staticTriangulation.cpp

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/***************************************************************************
 *   Copyright (C) 2005 by Christophe GONZALES and Pierre-Henri WUILLEMIN  *
 *   {prenom.nom}_at_lip6.fr                                               *
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 *   This program is free software; you can redistribute it and/or modify  *
 *   it under the terms of the GNU General Public License as published by  *
 *   the Free Software Foundation; either version 2 of the License, or     *
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 *   This program is distributed in the hope that it will be useful,       *
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of        *
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the         *
 *   GNU General Public License for more details.                          *
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 *   Free Software Foundation, Inc.,                                       *
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#include <agrum/core/priorityQueue.h>
#include <agrum/graphs/algorithms/triangulations/eliminationStrategies/defaultEliminationSequenceStrategy.h>
#include <agrum/graphs/algorithms/triangulations/junctionTreeStrategies/defaultJunctionTreeStrategy.h>
#include <agrum/graphs/algorithms/triangulations/staticTriangulation.h>

#ifdef GUM_NO_INLINE
#  include <agrum/graphs/algorithms/triangulations/staticTriangulation_inl.h>
#endif   // GUM_NO_INLINE

namespace gum {

  // default constructor
  StaticTriangulation::StaticTriangulation(
     const UndiGraph*                   theGraph,
     const NodeProperty< Size >*        domsizes,
     const EliminationSequenceStrategy& elimSeq,
     const JunctionTreeStrategy&        JTStrategy,
     bool                               minimality) :
      Triangulation(domsizes),
      _elimination_sequence_strategy(elimSeq.newFactory()),
      _junction_tree_strategy(JTStrategy.newFactory()), __original_graph(theGraph),
      __minimality_required(minimality) {
    // for debugging purposes
    GUM_CONSTRUCTOR(StaticTriangulation);

    // if the graph is not empty, resize several structures in order to speed-up
    // their fillings.
    if (theGraph != nullptr) {
      __elim_order.resize(theGraph->size());
      __reverse_elim_order.resize(theGraph->size());
      __elim_cliques.resize(theGraph->size());
      __node_2_max_prime_clique.resize(theGraph->size());
      __added_fill_ins.resize(theGraph->size());
    }

    // register the triangulation to its junction tree strategy
    _junction_tree_strategy->setTriangulation(this);
  }

  // default constructor
  StaticTriangulation::StaticTriangulation(
     const EliminationSequenceStrategy& elimSeq,
     const JunctionTreeStrategy&        JTStrategy,
     bool                               minimality) :
      _elimination_sequence_strategy(elimSeq.newFactory()),
      _junction_tree_strategy(JTStrategy.newFactory()),
      __minimality_required(minimality) {
    // for debugging purposes
    GUM_CONSTRUCTOR(StaticTriangulation);

    // register the triangulation to its junction tree strategy
    _junction_tree_strategy->setTriangulation(this);
  }

  // copy constructor
  StaticTriangulation::StaticTriangulation(const StaticTriangulation& from) :
      Triangulation(from), __original_graph(from.__original_graph),
      __triangulated_graph(from.__triangulated_graph), __fill_ins(from.__fill_ins),
      __elim_order(from.__elim_order),
      __reverse_elim_order(from.__reverse_elim_order),
      __elim_cliques(from.__elim_cliques), __elim_tree(from.__elim_tree),
      __max_prime_junction_tree(from.__max_prime_junction_tree),
      __node_2_max_prime_clique(from.__node_2_max_prime_clique),
      __has_triangulation(from.__has_triangulation),
      __has_triangulated_graph(from.__has_triangulated_graph),
      __has_elimination_tree(from.__has_elimination_tree),
      __has_junction_tree(from.__has_junction_tree),
      __has_max_prime_junction_tree(from.__has_max_prime_junction_tree),
      __has_fill_ins(from.__has_fill_ins),
      __minimality_required(from.__minimality_required),
      __added_fill_ins(from.__added_fill_ins),
      __we_want_fill_ins(from.__we_want_fill_ins) {
    // copy the strategies
    _elimination_sequence_strategy =
       from._elimination_sequence_strategy->copyFactory();
    _junction_tree_strategy = from._junction_tree_strategy->copyFactory(this);

    // if from has a junction tree, copy it
    if (from.__junction_tree != nullptr) {
      __junction_tree = &(_junction_tree_strategy->junctionTree());
    }

