PAT甲级专题|最短路

PAT甲级最短路

主要算法：dijkstra 求最短最长路、dfs图论搜索。

1018，dijkstra记录路径 + dfs搜索路径最值

25分，错误点暂时找不出。。

如果只用dijkstra没法做，只能得20分

#include<bits/stdc++.h>
using namespace std;

const int inf = 0x3f3f3f3f;
const int maxn = 510;
int cmax,n,ter,m;
int caps[maxn];

int g[maxn][maxn];
int vis[maxn];
vector<int> pre[maxn];
int dist[maxn];
int half;
vector<int> paths;
vector<int> temp;
int minsend = inf;
int mintake = inf;

//dfs
void dfs(int x){
	temp.push_back(x);
	if(x == 0){
		int need = 0;
		int take = 0;
		for(int i=0;i<temp.size();i++){
			if(temp[i] == 0) continue;
			if(caps[temp[i]] - half > 0){
				int d = caps[temp[i]] - half;
				if(need >= d) need -= d;
				else{
					need = 0;
					take += (d - need);
				}
			}else{
				need += (half - caps[temp[i]]);
			}
		}
		if(need < minsend){
			minsend = need;
			mintake = take;
			paths = temp;
		}else if(need == minsend){
			if(mintake > take){
				mintake = take;
				paths = temp;
			}
		}
		temp.pop_back();
		return;
	}
	for(int i=0;i<pre[x].size();i++){
		dfs(pre[x][i]);
	}
	temp.pop_back();
}

void dijkstra(){
    memset(vis,0,sizeof(vis));
    memset(dist,inf,sizeof(dist));
    int half = cmax/2;
    dist[0] = 0;
    for(int i=0;i<=n;i++){
        int v,min_w = inf;
        for(int j=0;j<=n;j++){
            if(!vis[j] && dist[j] < min_w){
                min_w = dist[j];
                v = j;
            }
        }
        if(min_w == inf) return;
        vis[v] = 1;
        for(int j=0;j<=n;j++){
			if(vis[j] || g[v][j] == inf) continue;
			if(g[v][j] + dist[v] < dist[j]){
				dist[j] = g[v][j] + dist[v];
				pre[j].clear();
				pre[j].push_back(v);
			}else if(g[v][j] + dist[v] == dist[j]){
				pre[j].push_back(v);
			}
        }
    }
}

int main(){
	memset(g,inf,sizeof(g));
    cin>>cmax>>n>>ter>>m;
    half = cmax/2;
    caps[0] = 0;
    for(int i=1;i<=n;i++) cin>>caps[i];
    for(int i=1;i<=m;i++){
        int u,v,w;
        cin>>u>>v>>w;
        g[u][v] = w;
        g[v][u] = w;
    }
    dijkstra();
	dfs(ter);
	cout<<minsend<<" ";
	for(int i=paths.size()-1;i>0;i--){
		cout<<paths[i]<<"->";
	}
	cout<<paths[0];
	cout<<" "<<mintake<<endl;
    return 0;
}

/*
10 3 3 5
6 7 10
0 1 1
0 2 1
0 3 2
1 3 1
2 3 1
*/

1030，多边权，多条更新

#include<bits/stdc++.h>
using namespace std;

/*
dijkstra:双边权
*/

const int inf = 0x3f3f3f3f;
const int maxn = 510;
int n,m,s,d;
int dist[maxn];
int path[maxn];
int cost[maxn];
int vis[maxn];
struct edge{
	int v,w,c;
	edge(int vv,int ww,int cc){
		v = vv;
		w = ww;
		c = cc;
	}
};
vector<int> paths;
vector<edge> g[maxn]; 

void dijkstra(int ss){
    memset(vis,0,sizeof(vis));
    memset(dist,inf,sizeof(dist));
    memset(cost,inf,sizeof(cost));
    dist[ss] = 0;
    cost[ss] = 0;
    path[ss] = -1;
    for(int i=0;i<n;i++){
        int v,min_w = inf;
        for(int j=0;j<n;j++){
            if(!vis[j] && dist[j] < min_w){
                min_w = dist[j];
                v = j;
            }
        }
        if(min_w == inf) return;
        vis[v] = 1;
        for(int j=0;j<g[v].size();j++){
            int x = g[v][j].v;
            if(!vis[x] && dist[v] + g[v][j].w < dist[x]){
            	cost[x] = cost[v] + g[v][j].c;
                dist[x] = dist[v] + g[v][j].w;
                path[x] = v;
            }else if(!vis[x] && dist[v] + g[v][j].w == dist[x]){
				if(cost[x] > cost[v] + g[v][j].c){
					path[x] = v;
					cost[x] = cost[v] + g[v][j].c;
				}
            }
        }
    }
}

