Boolean Cube Research Articles

Two results on the bit extraction problem

An ( n, s, t)-resilient function is a function f : {1,−1} n→{1,−1} s, such that every element in {1,−1} s has the same probability to occur when t arbitrary input variables are fixed by an adversary and the remaining n− t variables are assigned −1 or 1 uniformly and independently. A basic problem is to find the largest possible t given n and s such that an ( n, s, t)-resilient function exists. As mentioned in [5,2,3] resilient functions have been involving in some interesting cryptographic applications, such as amplifying privacy and generating shared random strings in the presence of faulty processors. The following coloring problem is closely related to resilient functions. A good coloring of the n-dimensional Boolean cube with c=2 s colors is such that in every k-dimensional subcube each color appears 2 k / c times. Here we are interested in the smallest k for which such a coloring exists. A coloring is locally symmetric if each vertex in the n-cube has the same number of neighbors from each color other than its own. An XOR-type coloring is one obtained by a linear map GF(2) n → GF(2) s . We answer the following problems proposed by Friedman [5]: (1) If c−1| n, are all optimal colorings necessarily locally symmetric? (2) Are there optimal colorings not obtainable as XOR type colorings? We provide positive answers to both questions. For (2), we show that there are infinitely many non- XOR-type optimal solutions, which is also proved by Stinson et al. with a different method [8]. Our main tool is an expression of the eigenvalues of the distance- i adjacency matrix of the n-cube via the Krawtchouk polynomials. With the technique, we are able to make the background of some claims in [5, p. 318] clearer. Along the way we prove some identities involving the distance distribution of color classes, in order to gain a better understanding of the structure of color classes.

Dimension of Projections in Boolean Functions

A projection is a subset of {0,1}n given by equations of the form xi = xj, xi = \bar{x}j, xi = 0, and xi = 1, where for $1\leq i \leq n$, xi are Boolean variables and $\bar{x}i are their complements. We study monochromatic projections in 2-colorings of an n-dimensional Boolean cube. We also study the dimension of the largest projection contained in a set specified by its density. We prove almost matching lower and upper bounds on the density of a set required to guarantee the existence of a d-dimensional projection. We also prove almost tight upper and lower bounds on the dimension of monochromatic projections in arbitrary Boolean functions. We then prove almost tight upper and lower bounds on the dimension of monochromatic projections in Boolean functions represented by low degree GF(2) polynomials. It follows from these lower bounds that low-degree GF(2) polynomials can define Boolean functions which are close to being extremal with respect to the property of having no large dimensional monochromatic projections.

On Generalized Fibonacci Cubes and Unitary Transforms

We present a new interconnection topology called generalized Fibonacci topology, which unifies a wide range of connection topologies such as the Boolean cube (or hypercube), classical Fibonacci cube, etc. Some basic topological properties of generalized Fibonacci cubes are established. Finally, we developed new classes of the discrete orthogonal transforms, based on the generalized Fibonacci recursions. They can be implemented efficiently by butterfly-type networks (like the Fourier, or the Haar transforms). A generalized Fibonacci cube based processor architecture (generalizing the known SIMD architecture — hypercube processor) can be efficiently used for hardware implementation of the proposed discrete orthogonal transforms.

Orbits of Boolean functions

The group of congruences and permutations of the two-colored N-dimensional Boolean cube is considered. The total number of orbits generated by these automorphisms are shown to scale as 2 2 n (2 n+1N when N tends to infinity. The probability that a randomly chosen function will belong to an orbit containing the maximum possible number of elements, 2 N+1 N, approaches one as N goes to infinity. Simulations for N ⩽ 6 are in agreement with the scaling predictions.

Bent functions and random boolean formulas

Let α be a nonlinear boolean connective with equal number of zero and unit entries in its table. We study an iterative process of combining randomly chosen boolean functions from some starting set G via the connective α. We suppose that all functions in G are defined on the same finite nonempty domain M. We are interested in the situations, when the process converges to the uniform distribution on {0, 1} M . We classify the boolean connectives α according to the asymptotic rate of this convergence. Although the probability of any function in our process converges to ( 1 2 ) ¦M , there are some differences in the terms of lower order of magnitude. If M is the boolean cube of an even dimension n and G is the set of all linear boolean functions of n variables and the connective α belongs to the class of the lowest possible rate of convergence in the above-mentioned classification, we can express the main nonconstant term of the asymptotic expansion of the probability of occurring of a single function f through the Fourier transform of f. Using this, we prove that the bent functions achieve asymptotically the minimal probability of occurring among all boolean functions. At the same time, the linear functions achieve asymptotically the maximal probability.

