Dimensional Subcube Research Articles

On Isoperimetric Stability

On Isoperimetric Stability, Discrete Analysis 2018:14, 11 pp. Let $A$ be a subset of the Hamming cube $\{0,1\}^n$. If we regard the cube as a graph in the usual way, by joining two points $x$ and $y$ if they differ in exactly one coordinate, then the _edge boundary_ of $A$ is defined to be the set of all edges that join a point in $A$ to a point in its complement $\{0,1\}^n\setminus A$. The edge-isoperimetric inequality in the cube tells us that if $A$ has cardinality $2^d$, then the size of the edge boundary of $A$ is minimized when $A$ is a subcube of dimension $d$, in which case the number of edges between $A$ and its complement is $2^d(n-d)$. We can regard the graph as a Cayley graph, where the group is $\mathbb F_2^n$ and the generating set $S$ consists of the $n$ standard basis vectors. Then we find that the size of the edge boundary is always at least $|A|(|S|-\log_2|A|)$, an inequality that continues to hold when the size of $A$ is not a power of 2. As a consequence, if the size of the edge boundary is at most $(1-\gamma)|A||S|$, then $A$ must have size at least $2^{\gamma|S|}$. The purpose of this paper is to prove a similar statement for Abelian groups in general. Given an Abelian group $G$, a subset $A\subset G$, and a generating set $S$ of $G$, define the edge boundary to be the set of pairs $(a,s)\in A\times S$ such that $a+s\notin A$. The precise statement proved is that if the size of the edge boundary is at most $(1-\gamma)|A||S|$, then $A$ must have size at least $4^{(1-1/d)\gamma|S|}$, where $d$ is the exponent of $G$ -- that is, the smallest order of any element of $G$. Note that when $G=\mathbb F_2^n$ , then $d=2$, and we recover the statement above. Note also that in this case we can rewrite the lower bound for the size of $A$ as $|G|^\gamma$, a bound that is shown to be valid for groups of exponent 3 or 4 as well. However, for larger exponents the lower bound of $|G|^\gamma$ does not hold, and examples show that we have to settle for the weaker bound $4^{(1-1/d)\gamma|S|}$ proved in this paper. The reason that groups of low exponent are a special case is connected with the notion of a _dissociated set_, which is a subset $X$ of an Abelian group $G$ with the property that if $Y$ and $Z$ are finite subsets of $X$ with $\sum_{x\in Y}x=\sum_{x\in Z}x$, then $Y=Z$. Equivalently, it is a subset $X$ with a kind of linear independence property: if $x_1,\dots,x_n$ are distinct elements of $X$ and $\epsilon_1,\dots,\epsilon_n$ are coefficients taken from the set $\{-1,0,1\}$ such that $\sum_i\epsilon_ix_i=0$, then $\epsilon_1=\dots=\epsilon_n=0$. For groups of exponent 2 or 3, all possible integer coefficients can be replaced by ones that belong to the set $\{-1,0,1\}$, so this is just normal independence over the integers (and therefore, when these groups are considered as vector spaces, linear independence), but for groups of higher exponent that ceases to be the case. For example, in the group $\mathbb F_5^n$, the set $\{e_1,\dots,e_n,2e_1,\dots,2e_n\}$ is dissociated but not independent. These notions lead naturally to notions of dimension for subsets of Abelian groups: given a finite subset $E$, we define $\mathrm{dim}_D(E)$ to be the size of the largest dissociated subset of $E$ and $\mathrm{dim}_I(E)$ to be the size of the largest independent subset. A corollary of the main results of this paper is a sharp bound for the dimension of the _popular difference set_ of a set $A$, which is defined as follows. Given a parameter $\gamma>0$, we set $P_\gamma(A)$ to be the set of all group elements $g$ such that $|A\cap(A-g)|\geq\gamma|A|$. Popular difference sets play an important role in additive combinatorics. A result of Shkredov and Yekhanin states that $\mathrm{dim}_D(P_\gamma(A))$ is at most $C\gamma^{-1}\log|A|$, for some absolute constant $C$. (To see why the factor $\gamma^{-1}$ is necessary, consider an arbitrary union of $\gamma^{-1}$ subspaces of $\mathbb F_2^n$ of the same dimension.) In this paper it is shown that if $p$ is the smallest order of a non-zero element of $G$, then $\mathrm{dim}_I(P_\gamma(A)) \le (2(1-1/p))^{-1}\gamma^{-1}\log_2|A|$, which is refined to $\mathrm{dim}_I(P_\gamma(A)) \le \gamma^{-1}\log_3|A|$ in the special case $p=3$. These estimates are sharp when $G$ is homocyclic of exponent $p=2$ or $3$, when the two notions of dimension coincide. One of the proof ingredients in the paper is a result that stands on its own. A _down-set_ in $\mathbb Z_{\geq 0}^n$ is a subset $A$ with the property that if $x\in A$ and $y_i\leq x_i$ for every $i$, then $y\in A$. The paper gives an upper bound of $\frac 12\log_2|A|$ for the average number of non-zero coordinates of a vector in a down-set. It is a simple exercise to prove that this bound is sharp.

