On the VLSI area and bisection width of star graphs and hierarchical cubic networks

We solve an open question posed by Akers and Krishnamurthy in 1986 concerning VLSI layout of star graphs. We show that the area of the optimal layout of an N-node star graph, hierarchical cubic network (HCN), or hierarchical folded-hypercube network (HFN) is N²/16±o(N²) under the Thompson model, or under the extended grid model where a node occupies a rectangle of sides that may range from n-1 to o(√N) for the n-star, log₂N+1 to o(√N) for the HCN, and log₂N+2 to o(√N) for the HFN. An n-dimensional star graph this requires less area than any possible layout of a similar-size hypercube, but more than that of the much smaller n-cube. We also derive multilayer layout for star graphs that has area N²/8[L²/2]±o(N ²/L²), where a node occupies a rectangle of sides ranging from [n-1/4] to o(√N/L) and the number L of wiring layers satisfies 2⩽L=o(√N/n). Finally we show that the bisection width of an N-node star graph is N/4±o(N) and the bisection width of an HCN or HFN is exactly N/4