VLSI Design

Research Article

Fine Control of Local Whitespace in Placement

Algorithm 3

Linear programming formulation for the horizontal direction to optimize HPWL of an existing placement. Further simplification is possible for 2- and 3-pin nets.
Let the set of cells and macros be denoted
       𝐶 = 𝑐 1 , 𝑐 2 , … , 𝑐 𝑖 ,
and let the set of nets be denoted
       𝑁 = 𝑛 1 , 𝑛 2 , … , 𝑛 𝑗 .
Using the variables:
       𝑥 𝑘 𝐿 f o r 1 ≤ 𝑘 ≤ 𝑖 , 𝑥 𝑘 , 𝑈 𝑥 𝑘 f o r 1 ≤ 𝑘 ≤ 𝑗 ,
and constants:
       𝑙 𝑒 𝑓 𝑡 _ 𝑏 𝑜 𝑢 𝑛 𝑑 𝑎 𝑟 𝑦 , 𝑟 𝑖 𝑔 ℎ 𝑡 _ 𝑏 𝑜 𝑢 𝑛 𝑑 𝑎 𝑟 𝑦 , 𝑝 𝑙 𝑎 𝑐 𝑒 𝑚 𝑒 𝑛 𝑡 ( 𝑘 ) . 𝑥 f o r 1 ≤ 𝑘 ≤ 𝑖 , 𝑤 𝑖 𝑑 𝑡 ℎ ( 𝑘 ) f o r 1 ≤ 𝑘 ≤ 𝑖 ,
we minimize
       𝑗 ∑ 𝑘 = 1 𝑈 𝑥 𝑘 − 𝐿 𝑥 𝑘
subject to constraints:
    𝐿 𝑥 𝑛 ≤ 𝑥 𝑘 ≤ 𝑈 𝑥 𝑛 w h e r e c e l l 𝑘 𝑖 𝑠 o n n e t 𝑛 , 𝑙 𝑒 𝑓 𝑡 _ 𝑏 𝑜 𝑢 𝑛 𝑑 𝑎 𝑟 𝑦 ≤ 𝑥 𝑘 𝑥 f o r 1 ≤ 𝑘 ≤ 𝑖 , 𝑘 𝑥 + 𝑤 𝑖 𝑑 𝑡 ℎ ( 𝑘 ) ≤ 𝑟 𝑖 𝑔 ℎ 𝑡 _ 𝑏 𝑜 𝑢 𝑛 𝑑 𝑎 𝑟 𝑦 f o r 1 ≤ 𝑘 ≤ 𝑖 , 𝑘 𝑥 = 𝑝 𝑙 𝑎 𝑐 𝑒 𝑚 𝑒 𝑛 𝑡 ( 𝑘 ) . 𝑥 w h e r e c e l l 𝑘 𝑖 𝑠 fi x e d , 𝑘 + 𝑤 𝑖 𝑑 𝑡 ℎ ( 𝑘 ) ≤ 𝑥 𝑙 w h e r e c e l l 𝑘 i s d i r e c t l y t o t h e l e f t o f c e l l 𝑙