F.F. Dragan (Kent State)

1
Provably Good Global Buffering Using an Available
Buffer Block Plan

• F.F. Dragan (Kent State)
• A.B. Kahng (UCSD)
• I. Mandoiu (Georgia Tech/UCLA)
• S. Muddu (Silicon Graphics)
• A. Zelikovsky (Georgia State)

2
Global Buffering via Buffer Blocks

• VDSM ? buffer / inverter insertion for all global
  nets
• 50nm technology ? 106 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 Cong99 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

3
Global Buffering via Buffer Blocks
4
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

5
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

6
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

7
Integer Program Formulation
8
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

9
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) lt min 1, ?(12?)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
• End For
• Output f/N

Run time for ?-approximation
10
Rounding to an Integer Solution

• Random walk algorithm RaghavanT87
• probability of routing a net proportional to
  nets 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

• 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

11
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 w1 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

12
Experimental Setup

• Test instances extracted from next-generation SGI
  microprocessor
• 4,000 nets
• U4,000 ?m, L500-2,000 ?m
• 50 buffer blocks
• BB capacity
• 400 (fully routable instances)
• 50 (hard instances, 50-60 routable)

13
Fully Routable Instance (4212 nets)
14
Fully Routable Instance (4212 nets)
15
Running Time vs. Solution Quality
16
57 Routable Instance (4212 nets)
17
Conclusions and Ongoing Work

• Provably good algorithm based on node-capacitated
  MCF approximation

• Extensions
• combine global buffering with BB planning
• combine with compaction

18
Combining with compaction
19
Combining with compaction
20
Combining with compaction

• Sum-capacity constraints cap(BB1) cap(BB2) ?
  const.

21
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)