MINIMUM AREA POLYOMINO VENN DIAGRAMS

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1 MINIMUM RE POLYOMINO VENN DIGRMS ette ultena, Matthew Klimesh, and Frank Ruskey bstract. Polyomino Venn (or polyvenn) diagrams are Venn diagrams whose curves are the perimeters of polyominoes drawn on the integer lattice. Minimum area polyvenn diagrams are those in which each of the 2 n intersection regions, in a diagram of n polyominoes, consists of exactly one unit square. We construct minimum area polyvenn diagrams in bounding rectangles of size 2 r 2 c whenever r, c 2. Our construction is inductive, and depends on two expansion results. First, a minimum area polyvenn in a c rectangle can be expanded to produce another one that fits into a c+3 bounding rectangle. Second, a minimum area polyvenn diagram in a 2 r 2 c rectangle can be expanded to produce another one that fits into a 2 r+1 2 c+1 rectangle. Finally, for even n we construct n-set polyvenn diagrams in bounding rectangles of size (2 n/2 1) (2 n/2 + 1) in which the empty set is not represented as a unit square. 1 Introduction Venn diagrams have been drawn with a greater variety of shapes than John Venn must have imagined. Much research has been concerned with the nature of the curves, for example, whether they have rotational symmetry or are convex ([3],[8]). In this paper, the curves are the perimeters of polyominoes and we further require that each region of the diagram be a unit square. Venn diagrams in which the area of each intersection region is in proportion to the size of the population that it represents are said to be area proportional, and this is now an active area of research; see, for example, [6] and [11]. It is also of interest to produce diagrams in which each region has the same area, particularly if the shape of each region is identical. Such diagrams might be useful because of their visual properties, or because of the ease of labelling them. For example, the article [4] uses a minimum area polyomino Venn diagram from [6] for data visualization in the field of microbiology. Our constructions can be viewed as embeddings of the hypercube into grid graphs (e.g., [1],[2]) and thus may be of interest to the graph embedding community. Perhaps the first polyomino Venn diagrams appeared around 2000 on a website of Mark Thompson [12], which introduced the idea of using the perimeters of polyominoes as the curves of Venn diagrams, with the additional property that each intersection region is The work of. ultena was supported in part by a Canada Research Scholarship. The work of F. Ruskey was supported in part by an NSERC Discovery Grant. Dept. of Computer Science, University of Victoria, CND Jet Propulsion Laboratory, California Institue of Technology, US JoCG 3(1), ,

2 a unit square. Chow and Ruskey [6] further investigated the problem of minimizing the number of unit squares in an n-set polyvenn, and asked, but did not resolve, the question of whether minimum area polyvenn diagrams exist for every number of curves. In this paper, we settle several conjectures/questions from [6]. Most notably we show how to construct minimum area polyomino diagrams in bounding rectangles of size 2 r 2 c whenever r, c 2. C C C C C C C C Figure 1: (1, 2)-polyVenn. Two depictions are shown: first, the three polyominoes and their positions within the base region, and second, a more compact depiction where each grid square contains the labels of the pieces that it intersects. 2 Denitions Consider the set of three L-shaped tetrominoes,, C, shown at the top of Figure 1. If these are overlapped in the obvious way, then all subsets of {,, C} occur as a unique unit square in the result, the 2 4 rectangle shown at the bottom of Figure 1. In other words, the curves that comprise the perimeters of these tetrominoes form a Venn diagram when overlaid as shown. Definition 1. polyomino [7] is set of unit squares whose corners lie on the integer lattice and whose perimeter forms a single simple closed curve. For our purposes, unit squares, and therefore polyominoes, are taken to include their interiors. Note that in our definition of a polyomino no holes are allowed. We use the term Venn diagram in the sense defined by Grünbaum in [9]. Definition 2. n n-venn diagram consists of n simple closed curves C 1, C 2,, C n drawn in the plane such that each of the 2 n intersections X 1 X 2 X n, (1) where X i is either the open interior or the open exterior of the curve C i, is both non-empty and connected. JoCG 3(1), ,

