F.F. Dragan Kent State

1
Provably Good Global Buffering by Multiterminal
Multicommodity Flow Approximation

• F.F. Dragan (Kent State)
• A.B. Kahng (UCSD)
• I. Mandoiu (UCLA)
• S. Muddu (Sanera Systems)
• A. Zelikovsky (Georgia State)

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Outline

• Buffer-block methodology for global buffering
• Global routing via buffer-blocks problem
• Integer node-capacitated multiterminal
  multicommodity flow (MTMCF) formulation
• Provably good approximation of fractional MTMCF
• Provably good rounding of fractional MTMCF
• Key implementation choices
• Experimental results
  
  
  
  
  
• Extensions conclusions

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Motivation

• VDSM ? buffer / inverter insertion for all global
  nets
• 50nm technology ? 1,000,000 buffers

• Solution insert buffers only in Buffer-Blocks
  (BBs)
• Simplified design isolates buffer insertion from
  circuit block implementations
• Efficient utilization of routing/area resources
  (RAR)
• RAR(cap. 2k buffer-block) ? ? RAR(cap. k
  buffer-block)
• For high-end designs, ? ? 1.6

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Buffer-Block Methodology

• Buffer-block planning Cong99 TangW00
• given placement of circuit blocks netlist
• find shape and location of BBs within available
  free space so that to maximize the number of
  routable nets
• Global buffering via given BBs ? This paper
• given nets BB locations and capacities
• find buffered routing for each net, subject to
  timing-driven and buffer-parity constraints

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Buffer-Block Methodology
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Problem Formulation

• Global Buffering via Buffer-Blocks (GRBB) Problem
• Given
• BB locations and capacities
• list of multi-pin nets, each net has
• upper-bound parity requirement on buffers for
  each source-sink path
• 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

Dragan00 2-pin nets

This paper multi-pin nets
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Our Contributions

• Integer node-capacitated MTMCF formulation
• Approximation algorithm for fractional MTMCF
• Extends GargK98,Fleischer99,Albrecht00,Dragan00
  to node-capacitated multiterminal case
• Provably good fractional MTMCF rounding
  algorithms,
• Provably good algorithm for GRBB Problem
• Practical rounding heuristics based on
  random-walks
• Computational study comparing alternative
  implementations

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Integer Program Formulation
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RelaxRound Approach

• Solve the fractional relaxation
• Relaxation node-capacitated multiterminal
  multicommodity flow
• Exact linear programming algorithms are
  impractical for large instances
• KEY IDEA use approximation algorithm
• can approximate optimum within a factor of (1-?)
  for any ?0
• allows continuous tradeoff between runtime and
  solution quality
• Round to integer solution
• Provably good rounding using RaghavanT87
• Practical rounding using random-walks

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The ?-MTMCF Algorithm

• w(v) ?, f 0
• For i 1 to N do
• For k 1, , nets do
• Find min weight valid routing tree T for net
  k
• While w(T)
• f(T) f(T) 1
• For every v ? T do
• w(v) ? ( 1 ? ?(T,v)/cap(v) ) w(v)
• End For
• Find min weight valid routing tree T for
  net k
• End While
• End For
• End For
• Output f/N

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Runtime of ?-MTMCF Algorithm

• Main step of ?-MTMCF algorithm
• computing min node-weight valid routing tree for
  a net
• ? min node-weight directed rooted Steiner tree
  (DRST) in a directed acyclic graph

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Implementation choices
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Provably Good Rounding

• Store fractional flows f(T) for every valid
  routing tree T
• Scale down each f(T) by 1-? for small ?
• Each net k routed with prob. f(k)? f(T) T
  routing for k
• Number of routed nets ? (1-? )OPT
• To route net k, choose tree T with probability
  f(T) / f(k)
• With high probability, no BB capacity is
  exceeded
• Problem Impractical to store all non-zero flow
  trees

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Random-Walk 2-TMCF Rounding

• Store fractional flows f(T) for every valid
  routing tree T
• Scale down each f(T) by 1-? for small ?
• Each net k routed with prob. f(k)? f(T) T
  routing for k
• Number of routed nets ? (1-? )OPT
• To route net k, choose tree T with probability
  f(T) / f(k)
• With high probability, no BB capacity is
  exceeded

Practical random walk requires storing only
flows on edges
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Random-Walk MTMCF Rounding
Source?Sinks
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Random-Walk MTMCF Rounding
Source?Sinks
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The MTMCF Rounding Heuristic

• Round each net k with probability f(k), using
  backward random walks
• No scaling-down, approximate MTMCF
• Resolve capacity violations by greedily deleting
  routed paths
• Few violations
• Greedily route remaining nets using unused BB
  capacity
• Further routing still possible

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Implemented Heuristics

• Greedy buffered routing
• For each net, route sinks sequentially along
  shortest path to source or node already connected
  to source
• After routing a net, remove fully used BBs
• MTMCF approximation randomized rounding
• 2TMCF Dragan00
• 3TMCF (3-pin decomposition ?-MTMCF rounding)
• 4TMCF (4-pin decomposition ?-MTMCF rounding)
• MTMCF (?-MTMCF w/ approximate DRST rounding)

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Experimental Setup

• Test instances extracted from next-generation SGI
  microprocessor
• Up to 5,000 nets, 6,000 sinks
• U4,000 ?m, L500-2,000 ?m
• 50 buffer blocks
• 200-400 buffers / BB

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Sinks Connected
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Runtime (sec.)
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Routed Nets vs. Runtime
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Resource Usage
nets 4,764 sinks 6,038 400
buffers/BB
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WL and Buffers for 100 Completion
nets 4,764 sinks 6,038
Flow-rounding wastes routing resources!
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Conclusions and Ongoing Work

• Provably good algorithms and practical heuristics
  based on node-capacitated MTMCF approximation
• Higher completion rates than previous algorithms

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

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Combining with compaction
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Combining with compaction
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Combining with compaction

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

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Conclusions and Ongoing Work

• Provably good algorithms and practical heuristics
  based on node-capacitated MTMCF approximation
• Higher completion rates than previous algorithms

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

• Enforce channel capacity constraints

• Improved resource usage
• smart release of resources