Geodesic
One-to-one correspondense between proper families of~Boolean functions and unique sink orientations of~cubes
Prikladnaâ diskretnaâ matematika, no. 2 (2020), pp. 16-21.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper, we study the relationship between proper families of Boolean functions and unique sink orientations of Boolean cubes. A family of Boolean functions $ F = (f_1(x_1, \ldots, x_n), \ldots, f_n(x_1, \ldots, x_n))$ is called proper if for every two binary vectors $\alpha, \beta$, $\alpha \ne \beta$, the following condition holds: $$ \exists i (\alpha_i \ne \beta_i\ \\ f_i(\alpha) = f_i(\beta)).$$ Unique sink orientation of Boolean cube $\mathbb{E}_n$ is such an orientation of edges of $\mathbb{E}_n$ that any subcube of $\mathbb{E}_n$ has a unique sink, i.e., a unique vertex without outgoing edges. The existence of one-to-one correspondence between two classes of objects is proved, and various properties are derived for proper families. The following boundary for the number $T(n)$ of proper families of given size $n$ is obtained: there exist two numbers $B$ and $A$, $B \ge A > 0$, such that $n^{A 2^n} \le T(n) \le n^{B 2^n}$ for $n \ge 2$. Also, coNP-completeness of the problem of recognizing properness is derived.
Keywords: proper families of Boolean functions
Mots-clés : unique sink orientations. 
@article{PDM_2020_2_a1,
     author = {K. D. Tsaregorodtsev},
     title = {One-to-one correspondense between proper families {of~Boolean} functions and unique sink orientations of~cubes},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {16--21},
     publisher = {mathdoc},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2020_2_a1/}
}
TY  - JOUR
AU  - K. D. Tsaregorodtsev
TI  - One-to-one correspondense between proper families of~Boolean functions and unique sink orientations of~cubes
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2020
SP  - 16
EP  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2020_2_a1/
LA  - ru
ID  - PDM_2020_2_a1
ER  - 
%0 Journal Article
%A K. D. Tsaregorodtsev
%T One-to-one correspondense between proper families of~Boolean functions and unique sink orientations of~cubes
%J Prikladnaâ diskretnaâ matematika
%D 2020
%P 16-21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2020_2_a1/
%G ru
%F PDM_2020_2_a1
K. D. Tsaregorodtsev. One-to-one correspondense between proper families of~Boolean functions and unique sink orientations of~cubes. Prikladnaâ diskretnaâ matematika, no. 2 (2020), pp. 16-21. http://geodesic.mathdoc.fr/item/PDM_2020_2_a1/

[1] Keedwell A., Denes J., Latin Squares and their Applications, 2nd Ed., North Holland, 2015, 438 pp. | MR

[2] Glukhov M. M., “Some applications of quasigroups in cryptography”, Prikladnaya Diskretnaya Matematika, 2008, no. 2(2), 28–32 (in Russian)

[3] Nosov V. A., “Constructing classes of Latin squares in the Boolean database”, Intelligent Systems, 4:3–4 (1999), 307–320 (in Russian)

[4] Nosov V. A., “Constructing a parametrical families of Latin squares in a vector database”, Intelligent Systems, 8:1–4 (2006), 517–529 (in Russian)

[5] Szabo T., Welzl E., “Unique sink orientations of cubes”, Proc. 42nd IEEE Symp. FOCS (Newport Beach, CA, USA, 2001), 547–555 | MR

[6] Schurr I., Unique Sink Orientations of Cubes, Doctoral Thesis, ETH Zurich, 2004 | MR

[7] Nosov V. A., Pankratyev A. E., “On functional representation of Latin squares”, Intelligent Systems, 12:1–4 (2008), 317–332 (in Russian) | MR

[8] Gärtner B., Thomas A., “The complexity of recognizing unique sink orientations”, Leibniz Intern. Proc. Informatics (LIPIcs), 30, 2015, 341–353 | MR

[9] Matousek J., “The number of unique-sink orientations of the hypercube”, Combinatorica, 26 (2006), 91–99 | DOI | MR | Zbl