Uniform random sampling of planar graphs in linear time

Abstract

This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon, and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Giménez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost; and the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for uniform approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with n vertices, which was a little over O(n 7 ).

An Experimental Study on Generating Planar Graphs

“…generator complexity complete uniform Fusy [13] O(n) yes yes Intersection [17] O(n log n) yes no * Kuratowski [17] O(n 2 log n) yes no * Markov [8] O(n 3 ) yes yes generator complexity complete uniform CHT [17] O(n + m) no * no Delaunay [17] …”

“…Fusy. The planar graph generator developed by Fusy [13] is based on the principles of a Boltzmann Sampler [11]. Labeled graphs of size n are drawn uniformly at random.…”

“…As the sheer number of planar graphs is way too large it is obvious to ask for a representative sample. Thus, planar graphs that are generated uniformly at random [13,8,5] may be an appropriate choice. The implementation of such a generator is a challenging issue.…”

Planar Preprocessing for Spring Embedders

“…We use six libraries of connected planar graphs for our experiments: I Prüfer trees (PRUFER) a set of uniformly sampled labeled trees obtained by using Prüfer's algorithm [12]; II Random trees (R-TREES) a set of random trees with unbounded maximum degree constructed by adding new pendant edges to randomly selected nodes in the tree constructed so far; III Fusy graphs (FUSY) a set of uniformly sampled unlabeled graphs created using the Fusy generator [8]; IV Expansion graphs (EXPAN) a set of random triconnected planar graphs using the expansion method that performs n − 4 split operations (starting with a K 4 ) on randomly selected nodes that randomly distributes neighbors between split nodes; V Rome graphs (ROME) a subset of the undirected Rome graphs from the GDToolkit consisting of all connected planar graphs; and VI AT&T graphs (AT&T) a subset of the directed AT&T graphs (also known as the Graph Catalog) consisting of all connected planar graphs. Libraries I-IV were constructed to each have 10 graphs per value n, whereas, V and VI have a variable number of graphs per value of n; see Table 1.…”

“…Hence, we would expect that a library consisting of uniformly sampled unlabeled trees would have a higher maximum degree on average than is the case with either PRUFER or R-TREES. The graphs in FUSY were constructed with the Fusy generator referred to in [8] written in Java. Unfortunately, this is a not an n-generator like the ones used to construct either R-TREES or EXPAN libraries where one only specifies the number of nodes n, and has an n-node tree or graph returned.…”