Quick Tour to LEMON

Let us first answer the question "What do I want to use LEMON for?". LEMON is a C++ library, so you can use it if you want to write C++ programs. What kind of tasks does the library LEMON help to solve? It helps to write programs that solve optimization problems that arise frequently when designing and testing certain networks, for example in telecommunication, computer networks, and other areas that I cannot think of now. A very natural way of modelling these networks is by means of a graph (we will always mean a directed graph by that and say undirected graph otherwise). So if you want to write a program that works with graphs then you might find it useful to use our library LEMON. LEMON defines various graph concepts depending on what you want to do with the graph: a very good description can be found in the page about graphs.

You will also want to assign data to the edges or nodes of the graph, for example a length or capacity function defined on the edges. You can do this in LEMON using so called maps. You can define a map on the nodes or on the edges of the graph and the value of the map (the range of the function) can be practically almost of any type. Read more about maps here.

In this quick tour we want to show you some facilities LEMON library can provide through examples (simple demo programs). The examples will only show part of the functionality, but links will always be given to reach complete details. You will find links next to the code fragments that help to download full demo programs: save them on your computer and compile them according to the description in the page about How to start using LEMON.

Have fun!

• The first thing to discuss is the way one can create data structures like graphs and maps in a program using LEMON. There are more graph types implemented in LEMON and you can implement your own graph type just as well: read more about this in the already mentioned page on graphs.

  First we show how to add nodes and edges to a graph manually. We will also define a map on the edges of the graph. After this we show the way one can read a graph (and perhaps maps on it) from a stream (e.g. a file). Of course we also have routines that write a graph (and perhaps maps) to a stream (file): this will also be shown. LEMON supports the DIMACS file formats to read network optimization problems, but more importantly we also have our own file format that gives a more flexible way to store data related to network optimization.

  1. The following code shows how to build a graph from scratch and iterate on its nodes and edges. This example also shows how to give a map on the edges of the graph. The type Listgraph is one of the LEMON graph types: the typedefs in the beginning are for convenience and we will assume them later as well.

     
     #include <iostream>
     #include <lemon/list_graph.h>
     
     int main()
     {
       typedef lemon::ListGraph Graph;
       typedef Graph::EdgeIt EdgeIt;
       typedef Graph::Edge Edge;
       typedef Graph::NodeIt NodeIt;
       typedef Graph::Node Node;
       typedef Graph::EdgeMap<int> LengthMap;
       using lemon::INVALID;
     
       Graph g;
       
       Node s=g.addNode();
       Node v2=g.addNode();
       Node v3=g.addNode();
       Node v4=g.addNode();
       Node v5=g.addNode();
       Node t=g.addNode();
     
       Edge s_v2=g.addEdge(s, v2);
       Edge s_v3=g.addEdge(s, v3);
       Edge v2_v4=g.addEdge(v2, v4);
       Edge v2_v5=g.addEdge(v2, v5);
       Edge v3_v5=g.addEdge(v3, v5);
       Edge v4_t=g.addEdge(v4, t);
       Edge v5_t=g.addEdge(v5, t);
       
       LengthMap length(g);
     
       length[s_v2]=10;
       length[s_v3]=10;
       length[v2_v4]=5;
       length[v2_v5]=8;
       length[v3_v5]=5;
       length[v4_t]=8;
       length[v5_t]=8;
     
       std::cout << "Hello World!" << std::endl;
       std::cout <<  std::endl;
       std::cout << "This is library LEMON here! We have a graph!" << std::endl;
       std::cout <<  std::endl;
     
       std::cout << "Nodes:";
       for (NodeIt i(g); i!=INVALID; ++i)
         std::cout << " " << g.id(i);
       std::cout << std::endl;
     
       std::cout << "Edges:";
       for (EdgeIt i(g); i!=INVALID; ++i)
         std::cout << " (" << g.id(g.source(i)) << "," << g.id(g.target(i)) << ")";
       std::cout << std::endl;
       std::cout <<  std::endl;
     
       std::cout << "There is a map on the edges (length)!" << std::endl;
       std::cout <<  std::endl;
       for (EdgeIt i(g); i!=INVALID; ++i)
         std::cout << "length(" << g.id(g.source(i)) << ","
                   << g.id(g.target(i)) << ")="<<length[i]<<std::endl;
     
       std::cout << std::endl;
     
     }
     

     See the whole program in file hello_lemon.cc in the demo subdir of LEMON package.

     If you want to read more on the LEMON graph structures and concepts, read the page about graphs.

  2. LEMON has an own file format for storing graphs, maps on edges/nodes and some other things. Instead of any explanation let us give a short example file in this format: read the detailed description of the LEMON graph file format and input-output routines here: Graph Input-Output.