Linear d-Polychromatic $$Q_{d-1}$$ Q d - 1 -Colorings of the Hypercube

Let $$n \ge d \ge \ell \ge 1$$ be integers, and denote the n-dimensional hypercube by $$Q_n$$ . A coloring of the $$\ell $$ -dimensional subcubes $$Q_\ell $$ in $$Q_n$$ is called a $$Q_\ell $$ -coloring. Such a coloring is d-polychromatic if every $$Q_d$$ in the $$Q_n$$ contains a $$Q_\ell $$ of every color. In this paper we consider a specific class of $$Q_\ell $$ -colorings that are called linear. Given $$\ell $$ and d, let $$p_{lin}^\ell (d)$$ be the largest number of colors such that there is a d-polychromatic linear $$Q_\ell $$ -coloring of $$Q_n$$ for all $$n \ge d$$ . We prove that for all $$d \ge 3$$ , $$p_{lin}^{d-1}(d) = 2$$ . In addition, using a computer search, we determine $$p_{lin}^\ell (d)$$ for some specific values of $$\ell $$ and d, in some cases improving on previously known lower bounds.

On the structure of subsets of the discrete cube with small edge boundary

On the structure of subsets of the discrete cube with small edge boundary, Discrete Analysis 2018:9, 29 pp. An isoperimetric inequality is a statement that tells us how small the boundary of a set can be given the of the set, for suitable notions of size and boundary. For example, one formulation of the classical isoperimetric inequality in $\mathbb R^n$ is as follows. Given a subset $X$ of $\mathbb R^n$, define the $\epsilon$-_expansion_ of $X$ to be the open set $X_\epsilon=\{y\in\mathbb R^n: d(y,X)<\epsilon\}$. If in addition $X$ is measurable, define the of its boundary to be $\lim\inf_{\epsilon\to 0} \epsilon^{-1}\mu(X_\epsilon\setminus X)$. (If $X$ is a set with a suitably smooth topological boundary $\partial X$, then this turns out to equal the surface measure of $\partial X$.) Then amongst all sets $X$ of a given measure, the one with the smallest boundary is an $n$-dimensional ball. Isoperimetric inequalities have been the focus of a great deal of research, partly for their intrinisic interest, but also because they have numerous applications. One particularly useful one is the _edge-isoperimetric inequality in the discrete cube_. This concerns subsets $X$ of the $n$-dimensional cube $\{0,1\}^n$, which we turn into a graph by joining two points $x$ and $y$ if they differ in exactly one coordinate. The of a set $X$ is simply its cardinality, the _edge-boundary_ of $X$ is defined to be the set of edges between $X$ and its complement, and the of the edge-boundary is the number of such edges. If $|X|=2^d$, then it is known that the edge-boundary is minimized when $X$ is a $d$-dimensional subspace of $\mathbb F_2^n$ generated by $d$ standard basis vectors. More generally, if $|X|=m$, then the edge-boundary is minimized when $X$ is an initial segment in the lexicographical ordering, which is the ordering where we set $x<y$ if $x_i<y_i$ for the first coordinate $i$ where $x_i$ and $y_i$ differ. (This coincides with the ordering we obtain if we think of the sequences as binary representations of integers.) For the two isoperimetric inequalities just mentioned, as well as many others, it is known that the extremal examples provided are essentially the only ones: a subset of $\mathbb R^n$ with a boundary that is as small as possible has to be an $n$-dimensional ball, and a subset of the discrete cube with edge-boundary that is as small as possible has to be an initial segment of the lexicographical ordering, up to the symmetries of the graph. Furthermore, there are _stablity_ results: a set with a boundary that is _almost_ as small as possible must be close to an extremal example. Such a result tells us that the isoperimetric inequalities are robust, in the sense that if you slightly perturb the condition on the set, then you only slightly perturb what the set has to look like. This paper is about an extremely precise stability result for the edge isoperimetric inequality in the discrete cube. There have been a number of papers on such results (see the introduction to the paper for details), but they have been mainly for sets of $2^d$ for some $d$, where the goal is to prove that they must be close to $d$-dimensional subcubes -- that is, subspaces (or their translates) generated by $d$ standard basis vectors. This paper considers sets of arbitrary and proves the following result. Suppose that $X$ is a subset of $\{0,1\}^n$ of $m$. Suppose that the of the edge-boundary of $X$ is at most $g_n(m)+l$, where $g_n(m)$ is the of the edge-boundary of the initial segment $I_m$ of $m$ in the lexicographical order. Then there is an automorphism $\phi$ of $\{0,1\}^n$ (meaning a bijection that takes neighbouring points to neighbouring points) such that $|X\Delta\phi(I_m)|\leq Cl$, where $C$ is an absolute constant. They give an example to show that $C$ must be at least 2, and thus that their result is best possible up to the value of the constant $C$. Previous proofs of stability versions of the edge-isoperimetric inequality in the cube have used Fourier analysis. The proof in this paper uses purely combinatorial methods, such as induction on the dimension, and compressions. To get these methods to work, several interesting ideas are needed, including some new results about the influence of variables.

