Tiling pictures of the plane with dominoes

AbstractWe consider the problem of tiling with dominoes pictures of the plane, in theoretical and algorithmic aspects. For generalities and other tiling problems, see for example Refs. Beauquier et al. (1995), Conway and Lagarias (1990), Kannan and Soroker (1992), Kenyon (1992), and Beauquier (1991). The pictures which are considered here may have holes, but uniquely balanced holes, that is every hole, if chessboard-like coloured, has an equal number of black squares and of white ones. We give an algorithmic characterization of tilable pictures and a canonical decomposition into ‘strongly’ tilable subpictures. The given algorithm is linear as far the considered pictures have a finite number of (balanced) holes. In the same hypothesis there is a good parallel algorithm (in class NC). Graphical extension of the used method (heights' method) is applied to a class of bipartite planar graphs. The particular case of without holes pictures is developed in Fournier (1996).As far as I know, the results in this paper are new, except the notions and the theorem in Section 2, which are substantially present in Thurston (1990)

Hammock-on-ears decomposition: a technique for the efficient parallel solution of shortest paths and other problems

We show how to decompose efficiently in parallel {\em any} graph into a number, $\tilde{\gamma}$, of outerplanar subgraphs (called {\em hammocks}) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G.~Frederickson and the parallel ear decomposition technique, thus we call it the {\em hammock-on-ears decomposition}. We mention that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in $O(\log n\log\log n)$ time using $O(n+m)$ CREW PRAM processors, for an $n$-vertex, $m$-edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of {\em sparse (di)graphs}. This class consists of all (di)graphs which have a $\tilde{\gamma}$ between $1$ and $\Theta(n)$, and includes planar graphs and graphs with genus $o(n)$. We improve previous bounds for certain instances of shortest paths and related problems, in this class of graphs. These problems include all pairs shortest paths, all pairs reachability