diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/Containers_filler.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/Containers_filler.h
index 2cbb2949017..9e9d5422024 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/Containers_filler.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/internal/Containers_filler.h
@@ -14,11 +14,13 @@
 
 #include <CGAL/license/Surface_mesh_parameterization.h>
 
-#include <CGAL/disable_warnings.h>
+#include <CGAL/boost/graph/internal/initialized_index_maps_helpers.h>
+#include <CGAL/Polygon_mesh_processing/connected_components.h>
 
-#include "boost/tuple/tuple.hpp"
+#include <boost/tuple/tuple.hpp>
 #include <boost/unordered_set.hpp>
 #include <boost/graph/graph_traits.hpp>
+
 #include <vector>
 
 namespace CGAL {
@@ -56,7 +58,7 @@ public:
     : mesh(mesh_), vertices(vertices_), faces(nullptr)
   { }
 
-  void operator()(face_descriptor fd)
+  void operator()(const face_descriptor fd)
   {
     halfedge_descriptor hd = halfedge(fd, mesh);
     for(vertex_descriptor vd : vertices_around_face(hd, mesh)) {
@@ -78,10 +80,10 @@ struct Index_map_filler
     : mesh(mesh), map(&map), index(0)
   { }
 
-  void operator()(const face_descriptor& fd)
+  void operator()(const face_descriptor fd)
   {
-    for(vertex_descriptor vd :
-                  vertices_around_face(halfedge(fd, mesh), mesh)) {
+    for(vertex_descriptor vd : vertices_around_face(halfedge(fd, mesh), mesh))
+    {
       typename Map::iterator it;
       bool new_element;
       boost::tie(it,new_element) = map->insert(std::make_pair(vd,1));
@@ -96,12 +98,47 @@ struct Index_map_filler
   int index;
 };
 
+template <typename TriangleMesh, typename VertexIndexMap>
+void fill_index_map_of_cc(const typename boost::graph_traits<TriangleMesh>::halfedge_descriptor bhd,
+                          const TriangleMesh& mesh,
+                          VertexIndexMap vimap)
+{
+  namespace PMP = CGAL::Polygon_mesh_processing;
+
+  typedef typename boost::graph_traits<TriangleMesh>::vertex_descriptor        vertex_descriptor;
+  typedef typename boost::graph_traits<TriangleMesh>::face_descriptor          face_descriptor;
+
+  std::vector<face_descriptor> CC_faces;
+
+  // 'reserve' might cause a huge amount of memory to be used for a tiny CC,
+  // but if this is a problem as a user, one could simply parameterize a Face_filtered_graph instead.
+  CC_faces.reserve(num_faces(mesh));
+
+  PMP::connected_component(face(opposite(bhd, mesh), mesh), mesh, std::back_inserter(CC_faces));
+
+  // If all vertices are involved, avoid walking all the faces
+  if(CC_faces.size() == faces(mesh).size())
+  {
+    BGL::internal::Index_map_initializer<VertexIndexMap, TriangleMesh> id_initializer;
+    id_initializer(CGAL::internal_np::vertex_index, vimap, mesh);
+  }
+  else
+  {
+    for(vertex_descriptor v : vertices(mesh))
+      put(vimap, v, -1);
+
+    int index = 0;
+    for(face_descriptor f : CC_faces)
+      for(vertex_descriptor v : vertices_around_face(halfedge(f, mesh), mesh))
+        if(get(vimap, v) == -1)
+          put(vimap, v, index++);
+  }
+}
+
 } // namespace internal
 
 } // namespace Surface_mesh_parameterization
 
 } // namespace CGAL
 
-#include <CGAL/enable_warnings.h>
-
 #endif // CGAL_SURFACE_MESH_PARAMETERIZATION_INTERNAL_CONTAINERS_FILLER_H
diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/parameterize.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/parameterize.h
index 53991908393..fee4f1ed94f 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/parameterize.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/parameterize.h
@@ -15,19 +15,10 @@
 
 #include <CGAL/license/Surface_mesh_parameterization.h>
 
-#include <CGAL/disable_warnings.h>
-
-#include <CGAL/Surface_mesh_parameterization/internal/Bool_property_map.h>
-
 #include <CGAL/Surface_mesh_parameterization/Error_code.h>
 #include <CGAL/Surface_mesh_parameterization/Mean_value_coordinates_parameterizer_3.h>
 
-#include <CGAL/Polygon_mesh_processing/connected_components.h>
-
-#include <boost/function_output_iterator.hpp>
 #include <boost/property_map/property_map.hpp>
-#include <boost/unordered_set.hpp>
-#include <boost/unordered_map.hpp>
 