An isoperimetric inequality for antipodal subsets of the discrete cube

We say a family of subsets of {1,2,…,n} is antipodal if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of {1,2,…,n} (of any size). Our inequality implies that for any k∈N, among all such families of size 2k, a family consisting of the union of two antipodal (k−1)-dimensional subcubes has the smallest possible edge boundary.

On the Entropy of a Noisy Function

Let $0 be a noise parameter, and let $T_{ \epsilon }$ be the noise operator acting on functions on the Boolean cube $ \{0,1\}^{n}$ . Let $f$ be a nonnegative function on $ \{0,1\}^{n}$ . We upper bound the entropy of $T_{ \epsilon } f$ by the average entropy of conditional expectations of $f$ , given sets of roughly $(1-2 \epsilon )^{\vphantom {R^{l}}2} \cdot n$ variables. In information-theoretic terms, we prove the following strengthening of Mrs. Gerber’s Lemma: let $X$ be a random binary vector of length $n$ , and let $Z$ be a noise vector, corresponding to a binary symmetric channel with crossover probability $ \epsilon $ . Then, setting $v = (1-2 \epsilon )^{2} \cdot n$ , we have (up to lower order terms): $H\big (X \oplus Z\big ) \ge n \cdot H_{2} \big ( \epsilon + (1-2 \epsilon ) \cdot H_{2}^{-1}\big (({ \mathop {\mathbb E}_{|B| = v} H\big (\{X_{i}\}_{i\in B}\big )}/{v})\big )\big )$ . Assuming $ \epsilon \ge 1/2 - \delta $ , for some absolute constant $\delta > 0$ , this inequality, combined with a strong version of a theorem of Friedgut et al. , due to Jendrej et al. , shows that if a Boolean function $f$ is close to a characteristic function $g$ of a subcube of dimension $n-1$ , then the entropy of $T_{ \epsilon } f$ is at most that of $T_{ \epsilon } g$ . Taken together with a recent result of Ordentlich et al. , this shows that the most informative Boolean function conjecture of Courtade and Kumar holds for high noise $ \epsilon \ge 1/2 - \delta $ . Namely, if $X$ is uniformly distributed in $ \{0,1\}^{n}$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ \epsilon $ , then, provided $ \epsilon \ge 1/2 - \delta $ , for any Boolean function $f$ holds $I \left ({ f(X);Y }\right )\le 1 - H( \epsilon )$ .

Fault tolerance in [formula omitted]-ary [formula omitted]-cube networks

The k-ary n-cube Qnk is one of the most commonly used interconnection topologies for parallel and distributed computing systems. Let f(n,m) be the minimum number of faulty nodes that make every (n−m)-dimensional subcube Qn−mk faulty in Qnk under node-failure models. In this paper, we prove that f(n,0)=1, f(n,1)=k for odd k≥3, f(n,n−1)=kn−1 for odd k≥3, and km≤f(n,m)≤(n−1m−1)km−(n−2m−1)km−1 for odd k≥3.