    // for debugging purposes
    GUM_CONS_CPY(StaticTriangulation);
  }

  // move constructor
  StaticTriangulation::StaticTriangulation(StaticTriangulation&& from) :
      Triangulation(std::move(from)),
      _elimination_sequence_strategy(from._elimination_sequence_strategy),
      _junction_tree_strategy(from._junction_tree_strategy),
      __original_graph(from.__original_graph),
      __triangulated_graph(std::move(from.__triangulated_graph)),
      __fill_ins(std::move(from.__fill_ins)),
      __elim_order(std::move(from.__elim_order)),
      __reverse_elim_order(std::move(from.__reverse_elim_order)),
      __elim_cliques(std::move(from.__elim_cliques)),
      __elim_tree(std::move(from.__elim_tree)),
      __max_prime_junction_tree(std::move(from.__max_prime_junction_tree)),
      __node_2_max_prime_clique(std::move(from.__node_2_max_prime_clique)),
      __has_triangulation(from.__has_triangulation),
      __has_triangulated_graph(from.__has_triangulated_graph),
      __has_elimination_tree(from.__has_elimination_tree),
      __has_junction_tree(from.__has_junction_tree),
      __has_max_prime_junction_tree(from.__has_max_prime_junction_tree),
      __has_fill_ins(from.__has_fill_ins),
      __minimality_required(from.__minimality_required),
      __added_fill_ins(std::move(from.__added_fill_ins)),
      __we_want_fill_ins(from.__we_want_fill_ins) {
    // move the factories contained in from
    from._elimination_sequence_strategy = new DefaultEliminationSequenceStrategy;
    from._junction_tree_strategy = new DefaultJunctionTreeStrategy;
    _junction_tree_strategy->moveTriangulation(this);

    // if from has a junction tree, copy it
    if (from.__junction_tree != nullptr) {
      __junction_tree = &(_junction_tree_strategy->junctionTree());
    }

    // for debugging purposes
    GUM_CONS_MOV(StaticTriangulation);
  }

  StaticTriangulation::~StaticTriangulation() {
    // for debugging purposes
    GUM_DESTRUCTOR(StaticTriangulation);

    delete _elimination_sequence_strategy;
    delete _junction_tree_strategy;

    // no need to deallocate __original_graph nor __junction_tree because
    // they are not owned by the static triangulation class
  }

  void StaticTriangulation::clear() {
    // clear the factories
    _elimination_sequence_strategy->clear();
    _junction_tree_strategy->clear();

    // remove the current graphs
    __original_graph = nullptr;
    __junction_tree = nullptr;
    __triangulated_graph.clear();
    __elim_tree.clear();
    __max_prime_junction_tree.clear();
    __elim_cliques.clear();
    __node_2_max_prime_clique.clear();

    // remove all fill-ins and orderings
    __fill_ins.clear();
    __added_fill_ins.clear();
    __elim_order.clear();
    __reverse_elim_order.clear();

    // indicates that the junction tree, the max prime tree, etc, are updated
    __has_triangulation = true;
    __has_triangulated_graph = true;
    __has_elimination_tree = true;
    __has_junction_tree = true;
    __has_max_prime_junction_tree = true;
    __has_fill_ins = true;
  }

  // removes redondant fill-ins and compute proper elimination cliques and order
  void StaticTriangulation::__computeRecursiveThinning() {
    // apply recursive thinning (fmint, see Kjaerulff (90), "triangulation of
    // graphs - algorithms giving small total state space")
    NodeId                       node1, node2;
    bool                         incomplete;
    std::vector< NodeId >        adj;
    EdgeSet                      T_prime;
    NodeProperty< unsigned int > R(__triangulated_graph.size());

    for (const auto node : __triangulated_graph)
      R.insert(node, 0);

    // the FMINT loop
    if (__added_fill_ins.size() > 0) {
      for (auto iter = __added_fill_ins.rbegin(); iter != __added_fill_ins.rend();
           ++iter) {
        if (iter->size()) {
          // here apply MINT to T_i = __added_fill_ins[i]
          EdgeSet& T = *iter;

          // compute R: by default, R is equal to all the nodes in the graph
          // when R[...] >= j (see the j in the loop below), this means that the
          // node belongs to an extremal edge in R
          for (std::size_t k = 0; k < __elim_order.size(); ++k)
            R[__elim_order[k]] =
               0;   // WARNING: do not replace R[__elim_order[k]] by