     So here is a file describing a graph of 6 nodes (0 to 5), two nodemaps (called coordinates_x and coordinates_y), several edges, an edge map called capacity and two designated nodes (called source and target).

     @nodeset
     id      coordinates_x   coordinates_y
     5       796.398 208.035
     4       573.002 63.002
     3       568.549 401.748
     2       277.889 68.476
     1       288.248 397.327
     0       102.239 257.532
     @edgeset
                     id      capacity
     4       5       6       8
     3       5       5       8
     2       4       4       5
     1       4       3       8
     1       3       2       5
     0       2       1       10
     0       1       0       10
     #This is a comment here
     @nodes
     source 0
     target 5
     @edges 
     @attributes 
     author "Attila BERNATH"
     @end
     

     Finally let us give a simple example that reads a graph from a file and writes it to the standard output.

     
     #include <iostream>
     #include <lemon/smart_graph.h>
     #include <lemon/graph_reader.h>
     #include <lemon/graph_writer.h>
     
     
     using namespace lemon;
     
     int main() {
       SmartGraph graph;
     
       try {
         std::string filename="sample.lgf";
         std::string name;
         GraphReader<SmartGraph> reader(filename,graph);
         SmartGraph::EdgeMap<int> cap(graph);
         reader.readEdgeMap("capacity",cap);
         reader.readAttribute("name",name);
         reader.run();
     
         std::cout << "Hello! We have read a graph from file " << filename<< 
           " and some maps on it:\n now we write it to the standard output!" << 
           std::endl;
     
     
         GraphWriter<SmartGraph> writer(std::cout, graph);
         writer.writeEdgeMap("multiplicity", cap);
         writer.writeAttribute("name",name);
         writer.run();
          
       } catch (DataFormatError& error) {
         std::cerr << error.what() << std::endl;
       }
     
     
       return 0;
     }
     

     See the whole program in file reader_writer_demo.cc.

  3. The following code shows how to read a graph from a stream (e.g. a file) in the DIMACS file format (find the documentation of the DIMACS file formats on the web).

     Graph g;
     std::ifstream f("graph.dim");
     readDimacs(f, g);
     

     One can also store network (graph+capacity on the edges) instances and other things (minimum cost flow instances etc.) in DIMACS format and read these in LEMON: to see the details read the documentation of the Dimacs file format reader.

• If you want to solve some transportation problems in a network then you will want to find shortest paths between nodes of a graph. This is usually solved using Dijkstra's algorithm. A utility that solves this is the LEMON Dijkstra class. The following code is a simple program using the LEMON Dijkstra class: it calculates the shortest path between node s and t in a graph g. We omit the part reading the graph g and the length map len.

      typedef ListGraph Graph;
      typedef Graph::Node Node;
      typedef Graph::Edge Edge;
      typedef Graph::EdgeMap<int> LengthMap;
  
      Graph g;
  

  ...

      std::cout << "Dijkstra algorithm demo..." << std::endl;
  
      Dijkstra<Graph, LengthMap> dijkstra_test(g,len);
      
      dijkstra_test.run(s);
      
      std::cout << "The distance of node t from node s: "
                << dijkstra_test.dist(t) << std::endl;
  
      std::cout << "The shortest path from s to t goes through the following "
                << "nodes (the first one is t, the last one is s): "
                << std::endl;
  
      for (Node v=t;v != s; v=dijkstra_test.predNode(v)) {
        std::cout << g.id(v) << "<-";
      }
      
      std::cout << g.id(s) << std::endl;  
  

  See the whole program in dijkstra_demo.cc.

  Some explanation: after instantiating a member of the Dijkstra class we run the Dijkstra algorithm from node s. After this we read some of the results. You can do much more with the Dijkstra class, for example you can run it step by step and gain full control of the execution. For a detailed description, see the documentation of the LEMON Dijkstra class.

• If you want to design a network and want to minimize the total length of wires then you might be looking for a minimum spanning tree in an undirected graph. This can be found using the Kruskal algorithm: the function LEMON Kruskal does this job for you.

  First make a graph g and a cost map edge_cost_map, then make a bool edgemap tree_map or a vector tree_edge_vec for the algorithm output. After calling the function it gives back the weight of the minimum spanning tree and the tree_map or the tree_edge_vec contains the edges of the tree.

  If you want to store the edges in a bool edgemap, then use the function as follows:

    // Kruskal with boolmap;
    std::cout << "The weight of the minimum spanning tree is " << 
      kruskal(g, edge_cost_map, tree_map)<<std::endl;
  

  And if you rather use a vector instead of a bool map:

    // Kruskal with vector;
    std::cout << "The weight of the minimum spanning tree again is " << 
      kruskal(g, edge_cost_map, std::back_inserter(tree_edge_vec)) <<std::endl;
  

  See the whole program in kruskal_demo.cc.

