Provably Good Global Buffering Using an Available Buffer Block Plan

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Provably Good Global Buffering Using an Available
Buffer Block Plan

• F. F. Dragan (Kent)
• A. B. Kahng (UCLA)
• I. Mandoiu (Gatech)
• S. Muddu (Silicon graphics)
• A. Zelikovsky (GSU)

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Outline

• Global routing via buffer blocks
• Global buffering problem
• Integer multicommodity flow formulation (MCF)
• Approximation of node-capacitated MCF
• Rounding fractional MCF
• Implemented heuristics
• Experimental results
  
  
  
  
  
• Extensions Conclusions

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Global routing via buffer blocks

• (V)DSM ? buffering all global nets
• Block methodology ? buffers outside blocks
• Buffer blocks use less routing/area resources
  (RAR)
• RAR(2-buffer block) ? ? RAR(buffer)
• ? ? 0.8 for high-end designs
• Buffer block planning
• Given circuit block placement and global netlist
• Plan shape/location of buffer blocks minimally
  impacting existing floorplan

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Global buffering problem
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Global buffering problem

• Given
• planar region with rectangular obstacles
  containing buffer blocks with given capacity
• set of 2-pin nets (s(k),t(k)), each net has
• non-negative importance (criticality) coefficient
• parity requirement parity on buffers b/w
  source and sink
• maximum buffers b/w source and sink
• Route given nets maximizing total importance s.t.
• distances b/w repeaters (pins) are in given
  interval L,U
• buffers for any net satisfy given constraints
• of nets passing through buffer block ? capacity

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Global buffering problem
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Integer MCF formulation

• Graph G(V,E), VpinsBBs, E legal edges
• P(k) set of legal (s(k),t(k)) paths
• P union P(k)
• q(p,v) 0 if v ?p 1 if v ? p 2 if loop vv
  ? p
• maximize ? f(p) p?P
• subject to ? q(p,v) ? f(p) p?P ? c(v) v
  ?V
• f(p) ? 0,1
  p?P

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Approximation of node-capacitated MCF

• Garg/Konemann Fleisher
• ? -MCF algorithm
• w(v) ?, f(k,v) 0 for all v in V and k
  1,,K
• For i 1 to N do
• for k 1,,K do
• find shortest path p in P(k)
• while w(p) lt min 1, ?(12?)i do
• f f1
• for all v in p f(k, v) f(k,v)1 (2 if v is
  loop in p)
• find p shortest path in P(k)
• output f/N and f(k,v)/N for all v in V and k
  1,,K
• ? -MCF algorithm is (1 8?)-approximation

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Rounding fractional MCF

• Raghavan-Thompson random walk from source
• probability of choosing an arc/node proport. node
  flow
• Probability of routing net proportional net flow
• Algorithm
• decrease flow by (1-?)
• route nets with randomized rounding
• With high probability no node capacity violations

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Greedy deletion/addition

• Greedy addition routing
• if there exists (s(i)-t(i))-path satisfying
  constraints
• find shortest path
• decrement capacity of all nodes on path
• if capacity of node 0, then delete it from
  graph
• Greedy deletion opposite to addition

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Implemented heuristics

• ?-MCF algorithm with greedy enhancement
• solve fractional MCF with ? approximation
• round fractional solution via random walks
• apply greedy deletion/addition to get feasible
  solution
• 1-shot heuristic
• assign weight w1 to each BB
• repeat until total overused capacity does not
  decrease
• for each pair find shortest path
• for each BB r increase weight by w?c(r) / C(r)
• apply greedy deletion/addition to get feasible
  solution

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Experimental results

• nets Greedy 1-shot ?-MCF
• 
  ?0.16 ?0.02
• 4212 4101 4207 4212
  4212
• 3962 2148 2179 2325
  2347
• 4191 2216 2236 2378
  2394
• 4212 2223 2232 2378
  2392
• Catching fully routable instances
• Reduce manual work when unroutable

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Extensions/Conclusions

• Enhancing with edge capacities channel capacity
• Multi-pin nets gt Steiner trees in dags
• approximate directed Steiner trees
• rounding of trees reduction to random walks
• Combining with compaction
• increasing/decreasing capacities
• sum-capacity constraints
• Covering LP approximation improves drastically
  many routing parameters

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Combining with compaction