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)

Fournier, J.C.

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ca __ __ l!iB DISCRETE MATHEMATICS 
ELSEVIER Discrete Mathematics 165 166 (1997) 3 13 320 
Tiling pictures of the plane with dominoes 
J.C. Fournier” 
Abstract 
We 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 
haltrnced 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 
(Izeiyhts’ mc~thf) 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). 
1. The problem 
We first consider the problem of tiling with dominoes bvithout holes pictures of the 
plane. Here, a picture is a subset of unit squares of the plane (considered as 2 x Z), see. 
for example, Fig. 2(b). 
It is easy to see that a necessary condition for the existence of a tiling for 
a chessboard-like coloured picture is that there is in the picture an equal number 01 
black squares and of white ones (~&UK& picture). 
This condition is not sufficient as we can see, while analysing the picture in Fig. 1. 
More deeply. the impossibility of tiling this picture is due to the existence of the line 
LEVY which delimits a subpicture, under ahc, with the following properties: 
(1) there are more black squares than white ones, 
(2) the line ubc runs alongside white squares of the subpicture. 
* E-mall: fournier~~iuniv-parisl2.fr. 
OOl?-365X:‘97’$17.00 Copyright c 1997 Elsewer Science B.V. All ri_ghts reserved 
PII SOO12-365X(96)00179-3 
314 J.C. Foumier / Discrete Mathematics 1651166 (1997) 313-320 
Fig. 1. Non tilable balanced picture and obstruction to tiling. 
Such a subpicture is called an obstruction to tiling. If a picture has no obstruction to 
tiling, then it is tilable. In fact, this condition corresponds to the classical Hall’s 
condition for matchings in bipartite graphs. We shall express this condition in another 
way. 
2. Characterization of tilable without holes pictures 
The boundary cycle of a without holes picture F determines a function h on its 
vertices, up to constants: if (SO, sl, . . . , sp) is the sequence of vertices of the boundary 
cycle, with sP = so, let us define: 
- h(s,) = k, where k is some integer, 
-for i = 1, . . . ,p: 
h(si) = 
i 
h(si_ 1) + 1 if the edge Si_ 1 Si delimits a white square of F, 
h(si_ 1) - 1 otherwise. 
Such a function will be called a height function on the boundary of the picture. 
Let F be a picture and C its boundary. We associate with F the white on the 
left-directed graph G in the following way: the vertices of G are the vertices of F and 
the arcs of G correspond to edges of F not in C and oriented in such a way that a white 
square is always on the left-hand side of each arc. We denote d(s, t) the distance from 
a vertex s to a vertex t in G (the length of a shortest path from s to t in G). 
We define a function h on the vertices of F, associated with a height function g on 
the boundary C of F, in the following way: 
For every vertex t of F, h(t) = min{g(s) + d(s, t)/ s vertex of C>. Such a function is 
called a height function on the picture. 
Note that for every vertex t of C we have h(t) < g(t). The inequality h(t) < g(t) is 
possible and it corresponds to the existence of an obstruction to tiling in the picture. 
The main result is: 
J.C. FournieriDiscrete Mathematics 165/l&5 11997) 313-320 315 
Theorem. Let F be a picture, C the boundary of F, g a heightfunction on C, h a height 
function on F ussociated with g. The picture is tilable lf and only, iffor every s E C. we 
have h(s) = g(s). Moreover, ifF is tilahle. the edges tt’ such that 1 h(t) - h(t) 1 = 1 define 
a tiling of F. 
Example. See Fig. 2. 
(a) height function 
- 
, 
I 
(b) deduced tiling 
Fig. 2. Tdable picture. 
316 J.C. Fourniev/Discrete Mathematics 165/166 (1997;) 313-320 
This theorem is informally given in [12], with a proof based on combinatorial 
group theory. We give in [S] a different proof, based on Hall’s condition for matchings 
in bipartite graphs. 
3. Algorithmic point of view 
This result allows us to find again the linear Thurston’s algorithm [12]. 
In fact, the height function is equal to a distance in an extension of the white on the 
left graph (for details on this extension, see [S]). It is then possible to calculate the 
height function h on F, and so to calculate a tiling of F, if the condition h(s) = g(s) for 
every s E C is true, by a classical Breadth First Search algorithm (note that the height 
function g on C is easy to calculate). The whole algorithm is linear. 
