Conjugation of Trees and Random Maps
LaBRI, UniversitÚ Bordeaux I
February 1, 1999
[summary by Alain Denise]
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We present a general scheme to generate uniformly at random planar
maps of various kinds. Our algorithms rely on a combinatorial and
bijective approach: we encode the planar maps by some classes of
particular trees. This encoding is the basis of efficient random
generators of maps of the most simple classes of planar maps. For more
complex structures, we proceed by using a new general probabilistic
scheme, that we call extraction/rejection algorithm.
A planar map is an embedding of a graph in the plane, considered
up to continuous deformations of the plane. Maps are allowed to have
multiple edges and loops; otherwise there are called simple
maps. The maps we consider are rooted: there is an oriented
edge, called the root.
We present here two families of algorithms for the random generation
of rooted planar maps. The first one applies to the few classes of
maps which are counted by multiplicative closed formulas. Here are
two examples from [2, 5]:
Our approach consists in finding new bijections between these maps and
some particular families of trees. More precisely, there exists, for
each family of maps as above, a family of trees such that
and the bijection is given by the closure of the trees (terms
like free leaves and closure will be explained below).
Then the generation scheme is the following: i) generate uniformly at random a
tree, by known efficient algorithms; ii) ``decode'' the tree to
get its associated map. Table 1 gives the basic
families that benefit from this approach, leading to linear generation
algorithms. The method is illustrated in Section 2
on the family of bipartite cubic maps.
|Number of planar maps with n edges:
|Number of bipartite cubic maps with 3n vertices:
However, a number of families do not present simple formulas as
above. In these cases we use a new method: extraction/rejection. It combines the usual rejection principle with
composition schemes that are common for maps. See in
Table 2 the new families reached with this new
approach, which is presented in Section 3.
||quartic (all degree 4)
|bipartite (or Eulerian)
||quartic without 2-cocycle
||non separable cubic
Table 1: Maps (with n edges, loops and multiple edges allowed)
generated in linear time
Table 2: Some extra properties reachable via extraction/rejection. (Non
separable means loopless 2-connected, smooth means without vertices of
2 Bijective Approach
We present here the method through the example of bipartite cubic
maps: maps whose vertices have degree 3 and are colored in black or
white in such a way that no edge joins two vertices of the same color
These maps are in bijection with blossom trees, as we will see
below. A planted plane binary tree is a plane binary tree whose
root has degree one. Figure 2 presents
such a tree (the leaves are omitted in the figure, except the root
leaf at the left). A blossom tree is a particular tree which can
be constructed from a planted plane tree as follows: in the middle of
each internal edge, we put a white-colored vertex with a bud on
one of the two sides, as in Figure 3.
Figure 1: A bipartite cubic map with 17 Î 2 vertices
Now we call partial closure of a blossom tree the structure
obtained after the following treatment: turn anti-clockwise around the
tree, and join each bud to the nearest leaf so that no crossing
occurs. (In this way, buds and leaves can be seen as a system of
nested parentheses.) See in Figure 4 the
partial closure of the blossom tree of Figure 3.
Since there are 3 more leaves than buds in any blossom tree, there
are necessarily 3 leaves which remain alone; we call them single leaves. We say that a blossom tree is balanced if its
root is a single leaf in its partial closure. Finally the complete closure of a blossom tree is obtained by joining the three
single leaves to a new vertex and rooting the resulting map towards
the root leaf (Figure 5). This gives the map
of Figure 1.
Figure 2: A planted plane tree with 17 internal nodes
Figure 3: A blossom tree
Figure 4: Partial closure of the blossom tree
Figure 5: Complete closure of the blossom tree
The complete closure defines a one-to-one correspondence between
balanced blossom trees with n nodes and bipartite cubic maps with
Now here is the way to generate uniformly at random a bipartite cubic map with
generate uniformly at random a planted binary tree with n nodes; this can be
done in linear complexity with well known algorithms (see
 for example);
- toss a coin independently for each edge to place buds;
- choose with probability 1/3 one of the three possible balanced
blossom trees and achieve the complete closure.
3 Extraction/Rejection Algorithms
Suppose that we have to generate uniformly at random a bicolored triangulation
like the one in Figure 6, with a given
number of faces.
By duality, this is equivalent to generate a bipartite cubic map as
the one in Figure 7: each black
(resp. white) vertex in the map gives a black (resp. white) triangle
in the triangulation, each face gives a vertex.
But the map must be without separator, i.e., without configuration like
the one on the left of Figure 8; otherwise the
associated triangulation might present multiple edges, like in the
right of the figure.
Figure 6: A bicolored triangulation
Figure 7: A bipartite cubic map without separator
A natural idea would be to use a rejection algorithm in order to
generate maps without separators: draw uniformly at random bipartite cubic maps
until we get a map without separator. Unfortunately, bipartite cubic
maps without separators are very rare, so this approach is
practically intractable. So we need another idea. Here is the
An important problem appears here: if we generate a map with n
vertices and remove the separators, the resulting map is likely to
have less than n vertices! To fix this problem, we will generate
maps with too many vertices: we can prove, using asymptotic
analysis, that drawing bipartite cubic maps with m=3n vertices
gives, with a good probability, cores having n vertices. This leads
to an algorithm with complexity O(n5/3).
Generate uniformly at random a bipartite cubic map with the algorithm of
- remove the possible separators (we get what we call the core of the map);
- construct the triangulation from the core by duality.
This approach can be generalized for a number of families, as seen in
Table 2. Details are given in
Figure 8: A separator in a bipartite cubic map and its result by
Figure 9: A map with separators
Alonso (Laurent) and Schott (RenÚ). --
Random generation of trees. --
Kluwer academic publishers, 1995.
Brown (W. G.) and Tutte (W. T.). --
On the enumeration of rooted non-separable planar maps. Canadian
Journal of Mathematics, vol. 16, 1964, pp. 572--577.
Schaeffer (Gilles). --
Conjugaison d'arbres et cartes combinatoires alÚatoires. --
PhD thesis, UniversitÚ Bordeaux I, 1998.
Schaeffer (Gilles). --
Random sampling of large planar maps and convex polyhedra. In Proceedings of STOC'99. --
Tutte (W. T.). --
On the enumeration of planar maps. Bulletin of the American
Mathematical Society, vol. 74, 1968, pp. 64--74.
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