1030 Travel Plan

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#算法 #图论 #PTA

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本文介绍了一种寻找无向图中两点间最短距离且权重最小路径的方法，提供了两种实现方式：暴力动态规划和迪杰斯特拉算法，并附带完整的C/C++代码示例。

A traveler's map gives the distances between cities along the highways, together with the cost of each highway. Now you are supposed to write a program to help a traveler to decide the shortest path between his/her starting city and the destination. If such a shortest path is not unique, you are supposed to output the one with the minimum cost, which is guaranteed to be unique.

Input Specification:

Each input file contains one test case. Each case starts with a line containing 4 positive integers N, M, S, and D, where N (≤500) is the number of cities (and hence the cities are numbered from 0 to N−1); M is the number of highways; S and D are the starting and the destination cities, respectively. Then M lines follow, each provides the information of a highway, in the format:

City1 City2 Distance Cost

where the numbers are all integers no more than 500, and are separated by a space.

Output Specification:

For each test case, print in one line the cities along the shortest path from the starting point to the destination, followed by the total distance and the total cost of the path. The numbers must be separated by a space and there must be no extra space at the end of output.

Sample Input:

Sample Output:

0 2 3 3 40

题目大意

求无向图中，某两点间最短距离且权重最小的路径

思路

暴力动规 or 帝洁磕四

C/C++ （暴力动规）

#include<bits/stdc++.h>
using namespace std;
vector<int> road[501],key,result;
int N,M,st,ed,len[501][501],cost[501][501],lenKey=99999999,costKey;
void findRoad(int now,int sumLen,int sumCost);
int main()
{
    int a,b,c,d;
    cin >> N >> M >> st >> ed;
    while (M--){
        cin >> a >> b >> c >> d;
        road[a].push_back(b);
        road[b].push_back(a);
        len[a][b] = len[b][a] = c;
        cost[a][b] = cost[b][a] = d;
    }
    findRoad(st,0,0);
    for(int x:result) cout << x << " ";
    cout << ed << " " << lenKey << " " << costKey;
    return 0;
}
bool apr[501];
void findRoad(int now,int sumLen,int sumCost)
{
    if(apr[now] || sumLen>lenKey) return;
    if(now==ed){
        if(sumLen<lenKey || (sumLen==lenKey && sumCost<costKey)){
            lenKey = sumLen;
            costKey = sumCost;
            result.assign(key.begin(),key.end());
        }

        return;
    }
    apr[now] = true;
    key.push_back(now);
    for(int x:road[now]) findRoad(x,sumLen+len[now][x],sumCost+cost[now][x]);
    key.pop_back();
    apr[now] = false;
}

C/C++ (Dijkstra)

#include<bits/stdc++.h>
using namespace std;
int N,M,road[501][501],cost[501][501],edC,st,ending;
int ed[501],last[501],edCost[501];
void Dijkstra(){
    memset(ed,0x3f,sizeof ed);
    memset(edCost,0,sizeof edCost);
    last[st] = -1;
    ed[st] = 0;
    bool apr[501] = {false};
    for(int z=0;z<N;z++){
        int key = -1;
        for(int z1=0;z1<N;z1++){
            if(!apr[z1] && (key==-1 || ed[z1]<ed[key])) key = z1;
        }
        apr[key] = true;
        for(int z1=0;z1<N;z1++){
            int len = ed[key]+road[key][z1];
            int money = edCost[key] + cost[key][z1];
            if(ed[z1]>len || (ed[z1]==len && edCost[z1]>money)){
                ed[z1] = len;
                last[z1] = key;
                edCost[z1] = money;
            };
        }
    }
}
void print(int now){
    if(now!=-1) print(last[now]);
    else return;
    printf("%d ",now);
    if(now==ending) printf("%d %d",ed[now],edCost[now]);
}
int main()
{
    memset(road,0x3f,sizeof road);
    cin >> N >> M >> st >> ending;
    while (M--){
        int a,b,c,d;
        cin >> a >> b >> c >> d;
        road[a][b] = road[b][a] = c;
        cost[a][b] = cost[b][a] = d;
    }
    Dijkstra();
    print(ending);
    return 0;
}