More interesting is the possibility of a parallelization of this algorithm, by means of 
a parallelization of Breadth first search, which is the main part of the algorithm. 
A classical parallel algorithm views the graph as an incidence matrix M and repeat- 
edly squares M, returning M” and so the BFS numbering of the graph. This yields an 
algorithm using O(log n) time and n3 processors on the concurrent-read concurrent- 
write (CRCW) parallel random access machine (PRAM). An improved parallel 
algorithm that computes the BFS numbering of a directed graph is given in [6]. 
Finally, we have an efficient algorithm for tiling pictures without holes (this problem is 
in the class NC). This last result is rather unexpected, apart from the new point of view 
given here. 
4. Canonical decomposition of tilings 
Let F be a without holes tilable picture with boundary C. We define a fracture path of 
F as a path in the white on the left-directed graph from a vertex s E C to a vertex s’ E C 
such that g(s’) = g(s) + 1, where I is the length of the path. A fracture edge is a picture’s 
edge which is on a fracture path, and thefiacture graph is the graph induced by fracture 
edges. It is to note that a fracture edge is never covered by a domino in a picture’s tiling. 
We define a strongly tilable picture as a tilable picture for which every edge, not on 
the boundary, can be covered in some tiling. Clearly, if a strongly tilable picture has 
a number of squares > 2, it is not tilable in an unique way. 
The main result here is: 
Theorem. The fracture graph of a without holes tilable picture divides the picture into 
without holes and strongly tilable subpictures. Furthermore, each picture’s tiling breaks 
up into subpicture’s tilings. 
We deduce many corollaries of this result, in particular, a characterization of 
uniqueness case of tiling: 
J.C. Fournier,/Discvete Mathematics 165’166 (1997) 313-320 : I I 
Corollary. A without holes picture is uniquely tilable if and only if its ,fracture graph 
divides it into dominoes. 
There is an interesting property of strongly tilable pictures which comes from 
properties of ‘elementary bipartite graphs’ (given in [9]): F or any tchite cuse und trn~’ 
bltrck square there exists a tiling of the picture which does not cover just these two cc~ses. 
5. Case of pictures with holes 
We can extend our algorithm to the case of pictures with balanced holes: that is. 
every hole, if chessboard-like coloured. has an equal number of black squares and of 
white ones. Let CO, C,, , C, be the boundaries cycles of the picture, C,, being the 
outer boundary cycle. For the calculation of a height function the inner boundaries 
cycles C,, . , C, are supposed to be clockwise oriented and CO is supposed to be 
anticlockwise oriented. 
The algorithm is based on two procedures. 
~~ Boundurie.s_,fit: For each boundary Ci with some given value on some of its 
vertices, this procedure calculates a height function on this boundary such that for 
every vertex s E Ci with a given value U, we have h(s) d c: and we have equality for 
some vertex of Ci (to do that, calculate any height function on Ci, and adjust it by 
means of a constant). 
~~- InsidePpropagation: This procedure calculates a height function on the picture as in 
the case of without holes pictures, that is from the values on the boundaries and in the 
white on the left-directed graph G, by means of the formula 
/r(t) = min (g(s) + d(.s, t)/s vertex of C), 
where d is the distance in G and C = COl,C1 _ i C, 
Algorithm 
begin 
fix some value on some vertex of CO; 
do p + 1 times 
houndaries_jit ; 
inside_propagation 
end do 
end. 
Result. The picture is tilable ifand only if the last iteration of‘the algorithm has no e#ect 
on the values of the height function. When the picture is tilable, we get a tiliny as in the 
case of without holes pictures, that is with edges tt’ such that 1 h(t) - h( t’) 1 = 1, where h is 
the obtained height function on the fiyure. 
318 J.C. FournierlDiscrete Mathematics 165/166 (1997) 313-320 
6. Weighted digraph associated with a picture 
Let there be given a picture, we consider the white on the left digraph augmented by 
the oriented boundaries’ edges, and then, we define for each arc a a weight 
: 
+ 1 if a is not on a boundary or if a is on a boundary and 
w(a) = 
has a white square of the picture on its left-hand side, 
- 1 if a is on some boundary and has a black square 
on its left-hand side. 
This digraph is called the associated weighted digraph. 
Theorem. A picture with balanced holes is tilable ifand only ifits associated weighted 
digraph does not have any negative cycle. 