3 Venn diagram definitions sometimes require that the curves intersect each other at a finite number of points, but for our purposes this requirement is too constraining. Let [n] = {1,..., n} and let P([n]) denote the power set of [n]. Let int(s) denote the interior of S R 2. Given polyominoes P 1,..., P n and a set U that contains the union of the polyominoes, for each I P([n]) it is convenient to define We may write X (U) I context. We call each X (U) I X (U) I (P 1,..., P n ) = i I int(p i ) i [n]\i int(u \ P i ). (2) instead of X (U) I (P 1,..., P n ) when the polyomino list is clear from an intersection region. Definition 3. polyomino Venn diagram, or polyvenn, consists of polyominoes P 1,..., P n such that for all I P([n]) the set X (R2 ) I is non-empty and connected. minimum area polyomino Venn diagram is a polyvenn together with a base region G that contains the union of all the polyominoes, such that for all I P([n]) the set X (G) I is a single unit square. Note that for any set of polyominoes that yield a polyvenn, their perimeters make up a Venn diagram. In the minimum area polyvenn definition the base region provides a kind of consistency, in that the empty set is represented by a bounded region in the same way as the other subsets of [n]. Definition 4. n (r, c)-polyvenn is an n-set minimum area polyvenn with n = r + c in which the base region G is a 2 r 2 c rectangle. y convention we will only consider (r, c)-polyvenns with r c, since if this is not the case the diagram can be rotated 90 to make it so. The polyvenn in Figure 1 is an example of a (1, 2)-polyVenn. Note that in an n-set polyvenn, each polyomino covers exactly 2 n 1 unit squares, and intersects with every other polyomino in exactly 2 n 2 squares. Proposition 1. (0, c)-polyvenn does not exist for any c > 2. Proof. Consider a (0, c)-polyvenn where c 2. Each of the c polyominoes must consist of 2 c 1 horizontally consecutive squares. Without loss of generality, let P 1 be the leftmost polyomino. ll other polyominoes must each intersect P 1 in its rightmost 2 c 2 squares. There is only one polyomino consisting of 2 c 1 horizontally consecutive squares that does this. polyvenn cannot have two totally overlapping polyominoes, so we must have c 2. 3 Expanding an Existing Diagram It is natural to ask whether n-set minimum area polyvenns exist for all n. If they do, is there a method to construct them? well-known question [10] and an open problem [13] JoCG 3(1), ,

4 ask if a simple Venn diagram of n curves can be extended to a simple Venn diagram of n+1 curves. We ask a similar question. We want to know if it is possible to extend a minimum area polyvenn by stretching the diagram and adding more polyomino pieces. Our primary tool for attacking this question is a technique we call expansion. The idea is to take an existing (r, c)-polyvenn V and expand it into an (r + r, c + c )-polyvenn by the addition of r + c polyominoes. The original polyominoes in V are expanded by uniformly stretching vertically by a factor of 2 r and horizontally by a factor of 2 c. We refer to this construction as a 2 r 2 c expansion. We also use the term expansion to refer to the stretching of the original polyvenn or its components; the meaning will be clear from context. The key element of a 2 r 2 c expansion is finding the r +c new polyominoes. The construction of these new polyominoes will depend on r, r, c, c, but not otherwise on V. Figure 2: (2, 3)-polyVenn created by expansion. Figure 2 shows an expansion of the (1, 2)-polyVenn of Figure 1 into a (2, 3)-polyVenn. Each polyomino is positioned on the 4 8 grid. This grid is comprised of 8 minigrids, which are the (in this case 2 2) expansions of the unit squares of the original polyvenn. The top three polyominoes are the expanded originals from Figure 1. The bottom two polyominoes have the property that they contain exactly one domino for each minigrid in the diagram. The two dominoes within each minigrid make up a single (1, 1)-polyVenn. lthough it occurs in this example (and in any 2 2 expansion), in a general 2 r 2 c expansion the intersection of a given minigrid with the added polyominoes does not necessarily need to form an (r, c )-polyvenn: the intersection of a given polyomino with the minigrid might not be connected even though the whole polyomino is simply connected. Suppose we want to form a 2 r 2 c expansion of an (r, c)-polyvenn by adding polyominoes E 1,..., E r +c. Let G ij be the 2 r 2 c miningrid expanded from the unit square in row i and column j of the original base rectangle (with the upper left square having indices (1, 1)). The relevant condition on the new polyominoes is that for every G ij JoCG 3(1), ,

