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F.F. Dragan (Kent State) A.B. Kahng (UCSD) I. Mandoiu (UCLA) S. Muddu (Sanera Systems) A. Zelikovsky (Georgia State) Provably Good Global Buffering by Multiterminal Multicommodity Flow Approximation

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2 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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3 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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4 1.Buffer-block planning [Cong+99] [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 2.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 Buffer-Block Methodology

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6 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 [Dragan+00]: 2-pin nets This paper: multi-pin nets

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7 Our Contributions Integer node-capacitated MTMCF formulation Approximation algorithm for fractional MTMCF –Extends [GargK98,Fleischer99,Albrecht00,Dragan+00] 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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8 Integer Program Formulation

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9 “Relax+Round” Approach 1.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 2.Round to integer solution –Provably good rounding using [RaghavanT87] –Practical rounding using random-walks

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10 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) < min{ 1, (1+2 )^i } do 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 Output f/N

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11 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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12 Implementation choices 2-Pin3,4-pinMulti-pin DecompositionStar, Minimum Spanning tree Matching, 3-restricted Steiner tree Not needed Min-weight DRSTShortest path (exact) Try all Steiner pts + shortest paths (exact) Very hard! heuristics RoundingRandom-walkBackward random-walks [Dragan+00]This paper

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13 1.Store fractional flows f(T) for every valid routing tree T 2.Scale down each f(T) by 1- for small 3.Each net k routed with prob. f(k)= { f(T) | T routing for k } Number of routed nets (1- )OPT 4.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 Provably Good Rounding

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14 1.Store fractional flows f(T) for every valid routing tree T 2.Scale down each f(T) by 1- for small 3.Each net k routed with prob. f(k)= { f(T) | T routing for k } Number of routed nets (1- )OPT 4.To route net k, choose tree T with probability = f(T) / f(k) With high probability, no BB capacity is exceeded Random-Walk 2-TMCF Rounding use random walk from source to sink Practical: random walk requires storing only flows on edges

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15 Random-Walk MTMCF Rounding S T1 T2 T3 Source Sinks

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16 Random-Walk MTMCF Rounding S T1 T2 T3 Source Sinks

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17 Our MTMCF Rounding Heuristic 1.Round each net k with probability f(k), using backward random walks –No scaling-down, approximate MTMCF < OPT 2.Resolve capacity violations by greedily deleting routed paths –Few violations 3.Greedily route remaining nets using unused BB capacity –Further routing still possible

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18 Implemented Heuristics Greedy buffered routing 1.For each net, route sinks sequentially along shortest path to source or node already connected to source 2.After routing a net, remove fully used BBs MTMCF approximation + randomized rounding –2TMCF [Dragan+00] –3TMCF (3-pin decomposition + -MTMCF + rounding) –4TMCF (4-pin decomposition + -MTMCF + rounding) –MTMCF ( -MTMCF w/ approximate DRST + rounding)

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19 Experimental Setup Test instances extracted from next-generation SGI microprocessor Up to 5,000 nets, ~6,000 sinks U=4,000 m, L=500-2,000 m 50 buffer blocks 200-400 buffers / BB

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20 % Sinks Connected #sinks/ #nets Greed 2TMCF3TMCF4TMCFMTMCF =.64 =.04 =.64 =.04 =.64 =.04 =.64 =.04 2958/ 2396 92.293.895.596.297.896.698.396.797.4 3077/ 2438 92.393.996.596.498.596.998.897.699.3 3099/ 2784 92.193.695.596.498.096.698.197.398.7 6038/ 4764 93.594.896.895.797.696.598.496.397.7 6296/ 4925 93.696.297.697.098.697.799.197.798.4 6321/ 4938 93.396.297.596.898.497.798.997.798.2

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21 Runtime (sec.) #sinks/ #nets Greed 2TMCF3TMCF4TMCFMTMCF =.64 =.04 =.64 =.04 =.64 =.04 =.64 =.04 2958/ 2396.301.633579.162,09098.9129,1902.33947 3077/ 2438.332.3535011.102,356128.3837,9702.87846 3099/ 2784.331.8039212.562,364132.8138,3412.86877 6038/ 4764.532.8460016.573,166182.5560,4504.981,866 6296/ 4925.554.3569019.53,721265.7877,6715.381,828 6321/ 4938.543.3773018.993,813255.3779,1235.431,833

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22 % Routed Nets vs. Runtime

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23 Resource Usage Greed 2TMCF3TMCF4TMCFMTMCF =.64 =.04 =.64 =.04 =.64 =.04 =.64 =.04 # Conn. Sinks 5,6455,7255,8425,7795,8965,8275,9425,8135,897 % Conn. Sinks 93.594.896.895.797.696.598.496.397.7 Wirelength (meters) 42.2245.1847.8044.4847.6644.1847.4945.3347.51 WL/sink (microns) 7,4797,8918,1827,6978,0837,5827,9927,7988,057 #Buffers90379,86010,6769,59110,6109,49710,5079,86010,647 #Buff/sink1.601.721.831.661.801.631.771.701.81 #nets = 4,764 #sinks = 6,038 400 buffers/BB

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24 WL and #Buffers for 100% Completion #nets = 4,764 #sinks = 6,038 Flow-rounding wastes routing resources! BB Cap. Greed 2TMCF3TMCF4TMCFMTMCF =.64 =.04 =.64 =.04 =.64 =.04 =.64 =.04 500——— 51.28 ——— 50.07 ——— 49.45 ——— 49.54 11,73811,31211,07911,161 600——— 50.4651.1348.9349.9548.0249.5848.3449.27 11,33011,68810,80211,26710,51211,11510,63111,075 1000 47.8950.5950.7649.0549.9348.0149.9848.2848.27 10,33011,33411,55810,80211,28410,51211,37310,61910,783 8000 47.8950.6250.2848.9751.2848.0751.4048.3348.44 10,33011,3341134010,79411,78810,50311,80310,61910,625

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25 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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26 Combining with compaction

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27 Combining with compaction

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28 Combining with compaction Sum-capacity constraints: cap(BB1) + cap(BB2) const.

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29 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

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