Find a path within a specific cost

问题

There are many algorithms or policies for finding a path with minimum or maximum costs. But, it is hard to find an approach that can find a path within (or below) a required cost (RC), i.e., such an RC is not a minimum or maximum one, and the actual cost should less than such an RC.

I am looking for a feasible algorithm to find a path satisfying the two constraints:

1. The cost of such a path should be lower than the required cost.
2. The path from source to destination should contain as many hops as possible.

One example is as specified below, e.g.,

The source is node A, the destination is node B; the required cost is 10. There are three pathes can be found from A to B:

 1. A --> C --> B;               cost is 5
 2. A --> C --> D --> B;         cost is 8
 3. A --> C --> D --> E --> B;   cost is 12

According to two mentioned constraints, path 2 (A --> C --> D --> B; cost is 8) is the best one, since the cost is 8 that less than the required cost 10, and path 2 is longer than path 1.

Hope I clearly explain my question. Are there any released algorithms or methods to address this issue?

Thank you in advance.

回答1:

I don't think there is well-known algorithm for this problem.

Let me just show you my snippet.

#include <iostream>
#include <algorithm>
#include <queue>
#include <vector>
using namespace std;

class info
{
    public:
        int hopcount;
    vector<int> path;
    int cost;
};

int main()
{
    int n;
    int s, e;
    vector<vector < int>> arr;
    cin >> n;
    arr.resize(n + 1);
    for (int i = 1; i <= n; i++)
    {
        arr[i].resize(n + 1);
    }
    cin >> s >> e;
    int pc;
    cin >> pc;
    for (int i = 0; i < pc; i++)
    {
        int source, def, cost;
        cin >> source >> def >> cost;
        arr[source][def] = cost;
    }
    int maxcost;
    cin >> maxcost;
    queue<info> q;
    q.push({1, {s}, 0 });
    int maxhopcount = -1;
    vector<int> hoppath;
    while (!q.empty())
    {
        auto qel = q.front();
        q.pop();
        int currentN = qel.path[qel.path.size() - 1];
        if (currentN == e)
        {
            if (qel.hopcount > maxhopcount)
            {
                maxhopcount = qel.hopcount;
                hoppath = qel.path;
            }
        }
        for (int i = 1; i <= n; i++)
        {
            int sign = 0;
            for (auto c: qel.path)
            {
                if (c == i)
                {
                    sign = 1;
                    break;
                }
            }
            if (sign == 1) continue;
            if (arr[currentN][i] > 0)
            {
                info ni = qel;
                ni.path.push_back(i);
                ni.hopcount += 1;
                ni.cost += arr[currentN][i];
                if (ni.cost <= maxcost)
                {
                    q.push(ni);
                }
            }
        }
    }
    cout << maxhopcount << endl;
    for (auto c: hoppath)
    {
        cout << c << " ";
    }
    return 0;
}
/*
testcases
5
1 2
6
1 3 2
3 2 3
3 4 3
4 2 3
4 5 4
5 2 3
10

1 3 4 2
4
*/

I wrote a code with simple bfs method.

Writing Information about each step will solve this problem.

Let me know if there is any corner cases.

来源：https://stackoverflow.com/questions/63609925/find-a-path-within-a-specific-cost