Yu. L. Vasil’ev

On the Length of a Cycle in an \(n\)-Dimensional Unit Cube

(Presented by Academician S. L. Sobolev on 13 VII 1962)

1. A cycle in the \(n\)-dimensional unit cube* \(E^n\) is a subset \(S\) of the set \(E^n\) possessing the following properties: first, it is connected, i.e., for any two vertices \(\alpha\) and \(\beta\) from \(S\) one can specify a number \(m\) and \(m\) vertices \({}^1\sigma,\ldots,{}^m\sigma\) such that the vertices \(\alpha\) and \({}^1\sigma\), \({}^1\sigma\) and \({}^2\sigma,\ldots,{}^{m-1}\sigma\) and \({}^m\sigma\), \({}^m\sigma\) and \(\alpha\) are adjacent; second, it is “one-dimensional” and “closed,” i.e., the neighborhood of an arbitrary vertex of the set \(S\) contains exactly two vertices of the set \(S\). The length of a cycle is the number of vertices in the set \(S\).

The problem of the maximal length of a cycle in the \(n\)-dimensional unit cube was formulated by Yu. I. Zhuravlev \((^1)\) in connection with the study of algorithms for simplifying disjunctive normal forms (d.n.f.).** Yu. I. Zhuravlev observed that, in order to estimate the dependence between the “performance” of an algorithm \(A\) for simplifying d.n.f.’s and the “volume” of that “part” of the reduced d.n.f. of a given function which is “surveyed” by the algorithm \(A\) “in one step,” it is essential to determine how the numerical function \(C(n)\)—the greatest length of a cycle in the \(n\)-dimensional unit cube \(E^n\)—behaves. Being related to the well-known “coding problem,” the question of constructing a cycle of greatest length is also of independent interest. By the definition of a cycle, \(C(n) \leq 2^n\).

Theorem 1. For every \(n=2^m,\ m=3,4,\ldots,\) one can construct a cycle in \(E^n\) whose length is \(2^n/n\).

Theorem 2. For arbitrary \(n=3,4,\ldots,\) one can construct a cycle in \(E^n\) whose length is \((1-\varepsilon(n))\,2^{\,n-1}/n,\ \varepsilon(n)\to0\) as \(n\to\infty\).

Theorem 3. Let \(M(n)\) be the number of distinct cycles in \(E^n\) of length not less than \(2^n/n,\ n\geq 2^{-4}\). Then \(M(n)\geq 2^{\,2^n(1-\varepsilon)},\ \varepsilon\to0\) as \(n\to\infty\).

Below we shall consider the main points in the proof of Theorem 1, carried out by a direct construction of a cycle of the indicated length. This construction rests on properties of splittings (see below) of certain d.n.f.’s and arose in connection with the study of the question of the maximal possible difference between the complexity of minimal and dead-end d.n.f.’s \((^3)\). In § 2, in order to describe the construction from given cycles of a cycle of greater length, the notion of a pseudocycle is introduced. In § 3 the means by which the pseudocycle will be constructed are considered. In § 4 an outline is given of the proof of Theorem 1.

* By definition, \(E^n\) is the set of all \(n\)-digit strings composed of zeros and ones. Each such string is called a vertex of the cube \(E^n\). Two vertices of the cube \(E^n\) are called adjacent if they differ from one another in only one digit. The neighborhood of the vertex \(\sigma=(\sigma_1,\ldots,\sigma_n)\) of the cube \(E^n\) is the set of all vertices of the cube \(E^n\) adjacent to the vertex \(\sigma\). An unordered pair of adjacent vertices of the cube \(E^n\) will be called an edge and denoted by \(p\); the set of all edges of the cube \(E^n\) will be denoted by \(\mathfrak R^p\). For a vertex \(\sigma=(\sigma_1,\ldots,\sigma_n)\) of the cube \(E^n\) we define \(|\sigma|=\sigma_1\oplus\cdots\oplus\sigma_n\), where \(\oplus\) denotes addition mod 2.

