j] >= n) return true;
                if (!st[j])
                {
                    q.push(j);
                    st[j] = true;
                }
            }
        }
    }

    return false;
}

int main()
{
    scanf("%d%d", &n, &m);

    memset(h, -1, sizeof h);

    while (m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        add(a, b, c);
    }

    if (spfa()) puts("Yes");
    else puts("No");

    return 0;
}

5.spfa

#include <iostream>
#include <cstring>
#include <algorithm>

using namespace std;

const int N = 210, INF = 1e9;

int d[N][N];
int n, m ,Q;

void floyd()
{
    for (int k = 1; k <= n; k ++ )
        for (int i = 1; i <= n; i ++ )
            for (int j = 1; j <= n; j ++ )
                d[i][j] = min(d[i][j], d[i][k] + d[k][j]);
}

int main()
{
    scanf("%d%d%d", &n, &m, &Q);
    
    for (int i = 1; i <= n; i ++ )
        for (int j = 1; j <= n; j ++ )
            if(i == j) d[i][j] = 0;
            else d[i][j] = INF;
    
    while (m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        d[a][b] = min(d[a][b], c);
    }
    
    floyd();
    
    while (Q -- )
    {
        int a, b;
        scanf("%d%d", &a, &b);
        
        if(d[a][b] > INF / 2) puts("impossible");
        else printf("%d\n", d[a][b]);
    }
    
    return 0;
}

6.朴素版的prim

#include <iostream>
#include <algorithm>
#include <cstring>

using namespace std;

const int N = 510, INF = 0x3f3f3f3f;

int g[N][N], dist[N];
bool st[N];
int n, m;

int prim()
{
    memset(dist, 0x3f, sizeof dist);
    
    int res = 0;
    for (int i = 0; i < n; i ++ )
    {
        int t = -1;
        for (int j = 1; j <= n; j ++ )
            if(!st[j] && (t == -1 || dist[t] > dist[j]))
                t = j;
        
        if(i && dist[t] == INF) return INF;
        
        if(i) res += dist[t];
        st[t] = true;
        
        for (int j = 1; j <= n; j ++ ) dist[j] = min(dist[j], g[t][j]);
    }
    
    return res;
}

int main()
{
    memset(g, 0x3f, sizeof g);
    
    scanf("%d%d", &n, &m);
    while (m -- )
    {
        int a, b, c;
        scanf("%d%d%d", &a, &b, &c);
        
        g[a][b] = g[b][a] = min(g[a][b], c);
    }
    
    int t = prim();
    
    if(t == INF) puts("impossible");
    else printf("%d\n", t);
    
    return 0;
}

7.Kruskal

#include <iostream>
#include <algorithm>
#include <cstring>

using namespace std;

const int N = 100010, M = 200010, INF = 0x3f3f3f3f;

struct Edge
{
    int a, b, w;
    
    bool operator< (const Edge & W)const
    {
        return w < W.w;
    }
}edges[M];

int p[N];
int n, m;

int find(int x)
{
    if(p[x] != x) p[x] = find(p[x]);
    return p[x];
}

int Kruskal()
{
    sort(edges, edges + m);
    
    for (int i = 1; i <= n; i ++ ) p[i] = i;
    
    int res = 0, cnt = 0;
    for (int i = 0; i < m; i ++ )
    {
        int a = edges[i].a, b = edges[i].b, w = edges[i].w;
        
        a = find(a), b = find(b);
        if(a != b)
        {
            p[a] = b;
            res += w;
            cnt ++ ;
        }
    }
    
    if(cnt < n - 1) return INF;
    return res;
}

int main()
{
    scanf("%d%d", &n, &m);
    
    for (int i = 0; i < m; i ++ )
    {
        int a, b, w;
        scanf("%d%d%d", &a, &b, &w);
        edges[i] = {a, b, w};
    }
    
    int t = Kruskal();
    
    if(t == INF) puts("impossible");
    else printf("%d\n", t);
    
    return 0;
}

8.染色法判断二分图

#include <iostream>
#include <cstring>
#include <algorithm>

using namespace std;

const int N = 100010, M = 200010;

int h[N], e[M], ne[M], idx;
int color[N];
int n, m;

void add(int a, int b)
{
    e[idx] = b, ne[idx] = h[a], h[a] = idx ++ ;
}

bool dfs(int u, int c)
{
    color[u] = c;
    
    for (int i = h[u]; i != -1; i = ne[i])
    {
        int j = e[i];
        if(!color[j])
        {
            if(!dfs(j, 3 - c)) return false;
        }
        else if(color[j] == c) return false;
    }
    
    return true;
}

int main()
{
    scanf("%d%d", &n, &m);
    
    memset(h, -1, sizeof h);
    
    for (int i = 0; i < m; i ++ )
    {
        int a, b;
        scanf("%d%d", &a, &b);
        
        add(a, b), add(b, a);
    }
    
    bool flag = true;
    for (int i = 1; i <= n; i ++ )
        if(!color[i])
        {
            if(!dfs(i, 1))
            {
                flag = false;
                break;
            }
        }
    
    if(flag) puts("Yes");
    else puts("No");
    
    return 0;
}

9.匈牙利算法求二分图最大匹配

#include <iostream>
#include <algorithm>
#include <cstring>

using namespace std;

const int N = 510, M = 100010;

int n1, n2, m;
int h[N], e[M], ne[M], idx;
int match[N];
bool st[N];

void add(int a, int b)
{
    e[idx] = b, ne[idx] = h[a], h[a] = idx ++ ;
}

bool find(int x)
{
    for (int i = h[x]; i != -1; i = ne[i])
    {
        int j = e[i];
        if (!st[j])
        {
            st[j] = true;
            if (match[j] == 0 || find(match[j]))
            {
                match[j] = x;
                return true;
            }
        }
    }

    return false;
}

int main()
{
    scanf("%d%d%d", &n1, &n2, &m);
    
    memset(h, -1, sizeof h);
    
    while (m -- )
    {
        int a, b;
        scanf("%d%d", &a, &b);
        add(a, b);
    }
    
    int res = 0;
    for (int i = 1; i <= n1; i ++ )
    {
        memset(st, false, sizeof st);
        if(find(i)) res ++ ;
    }
    
    printf("%d\n", res);
    
    return 0;
}