Hamiltonian cycles in hypercubes with [formula omitted] faulty edges

In this paper we consider Hamiltonian cycles in hypercubes with faulty edges and present a simple proof that the hypercube Qn is not Hamiltonian if it contains a subgraph T such that: (i) exactly half of the vertices in T have parity 0 (the other half have parity 1) and (ii) all edges joining the nodes of parity 0 (or 1) with nodes outside T, are faulty. We will call such a subgraph a trap disconnected halfway. Moreover, we show that for n⩾5, the cube Qn with 2n-4 faulty edges is not Hamiltonian only if it contains a trap T disconnected halfway with T being a subcube of dimension 1 or 2.

Polychromatic Colorings of Subcubes of the Hypercube

Alon, Krech, and Szabó [SIAM J. Discrete Math., 21 (2007), pp. 66–72] called an edge-coloring of a hypercube with p colors such that every subcube of dimension d contains every color a d-polychromatic p-coloring. Denote by $p_d$ the maximum number of colors with which it is possible to d-polychromatically color any hypercube. We find the exact value of $p_d$ for all values of d.

The number of 2-sat functions

Our aim in this paper is to address the following question: of the 22 n Boolean functions onn variables, how many are expressible as 2-SAT formulae? In other words, we wish to count the number of different instances of 2-SAT, counting two instances as equivalent if they have the same set of satisfying assignments. Viewed geometrically, we are asking for the number of subsets of then-dimensional discrete cube that are unions of (n-2)-dimensional subcubes. There is a trivial upper bound of 24(n/2), the number of 2-SAT formulae. There is also an obvious lower bound of 2(n/2), corresponding to the monotone 2-SAT formulae. Our main result is that, rather surprisingly, this lower bound gives the correct speed: the number of 2-SAT functions is 2(1+0(1)) n 2 2.

Real-time fault-tolerant hypercube multicomputer

A real-time fault-tolerant design for a d-dimensional hypercube multiprocessor with two modes of operation is presented and its reconfigurability is examined. The augmented hypercube, at stage one, has a spare node connected to each node of a subcube of dimension i, and the spare nodes are also connected as a (d−i)-dimensional hypercube. At stage two, the process is repeated by assigning one spare node to each (d−i−j)-dimensional spare subcube of stage one. Two modes of operations are considered, one under heavy computation or hard deadline and the other under light computation or soft deadline. By utilising the capabilities of wave-switching communication modules of the spare nodes, faulty nodes and faulty links can be tolerated. Both theoretical and experimental results are presented. Compared with other proposed schemes, the proposed approach can tolerate significantly more faulty components with a low overhead and no performance degradation.

Highly fault-tolerant hypercube multicomputer

A strongly fault-tolerant design for a d-dimentrun aucountsional hypercube multiprocessor is presented and its reconfigurability examined. The augmented hypercube has a spare node connected to each node of a subcube of dimension i, and the spare nodes are also connected as a (d - i)-dimensional hypercube. By utilising the circuit-switched capabilities of the communication modules of the spare nodes, a large number of faulty nodes and faulty links can be tolerated. Both theoretical and experimental results are presented. Compared with other proposed schemes the approach can tolerate significantly more faulty nodes and faulty links with a low overhead and no performance degradation.

Optimal Subcube Allocation in a Circuit-Switched Faulty Hypercube

In this paper, we present a scheme where a (d - 1)-dimensional subcube is allocated in a faulty d-dimensional circuit-switched hypercube in the presence of up to 2(d - 1) faulty nodes. The scheme is then extended to allocate a (d - 1)-dimensional subcube in the presence of a combination of faulty nodes and faulty links. Theoretical proofs and simulation results are presented to analyze the performance of the scheme.

Split sequence generation algorithms for efficient identification of operational subcubes in faulty hypercubes

Two new heuristic algorithms are proposed for enhancing operational subcube identification in hypercubes with faulty nodes. Either of the algorithms can adaptively construct a complete binary tree based on the fault set such that subcube identification is greatly improved. They can be used to modify the standard Gray Code or Buddy strategy to provide performance improvement in the identification of higher dimensional operational subcubes with practical time complexities that make them attractive. In essense, the algorithms provide preprocessing techniques based on recursive splitting of the faulty hypercube so that a large number of high dimensional subcubes can be identified. For the identification part, existing schemes can be used to actually harvest the subcubes.

