Wire Length Distribution for Placements of Computer Logic

It is shown from simple theoretical considerations that the distribution ƒ
k
of wire lengths for a good two-dimensional placement on a square Manhattan grid should be of the form ƒ
k
= g/k
^γ
(1 ≤ k ≤ L) and ƒ
k
≈ 0 (k > L), where γ is related to the Rent partitioning exponent p by the equation 2p + γ ≈ 3. Three placements were investigated and the distribution functions for wire length were found to follow the above relationships.