** The basic concepts of the theory of d.n.f.’s are set forth in \((^2)\). We shall say that the vertex \(\sigma=(\sigma_1,\ldots,\sigma_n)\) is a vertex of the d.n.f. \(\Phi(x^n)\) if \(\Phi(\sigma_1,\ldots,\sigma_n)\equiv1\). The distance \(\rho\) between vertices \(\sigma\) and \(\tau\) of the cube \(E^n\) is the number of positions in which these vertices do not coincide. The distance between d.n.f.’s \(\Phi_1(x^n)\) and \(\Phi_2(x^n)\) is the number
\[ \min_{\sigma\in N_1,\ \tau\in N_2}\rho(\sigma,\tau), \]
where \(N_1,N_2\) are, respectively, the sets of vertices of the d.n.f.’s \(\Phi_1\) and \(\Phi_2\).

1. Definition. A D.N.F. \(\Phi\) is called connected if, for any two of its vertices \(\alpha\) and \(\beta\), one can indicate a number \(m\) and \(m\) conjunctions \(K_1,\ldots,K_m\) from \(\Phi\) such that \(\alpha\) and \(\beta\) belong respectively to the conjunctions \(K_1\) and \(K_m\), and \(K_i K_{i+1}\ne 0\) for all \(i=1,\ldots,m-1\).

The following definition of a cycle differs from the definition given above only in form.

Definition. A cycle is a connected D.N.F. \(S(x^n)\), all conjunctions of which are one-dimensional, and, moreover, the neighborhood of each vertex of the D.N.F. \(S(x^n)\) contains exactly two vertices of the D.N.F. \(S(x^n)\). The length* of the cycle \(S\) is the number of conjunctions of the D.N.F. \(S\); it will be denoted by \(l(S)\). It is not difficult to verify that a cycle is a reduced D.N.F. and that the length of a cycle is always even.

Definition. A path between the vertices \(\alpha\) and \(\beta\) of the cube \(E^n\) is a connected D.N.F. \(P(x^n)\), all conjunctions of which are one-dimensional and to which the vertices \(\alpha\) and \(\beta\) belong; moreover, the neighborhood of an arbitrary vertex \(v\) of the D.N.F. \(P(x^n)\) contains only one vertex of the D.N.F. \(P(x^n)\) when \(v\) is one of the vertices \(\alpha\) or \(\beta\), and contains exactly two vertices of the D.N.F. \(P(x^n)\) when \(v\ne\alpha,\ v\ne\beta\). The vertices \(\alpha\) and \(\beta\) of the path \(P(x^n)\) will be called boundary vertices, and the other vertices of the path \(P(x^n)\)—internal vertices. The length of the path \(P\) is the number of conjunctions of the D.N.F. \(P\), and will be denoted by \(l(P)\).

Definition. Let \({}^{1}S,\ldots,{}^{t}S\) be nonintersecting cycles. Suppose that in the cycle \({}^{i}S\) two vertices \({}^{i}\alpha\) and \({}^{i}\beta\) are marked, which are not adjacent, and suppose that \({}^{i}P\) is a path between the vertex \({}^{i}\beta\) of the cycle \({}^{i}S\) and the vertex \({}^{i+1}\alpha\) of the cycle \({}^{i+1}S,\ i=1,\ldots,t\)**. The D.N.F.

\[ \bigvee_{i=1}^{t} {}^{i}S \vee \bigvee_{i=1}^{t} {}^{i}P \]

will be called a pseudocycle. The vertices \({}^{i}\alpha,{}^{i}\beta,\ i=1,\ldots,t\), will be called the nodes of the pseudocycle. A pseudocycle \(\Pi\) will be called exact if the neighborhood of any of its vertices \(v\) contains exactly three vertices of \(\Pi\) when \(v\) is one of the node vertices of \(\Pi\), and contains exactly two vertices of \(\Pi\) when \(v\) is not a node vertex of \(\Pi\).