5 and every T P([r + c ]), the set int(g ij E t ) t T t [r +c ]\T int(g ij \ E t ) (3) is a single unit square. When this condition holds for a given G ij we will say the added polyominoes satisfy the expanded minigrid intersection (EMI) property for G ij. We now can state and prove a straightforward but useful expansion result. Theorem 2. Let V be an (r, c)-polyvenn in a rectangle G. Suppose r, c 0 and let X( ) stretch the plane by factors of 2 r vertically and 2 c horizontally, so that polyominoes P 1,..., P r+c in G become X(P 1 ),..., X(P r+c ) in X(G). Suppose also that E 1,..., E r +c are polyominoes contained within X(G) that satisfy the EMI property for each minigrid of X(G). Then augmenting X(V ) by E 1,..., E r +c results in an (r + r, c + c )-polyvenn that is a 2 r 2 c expansion of V. Proof. We need to show that the set of polyominoes S = {X(P 1 ),..., X(P r+c )} {E 1,..., E r +c } together with the bounding rectangle X(G) make up an (r + r, c + c )-polyvenn. Clearly each member of S is a polyomino: each X(P k ) is an expanded polyomino, and each E t is a polyomino by hypothesis. It remains to show that the required intersection regions all consist of a single unit square. Each such region is of the form ( ) ( ) int(x(p k )) int(x(g) \ X(P k )) int(e t ) int(x(g) \ E t ), k K k [r+c]\k } {{ } where K P([r + c]) and T P([r + c ]). t T t [r +c ]\T Note that the subexpression is equal to X(X (G) K (P 1,..., P r+c )). However, since V is an (r, c)-polyvenn, X (G) K (P 1,..., P r+c ) consists of a single unit square in G. Thus is an expanded unit square, which is a minigrid G ij for some 1 i 2 r and 1 j 2 c. Therefore (4) can be written as ( ) int(g ij ) int(e t ) int(x(g) \ E t ) t T which, since G ij X(G), is equal to int(g ij E t ) t T t [r +c ]\T t [r +c ]\T int(g ij \ E t ). ut this is just (3), which by the EMI property is a single grid square, so the proof is complete. (4) JoCG 3(1), ,

6 4 Two Expansions We have two expansion results for (r, c)-polyvenns, Theorem 3 and Theorem 4. Theorem 3. For c 0, a (2, c)-polyvenn can be expanded to a (2, c + 3)-polyVenn. Proof. Let V be a (2, c)-polyvenn. We will exhibit a 1 8 expansion by constructing polyominoes E 1, E 2, and E 3 such that the conditions of Theorem 2 are satisfied. It is convenient to describe E 1, E 2, and E 3 by their intersections with the expanded unit squares of V. Toward that end consider the following triples of subsets of a 1 8 minigrid: M 11 = ( KK JJ, MLML, KJJ K ), M 12 = ( JJ KK, MLML, KJJ K ), M 21 = ( JMKL, KJKJ, MLML ), M 22 = ( MKLJ, JKJK, MLML ), M 31 = ( JKJK, MKLJ, MLML ), M 32 = ( KJKJ. JMKL, MLML ), M 41 = ( MLML, JJ KK, JKK J ), M 42 = ( MLML, KK JJ, JKK J ). We remark that each component of each M i1 is a horizontal reflection of the same component of M i2. For a given minigrid G ij (with 1 i 2 2 and 1 j 2 c ), set the intersections (E 1 G ij, E 2 G ij, E 3 G ij ) to be M i1 when j is odd and M i2 when j is even. The EMI property for E 1, E 2, and E 3 with all of the 1 8 minigrids is readily verified by inspection; for each M ij, each of the needed 2 3 = 8 intersections (complement or not for each component) is a unit square. It must also be shown that E 1, E 2, and E 3 are polyominoes (specifically, that they are simply connected). This can be verified by inspection for c = 0, 1. For larger c, the resulting polyominoes are formed by repeatedly concatenating the c = 1 case polyominoes with themselves. gain by inspection, this pattern produces simply-connected shapes (e.g., see Figure 3). Thus Theorem 2 produces the desired conclusion. Theorem 4. For r, c 1, an (r, c)-polyvenn can be expanded to an (r + 1, c + 1)-polyVenn. Proof. Let V be an (r, c)-polyvenn. We will exhibit a 2 2 expansion by constructing polyominoes E 1 and E 2 such that the conditions of Theorem 2 are satisfied. We describe E 1 and E 2 by their intersections with the expanded grid squares of V. Each such intersection will correspond to one of the following pairs of subsets of a 2 2 minigrid: M nw = ( E, G ), M ne = ( G, D ), M sw = ( F, E ), M se = ( D, F ). For a given 2 2 minigrid square G ij in the expanded bounding rectangle, set the intersections (E 1 G ij, E 2 G ij ) to be M nw, M ne, M sw, or M se depending on whether G ij is in the upper left, upper right, lower left, or lower right quadrant of the expanded bounding rectangle. JoCG 3(1), ,

