1. 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)

2. 2 ISMP’2000 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

3. 3 ISMP’2000 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) RAR(2-buffer block) =   RAR(buffer)   0.8 for high-end designs   0.8 for high-end designs  Buffer block planning: Given circuit block placement and global netlist Given circuit block placement and global netlist Plan shape/location of buffer blocks minimally impacting existing floorplan Plan shape/location of buffer blocks minimally impacting existing floorplan

4. 4 ISMP’2000 Global buffering problem BB

5. 5 ISMP’2000 Global buffering problem  Given planar region with rectangular obstacles containing buffer blocks with given capacity planar region with rectangular obstacles containing buffer blocks with given capacity set of 2-pin nets (s(k),t(k)), each net has set of 2-pin nets (s(k),t(k)), each net has t non-negative importance (criticality) coefficient t parity requirement = parity on # buffers b/w source and sink t 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] distances b/w repeaters (pins) are in given interval [L,U] # buffers for any net satisfy given constraints # buffers for any net satisfy given constraints # of nets passing through buffer block  capacity # of nets passing through buffer block  capacity

6. 6 ISMP’2000 Global buffering problem BB

7. 7 ISMP’2000 Integer MCF formulation  Graph G=(V,E), V=pins+BB’s, 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 f(p)  {0,1} p  P

8. 8 ISMP’2000 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 w(v) = , f(k,v) = 0 for all v in V and k = 1,…,K For i = 1 to N do For i = 1 to N do t for k = 1,…,K do t find shortest path p in P(k) t while w(p) < min{ 1,  (1+2  )^i} do u f= f+1 u for all v in p f(k, v) = f(k,v)+1 (+2 if v is loop in p) u find p shortest path in P(k) output f/N and f(k,v)/N for all v in V and k = 1,…,K output f/N and f(k,v)/N for all v in V and k = 1,…,K   -MCF algorithm is (1+ 8  )-approximation

9. 9 ISMP’2000 Rounding fractional MCF  Raghavan-Thompson: random walk from source probability of choosing an arc/node proport. node flow probability of choosing an arc/node proport. node flow  Probability of routing net proportional net flow  Algorithm decrease flow by (1-  ) decrease flow by (1-  ) route nets with randomized rounding route nets with randomized rounding  With high probability no node capacity violations

10. 10 ISMP’2000 Greedy deletion/addition  Greedy addition routing if there exists (s(i)-t(i))-path satisfying constraints if there exists (s(i)-t(i))-path satisfying constraints t find shortest path t decrement capacity of all nodes on path t if capacity of node = 0, then delete it from graph  Greedy deletion = opposite to addition

11. 11 ISMP’2000 Implemented heuristics   -MCF algorithm with greedy enhancement solve fractional MCF with  approximation solve fractional MCF with  approximation round fractional solution via random walks round fractional solution via random walks apply greedy deletion/addition to get feasible solution apply greedy deletion/addition to get feasible solution  1-shot heuristic assign weight w=1 to each BB assign weight w=1 to each BB repeat until total overused capacity does not decrease repeat until total overused capacity does not decrease t for each pair find shortest path t for each BB r increase weight by w  c(r) / C(r) apply greedy deletion/addition to get feasible solution apply greedy deletion/addition to get feasible solution

12. 12 ISMP’2000 Experimental results # nets Greedy 1-shot  -MCF  =0.16  =0.02  =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

13. 13 ISMP’2000 Extensions/Conclusions  Enhancing with edge capacities = channel capacity  Multi-pin nets => Steiner trees in dags approximate directed Steiner trees approximate directed Steiner trees rounding of trees: reduction to random walks rounding of trees: reduction to random walks  Combining with compaction increasing/decreasing capacities increasing/decreasing capacities sum-capacity constraints sum-capacity constraints  Covering LP approximation improves drastically many routing parameters

14. 14 ISMP’2000 Combining with compaction BB