Subcube Fault-Tolerance in Hypercubes

We consider the problem of determining the minimum number of faulty processors, K ( n , m ), and of faulty links, λ( n , m ), in an n -dimensional hypercube computer so that every m -dimensional subcube is faulty. Best known lower bounds for K ( n , m ) and λ( n , m ) are proved, several new recursive inequalities and new upper bounds are established, their asymptotic behavior for fixed m and for fixed n − m is analyzed, and their exact values are determined for small n and m . Most of the methods employed show how to construct sets of faults attaining the bounds. An extensive survey of related work is also included, showing connections to resource allocation, k -independent sets, and exhaustive testing.

A RECONFIGURATION TECHNIQUE FOR FAULT TOLERANCE IN A HYPERCUBE

The hypercube architecture is a popular topology for many parallel processing applications. For continued operation of the hypercube multiprocessors after the failure of one or more i-subcubes and/or links, fault tolerance by reconfiguration is an important problem. This paper considers the reconfiguration issue and presents an algebraic technique to analyze the problem extending the concepts in [9]. The technique uses algebraic operators to identify the maximum dimensional fault-free subcube, and thus helps in achieving graceful degradation of the system. We analyze the complexity of our algorithm and show that it is efficient as compared to previous algorithms [1, 8, 14].

Scheduling independent jobs on partitionable hypercubes

We consider the problem of scheduling k independent jobs on an n-dimensional hypercube system to minimize finishing time, where each job J i is associated with a dimension d i and a processing time t i , meaning that J i requires a d i -dimensional subcube for t i units of time. This problem is NP-hard if no preemption is allowed. We propose a simple heuristic called LDLPT (largest dimension longest processing time) and analyze its worst-case performance; the ratio of the heuristic to the optimal finishing time does not exceed 2 − 1 2 n−1 .

A top-down processor allocation scheme for hypercube computers

An efficient processor allocation policy is presented for hypercube computers. The allocation policy is called free list since it maintains a list of free subcubes available in the system. An incoming request of dimension k (2/sup k/ nodes) is allocated by finding a free subcube of dimension k or by decomposing an available subcube of dimension greater than k. This free list policy uses a top-down allocation rule in contrast to the bottom-up approach used by the previous bit-map allocation algorithms. This allocation scheme is compared to the buddy, gray code (GC), and modified buddy allocation policies reported for the hypercubes. It is shown that the free list policy is optimal in a static environment, as are the other policies, and it also gives better subcube recognition ability compared to the previous schemes in a dynamic environment. The performance of this policy, in terms of parameters such as average delay, system utilization, and time complexity, is compared to the other schemes to demonstrate its effectiveness. The extension of the algorithm for parallel implementation, noncubic allocation, and inclusion/exclusion allocation is also given. >

Preemptive scheduling of independent jobs on a hypercube

We consider the problem of scheduling n independent jobs on an m-dimensional hypercube system to minimize the finishing time, where each job J i is associated with a dimension d i and a processing time t i , meaning that J i requires a d i -dimensional subcube for t i units of time. An O( n 2) algorithm is presented that decides if all n jobs can be finished by a given deadline T. Using this algorithm, one may obtain a minimum-finishing-time schedule in polynomial time.

How robust is the n-cube?

The n -cube network is called faulty if it contains any faulty processor or any faulty link. For any number k we want to compute the minimum number f(n, k) of faults which is necessary for an adversary to make every ( n − k )-dimensional subcube faulty. Reversely formulated: The existence of an ( n − k )-dimensional non-faulty subcube can be guaranteed, if there are less than f(n, k) faults in the n -cube. In this paper several lower and upper bounds for f(n, k) are derived such that the resulting gaps are “small.” For instance if k ≥ 2 is constant, then f(n, k) = θ(logn). Especially for k = 2 and large n : f(n, 2) ∈ [[α n ⌉]: [α n ]⌉ + 2], where α n =logn + ½ log log n + ½. Or if k = ω(log log n ) then 2 k < f(n, k) < 2 (1 + ɛ)k , with ɛ chosen arbitrarily small. The aforementioned upper bounds are obtained by analysing the behaviour of an adversary who makes “worst-case” distributions of a given number of faulty processors. For k = 2 the “worst-case” distribution is obtained constructively. In the general case the constructive methods presented in this paper lead to a (rather “bad”) upper bound which can be significantly improved by probabilistic arguments. The bounds mentioned above change if the notions are relativized with respect to some given parallel fault-checking procedure P . In this case only subcubes which are possible outputs of P must be made faulty by the adversary. The notion of directed chromatic index is defined in order to analyse the case k = 2. Relations between the directed chromatic index and the chromatic number are derived, which are of interest in their own right.