Improved boolean formulas for the Ramsey graphs

AbstractIn the theory of the random graphs, there are properties of graphs such that almost all graphs satisfy the property, but there is no effective way to find examples of such graphs. By the well‐known results of Razborov, for some of these properties, e.g., some Ramsey property, there are Boolean formulas in ACC representing the graphs satisfying the property and having exponential number of vertices with respect to the number of variables of the formula. Razborov's proof is based on a probabilistic distribution on formulas of n variables of size approximately nd2 log d, where d is a polynomial in n, and depth 3 in the basis { ∧, ⊕} with the following property: The restriction of the formula randomly chosen from the distribution to any subset A of the Boolean cube {0, 1}n of size at most d has almost uniform distribution on the functions A → {0, 1}. We show a modified probabilistic distribution on Boolean formulas which is defined on formulas of size at most nd log2 d and has the same property of the restrictions to sets of size at most d as the original one. This allows us to obtain formulas the complexity of which is a polynomial of a smaller degree in n than in Razborov's paper while the represented graphs satisfy the same properties.

Stable networks and product graphs

A network is a collection of gates, each with many inputs and many outputs, where links join individual outputs to individual inputs of gates; the unlinked inputs and outputs of gates are viewed as inputs and outputs of the network. A stable configuration assigns values to inputs, outputs, and links in a network, so as to ensure that the gate equations are satisfied. The problem of finding stable configurations in a network is computationally hard, even when all values are boolean and all input values are specified in advance; in general, the difficulty of a stability problem seems to depend on the kinds of mappings associated with the gates present in the network. The study can be restricted to gates that satisfy a nonexpansiveness condition requiring small perturbations at the inputs of a gate to have only a small effect at the outputs of the gate. The stability question on the class of networks satisfying this local nonexpansiveness condition contains stable matching as a main example, and defines the boundary between tractable and intractable versions of network stability. The structural and algorithmic study of stability in nonexpansive networks is based on a representation of the possible assignments of boolean values for a network as vertices in a boolean cube under the associated Hamming metric. This global view takes advantage of the median properties of the cube, and extends to metric networks, where individual values are now chosen from finite metric spaces and combined by means of an additive product operation. The relationship between products of metric spaces and products of graphs establishes then a connection between isometric representations in graphs and nonexpansiveness in metric networks.

Hot-potato algorithms for permutation routing

We develop a methodology for the design of hot-potato algorithms for routing permutations. The basic idea is to convert existing store-and-forward routing algorithms to hot-potato algorithms. Using it, we obtain the following complexity bounds for permutation routing: n/spl times/n Mesh: 7n+o(n) steps; 2/sup n/ hypercube: O(n/sup 2/) steps; n/spl times/n Torus: 4n+o(n) steps. The algorithm for the two-dimensional grid is the first to be both deterministic and asymptotically optimal. The algorithm for the 2/sup n/-nodes Boolean cube is the first deterministic algorithm that achieves a complexity of o(2/sup n/) steps.

Embedding hyperpyramids into hypercubes

Abstract : Hyper-pyramids contain pyramids as proper subgraphs. We show that a P(k.d) hyper-pyramid can be embedded in a Boolean cube with minimal expansion and dilation. In addition to expansion, dilation, and congestion we also characterize the embedding with the active-degree, and the node-load.

Local algorithms for circuits in a Boolean cube

Local algorithms for circuits in a Boolean cube

Fibonacci cubes-a new interconnection Topology

A novel interconnection topology called the Fibonacci cube is shown to possess attractive recurrent structures in spite of its asymmetric and relatively sparse interconnections. Since it can be embedded as a subgraph in the Boolean cube (hypercube) and it is also a supergraph of other structures, the Fibonacci cube may find applications in fault-tolerant computing. For a graph with N nodes, the diameter, the edge connectivity, and the node connectivity of the Fibonacci cube are in the logarithmic order of N. It is also shown that common system communication primitives can be implemented efficiently.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Generalized shuffle permutations on boolean cubes

In a generalized shuffle permutation an address ( a q−1 a q−2 … a 0) receives its content from an address obtained through a cyclic shift on a subset of the q dimensions used for the encoding of the addresses. Bit-complementation may be combined with the shift. We give an algorithm that requires K 2 + 2 exchanges for K elements per processor, when storage dimensions are part of the permutation, and concurrent communication on all ports of every processor is possible. The number of element exchanges in sequence is independent of the number of processor dimensions σ r in the permutation. With no storage dimensions in the permutation our best algorithm requires ( σ 4 r + 1)[ K 2σ r ] element exchanges. We also give an algorithm for σ r = 2, or the real shuffle consists of a number of cycles of length two, that requires K 2 + 1 element exchanges in sequence when there is no bit complement. The lower bound is K 2 for both real and mixed shuffles with no bit-complementation. The minimum number of communication start-ups is σ r for both cases, which is also the lower bound. The data transfer time for communication restricted to one port per processor is σ r ( K 2 ) , and the minimum number of start-ups is σ r. The analysis is verified by experimental results on the Intel iPSC/1, and for one case also on the Connection Machine model CM-2.