7 P 1 P 2 P 3 P 4 X(P 1 ) X(P 2 ) X(P 3 ) X(P 4 ) E 1 E 2 E 3 Figure 3: Example of the 1 8 expansion used in the proof of Theorem 3, applied to the (2, 2)-polyVenn with polyominoes P 1, P 2, P 3, P 4. s in the proof of Theorem 3, the EMI property is easily verified by inspection of the minigrid intersections with the added polyominoes. In this case these intersections are M nw, M ne, M sw, and M se, which are all (1, 1)-polyVenns. It is also required to show that E 1 and E 2 are polyominoes. It is clear from Figure 4 that E 1 and E 2 are simply connected for square grids up to One can imagine the pattern for all rectangles with height greater than four. We do not pursue a more rigorous argument, but we note that one could be produced by showing, for k = 1, 2, that: (1) the construction does not produce any diagonal-only connections in E k, (2) the exterior of E k is connected, since from any square that is not a member of E k, one can exit the bounding rectangle with a straight path that does not intersect E k, and (3) E k is connected since, from any square in E k, one can reach the center 2 2 region of E k with no more than a two line-segment path with E k. From these facts, one could conclude that the perimeters of E 1 and E 2 are simple closed curves and thus E 1 and E 2 are polyominoes. pplying Theorem 2 completes the proof. JoCG 3(1), ,

8 Figure 4: The constructed E 1 and E 2 described in the proof of Theorem 4, shown for (r, c) = (1, 1), (2, 2), (3, 3), and (4, 4). Note that the construction is valid for any r, c 1. Figure 5: lternative E 1 and E 2 polyominoes for producing 2 2 expansions, shown for (r, c) = (1, 1), (2, 2), (3, 3), and (4, 4). The general construction is valid for any r, c 1. JoCG 3(1), ,

9 lternative constructions of E 1 and E 2 are possible in the proof of Theorem 4. Figure 5 illustrates an interesting construction that is also defined on a 2 r+1 2 c+1 rectangle when r, c 1. Let E 1 and E 2 be the polyominoes in this case, while E 1 and E 2 are the polyominoes described in the proof of Theorem 4. This construction is very similar: The center 4 4 regions of E 1 and E 2 are identical to the same regions of E 1 and E 2. lso the same 2 2 pairs of subsets are applied to the same quadrants. However, instead of using only the first element of each subset for E 1 and the second for E 2, the pairs are alternating in E 1 and E 2, so adjacent minigrids in a quadrant always contain both the first and second element of the pair. lthough the simple connectivity is clear to the eye from the examples in Figure 5, the argument that E 1 and E 2 are polyominoes would be laid out in the same manner as the description provided in the proof of Theorem 4. 5 Summary of Existence Results Figure 6: The trivial polyvenns. Theorems 3 and 4 provide a way of inductively determining ranges of (r, c) pairs for which an (r, c)-polyvenn exists. ut in order to begin we must establish some base cases. It turns out to be sufficient to consider base cases with r = 0 and r = 1. Figure 6 exhibits (trivial) (0, c)-polyvenns for c = 0, 1, 2 and the (1, 1)-polyVenn. The trivial (1, 2)- polyvenn was shown in Figure 1. In Figures 7 and 8, we show a (1, 3)-polyVenn and a (1, 4)-polyVenn, respectively. For c > 4 we do not know of any (1, c)-polyvenns and we make the following conjecture: Conjecture 5. (1, c)-polyvenn does not exist for c > 4. motivation for this conjecture comes from the observation that having only two rows is very constraining in the construction of polyvenns; for example no (0, 1) expansion seems possible on an existing (1, c)-polyvenn. lso, a partial computer search has not produced a (1, 5)-polyVenn. However, given the base cases and the expansion theorems, we summarize the known polyvenns in the following theorem: Theorem 6. For every n > 0 and for every r, c > 2 where r+c = n, there exists a minimum polyomino Venn diagram bound in a 2 r 2 c bounding rectangle. JoCG 3(1), ,

