We define an interconnection network AQn,k which we call the augmented kary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube AQn,k has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n...

Consider a hypergraph H n where the vertices are points of the d-dimensional combinatorial cube n and the edges are all sets of n points such that they are in one line. We study the structure of the group of automorphisms of H n, i.e., permutations of points of n d preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism prob...

The Rubik’s Cube is perhaps the world’s most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik’s Cube also has a rich underlying algorithmic structure. Specifically, we show that the n×n×n Rubik’s Cube, as well as the n×n×1 variant, has a “God’s Number” (diameter of the configuration space) of Θ(n/ logn). ...

Families of non-lattice tilings of R n by unit cubes are constructed. These tilings are specializations of certain families of nonlinear codes over GF( 2 ). These cube-tilings provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common. We construct cube-tilings of R n such that no two cubes have a common face of dimension exceeding...

An (n, a, b)-perfect double cube is a b × b × b sized n-ary periodic array containing all possible a × a × a sized n-ary array exactly once as subarray. A growing cube is an array whose cj × cj × cj sized prefix is an (nj, a, cj)-perfect double cube for j = 1, 2, . . ., where cj = n v/3 j , v = a 3 and n1 < n2 < · · ·. We construct the smallest possible perfect double cube (a 256×256×256 sized ...

A Lucas cube 3^ can be defined as the graph whose vertices are the binary strings of length n without either two consecutive l's or a 1 in the first and in the last position, and in which the vertices are adjacent when their Hamming distance is exactly 1. A Lucas cube 5E„ is very similar to the Fibonacci cube Tn which is the graph defined as 2J, except for the fact that the vertices are binary ...