On the complexity of representing sets of vertices in the n-cube
Miguel Couceiro∗ Stephan Foldes† Erkko Lehtonen‡
Abstract
Representations of an arbitrary set of vertices of an n-dimensional cube in terms of convex sets are compared with respect to their complexity. The notion of complex-ity used in the representations is based on the union, symmetric difference and ternary median operations. Convexity of a set of vertices refers to Hamming dis-tance or, equivalently, to the intersection of cubes with supporting hyperplanes.
We denote by Vnthe set of vertices of the n-dimensional hypercube. With reference to a general notion of con-vexity, the theory of which is presented in [3] by van de Vel, we call convex set any set of vertices that is convex under Hamming distance. These are exactly ∅, Vn, and the intersections of Vn with supporting hyperplanes of the n-cube.
Any subset S of Vn can be represented as the union of convex sets. Similarly, it can be built up from convex sets by repeated median combinations (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C). Moreover, it can be represented using symmetric differences A ⊕ B = (A ∪ B) \ (A ∩ B) of intersecting convex sets:
Theorem 1 Let w be a vertex of the n-cube, w ∈ Vn.
For every S ⊆ Vn, there exists a unique m ≥ 0 and a
unique set of m convex sets C1, . . . , Cm containing w
such that S = C1⊕ · · · ⊕ Cm.
Sketch of proof. By induction on the dimension n of the cube. The base case is trivial. Partition Vn in two discrete half-cubes H1, H2(complementary convex sets of 2n−1vertices); these have dimension n − 1. Without loss of generality, let w ∈ H1. For each v ∈ H1, let v0 be the only vertex in H
2 adjacent (distance 1) to v, and for any T ⊆ H1, let T0 = {v0 : v ∈ T }. Note
∗Department of Mathematics, Statistics and Philosophy,
Uni-versity of Tampere, FI-33014 Tampereen yliopisto, Finland, miguel.couceiro@uta.fi. Research partially supported by the Graduate School in Mathematical Logic MALJA and by grant #28139 from the Academy of Finland.
†Institute of Mathematics, Tampere University of Technology,
P.O. Box 553, FI-33101 Tampere, Finland, stephan.foldes@tut. fi
‡Institute of Mathematics, Tampere University of Technology,
P.O. Box 553, FI-33101 Tampere, Finland, erkko.lehtonen@tut. fi
that T 7→ T0 is a convexity-preserving bijection from
P(H1) to P(H2). The inverse bijection Q 7→ Q∗ is also convexity-preserving.
For a given S ⊆ Vn, let Si = S ∩ Hi, i = 1, 2. By induction hypothesis, S1= C1⊕ · · · ⊕ Cmwith Cj ⊆ H1 and w ∈ Cj for all j, and S2 = K1⊕ · · · ⊕ Kt with
Kj ⊆ H2 and w0∈ Kj for all j, where each Cj and Kj is convex. We have that
S∗2= K1∗⊕ · · · ⊕ Kt∗ with K∗
j ⊆ H1, w ∈ Kj∗ for all j, and
S2∪ S2∗= (K1∪ K1∗) ⊕ · · · ⊕ (Kt∪ Kt∗) with w ∈ Kj∪ Kj∗ for all j, and every set Kj∪ Kj∗ is convex. Combining these representations we obtain
S = S1⊕ S∗2⊕ (S2∪ S2∗) =
C1⊕ · · · ⊕ Cm⊕ K1∗⊕ · · · ⊕ Kt∗
⊕ (K1∪ K1∗) ⊕ · · · ⊕ (Kt∪ Kt∗), where each convex set appearing in the representation contains w.
A proof of uniqueness can be similarly based on the
convexity-preserving map T 7→ T0. ¤
Based on these facts, we consider three ways of con-structing families of sets of vertices from any initial fam-ily B1, . . . , Bm of convex sets by the following alterna-tive procedures.
U Let U1= {B1, . . . , Bm}. For i > 1, let Ui= Ui−1∪
{F }, where F = A ∪ B with A, B ∈ Ui−1.
P We require that Tmi=1Bi 6= ∅, and we let P1 =
{B1, . . . , Bm}. For i > 1, let Pi = Pi−1∪ {F }, where F = A ⊕ B with A, B ∈ Pi−1.
M Let M1 = {B1, . . . , Bm}. For i > 1, let Mi =
Mi−1∪ {F }, where F = (A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C) with A, B, C ∈ Mi−1.
For a given subset S of Vn, its U-complexity is the smallest number i such that S ∈ Ui over all possible sequences constructed by procedure U from all possi-ble initial families U1 of convex sets. We define the
P-complexity and the M-P-complexity of S in a similar way. These complexity measures are integer-valued functions
on the setSn≥1P(Vn), and thus it makes sense to com-pare any two of these complexities by asking whether one is within a polynomial bound of another.
The three construction procedures correspond to three different algebraic representations of Boolean functions, namely the disjunctive normal form (DNF), the Zhegalkin (Reed–Muller) polynomial, and the me-dian normal form. The geometric approach to DNF (and CNF) was taken, e.g., by Foldes and Hammer [2] (where further references to such approaches were also given).
A comparative study of representation complexities was carried out in a framework using formulas of spe-cific algebraic syntax in [1]. In the present geometric framework, we have the following:
Theorem 2 The U- and P-complexities are not within any polynomial bound of the M-complexity nor within any polynomial bound of each other, but the M-complexity is within a quadratic bound of the U- and P-complexities.
Sketch of proof. It follows from what was shown in [1] that the U- and P-complexities are not within any polynomial bound of the M-complexity, the U-complexity is not within any polynomial bound of the P-complexity, and the M-complexity is within a quadratic bound of the U- and P-complexities.
To see that the P-complexity is not within any poly-nomial bound of the U-complexity, let w1, w2 be two antipodal vertices of the n-cube and consider S = {w1, w2}. Obviously the U-complexity of S is at most 2, regardless of the dimension n.
Let S = C1⊕ · · · ⊕ Cm, where the Cj’s are distinct convex sets having a common vertex w. For i = 1, 2, let Ni be the set of those neighbours of wi(at distance 1 of wi) that are in the convex hull of w and wi. We have Card N1+ Card N2= n.
Assume that n > 1. For each i = 1, 2, each vertex x ∈ Ni must be contained in some Cj that does not contain wi. Choose one such Cj and write j = f (x). The function f is injective. It follows that m ≥ n/2 and the P-complexity of S is at least n/2. ¤ We note that the efficiency advantage of procedure M over U and P is not simply due to the ternary char-acter of the operation, as the replacement of the binary union or symmetric difference by corresponding ternary operations A ∪ B ∪ C and A ⊕ B ⊕ C would result in no decrease in order of complexity.
References
[1] M. Couceiro, S. Foldes, E. Lehtonen, Composition
of Post classes and normal forms of Boolean functions,
RUTCOR Research Report 5–2005, Rutgers University, 2005, http://rutcor.rutgers.edu/∼rrr/.
[2] S. Foldes, P. L. Hammer, Disjunctive and conjunctive representations in finite lattices and convexity spaces,
Discrete Math. 258 (2002) 13–25.
[3] M. L. J. van de Vel, Theory of Convex Structures, El-sevier, Amsterdam, 1993.