Unfortunately, this result does not give an interesting algorithm: in fact, the best 
Algorithm I knows which can find a negative cycle in a directed graph is running in 
O(n, e) [ll]. This algorithm gives here a complexity O(n2). 
Given a tilable picture with balanced holes, let us define the fracture graph as the 
graph induced by edges which are on some null cycle (cycle of length 0). We have the 
following extension of the decomposition theorem: 
Theorem. The fracture graph of a tilable picture with balanced holes divides the picture 
into subpictures with balanced holes which are strongly tilable. Furthermore, each of 
picture’s tiling breaks up into subpictures’ tilings. 
7. Extensions 
The previous algorithm, which we call here the heights’ method, also works for more 
general pictures: that is, the without holes pictures which are composed of d-regular 
cells and whose degrees of its inner vertices are even [3]. 
We can even more extend the heights’ method by considering the dual graph of the 
picture. 
Let G be a bipartite graph, (X, Y) a bicoloration of its vertices, A its maximum 
degree. 
We define the A-augmented graph of G as the graph G with a new vertex o and for 
each vertex x of G, A-d(x) edges between w and x (d(x) being the degree of x in G). 
A balanced splitting of w is a splitting of the vertex o into vertices wl, . . , wq (q 2 l), 
where these vertices share between themselves the edges of w in a balanced way, that is 
each vertex Wi gets an equal number of edges (w, x) with x E X and edges (w, y) with 
YE Y. 
J.C. Fournier/Discrete Muthernatics If5166 (1997) 313- 320 319 
Theorem. The heights’ method extends to bipartite planar graphs which admit a planar 
balanced splitting qf its A-augmented graph. 
Consider a plane representation of a planar balanced splitting of the A-augmented 
graph. Each vertex coi, for i = 1, . , q, defines a face which corresponds, by duality in 
the plane, to a balanced hole of a picture. One of these holes is in fact the exterior of 
the picture. In particular, the case q = 1 corresponds to without holes pictures. 
Complexity of the obtained algorithm: O(nq). 
8. Open questions 
~- Known algorithms on matchings in graphs, in particular bipartite graphs, give 
algorithms of complexity O(IZ’.~) for tiling any pictures. 
We gave here a linear algorithm for the case of without holes pictures, or pictures 
with a bounded number of balanced holes. 
Is there a linear algorithm for the general case? Or an algorithm with complexity 
between O(n) and O(n’.5 )? 
- Instead of perfect matchings, we can consider maximum matchings: that is partial 
tilings with a maximum number of dominoes. 
Is there a linear algorithm which can find a maximal matching in without holes 
pictures’? 
~~ It is natural to consider the following general problem: instead of considering 
dominoes, we consider ‘bars’ of some length, horizontal bars h, of length p. and 
vertical bars L’~ of length q. 
Robson [lo] proved that the question of tiling a picture by /I, and u.:4 is NP- 
complete when p 3 3 or q 3 3. Thus, the problem is closed.. unless P = NP! 
~ Parallel algorithmic aspect gives other open questions, for instance: 
Is the problem of finding a maximum tiling in class NC? 
References 
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VI 
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M 
I51 
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D. Beauquier, kd., Polyominos et Pavages, Publication interne de 1’ universitk Paris I2 (Crkteil. 199 I). 
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Geom. 5 (1995) I-25. 
T. Chaboud, Pavages et graphes de Cayley planaires. Thkse E.N.S. Lyon, 1995. 
J.H. Conway and J.C. Lagarias, Tiling with Polyominoes and combinatorial group theory, J. Combin 
Theory A 53 (1990) 183-208. 
J.C. Fournier, Pavage des figures planes sans trous par des dominos: Fondement graphique de 
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[7] S. Kannan and D. Soroker, Tiling polygons with parallelograms, Discrte Comput. Geom. 7 (1992) 
1755188. 
[S] C. Kenyon and K. Kenyon, Tiling a polygon with rectangles, FOCS 92 (1992) 610-619. 
[9] L. Lovasz. Combinatorial Problems and Exercices (North Holland, Amsterdam, 1979). 
[lo] J.M. Robson, Sur le pavage de figures du plan par des barres, in [I] (1991) 59 -107. 
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Tiling pictures of the plane with dominoes 

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Tiling pictures of the plane with dominoes

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