int main(){
	cin>>n>>m>>s>>d;
	for(int i=1;i<=m;i++){
		int u,v,w,c;
		cin>>u>>v>>w>>c;
		g[u].push_back(edge(v,w,c));
		g[v].push_back(edge(u,w,c));
	}
	dijkstra(s);
	int cur = d;
	while(cur != -1){
		paths.push_back(cur);
		cur = path[cur];
	}
	for(int i=paths.size()-1;i>=0;i--){
		cout<<paths[i]<<" ";
	}
	cout<<dist[d]<<" "<<cost[d]<<endl;
	return 0;
}

1087，记录所有路径，dfs搜索路径最值

#include<bits/stdc++.h>
using namespace std;

/*
map映射:string <-> cityId  Name <-> Int
dijkstra:找出最短路的长度 以及所有最短路径(存入到pre容器中)
dfs:统计最短路的数量 从终点触发在出口判断更新所需的权值
*/
const int inf = 0x3f3f3f3f;
const int maxn = 210;
map<string, int> mp;
map<int,string> mp2;
int n,k;
string start;
int haps[maxn];
int g[maxn][maxn];
int nums;
int vis[maxn];
int dist[maxn];
int ter = 0;
vector<int> pre[maxn];
vector<int> temp,path;
int maxhap = 0;
double maxave = 0;

void dfs(int x){
	temp.push_back(x);
	if(x == n){
		int curhap = 0;
		double curave = 0;
		for(int i=temp.size()-1;i>=0;i--){
			curhap += haps[temp[i]];
		}
		if(temp.size() == 1){
			curave = 0;
		}else{
			curave = curhap*1.0/(temp.size()-1);
		}
		if(maxhap < curhap){
			maxhap = curhap;
			maxave = curave;
			path = temp;
		}else if(maxhap == curhap){
			if(maxave < curave){
				maxave = curave;
				path = temp;
			}
		}
		nums++;
		temp.pop_back();
		return;
	}
	for(int i=0;i<pre[x].size();i++){
		dfs(pre[x][i]);
	}
	temp.pop_back();
}

void dijkstra(int s){
    memset(vis,0,sizeof(vis));
    memset(dist,inf,sizeof(dist));
    dist[s] = 0;
    for(int i=1;i<=n;i++){
        int v,min_w = inf;
        for(int j=1;j<=n;j++){
            if(!vis[j] && dist[j] < min_w){
                v = j;
                min_w = dist[j];
            }
        }
        vis[v] = 1;
        if(min_w == inf) return;
        for(int j=1;j<=n;j++){
            if(!vis[j] && g[v][j] != inf){
                if(dist[j] > dist[v] + g[v][j]){
                	dist[j] = dist[v] + g[v][j];
                	pre[j].clear();
                	pre[j].push_back(v);
                }else if(dist[j] == dist[v] + g[v][j]){
                	pre[j].push_back(v);
                }
            }
        }
    }
}

int main(){
    memset(g,inf,sizeof(g));
    cin>>n>>k>>start;
    for(int i=1;i<=n-1;i++){
        string city;
        int hap;
        cin>>city>>hap;
        mp[city] = i;
        mp2[i] = city;
        haps[i] = hap;
    }
    mp[start] = n;
    mp2[n] = start;
    haps[n] = 0;
    for(int i=1;i<=k;i++){
        string c1,c2;
        int cost;
        cin>>c1>>c2>>cost;
        int u = mp[c1];
        int v = mp[c2];
        g[u][v] = cost;
        g[v][u] = cost;
    }
    ter = mp["ROM"];
    dijkstra(n);
    dfs(ter);
    cout<<nums<<" "<<dist[ter]<<" "<<maxhap<<" "<<int(maxave)<<endl;
    cout<<start;
    for(int i=path.size()-2;i>0;i--){
    	cout<<"->"<<mp2[path[i]];
	}
	cout<<"->ROM"<<endl;
    return 0;
}

1111，多边权、记录路径、多条件更新

#include<bits/stdc++.h>
using namespace std;