          // R[k] because the node ids may not be
          // consecutive numbers

          // apply MINT while some edges can possibly be deleted
          bool require_mint = true;

          for (unsigned int j = 0; require_mint; ++j) {
            // find T' (it is equal to the edges (v,w) of T such that
            // the intersection of adj(v,G) and adj(w,G) is complete and such
            // that
            // v and/or w belongs to an extremal node in R
            for (const auto& edge : T) {
              node1 = edge.first();
              node2 = edge.second();

              // check if at least one extremal node belongs to R
              if ((R[node1] < j) && (R[node2] < j)) continue;

              // check if the intersection of adj(v,G) and adj(w,G) is a
              // complete subgraph
              if (__triangulated_graph.neighbours(node2).size()
                  < __triangulated_graph.neighbours(node1).size()) {
                NodeId tmp = node1;
                node1 = node2;
                node2 = tmp;
              }

              incomplete = false;

              // find the nodes belonging to the intersection of adj(v,G)
              // and adj(w,G)
              for (const auto adjnode : __triangulated_graph.neighbours(node1))
                if (__triangulated_graph.existsEdge(node2, adjnode))
                  adj.push_back(adjnode);

              // check if the intersection is complete
              for (unsigned int k = 0; k < adj.size() && !incomplete; ++k) {
                for (unsigned int m = k + 1; m < adj.size(); ++m) {
                  if (!__triangulated_graph.existsEdge(adj[k], adj[m])) {
                    incomplete = true;
                    break;
                  }
                }
              }

              adj.clear();

              if (!incomplete) {
                T_prime.insert(edge);
                R[node1] = j + 1;
                R[node2] = j + 1;
              }
            }

            // compute the new value of T (i.e. T\T') and update the
            // triangulated graph
            for (const auto& edge : T_prime) {
              T.erase(edge);
              __triangulated_graph.eraseEdge(edge);

              if (__has_fill_ins) __fill_ins.erase(edge);
            }

            if (T_prime.size() == 0) require_mint = false;

            T_prime.clear();
          }
        }
      }
    }

    // here, the recursive thinning has removed all the superfluous edges.
    // Hence some cliques have been split and, as a result, the elimination
    // order has changed. We should thus compute a new perfect ordering. To
    // do so, we use a Maximal Cardinality Search: it has indeed the nice
    // property that, when the graph is already triangulated, it finds a
    // compatible order of elimination that does not require any edge addition

    // a structure storing the number of neighbours previously processed
    PriorityQueue< NodeId, unsigned int, std::greater< unsigned int > >
       numbered_neighbours(std::greater< unsigned int >(),
                           __triangulated_graph.size());

    for (Size i = 0; i < __elim_order.size(); ++i)
      numbered_neighbours.insert(__elim_order[i], 0);

    // perform the maximum cardinality search
    if (__elim_order.size() > 0) {
      for (Size i = Size(__elim_order.size()); i >= 1; --i) {
        NodeId ni = NodeId(i - 1);
        NodeId node = numbered_neighbours.pop();
        __elim_order[ni] = node;
        __reverse_elim_order[node] = ni;

        for (const auto neighbour : __triangulated_graph.neighbours(node)) {
          try {
            numbered_neighbours.setPriority(
               neighbour, 1 + numbered_neighbours.priority(neighbour));
          } catch (NotFound&) {}
        }
      }
    }

    // here the elimination order is ok. We now need to update the
    // __elim_cliques
    for (Size i = 0; i < __elim_order.size(); ++i) {
      NodeSet& cliques = __elim_cliques.insert(__elim_order[i], NodeSet()).second;
      cliques << __elim_order[i];

      for (const auto neighbour : __triangulated_graph.neighbours(__elim_order[i]))
        if (__reverse_elim_order[neighbour] > i) cliques << neighbour;
    }
  }

  void StaticTriangulation::__computeEliminationTree() {
    // if there already exists an elimination tree, no need to compute it again
    if (__has_elimination_tree) return;

    // if the graph is not triangulated, triangulate it
    if (!__has_triangulation) __triangulate();

    // create the nodes of the elimination tree
    __elim_tree.clear();

    Size size = Size(__elim_order.size());
    for (NodeId i = NodeId(0); i < size; ++i)
      __elim_tree.addNode(i, __elim_cliques[__elim_order[i]]);

