Extended Fibonacci cubes

The Fibonacci Cube is an interconnection network that possesses many desirable properties that are important in network design and application. The Fibonacci Cube can efficiently emulate many hypercube algorithms and uses fewer links than the comparable hypercube, while its size does not increase as fast as the hypercube. However, most Fibonacci Cubes (more than 2/3 of all) are not Hamiltonian. In this paper, we propose an Extended Fibonacci Cube (EFC1) with an even number of nodes. It is defined based on the same sequence F(i ) = F(i 1) + F(i 2) as the regular Fibonacci sequence; however, its initial conditions are different. We show that the Extended Fibonacci Cube includes the Fibonacci Cube as a subgraph and maintains its sparsity property. In addition, it is Hamiltonian and is better in emulating other topologies. Specifically, the Extended Fibonacci Cube can embed binary trees more efficiently than the regular Fibonacci Cube and is almost as efficient as the hypercube, even though the Extended Fibonacci Cube is a much sparser network than the hypercube. We also propose a series of Extended Fibonacci Cubes with even number of nodes. Any Extended Fibonacci Cube (EFCk, with k ≥ 1) in the series contains the node set of any other cube that precedes EFCk in the series. We show that any Extended Fibonacci Cube maintains virtually all the desirable properties of the Fibonacci Cube. The EFCks can be considered as flexible versions of incomplete hypercubes, which eliminates their restriction on the number of nodes, and, thus, makes it possible to construct parallel machines with arbitrary sizes.