Starting from a given exact pseudocycle, it is not difficult to construct a new cycle. To this end, observe that the vertices \({}^{i}\alpha\) and \({}^{i}\beta\) generate a decomposition of the cycle \({}^{i}S\) into two paths, which we shall denote by \({}^{i1}S\) and \({}^{i2}S\), and we agree that \(l({}^{i1}S)\le l({}^{i2}S)\). Namely, the paths \({}^{i1}S\) and \({}^{i2}S\) are composed of conjunctions of the cycle \({}^{i}S\), have boundary vertices \({}^{i}\alpha\) and \({}^{i}\beta\), and have no common internal vertices,

\[ l({}^{i1}S)+l({}^{i2}S)=l({}^{i}S). \]

Proposition 1. If the pseudocycle

\[ \Pi=\bigvee_{i=1}^{t} {}^{i}S \vee \bigvee_{i=1}^{t} {}^{i}P \]

is exact, then the D.N.F.

\[ C=\bigvee_{i=1}^{t} {}^{i2}S \vee \bigvee_{i=1}^{t} {}^{i}P \]

is a cycle.

1. Definition. A development of the cube \(E^n\) is a connected D.N.F. of the function \(1(x^n)\equiv 1\), all conjunctions of which are one-dimensional and each vertex of which belongs to exactly two conjunctions.

By induction on \(n=1,2,\ldots\) it is not difficult to verify that a development of the cube \(E^n\) exists. The number of conjunctions in a development of the cube is equal to \(2^n\). For example, the D.N.F.

\[ x_1 \vee \bar{x}_1 \vee x_2 \vee \bar{x}_2 \]

is a development of the cube \(E^2\).

Recall that a set \(M\) of vertices of the cube \(E^p\) is called a densely packed (d.p.) \((p,2l+1)\)-code if the distance between any two arbitrary vertices from \(M\) is not less than \(2l+1\), and if, for every vertex \(\sigma\) of the cube \(E^p\), there is a vertex \(\tau\) from \(M\) such that \(\rho(\sigma,\tau)\le l\). As is known \((^{4})\), d.p. \((p,3)\)-codes exist for values

\[ \text{* That is, all variables } x_1,\ldots,x_n \text{ enter into each conjunction except one of these variables.} \]

\[ \text{** We agree that by the }(t+1)\text{-st element of an ordered set }M\text{ of }t\text{ elements we shall mean the first element of }M. \]

\(p=2^q-1,\ q=2,3,\ldots,\) and only for these values of \(p\); the number of vertices in the p.u. \((p,3)\)-code is equal to \(2^p/(p+1)\).

Definition. We shall call the numbering of the vertices of a p.u. \((p,3)\)-code regular if the distance between its \(i\)-th and \((i+1)\)-st vertices is equal to 3, \(i=1,\ldots,2^p/(p+1)\).

Proposition 2. For any \(p=2^q-1,\ q=2,3,\ldots,\) one can construct a p.u. \((p,3)\)-code having a regular numbering of the vertices.

The proof follows, for example, from the inductive method of construction of p.u. \((p,3)\)-codes proposed in \((^5)\).

Definition. Let the variables \(u^s, z^p, w^p\) be pairwise distinct, \(1\le s\le p\), and let \(\tau=(\tau_1,\ldots,\tau_p)\) be an arbitrary vertex of \(E^p\). By the splitting of the conjunction \(K=u_1^{\sigma_1}\cdots u_s^{\sigma_s}\) with respect to \(\tau\) we shall mean the conjunction
\[ R_\tau[K]=z_1^{\sigma_1}w_1^{\sigma_1\oplus\tau_1}\cdots z_s^{\sigma_s}w_s^{\sigma_s\oplus\tau_s}. \]
By the splitting of a d.n.f. \(\Phi(u^p)\) with respect to \(\tau\) we shall mean the d.n.f. \(R_\tau[\Phi]\), made up of the splittings of the conjunctions of the d.n.f. \(\Phi(u^p)\) with respect to \(\tau\).