    // create the edges of the elimination tree: join a node to the one in
    // its clique that is eliminated first
    for (NodeId i = NodeId(0); i < size; ++i) {
      NodeId         clique_i_creator = __elim_order[i];
      const NodeSet& list_of_nodes = __elim_cliques[clique_i_creator];
      Idx            child = __original_graph->bound() + 1;

      for (const auto node : list_of_nodes) {
        Idx it_elim_step = __reverse_elim_order[node];

        if ((node != clique_i_creator) && (child > it_elim_step))
          child = it_elim_step;
      }

      if (child <= __original_graph->bound()) {
        // WARNING: here, we assume that the nodes of the elimination tree are
        // indexed from 0 to n-1
        __elim_tree.addEdge(i, child);
      }
    }

    __has_elimination_tree = true;
  }

  void StaticTriangulation::__computeMaxPrimeMergings(
     const NodeId        node,
     const NodeId        from,
     std::vector< Arc >& merged_cliques,
     NodeSet&            mark) const {
    mark << node;

    for (const auto other_node : __junction_tree->neighbours(node))
      if (other_node != from) {
        const NodeSet& separator = __junction_tree->separator(node, other_node);
        // check that the separator between node and other_node is complete
        bool complete = true;

        for (auto iter_sep1 = separator.begin();
             iter_sep1 != separator.end() && complete;
             ++iter_sep1) {
          auto iter_sep2 = iter_sep1;

          for (++iter_sep2; iter_sep2 != separator.end(); ++iter_sep2) {
            if (!__original_graph->existsEdge(*iter_sep1, *iter_sep2)) {
              complete = false;
              break;
            }
          }
        }

        // here complete indicates whether the separator is complete or not
        if (!complete) merged_cliques.push_back(Arc(other_node, node));

        __computeMaxPrimeMergings(other_node, node, merged_cliques, mark);
      }
  }

  void StaticTriangulation::__computeMaxPrimeJunctionTree() {
    // if there already exists a max prime junction tree, no need
    // to recompute it
    if (__has_max_prime_junction_tree) return;

    // if there is no junction tree, create it
    if (!__has_junction_tree) junctionTree();

    // the maximal prime subgraph join tree is created by aggregation of some
    // cliques. More precisely, when the separator between 2 cliques is not
    // complete in the original graph, then the two cliques must be merged.
    // Create a hashtable indicating which clique has been absorbed by some
    // other
    // clique.
    NodeProperty< NodeId > T_mpd_cliques(__junction_tree->size());

    for (const auto clik : __junction_tree->nodes())
      T_mpd_cliques.insert(clik, clik);

    // parse all the separators of the junction tree and test those that are not
    // complete in the orginal graph
    std::vector< Arc > merged_cliques;

    NodeSet mark;

    for (const auto clik : __junction_tree->nodes())
      if (!mark.contains(clik))
        __computeMaxPrimeMergings(clik, clik, merged_cliques, mark);

    // compute the transitive closure of merged_cliques. This one will contain
    // pairs (X,Y) indicating that clique X must be merged with clique Y.
    // Actually clique X will be inserted into clique Y.
    for (unsigned int i = 0; i < merged_cliques.size(); ++i) {
      T_mpd_cliques[merged_cliques[i].tail()] =
         T_mpd_cliques[merged_cliques[i].head()];
    }

    // now we can create the max prime junction tree. First, create the cliques
    for (const auto& elt : T_mpd_cliques)
      if (elt.first == elt.second)
        __max_prime_junction_tree.addNode(elt.second,
                                          __junction_tree->clique(elt.second));

    // add to the cliques previously created the nodes of the cliques that were
    // merged into them
    for (const auto& elt : T_mpd_cliques)
      if (elt.first != elt.second)
        for (const auto node : __junction_tree->clique(elt.first)) {
          try {
            __max_prime_junction_tree.addToClique(elt.second, node);
          } catch (DuplicateElement&) {}
        }

    // add the edges to the graph
    for (const auto& edge : __junction_tree->edges()) {
      NodeId node1 = T_mpd_cliques[edge.first()];
      NodeId node2 = T_mpd_cliques[edge.second()];

      if (node1 != node2) {
        try {
          __max_prime_junction_tree.addEdge(node1, node2);
        } catch (DuplicateElement&) {}
      }
    }

