F.F. Dragan (Kent State) A.B. Kahng (UCSD) I. Mandoiu (Georgia Tech/UCLA) S. Muddu (Silicon Graphics) A. Zelikovsky (Georgia State) Provably Good Global Buffering Using an Available Buffer Block Plan

Global Buffering via Buffer Blocks VDSM  buffer / inverter insertion for all global nets –50nm technology  10^6 buffers Buffer Block (BB) methodology –isolate buffer insertion from block implementations –improve routing area resources (RAR) utilization RAR(2k-buffer block) =   RAR(k-buffer block) For high-end designs   1.6 Buffer block planning [Cong+99] [TangW00] –given block placement + nets –find shape and location of BBs Global buffering via BBs –given nets + BB locations and capacities –find buffered routing for each net

Global Buffering via Buffer Blocks

Global Buffering Problem Given: Pin & BB locations, BB capacities list of 2-pin nets, each net has upper-bound on #buffers parity requirement on #buffers [non-negative weight (criticality coefficient)] L/U bounds on wirelength b/w consecutive buffers/pins Find: buffered routing of a maximum [weighted] number of nets subject to the given constraints

Global Buffering Problem Given: Pin & BB locations, BB capacities list of 2-pin nets, each net has upper-bound on #buffers new parity requirement on #buffers new [non-negative weight (criticality coefficient)] new L/U bounds on wirelength b/w consecutive buffers/pins Find: buffered routing of a maximum [weighted] number of nets subject to the given constraints Previous work: 1 buffer per connection, no weights

Outline of Results Provably good algorithm for the Global Buffering Problem –integer node-capacitated multi-commodity flow (MCF) formulation –approximation algorithm for solving fractional relaxation –provably good randomized rounding based on [RaghavanT87] –allows tradeoff between run-time and solution quality Fast heuristic based on ideas from the approximation algorithm superior to simpler greedy approaches almost matches the provably good algorithm for loosely constrained instances

Integer Program Formulation

High-Level Approach Solve fractional relaxation + rounding –first introduced for global routing [RaghavanT87] –fractional relaxation = node-capacitated multi-commodity flow (MCF) can be solved exactly using Linear Programming (LP) techniques exact LP algorithms are not practical for large instances Key idea: approximate solution to the relaxation –we generalize edge-capacitated MCF approximation of [GargK98, F99] –[GargK98] successfully applied to global routing by [Albrecht00]

Approximating the Fractional MCF  -MCF algorithm w(v) = , f = 0 For i = 1 to N do For k = 1, …, #nets do Find a shortest path p  P for net k While w(p) < min{ 1,  (1+2  )^I } do f(p)= f(p) + 1 For every v  p do w(v)  ( 1 +  /c(v) ) * w(v) End For End While End For Output f/N Run time for  -approximation =

Random walk algorithm [RaghavanT87] –probability of routing a net proportional to net’s flow –probability of choosing an arc proportional to fractional flow along arc –run time = O( #inserted buffers ) –To avoid BB overuse, scale-down fractional flow by 1-  before rounding Rounding to an Integer Solution Modifications approximate MCF underestimates optimum  few violations + unused BB capacity for large  resolve capacity violations by greedily deleting paths greedily route remaining nets using unused BB capacity

Implemented Heuristics  -MCF w/ greedy enhancement –solve fractional MCF with  approximation –round fractional solution via random walks –apply greedy deletion/addition to get feasible solution Greedy –sequentially route nets along shortest available paths 1-shot integer MCF –assign weight w=1 to each BB –repeat until total overused capacity does not decrease for each net find shortest path for each BB r increase weight by factor (1 +  usage(r) / cap(r)) – apply greedy deletion/addition to get feasible solution

Experimental Setup Test instances extracted from next-generation SGI microprocessor ~4,000 nets U=4,000  m, L=500-2,000  m 50 buffer blocks BB capacity –400 (fully routable instances) –50 (hard instances, 50-60% routable)

Fully Routable Instance (4212 nets)

Greedy1-Shot

Running Time vs. Solution Quality

~57% Routable Instance (4212 nets) Greedy1-Shot

Conclusions and Ongoing Work Provably good algorithm based on node- capacitated MCF approximation Extensions: –combine global buffering with BB planning combine with compaction

Combining with compaction

Sum-capacity constraints: cap(BB1) + cap(BB2)  const.

Conclusions and Ongoing Work Provably good algorithm based on node- capacitated MCF approximation Extensions: –combine global buffering with BB planning combine with compaction –enforce channel capacity constraints –multi-terminal nets (ASPDAC-01)