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I'm trying to solve the Oil Well problem on Hackerrank using dynamic programming and it works. However, it times out for some of the test cases. I wanted to know how this program can be improved so that it runs faster.

The challenge is, given a matrix \$M\$ of size \$r \times c\$ (at most \$50 \times 50\$), where each \$M_{x,y}\$ is either 0 or 1, find the minimum cost path \$E\$ for a spanning tree that connects all the entries where \$M_{x,y} = 1\$. The cost metric is based on ACME distance: \$\max(\left|x - x_1\right|, \left|y - y_1\right|)\$ where \$x, y\$ is the position of the existing wells, and \$x_1, y_1\$ is the position of the new well.

My basic idea is:

$$f(\mathrm{vertices}) = \min( \max(v_i \to \{\mathrm{vertices} - v_i\}) + f(\mathrm{vertices} - v_i))$$

for all vertices \$v_i\$.

The Hackerrank editorial shows a different logic but I'm trying to optimize my solution.

#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
#include <queue>
#include <set>
#include <climits>
#include <map>
#include <sstream>

using namespace std;
map<string,int> cache;
map<pair<int,int>,int> pd;
string has(vector<int>& vertices)
{
    ostringstream s;
    for(int i = 0; i < vertices.size(); i++)
    {
        s << vertices[i] << "|";
    }
    return s.str();
}
int dist(vector<int>& vertices)
{
    if(vertices.size() <= 1)
        return 0;
    string h = has(vertices);
    if(cache.count(h) != 0) {
        return cache[h];
    }
    int mind = INT_MAX;
    for(int i = 0; i < vertices.size(); i++) {
        int iv = vertices[i];        
        int x = 0;
        for(int j = 0; j < vertices.size(); j++) {
            int ov = vertices[j];            
            x = max(x,pd[make_pair(min(iv,ov),max(iv,ov))]);
        }        
        vertices.erase(vertices.begin()+i);
        mind = min(x + dist(vertices), mind);
        vertices.insert(vertices.begin()+i,iv);
    }
    cache[h]=mind;
    return mind;
}
int main() {
    int r, c;
    cin >> r >> c;
    vector<pair<int,int>> vertices;
    int a;
    for(int i = 0; i < r; i++) {
        for(int j = 0; j < c; j++) {
            cin >> a;
            if(a) {
                vertices.push_back(make_pair(i,j));
            }
        }
    }
    vector<int> vv;
    for(int i = 0; i < vertices.size(); i++) {
        vv.push_back(i);
        for(int j = i+1; j < vertices.size();j++) {
            pair<int,int> ip = vertices[i];
            pair<int,int> op = vertices[j];
            int d = max(abs(ip.first - op.first),abs(ip.second - op.second));
            pd[make_pair(i,j)] = d;
        }
    }
    cout << dist(vv) << endl;
    return 0;
}
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  • 2
    \$\begingroup\$ The site does not display the challange for me. \$\endgroup\$ – nwp Jan 7 '15 at 8:38
  • \$\begingroup\$ @nwp I clicked on the same hyperlink in your comment but it shows me the problem. The question has been edited to include the problem. \$\endgroup\$ – ssh Jan 7 '15 at 15:28
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It is not possible to make this solution fast enough. The number of states in your dynamic programming is O(2^numberOfVertices)(the number of subsets of a set with numberOfVertices elements), which is 2^2500 in the worst case under given constraints. So a more efficient algorithm is required.

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