    // compute for each node which clique of the max prime junction tree was
    // created by the elimination of the node
    const NodeProperty< NodeId >& node_2_junction_clique =
       _junction_tree_strategy->createdCliques();

    for (const auto& elt : node_2_junction_clique)
      __node_2_max_prime_clique.insert(elt.first, T_mpd_cliques[elt.second]);

    __has_max_prime_junction_tree = true;
  }

  const UndiGraph& StaticTriangulation::triangulatedGraph() {
    if (!__has_triangulation) __triangulate();

    // if we did not construct the triangulated graph during the triangulation,
    // that is, we just constructed the junction tree, we shall construct it now
    if (!__has_triangulated_graph) {
      // just in case, be sure that the junction tree has been constructed
      if (!__has_junction_tree) junctionTree();

      // copy the original graph
      __triangulated_graph = *__original_graph;

      for (const auto clik_id : *__junction_tree) {
        // for each clique, add the edges necessary to make it complete
        const NodeSet&        clique = __junction_tree->clique(clik_id);
        std::vector< NodeId > clique_nodes(clique.size());
        unsigned int          i = 0;

        for (const auto node : clique) {
          clique_nodes[i] = node;
          i += 1;
        }

        for (i = 0; i < clique_nodes.size(); ++i) {
          for (unsigned int j = i + 1; j < clique_nodes.size(); ++j) {
            try {
              __triangulated_graph.addEdge(clique_nodes[i], clique_nodes[j]);
            } catch (DuplicateElement&) {}
          }
        }
      }

      __has_triangulated_graph = true;
    }

    return __triangulated_graph;
  }

  // initialize the triangulation algorithm for a new graph
  void StaticTriangulation::setGraph(const UndiGraph*            graph,
                                     const NodeProperty< Size >* domsizes) {
    // remove the old graph
    clear();

    if (graph != nullptr) {
      // prepare the size of the data for the new graph
      __elim_order.resize(graph->size());
      __reverse_elim_order.resize(graph->size());
      __elim_cliques.resize(graph->size());
      __added_fill_ins.resize(graph->size());
      __node_2_max_prime_clique.resize(graph->size());
    }

    // keep track of the variables passed in argument
    __original_graph = graph;
    _domain_sizes = domsizes;

    // indicate that no triangulation was performed for this graph
    __has_triangulation = false;
    __has_triangulated_graph = false;
    __has_elimination_tree = false;
    __has_junction_tree = false;
    __has_max_prime_junction_tree = false;
    __has_fill_ins = false;
  }

  void StaticTriangulation::__triangulate() {
    // if the graph is already triangulated, no need to triangulate it again
    if (__has_triangulation) return;

    // copy the graph to be triangulated, so that we can modify it
    UndiGraph tmp_graph = *__original_graph;

    _initTriangulation(tmp_graph);
    _elimination_sequence_strategy->askFillIns(__we_want_fill_ins);

    // if we are to do recursive thinning, we will have to add fill-ins to the
    // triangulated graph each time we eliminate a node. Hence, we shall
    // initialize the triangulated graph by a copy of the original graph
    if (__minimality_required) {
      __triangulated_graph = *__original_graph;
      __has_triangulated_graph = true;
    }

    // perform the triangulation
    NodeId removable_node = 0;

    for (unsigned int nb_elim = 0; tmp_graph.size() != 0; ++nb_elim) {
      removable_node = _elimination_sequence_strategy->nextNodeToEliminate();

      // when minimality is not required, i.e., we won't apply recursive
      // thinning, the cliques sets can be computed
      if (!__minimality_required) {
        NodeSet& cliques = __elim_cliques.insert(removable_node, NodeSet()).second;
        cliques.resize(tmp_graph.neighbours(removable_node).size() / 2);
        cliques << removable_node;
        for (const auto clik : tmp_graph.neighbours(removable_node))
          cliques << clik;
      } else {
        // here recursive thinning will be applied, hence we need store the
        // fill-ins added by the current node removal
        EdgeSet& current_fill = __added_fill_ins[nb_elim];
        NodeId   node1, node2;

        const NodeSet& nei = tmp_graph.neighbours(removable_node);

        for (auto iter_edges1 = nei.begin(); iter_edges1 != nei.end();
             ++iter_edges1) {
          node1 = *iter_edges1;
          auto iter_edges2 = iter_edges1;

          for (++iter_edges2; iter_edges2 != nei.end(); ++iter_edges2) {
            node2 = *iter_edges2;
            Edge edge(node1, node2);

            if (!tmp_graph.existsEdge(edge)) {
              current_fill.insert(edge);
              __triangulated_graph.addEdge(node1, node2);
            }
          }
        }
      }