 /// \file parameterize.h
 
@@ -57,7 +48,7 @@ namespace Surface_mesh_parameterization {
 /// \param mesh a triangulated surface.
 /// \param parameterizer a parameterizer.
 /// \param bhd a halfedge descriptor on the boundary of `mesh`.
-/// \param uvm an instanciation of the class `VertexUVmap`.
+/// \param uvmap an instanciation of the class `VertexUVmap`.
 ///
 /// \pre `mesh` must be a triangular mesh.
 /// \pre The mesh border must be mapped onto a convex polygon
@@ -68,23 +59,18 @@ template <class TriangleMesh, class Parameterizer, class HD, class VertexUVmap>
 Error_code parameterize(TriangleMesh& mesh,
                         Parameterizer parameterizer,
                         HD bhd,
-                        VertexUVmap uvm)
+                        VertexUVmap uvmap)
 {
-  typedef typename boost::graph_traits<TriangleMesh>::vertex_descriptor vertex_descriptor;
+  typedef CGAL::dynamic_vertex_property_t<int>                                 Vertex_int_tag;
+  typedef typename boost::property_map<TriangleMesh, Vertex_int_tag>::type     Vertex_int_map;
+  Vertex_int_map vimap = get(Vertex_int_tag(), mesh);
+  internal::fill_index_map_of_cc(bhd, mesh, vimap);
 
-  typedef boost::unordered_map<vertex_descriptor, int> Indices;
-  Indices indices;
-  CGAL::Polygon_mesh_processing::connected_component(
-         face(opposite(bhd, mesh), mesh),
-         mesh,
-         boost::make_function_output_iterator(
-         internal::Index_map_filler<TriangleMesh, Indices>(mesh, indices)));
-  boost::associative_property_map<Indices> vipm(indices);
+  typedef CGAL::dynamic_vertex_property_t<bool>                                Vertex_bool_tag;
+  typedef typename boost::property_map<TriangleMesh, Vertex_bool_tag>::type    Vertex_bool_map;
+  Vertex_bool_map vpmap = get(Vertex_bool_tag(), mesh);
 
-  boost::unordered_set<vertex_descriptor> vs;
-  internal::Bool_property_map<boost::unordered_set<vertex_descriptor> > vpm(vs);
-
-  return parameterizer.parameterize(mesh, bhd, uvm, vipm, vpm);
+  return parameterizer.parameterize(mesh, bhd, uvmap, vimap, vpmap);
 }
 
 /// \ingroup  PkgSurfaceMeshParameterizationMainFunction
@@ -106,7 +92,7 @@ Error_code parameterize(TriangleMesh& mesh,
 ///
 /// \param mesh a triangulated surface.
 /// \param bhd a halfedge descriptor on the boundary of `mesh`.
-/// \param uvm an instanciation of the class `VertexUVmap`.
+/// \param uvmap an instanciation of the class `VertexUVmap`.
 ///
 /// \pre `mesh` must be a triangular mesh.
 /// \pre The vertices must be indexed (vimap must be initialized).
@@ -114,16 +100,14 @@ Error_code parameterize(TriangleMesh& mesh,
 template <class TriangleMesh, class HD, class VertexUVmap>
 Error_code parameterize(TriangleMesh& mesh,
                         HD bhd,
-                        VertexUVmap uvm)
+                        VertexUVmap uvmap)
 {
   Mean_value_coordinates_parameterizer_3<TriangleMesh> parameterizer;
-  return parameterize(mesh, parameterizer, bhd, uvm);
+  return parameterize(mesh, parameterizer, bhd, uvmap);
 }
 
 } // namespace Surface_mesh_parameterization
 
 } // namespace CGAL
 
-#include <CGAL/enable_warnings.h>
-
 #endif // CGAL_PARAMETERIZE_H
diff --git a/Surface_mesh_parameterization/test/Surface_mesh_parameterization/extensive_parameterization_test.cpp b/Surface_mesh_parameterization/test/Surface_mesh_parameterization/extensive_parameterization_test.cpp
index 2b5d8aa72bb..7a10646669f 100644
--- a/Surface_mesh_parameterization/test/Surface_mesh_parameterization/extensive_parameterization_test.cpp
+++ b/Surface_mesh_parameterization/test/Surface_mesh_parameterization/extensive_parameterization_test.cpp
@@ -143,17 +143,15 @@ int main(int, char**)
                             boost::hash<PM_vertex_descriptor> > > uvpm(uvhm);
 
     // Indices map
-    typedef boost::unordered_map<PM_vertex_descriptor, int> Indices;
-    Indices indices;
-    PMP::connected_component(face(opposite(hd, pm), pm), pm,
-                             boost::make_function_output_iterator(
-                               SMP::internal::Index_map_filler<PMesh, Indices>(pm, indices)));
-
-    boost::associative_property_map<Indices> vipm(indices);
+    typedef CGAL::dynamic_vertex_property_t<int>                                 Vertex_int_tag;
+    typedef typename boost::property_map<PMesh, Vertex_int_tag>::type            Vertex_int_map;
+    Vertex_int_map vipm = get(Vertex_int_tag(), pm);
+    CGAL::Surface_mesh_parameterization::internal::fill_index_map_of_cc(hd, pm, vipm);
 
     // Vertex parameterized map
-    boost::unordered_set<PM_vertex_descriptor> vs;
-    SMP::internal::Bool_property_map<boost::unordered_set<PM_vertex_descriptor> > vpm(vs);
+    typedef CGAL::dynamic_vertex_property_t<bool>                                Vertex_bool_tag;
+    typedef typename boost::property_map<PMesh, Vertex_bool_tag>::type           Vertex_bool_map;
+    Vertex_bool_map vpm = get(Vertex_bool_tag(), pm);
 
     // Parameterizer
     SMP::ARAP_parameterizer_3<PMesh> parameterizer;