4. Proof of Theorem 1. Starting from a partition of the set of vertices of the cube \(E^p\) and a p.u. \((p,3)\)-code with regular numbering of the vertices, by means of splittings there are constructed in the cube \(E^{2p+2}\) “sufficiently many” “sufficiently long” cycles, and a method is given for joining them into a pseudocycle. Then, according to item 2, the pseudocycle is transformed into a cycle of the length indicated in Theorem 1. The construction is divided into six parts.

I. Construction in the cube \(E^{2p}\) of \(2^p/(p+1)\) nonintersecting cycles of length \(2^{p+1}\). Denote by \(r_p\) the conjunction whose vertices are the edges \(p\) of the cube \(E^p\), and only these vertices. Denote by \(\mathfrak R^p\) such an arbitrary subset of the set \(\mathfrak R^p\) of edges of the cube \(E^p\) that the d.n.f.
\[ \bigvee_{\mathrm p\in\widetilde{\mathfrak R}^p} r_p \]
is a partition of the cube \(E^p\); by \(\tau=(\tau_1,\ldots,\tau_p)\) denote an arbitrary vertex of the cube \(E^p\). Consider the splitting of the d.n.f.
\[ \bigvee_{\mathrm p\in\widetilde{\mathfrak R}^p} r_p \]
—the d.n.f.
\[ \Psi_\tau(z_p,w^p)=\bigvee_{\mathrm p\in\widetilde{\mathfrak R}^p} R_\tau[r_p]. \]

The d.n.f. \(\Psi_\tau\) is connected, \(I(\Psi_\tau)=2^p\). A vertex \(\sigma\) of the d.n.f. \(\Psi_\tau\) will be called a breaking vertex (respectively plane), if \(\sigma\) belongs to more than one (respectively only one) conjunction of the d.n.f. \(\Psi_\tau\). Each breaking vertex of the d.n.f. \(\Psi_\tau\) belongs to exactly two conjunctions of the d.n.f. \(\Psi_\tau\). Of the four vertices of an arbitrary conjunction of the d.n.f. \(\Psi_\tau\), two vertices (not being adjacent) are breaking, the other two vertices are plane.

Pass from the d.n.f. \(\Psi_\tau\) to a new d.n.f., which we denote by \(\Phi_\tau\). Let \(R_\tau[r_p]\) be an arbitrary conjunction of the d.n.f. \(\Psi_\tau\), and let \(C_{p,\tau}\) be either one of the two paths of length 2 between the breaking vertices of the conjunction \(R_\tau[r_p]\). The internal vertex of this path is a plane vertex of the conjunction \(R_\tau[r_p]\). Put
\[ \Phi_\tau(z^p,w^p)=\bigvee_{\mathrm p\in\widetilde{\mathfrak R}^p} C_{p,\tau}. \]

Proposition 3. The d.n.f. \(\Phi_\tau\) is a cycle, \(I(\Phi_\tau)=2^{p+1}\). In what follows we shall assume that \(p=2^q-1,\ q=2,3,\ldots\).

Proposition 4. Let \(\tau\) and \(\nu,\ \tau\ne\nu,\) be vertices of a p.u. \((p,3)\)-code. Then \(\Phi_\tau\cdot\Phi_\nu\equiv 0\).

Propositions 3 and 4 mean that in the cube \(E^{2p}\) one can construct \(2^p/(p+1)\) nonintersecting cycles of length \(2^{p+1}\).