10 D C D C C C D D C C D D D C D C C D Figure 7: (1, 3)-polyVenn with nice symmetry. C C C C C C C C C C C C C C C D D D E D D D E D E D E E E E C D D D D E D E D E E D E D E E E E C D E Figure 8: (1, 4)-polyVenn. JoCG 3(1), ,

11 Proof. pplying Theorems 3 and 4 with the base cases completes the proof. Table 1 shows how the base cases are extended and summarizes our state of knowledge about for which r and c an (r, c)-polyvenn exists F6 F6 F6 X X X X X X 2 1 F6 F6 F7 F8???? 2 2 T4 T4 T4 T3 T3 T3 T3 2 3 T4 T4 T4 T4 T4 T4 2 4 T4 T4 T4 T4 T4 2 5 T4 T4 T4 T4 2 6 T4 T4 T4 2 7 T4 T4 2 8 T4.... Table 1: Our knowledge of existence of (r, c)-polyvenns. Rows and columns are labelled with bounding box dimensions (2 r and 2 c ). Fn indicates existence by example given in Figure n; X indicates nonexistence following from Proposition 1; T3 and T4 indicate existence implied by Theorems 3 and 4, respectively; and question marks are currently open cases. 6 Rectangles That Omit the Empty Set In this section we discuss the problem of finding n-set polyvenns in which each polyomino fits into an h w rectangle where 2 n 1 = hw, so that the region for the empty set (the intersection of the exterior of all of the polyominoes) is the exterior of the rectangle. We refer to such polyvenns as minimum area empty-set-omitting polyvenns or ESO-polyVenns. For n > 2 it is only possible for such polyvenns to exist when M n = 2 n 1 is not a Mersenne prime, since otherwise the resulting diagram must have dimensions 1 M n and would imply the existence of a polyvenn that violates Proposition 1. For n > 2, the first 8 such numbers are = 15 = 3 5, = 63 = 3 2 7, = 255 = , = 511 = 7 73, = 1023 = , = 2047 = 23 89, and = 4095 = This problem was first mentioned in [6]. Theorem 7. For all r 0 there exists a minimum area ESO-polyVenn with a (2 r 1) (2 r + 1) base rectangle. Proof. We use a simple construction of a polyvenn of dimension (2 r 1) (2 r + 1), where r = n/2, based on the successive 2 2 expansions of the (1, 1)-polyVenn of Figure 6. We can use either the construction detailed in the proof of Theorem 4 (epitomized by Figure 4) or the alternative construction (epitomized by Figure 5). In either case the resulting (r, r)- polyvenns have two easily verified properties. First, the empty set is represented by the JoCG 3(1), ,