Performance modeling of distributed memory architectures

We provide performance models for several primitive operations on data structures distributed over memory units interconnected by a Boolean cube network. In particular, we model single-source and multiple-source concurrent broadcasting or reduction, concurrent gather and scatter operations, shifts along several axes of multidimensional arrays, and emulation of butterfly networks. We also show how the processor configuration, the data aggregation, and the encoding of the address space affect the performance for two important basic computations: the multiplication of arbitrarily shaped matrices and the Fast Fourier Transform. We also give an example of the performance behavior for local matrix operations for a processor with a single path to local memory and a set of processor registers. The analytic models are verified by measurements on the Connection Machine Model CM-2.

FTN topology and protocols

Abstract We have developed a network called FTN that satisfies the requirements for a large class of fault tolerant applications in distributed processing. (N denotes the number of devices and m = ⌈log2N⌉). The network has a high fault tolerance: up to m−2 faulty processors can be tolerated. The fault tolerance is high and optimal. The criteria of optimality is the following: the number of vertex disjoint paths is equal to the minimal degree in the graph (m−1). We have developed algorithms to provide routing in both faulty and nonfaulty systems. The paths generated by the algorithms are nearly the shortest paths for any two source and destination processors. The network can be constructed for a system of an arbitrary size. We developed an algorithm which can add a new processor at the small cost of modifying only O(m) edges. The O(m) diameter implies a low transit delay. A multiplexing/demultiplexing protocol can utilize up to m−1 nonblocking paths between any two processors to achieve a high throughput. In general the network has much better fault tolerance than the recently reported Incomplete Cube, can be designed for any number of processors in contrast to the Boolean Cube, and it has a significantly lower cost than the Generalized Hypercube network in terms of the link count.

Embeddings on a boolean cube

In this paper, we characterize a class of graphs which can be embedded on a boolean cube. Some of the graphs in this class are identified with the well known graphs such asmulti-dimensional mesh of trees, tree of meshes, etc. We suggest (i) an embedding of anr-dimensional mesh of trees ofnr(r+1)−rnr−1 nodes on a boolean cube of (2n)r nodes, and (ii) an embedding of a tree of meshes with 2n2 logn+n2 nodes on a boolean cube withn2 exp2 (log (2 logn+1)]) nodes.

Embedding meshes in Boolean cubes by graph decomposition

This paper explores the embeddings of multidimensional meshes into minimal Boolean cubes by graph decomposition. The dilation and the congestion of the product graph ( G 1 × G 2) → ( H 1 × H 2) is the maximum of the dilation and congestion for the two embeddings G 1 → H 1 and G 2 → H 2. The graph decomposition technique can be used to improve the average dilation and average congestion. The graph decomposition technique combined with some particular two-dimensional embeddings allows for minimal-expansion, dilation-two, congestion-two embeddings of about 87% of all two-dimensional meshes, with a significantly lower average dilation and congestion than by modified line compression. For three-dimensional meshes we show that the graph decomposition technique, together with two three-dimensional mesh embeddings presented in this paper and modified line compression, yields dilation-two embeddings of more than 96% of all three-dimensional meshes contained in a 512 × 512 × 512 mesh. The graph decomposition technique is also used to generalize the embeddings to meshes with wrap-around. The dilation increases by at most one compared to a mesh without wraparound. The expansion is preserved for the majority of meshes, if a wraparound feature is added to the mesh.

Spanning Balanced Trees in Boolean Cubes

A Spanning Balancedn-tree (SBnT) in a Boolean n-cube is a spanning tree in which the root has fanout n, and all the subtrees of the root have $O({{2^n } / n})$ nodes. The number of tree edges in each dimension of the n-cube is of order $O({{2^n } / n})$. The spanning balanced n-tree allows for scheduling disciplines that realize lower bound (within a factor of two) one-to-all personalized communication, all-to-all broadcasting, and all-to-all personalized communication on a Boolean n-cube [C.-T. Ho and S. L. Johnsson, Proc. 1986 International Conference on Parallel Processing, pp. 640–648, IEEE Computer Society, 1986; Tech. Report YALEU/DCS/RR–483, May 1986], [S. L. Johnsson and C.-T. Ho, Tech. Report YALEU/DCS/RR–610, Dept. of Computer Science, Yale Univ., New Haven, CT, November 1987]. The improvement in data transfer time over the familiar binomial tree routing is a factor of ${n / 2}$ for concurrent communication on all ports and one-to-all personalized communication and all-to-all broadcasting. For all-to-all personalized communication on all ports concurrently, the improvement is of order $O(\sqrt n )$. Distributed routing algorithms defining the spanning balanced n-tree are given. The balanced n-tree is not unique, and a few definitions of n-trees that are effectively edge-disjoint are provided. Some implementation issues are also discussed. Binary numbers obtained from each other through rotation form necklaces that are full if the period is equal to the length of the number; otherwise, they are degenerate. As an intermediary result, it is shown that the ratio between the number of degenerate necklaces and the total number of necklaces with l bits equal to one is at most ${4 / {(4 + n)}}$ for $1 \leqq l < n$.

