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Successive shortest paths in C++

This solves the assignment problem by solving the associated minimum cost flow using the successive shortest paths method.

#include <iostream>
#include <vector>
#include <fstream>
#include <algorithm>
#include <limits>

using std::vector;

using node = int;
using edge = int;
using capacity = int;
using flow = int;
using cost = int;

struct Edge {
  node from;   // from < 2^31
  node to;     // to   < 2^31
  cost c;      // cost < 2^31
  capacity u;  // capacity < 2^31 
  flow f;      // flow <= capacity
};

// In an EdgeCollection we store both edges and their inverses. 
// A edge is represented by an integer [0,edges.size) and its inverse edge 
// by the negative minus one [-edges.size, 0).
class EdgeCollection {
public:
  EdgeCollection(vector<Edge>& es): edges{es} {}

  edge inverse(edge e) {
    return -e-1;
  }
  node from(edge e) {
    return e >= 0 ? edges[e].from : edges[inverse(e)].to;
  }
  node to(edge e) {
    return e >= 0 ? edges[e].to : edges[inverse(e)].from;
  }
  cost get_cost(edge e) {
    return e >= 0 ? edges[e].c : - edges[inverse(e)].c;
  }
  capacity capacity_left(edge e) {
    return e >= 0 ? edges[e].u - edges[e].f : get_flow(inverse(e));
  }
  flow get_flow(edge e) {
    return e >= 0 ? edges[e].f : capacity_left(inverse(e));
  }
  void augment(edge e, flow f) {
    if(e >= 0) {
      edges[e].f += f;
    } else {
      edges[inverse(e)].f -= f;
    }
  }
  std::size_t size() {
    return edges.size();
  }
  vector<Edge>& get_edges() {
    return edges;
  }
private:
  vector<Edge> edges;
};

class AssignmentProblem {
public:
  AssignmentProblem(int num_nodes, vector<Edge>& edges)
    : ec(edges), pi(num_nodes + 2, 0), num_nodes{num_nodes}, s{num_nodes}, t{num_nodes + 1} {
    // Since all the edges go from A to B, 
    // we can calculate the initial potential pi in linear time:
    cost max = 0;
    for(size_t i = 0; i < edges.size(); ++i) {
      pi[edges[i].from] = - std::min( - pi[edges[i].from], edges[i].c);
      max = - std::min(edges[i].c, - max);
    }
    pi[s] = max;
    for(int i = 0; i < num_nodes/2; ++i) {
      edges.push_back(Edge {s,i,0,1,0});
    }
    for(int i = num_nodes/2; i < num_nodes; ++i) {
      edges.push_back(Edge {i,t,0,1,0});
    }
    ec = EdgeCollection(edges);
  }

  vector<Edge> succ_shortest_paths() {
    for(int b_of_s = num_nodes/2; b_of_s > 0; --b_of_s) {
      vector<int>  l(num_nodes + 2, std::numeric_limits<int>::max());
      vector<bool> r(num_nodes + 2, false);
      vector<edge> p(num_nodes + 2, -1);
      l[s] = 0;
      node next = s;
      vector<vector<size_t>> incident(num_nodes + 2);
      for(size_t i = 0; i < ec.size(); ++i) {
        if(ec.capacity_left(i) > 0) {
          incident[ec.from(i)].push_back(i);
        }
        if(ec.get_flow(i) > 0) {
          incident[ec.to(i)].push_back(ec.inverse(i));
        }
      }
      while(next != -1) { // Dijkstra
        r[next] = true;

        for(size_t i = 0; i < incident[next].size(); ++i) {
          edge e = incident[next][i]; node w = ec.to(e);
          cost c = l[next] + ec.get_cost(e) + pi[next] - pi[w];
          if(!r[w] and l[w] > c) {
            l[w] = c; p[w] = incident[next][i];
          }
        }

        next = -1; cost min = std::numeric_limits<int>::max();
        for(size_t i = 0; i < r.size(); ++i) {
          if(!r[i] and l[i] < min) {
            min = l[i];
            next = i;
          }
        }
      }

      node w = t;
      while(w != s) {
        edge way = p[w];
        ec.augment(way, 1); // In every iteration gamma will be 1.
        w = ec.from(way);
      }

      for(size_t i = 0; i < pi.size(); ++i) {
        pi[i] += l[i];
      }
    }
    return ec.get_edges();
  }
private:
  EdgeCollection ec;
  vector<int> pi;
  int num_nodes;
  node s, t;
};

void solve(int n, vector<Edge>& edges) {
  AssignmentProblem ap(n, edges);
  vector<Edge> f = ap.succ_shortest_paths();
  int total_cost = 0;
  for(size_t i = 0; i + n < f.size(); ++i) if(f[i].f == 1) total_cost += f[i].c;
  std::cout << total_cost << std::endl;
  for(size_t i = 0; i + n < f.size(); ++i) {
    Edge e = f[i];
    if(e.f == 1) {
      std::cout << e.from << " " << e.to << std::endl;
    }
  }
}

int main(int argc, char *argv[]) {
  if(argc != 2) {
    // We expect exactly one argument...
    std::cout << "Aufruf: " << argv[0] << " <filename>" << std::endl;
  } else {
    // which should be a filename...
    std::ifstream file(argv[1]);
    unsigned int n = 0;
    vector<Edge> edges(0);
    if(!file.is_open()) {
      // of a non-busy file.
      std::cout << "Konnte Datei nicht öffnen." << std::endl;
    } else {
      file >> n; // First we read the number of nodes...
      node a, b; cost c;
      while(file >> a >> b >> c) {
        // and then the edges...
        edges.push_back(Edge {a,b,c,1,0});
      }
      solve(n, edges);
    } 
  }
}

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