• Many problems in network optimization can be formalized by means of a linear programming problem (LP problem, for short). In our library we decided not to write an LP solver, since such packages are available in the commercial world just as well as in the open source world, and it is also a difficult task to compete these. Instead we decided to develop an interface that makes it easier to use these solvers together with LEMON. The advantage of this approach is twofold. Firstly our C++ interface is more comfortable than the solvers' native interface. Secondly, changing the underlying solver in a certain software using LEMON's LP interface needs zero effort. So, for example, one may try his idea using a free solver, demonstrate its usability for a customer and if it works well, but the performance should be improved, then one may decide to purchase and use a better commercial solver.

  So far we have an interface for the commercial LP solver software CPLEX (developed by ILOG) and for the open source solver GLPK (a shorthand for Gnu Linear Programming Toolkit).

  We will show two examples, the first one shows how simple it is to formalize and solve an LP problem in LEMON, while the second one shows how LEMON facilitates solving network optimization problems using LP solvers.

  1. The following code shows how to solve an LP problem using the LEMON lp interface. The code together with the comments is self-explanatory.

       //A default solver is taken
       Lp lp;
       typedef Lp::Row Row;
       typedef Lp::Col Col;
       
     
       std::cout<<"A program demonstrating the LEMON LP solver interface"<<std::endl; 
       std::cout<<"Solver used: "<<default_solver_name<<std::endl;
     
       //This will be a maximization
       lp.max();
     
       //We add coloumns (variables) to our problem
       Col x1 = lp.addCol();
       Col x2 = lp.addCol();
       Col x3 = lp.addCol();
     
       //Constraints
       lp.addRow(x1+x2+x3 <=100);  
       lp.addRow(10*x1+4*x2+5*x3<=600);  
       lp.addRow(2*x1+2*x2+6*x3<=300);  
       //Nonnegativity of the variables
       lp.colLowerBound(x1, 0);
       lp.colLowerBound(x2, 0);
       lp.colLowerBound(x3, 0);
       //Objective function
       lp.obj(10*x1+6*x2+4*x3);
       
       //Call the routine of the underlying LP solver
       lp.solve();
     
       //Print results
       if (lp.primalStatus()==LpSolverBase::OPTIMAL){
         std::cout<<"Optimal solution found!"<<std::endl;
         printf("optimum value = %g; x1 = %g; x2 = %g; x3 = %g\n", 
                lp.primalValue(), 
                lp.primal(x1), lp.primal(x2), lp.primal(x3));
       }
       else{
         std::cout<<"Optimal solution not found!"<<std::endl;
       }
     
       //End of LEMON style code
     

     See the whole code in lp_demo.cc.

  2. The second example shows how easy it is to formalize a max-flow problem as an LP problem using the LEMON LP interface: we are looking for a real valued function defined on the edges of the digraph satisfying the nonnegativity-, the capacity constraints and the flow-conservation constraints and giving the largest flow value between to designated nodes.

     In the following code we suppose that we already have the graph g, the capacity map cap, the source node s and the target node t in the memory. We will also omit the typedefs.

       //Define a map on the edges for the variables of the LP problem
       typename G::template EdgeMap<Lp::Col> x(g);
       lp.addColSet(x);
       
       //Nonnegativity and capacity constraints
       for(EdgeIt e(g);e!=INVALID;++e) {
         lp.colUpperBound(x[e],cap[e]);
         lp.colLowerBound(x[e],0);
       }
     
     
       //Flow conservation constraints for the nodes (except for 's' and 't')
       for(NodeIt n(g);n!=INVALID;++n) if(n!=s&&n!=t) {
         Lp::Expr ex;
         for(InEdgeIt  e(g,n);e!=INVALID;++e) ex+=x[e];
         for(OutEdgeIt e(g,n);e!=INVALID;++e) ex-=x[e];
         lp.addRow(ex==0);
       }
       
       //Objective function: the flow value entering 't'
       Lp::Expr obj;
       for(InEdgeIt  e(g,t);e!=INVALID;++e) obj+=x[e];
       for(OutEdgeIt e(g,t);e!=INVALID;++e) obj-=x[e];
       lp.obj(obj);
     
     
       //Maximization
       lp.max();
       //Solve with the underlying solver
       lp.solve();
     

     The complete program can be found in file lp_maxflow_demo.cc. After compiling run it in the form:

     ./lp_maxflow_demo < sample.lgf

     where sample.lgf is a file in the lemon format containing a maxflow instance (designated "source", "target" nodes and "capacity" map on the edges).