Topological aspects of immunological control networks

Immunological control networks can be modelled by bipartite signed directed graphs called influence diagrams. Possible flow topologies around unstable steady states in such systems can be determined by kinetic logic based on the directed Boolean cubes of switching circuit theory. This method can identify the stable “help” and “suppression” configurations of the basic Herzenberg core regulatory circuit as well as the “memory” and “non-responsive” configurations of the model of Kaufman, Urbain and Thomas. Furthermore, addition of a single vertex to the Herzenberg core regulatory circuit, so as to preserve the bipartite nature of the signed directed graph, leads to a network with five independent internal variables and two connected feedback circuits, either positive and negative circuits of length 4 or positive circuits of lengths 4 and 2. The flow topology of either of these systems indicates that addition of the fifth vertex switches the basic four-vertex core regulatory circuit into one of its two stable configurations.

Solving Tridiagonal Systems on Ensemble Architectures

The concurrent solution of tridiagonal systems on linear and 2-dimensional arrays, complete binary trees, shuffle-exchange and perfect shuffle networks, and boolean cubes by elimination methods are devised and analyzed. The methods can be obtained by symmetric permutations of some rows and columns, and amounts to cyclic reduction or a combination of Gaussian elimination and cyclic reduction (GECR). The ensembles have only local storage and no global control. Synchronization is accomplished via message passing to neighboring processors.The parallel arithmetic complexity of GECR for N equations on a K processor ensemble is $O({ N / K } + \log _2 K)$, and the communication complexity is $O(K)$ for the linear array, $O(\sqrt K )$ for the 2-dimensional mesh, and $O(\log _2 K)$ for the networks of diameter $O(\log _2 K)$. The maximum speed-up for the linear array is attained at $K \approx {({ N / \alpha })}^{1/2} $ and for the 2-d mesh at $K \approx ({N / 2\alpha })^{2/3} $, where $\alpha $ (the time to communicate one floating-point number)/(the time for a floating-point arithmetic operation). For the binary tree the maximum speed-up is attained at $K = N$, and for the perfect shuffle and boolean k-cube networks, $K = N/(1 + \alpha )$ yields the maximum speed-up. The minimum time complexity is of order $O(N^{1/2} )$ for the linear array, of order $O(N^{1/3} )$ for the mesh, and of order $O(\log _2 N)$ for the binary tree, the shuffle-exchange, the perfect shuffle and the boolean k-cube.The relative decrease in computational complexity due to a truncation of the reduction process in a highly concurrent system is much greater than on a uniprocessor. The reduction in the arithmetic complexity is proportional to the number of steps avoided, if the number of processing elements equals the number of equations. So also is the reduction in the communication complexity for ensembles configured as binary trees, shuffle-exchange and perfect shuffle networks, and boolean cubes.Partitioning the ensemble into subsets of processors is shown to be more efficient for the solution of multiple independent problems than pipelining the solutions over the entire ensemble. A balanced cyclic reduction algorithm is presented for the case where each system is spread uniformly over the processing elements, and its complexity is compared with Gaussian elimination.

Performance evaluation of spanning multiaccess channel hypercube interconnection network

The spanning multiaccess channel hypercube (SMCH) is a distributed computer interconnection structure, based on a hypercube topology, where processors are connected using multiaccess channels spanning all dimensional axes. The channel access arbitration is achieved through demand assignment multiple access (DAMA) protocols. We compare the SMCH to the generalised hypercube (GHC), the nearest neighbour mesh hypercube (NNMH), cube connected cycles (CCC), and the Boolean cube (BCube). The comparison shows the SMCH to have excellent performance characteristics, at a cost significantly lower than comparable structures. This is achieved by combining the recent advances in processor interconnection architectures, the economies of scale available with highcapacity channels, and demand assignment multiple-access protocols, to form a highly efficient, cost effective, distributed computer system.