      // if we want fill-ins but the elimination sequence strategy does not
      // compute them, we shall compute them here
      if (__we_want_fill_ins
          && !_elimination_sequence_strategy->providesFillIns()) {
        NodeId node1, node2;

        // add edges between removable_node's neighbours in order to create
        // a clique
        const NodeSet& nei = tmp_graph.neighbours(removable_node);

        for (auto iter_edges1 = nei.begin(); iter_edges1 != nei.end();
             ++iter_edges1) {
          node1 = *iter_edges1;
          auto iter_edges2 = iter_edges1;

          for (++iter_edges2; iter_edges2 != nei.end(); ++iter_edges2) {
            node2 = *iter_edges2;
            Edge edge(node1, node2);

            if (!tmp_graph.existsEdge(edge)) { __fill_ins.insert(edge); }
          }
        }

        // delete removable_node and its adjacent edges
        tmp_graph.eraseNode(removable_node);
      }

      // inform the elimination sequence that we are actually removing
      // removable_node (this enables, for instance, to update the weights of
      // the nodes in the graph)
      _elimination_sequence_strategy->eliminationUpdate(removable_node);

      if (!_elimination_sequence_strategy->providesGraphUpdate()) {
        NodeId node1, node2;

        const NodeSet& nei = tmp_graph.neighbours(removable_node);

        for (auto iter_edges1 = nei.begin(); iter_edges1 != nei.end();
             ++iter_edges1) {
          node1 = *iter_edges1;
          auto iter_edges2 = iter_edges1;

          for (++iter_edges2; iter_edges2 != nei.end(); ++iter_edges2) {
            node2 = *iter_edges2;
            Edge edge(node1, node2);

            if (!tmp_graph.existsEdge(edge)) { tmp_graph.addEdge(node1, node2); }
          }
        }

        tmp_graph.eraseNode(removable_node);
      }

      // update the elimination order
      __elim_order[nb_elim] = removable_node;
      __reverse_elim_order.insert(removable_node, nb_elim);
    }

    // indicate whether we actually computed fill-ins
    __has_fill_ins = __we_want_fill_ins;

    // if minimality is required, remove the redondant edges
    if (__minimality_required) __computeRecursiveThinning();

    __has_triangulation = true;
  }

  const EdgeSet& StaticTriangulation::fillIns() {
    // if we did not compute the triangulation yet, do it and commpute
    // the fill-ins at the same time
    if (!__has_triangulation) {
      bool want_fill_ins = __we_want_fill_ins;
      __we_want_fill_ins = true;
      __triangulate();
      __we_want_fill_ins = want_fill_ins;
      __has_fill_ins = true;
    }

    // here, we already computed a triangulation and we already computed
    // the fill-ins, so return them
    if (__has_fill_ins) {
      if (_elimination_sequence_strategy->providesFillIns())
        return _elimination_sequence_strategy->fillIns();
      else
        return __fill_ins;
    } else {
      // ok, here, we shall compute the fill-ins as they were not precomputed
      if (!__original_graph) return __fill_ins;

      // just in case, be sure that the junction tree has been constructed
      if (!__has_junction_tree) junctionTree();

      for (const auto clik_id : __junction_tree->nodes()) {
        // for each clique, add the edges necessary to make it complete
        const NodeSet&        clique = __junction_tree->clique(clik_id);
        std::vector< NodeId > clique_nodes(clique.size());
        unsigned int          i = 0;

        for (const auto node : clique) {
          clique_nodes[i] = node;
          i += 1;
        }

        for (i = 0; i < clique_nodes.size(); ++i) {
          for (unsigned int j = i + 1; j < clique_nodes.size(); ++j) {
            Edge edge(clique_nodes[i], clique_nodes[j]);

            if (!__original_graph->existsEdge(edge)) {
              try {
                __fill_ins.insert(edge);
              } catch (DuplicateElement&) {}
            }
          }
        }
      }

      return __fill_ins;
    }
  }

  void StaticTriangulation::_initTriangulation(UndiGraph& graph) {
    _elimination_sequence_strategy->setGraph(&graph, _domain_sizes);
  }

} /* namespace gum */