We pass to a refinement of the form of the initial partition and of the cycles. Let \(C^p\) be a p.u. \((p,3)\)-code with regular numbering of the vertices, \(t\) the number of vertices of the code \(C^p\); \(t=2^p/(p+1)\), \(i=1,\ldots,t\); \(i_\tau\) the \(i\)-th vertex of the code \(C^p\), with \({}^1\tau=(0,\ldots,0)\). By definition \(|{}^i\tau|=|{}^{i+1}\tau|\oplus 1\). It is not difficult to verify that one can construct such a partition of the cube \(E^p\), three conjunctions of which form a path \({}^iQ\) of length 3 between the vertices \((0,\ldots,0)\) and \({}^i\tau\oplus{}^{i+1}\tau\) of the cube \(E^p\). Denote such a partition by \({}^iR\). Denote

by \({}^i\mathfrak R\) such a subset of the set \(\mathfrak R^p\) of edges of the cube \(E^p\) that \(\bigvee_{\rho\in{}^i\mathfrak R} r_\rho\) is \({}^iR\). Put \({}^iS(z^p,w^p)=\bigvee_{\rho\in{}^i\mathfrak R} C_{\rho,\,{}^i\tau}\). In other words, the d.n.f. \({}^iS\) is such a cycle \(\Phi_\tau\), in the construction of which \({}^i\tau\) is taken as the vertex \(\tau\), and \({}^iR\) is taken as the initial unfolding; \(l({}^iS)=2p+1\), \(i=1,\ldots,t\).

Proposition 5. If \(i\ne j,\quad i,j=1,\ldots,t\), then \(\rho({}^iS,{}^jS)\ge 1\), if \(i\ne j\pmod 2\); \(\rho({}^iS,{}^jS)\ge 2\), if \(i\equiv j\pmod 2\).

II. Construction of nodal vertices. Put
\[ {}^i\alpha=(0,{}^i\tau)=(0,\ldots,0,{}^i\tau,\ldots,{}^i\tau); \]
\[ {}^i\beta=({}^i\tau\oplus{}^{i+1}\tau,{}^{i+1}\tau) =({}^i\tau_1\oplus{}^{i+1}\tau_1,\ldots,{}^i\tau_p\oplus{}^{i+1}\tau_p, {}^{i+1}\tau_1,\ldots,{}^{i+1}\tau_p). \]
By construction the vertices \({}^i\alpha\) and \({}^i\beta\) belong to the cycle \({}^iS\), \(\rho({}^i\beta,{}^{i+1}\alpha)=3,\ i=1,\ldots,t\). Further, the vertices \({}^i\alpha\) and \({}^i\beta\) generate a decomposition of the cycle \({}^iS\) into two paths, which we denote by \({}^iS\) and \({}^{i2}S\) (see the end of item 2). It turns out that \(l({}^{i1}S)=6,\ l({}^{i2}S)=2p+1-6\). This follows from the fact that, in constructing the cycle \({}^iS\), we started not from an arbitrary unfolding of the cube \(E^p\), but specifically from \({}^iR\). In this case the path \({}^iQ\) of length 3 between \((0,\ldots,0)\) and \({}^i\tau\oplus{}^{i+1}\tau\) becomes the path \({}^{i1}S\) of length 6 between \({}^i\alpha\) and \({}^i\beta\).

III. Construction of the paths \({}^iP\) between the vertices \({}^i\beta\) and \({}^{i+1}\alpha\) of the cycles \({}^iS\) and \({}^{i+1}S\). Consider the 3-dimensional subcube of the cube \(E^{2p}\) containing the vertices \({}^i\beta\) and \({}^{i+1}\alpha\). Denote this subcube by \({}^iK\). The vertices of any path joining \({}^i\beta\) and \({}^{i+1}\alpha\) and having length 3 belong to \({}^iK\).

Proposition 6. Among the vertices of the cube \({}^iK\) adjacent to \({}^{i+1}\alpha\), there is one which does not belong to any of the cycles \({}^jS,\ j=1,\ldots,t\) (we denote this vertex by \({}^i\xi\)).

