The starting gates Nstart used in the derivation of the interconnect length distribution for three-dimensional (3-D) circuits in [291] are described in this appendix. The starting gates are those gates that form manhattan hemispheres of radius l, where l is the interconnect length connecting two gates. For a 3-D circuit consisting of n tiers and including Nn = N/n gates per tier and those gates located close to the periphery of each tier, only partial manhattan hemispheres are formed. Consequently, the number of gates encircled by these partial manhattan hemispheres varies with the interconnect length:
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The function g[x,y] is the discrete quotient function while function f[x,y,z] is
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dv denotes the intertier distance between two adjacent tiers within a 3-D system.