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 [1, 3] 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(pN) for the n-star, log2 N + 1 to o(pN) for the HCN, and log2 N+ 2 to o(pN) for the HFN. An ndimensional star graph thus 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 N 2 8bL2=2c o(N2 L2 ), where a node occupies a rectangle of sides ranging from d n 1 4 e to o(pN=L) and the number L of wiring layers satisfies 2 L = o(pN=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.