12 Figure 9: Converting the base case (1, 1)-polyVenn to a 1 3 polyvenn that omits the empty set. Figure 10: Omiting the empty set: a polyvenn on the (2 n/2 1) (2 n/2 + 1) grid for n = 8. upper left square of the bounding rectangle. Second, if the top row of the diagram is removed, rotated 90 clockwise, and attached to the right side of the diagram (with the formerly rightmost square even with the bottom of the diagram), then the rearranged polyominoes continue to be simply connected (i.e., they are still polyominoes). Performing the rearrangement described in this second property results in a minimum area polyvenn whose base region is a (2 r 1) (2 r + 1) rectangle with an extra square attached at the top of the rightmost column. ut by the first property this extra square represents the empty set, so the resulting polyominoes actually form a minimum area ESO-polyVenn in the (2 r 1) (2 r + 1) rectangle. Figure 9 demonstrates how this construction converts the (1, 1)-polyVenn to a polyvenn on a 1 3 grid. Figure 10 shows the result of the construction when n = 8 and the (4, 4)- polyvenn was constructed using repeated 2 2 expansions of the type illustrated in Figure 5. We do not know of any minimum area ESO-polyVenns for grid dimensions not covered by this construction. The next few such dimensions are 63 = 3 21, 255 = 3 85 = 5 51, and 511 = Given the seemingly mysterious nature of the Mersenne primes, we feel that there will not be any general construction possible when n is odd and M n is not prime. JoCG 3(1), ,

13 7 Final Remarks and Open Problems If a (1, 5)-polyVenn does not exist, this might be established through either an exhaustive computer search or through a more mathematical proof. We have expended some effort in both of these directions. computer search can easily find a large number of (1, 4)- polyvenns but exhaustively searching for a (1, 5)-polyVenn may be beyond our current technique. Since our results show that (2, c)-polyvenns exist for all c, we do know that minimum area polyvenns exist with very long and thin bounding rectangles. The facts that (r, c)-polyvenns exist for all r, c 2 and that minimum area emptyset-omitting polyvenns exist for bounding rectangles of size (2 n/2 1) (2 n/2 +1) for even n suggest that additional interesting constructions of minimum area polyvenns may exist. In particular it would be interesting to see a construction of minimum area empty-set-omitting polyvenns for another infinite family of bounding rectangles. cknowledgements We thank the referees for their helpful comments, which have improved the presentation of this paper. References [1] S. ettayeb, Z. Miller, and I. H. Sudborough. Embedding grids into hypercubes. Journal of Computer and System Sciences, 45 no. 3 (December 1992): [2] S. L. ezrukov, J. D. Chavez, L. H. Harper, M. Röttger, and U.-P. Schroeder. The congestion of n-cube layout on a rectangular grid. Discrete Mathematics, 213 (2000): [3]. ultena,. Grünbaum, and F. Ruskey. Convex drawings of intersecting families of simple closed curves. 11 th Canadian Conference on Computational Geometry, (1999): [4] S. Casimiro, R. Tenreiro, and.. Monteiro. Identification of pathogenesis-related ESTs in the crucifer downy mildew oomycete Hyaloperonospora parasitica by highthroughput differential display analysis of distinct phenotypic interactions with rassica oleracea. Journal of Microbiological Methods, 66 no. 3 (September 2006): [5] S. Chow and P. Rodgers. Constructing area-proportional Venn and Euler diagrams with three circles. Euler Diagrams Workshop (ugust 2005). [6] S. Chow and F. Ruskey. Minimum area Venn diagrams whose curves are polyominoes. Mathematics Magazine, 80 no. 2 (pril 2007): [7] S. W. Golomb. Polyominoes: Puzzles, Patterns, Problems, and Packings. Princeton: Princeton University Press (1996). JoCG 3(1), ,

14 [8] J. Griggs, C. E. Killian, and C. D. Savage, Venn diagrams and symmetric chain decompositions in the oolean lattice. Electronic Journal of Combinatorics, 11 no. 1 (2004): R2. [9]. Grünbaum, Venn diagrams and independent families of sets. Mathematics Magazine, 48 no. 1 (Jan.-Feb. 1975): [10]. Grünbaum. Venn diagrams I. Geombinatorics, I no. 4 (1992): [11] G. Stapleton, J. Howse, and P. J. Rodgers, graph theoretic approach to general Euler diagram drawing. Theoretical Computer Science, 411 no. 1 (2010): [12] M. Thompson. Venn polyominoes. Math Recreations, (Original website no longer available, but a copy is available here: polyominos.html). [13] P. Winkler. Venn diagrams: Some observations and an open problem. Congressus Numerantium, 45 (1984): JoCG 3(1), ,