As the path \({}^iP\) we take an arbitrary path of length 3 between the vertices \({}^i\beta\) and \({}^{i+1}\alpha\), one of whose internal vertices is \({}^i\xi\). It is not difficult to verify that, for \(i\ne j\), \(\rho({}^iP,{}^jP)\ge 3\).

IV. The d.n.f.
\[ \Pi(z^p,w^p)=\bigvee_{i=1}^{t}{}^iS\vee\bigvee_{i=1}^{t}{}^iP \]
is a pseudocycle with nodal vertices \({}^i\alpha\) and \({}^i\beta,\ i=1,\ldots,t\). The pseudocycle \(\Pi\) is not exact.

V. Transformation of the pseudocycle \(\Pi(z^p,w^p)\). Put
\[ \Pi^*(z^p,w^p,x,y)=\bigvee_{i=1}^{t}{}^iS^*\vee\bigvee_{i=1}^{t}{}^iP^*, \]
where \({}^iS^*\) is the d.n.f. of the formula
\[ {}^iSxy^{|{}^i\tau|},\quad {}^iP^* \]
is the d.n.f. of the formula
\[ {}^iBy^{|{}^i\tau|}\vee{}^iBx\vee{}^iPxy^{|{}^{i+1}\tau|}\vee{}^{i+1}Ay^{|{}^{i+1}\tau|}, \]
\[ {}^iB=\bigwedge_{j=1}^{p} z_j^{{}^i\beta_j}w_j^{{}^i\beta_{p+j}},\quad {}^{i+1}A=\bigwedge_{j=1}^{p} z_j^{{}^{i+1}\alpha_j}w_j^{{}^{i+1}\alpha_{p+j}}. \]

The d.n.f. \(\Pi^*\) is an exact pseudocycle with nodal vertices
\[ {}^i\alpha^*=({}^i\alpha_1,\ldots,{}^i\alpha_{2p},1,|{}^i\tau|) \]
and
\[ {}^i\beta^*=({}^i\beta_1,\ldots,{}^i\beta_{2p},1,|{}^i\tau|),\quad i=1,\ldots,t. \]
The proof of the exactness of \(\Pi^*\) rests on Proposition 5 (item I) and on the properties of the paths \({}^iP\) (item III).

VI. Passage from the pseudocycle \(\Pi^*\) to a cycle. By virtue of item II of the present proof, the vertices \({}^i\alpha^*\) and \({}^i\beta^*\) generate a decomposition of the cycle \({}^iS^*\) into two paths \({}^{i1}S^*\) and \({}^{i2}S^*\), \(l({}^{i1}S^*)=6,\ l({}^{i2}S^*)=2p+1-6,\ i=1,\ldots,t\). According to Proposition 1, the d.n.f.
\[ C(z^p,w^p,x,y)=\bigvee_{i=1}^{t}{}^{i2}S^*\vee\bigvee_{i=1}^{t}{}^iP^* \]
is a cycle,
\[ l(C)=t\bigl(l({}^{i2}S^*)+l({}^iP^*)\bigr) =t(2p+1-6+6)=2^{2p+1}/(p+1)=2^n/n, \]
where \(n=2p+2=2^{q+1},\ q=2,3,\ldots\). Theorem 1 is proved.

We omit the consideration of Theorems 2 and 3.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
10 VII 1962

CITED LITERATURE

1. Yu. I. Zhuravlev, in: Problems of Cybernetics, issue 8, 1962.
2. S. V. Yablonskii, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 51, 5 (1958).
3. Yu. L. Vasil’ev, in: Problems of Cybernetics, issue 10 (in print).
4. R. W. Hamming, Bell Syst. Techn. J., 29, 147 (1950); R. Hamming, Codes with Error Detection and Correction, IL, 1956, p. 7.
5. Yu. L. Vasil’ev, in: Problems of Cybernetics, issue 8, 1962.