/*
30分
*/

const int inf = 0x3f3f3f3f;
const int maxn = 510;
int source,ter;
int n,m;
int vis1[maxn];
int vis2[maxn];
struct edge{
    int v;
    int length;
    int time;
    edge(int vv,int len,int ti){
        v = vv;
        length = len;
        time = ti;
    }
};
vector<edge> g[maxn];
int dist1[maxn];
int time1[maxn];
int time2[maxn];
int path1[maxn];
int path2[maxn];
int nums[maxn];
vector<int> ans1;
vector<int> ans2;

void dijkstra1(){
    memset(vis1,0,sizeof(vis1));
    memset(dist1,inf,sizeof(dist1));
    memset(time1,inf,sizeof(time1));
    dist1[source] = 0;
    time1[source] = 0;
    path1[source] = -1;
    for(int i=0;i<n;i++){
        int v,min_w = inf;
        for(int j=0;j<n;j++){
            if(!vis1[j] && dist1[j] < min_w){
                v = j;
                min_w = dist1[j];
            }
        }
        vis1[v] = 1;
        if(min_w == inf) return;
        for(int j=0;j<g[v].size();j++){
            int u = g[v][j].v;
            if(!vis1[u]){
                if(dist1[u] > dist1[v] + g[v][j].length){
                    time1[u] = time1[v] + g[v][j].time;
                    dist1[u] = dist1[v] + g[v][j].length;
                    path1[u] = v;
                }else if(dist1[u] == dist1[v] + g[v][j].length && time1[u] > time1[v] + g[v][j].time){
                    time1[u] = time1[v] + g[v][j].time;
                    path1[u] = v;
                }
            }
        }
    }
}

void dijkstra2(){
    memset(vis2,0,sizeof(vis2));
    memset(nums,inf,sizeof(nums));
    memset(time2,inf,sizeof(time2));
    nums[source] = 1;
    time2[source] = 0;
    path2[source] = -1;
    for(int i=0;i<n;i++){
        int v,min_w = inf;
        for(int j=0;j<n;j++){
            if(!vis2[j] && time2[j] < min_w){
                v = j;
                min_w = time2[j];
            }
        }
        vis2[v] = 1;
        if(min_w == inf) return;
        for(int j=0;j<g[v].size();j++){
            int u = g[v][j].v;
            if(!vis2[u]){
                if(time2[u] > time2[v] + g[v][j].time){
                    time2[u] = time2[v] + g[v][j].time;
                    nums[u] = nums[v] + 1;
                    path2[u] = v;
                }else if(time2[u] == time2[v] + g[v][j].time){
                    if(nums[u] >= nums[v] + 1){
                        path2[u] = v;
                        nums[u] = nums[v] + 1;
                    }
                }
            }
        }
    }
}

bool identical(){
    int cur = ter;
    while(cur != -1){
        ans1.push_back(cur);
        cur = path1[cur];
    }
    cur = ter;
    while(cur != -1){
        ans2.push_back(cur);
        cur = path2[cur];
    }
    reverse(ans1.begin(),ans1.end());
    reverse(ans2.begin(),ans2.end());
    if(ans1.size() != ans2.size()) return false;
    for(int i=0;i<ans1.size();i++){
        if(ans1[i] != ans2[i])return false;
    }
    return true;
}

int main(){
//	freopen("in.txt", "r", stdin);
//	freopen("out.txt", "w", stdout);
    cin>>n>>m;
    for(int i=1;i<=m;i++){
        int v1,v2,one,length,time;
        cin>>v1>>v2>>one>>length>>time;
        if(one == 0){
            g[v1].push_back(edge(v2,length,time));
            g[v2].push_back(edge(v1,length,time));
        }else{
            g[v1].push_back(edge(v2,length,time));
        }
    }
    cin>>source>>ter;
    dijkstra1();
    dijkstra2();
    if(identical()){
        printf("Distance = %d; Time = %d: %d",dist1[ter],time2[ter],source);
        for(int i=1;i<ans1.size();i++){
            printf(" -> %d",ans1[i]);
        }
        printf("\n");
    }else{
        printf("Distance = %d: %d",dist1[ter],source);
        for(int i=1;i<ans1.size();i++){
            printf(" -> %d",ans1[i]);
        }
        printf("\n");
        printf("Time = %d: %d",time2[ter],source);
        for(int i=1;i<ans2.size();i++){
            printf(" -> %d",ans2[i]);
        }
        printf("\n");
    }
    return 0;
}
/*
4 4
0 1 1 1 2
0 2 1 2 1
1 3 1 2 2
2 3 1 1 1
0 3
*/