Resource Sharing, Routing, Chip Design. Jens Vygen

Transcription

1 Resource Sharing, Routing, Chip Design Jens Vygen University of Bonn joint work with Markus Ahrens, Michael Gester, Stephan Held, Niko Klewinghaus, Dirk Müller, Tim Nieberg, Christian Panten, Sven Peyer, Klaus Radke, Daniel Rotter, Pietro Saccardi, Rudolf Scheifele, Christian Schulte, Gustavo Tellez, Vera Traub, et al.

2 Chip design: placement telecommunication chip with 1 billion transistors 8 million circuits

3 Chip design: routing 9 routing layers 30 million pins million nets million wire segments million vias 1 kilometer total wire length

4 Combinatorial optimization in chip design shortest paths, TSP, spanning trees, Steiner trees rectangle packing, knapsack problem, bin packing facility location, partitioning, clustering maximum flows, discrete time-cost tradeoff problem minimum mean cycles, parametric shortest paths transportation and minimum cost flows multicommodity flows, disjoint paths and trees, resource sharing

5 The BonnTools, I developed by our group at the University of Bonn, I cover all major areas of layout and timing optimization, I include libraries for combinatorial optimization, advanced, data structures, computational geometry, etc., I have more than one million lines of C++ code, I are being used worldwide by IBM and other companies, I have been used for the design of thousands of chips, I including several complete microprocessor series I and the most complex chips of major technology companies.

6 Detail of the routing (less than one millionth)

7 Detail of the routing (less than one millionth) Vias

8 Detail of the routing (less than one millionth) Vias

9 Routing: task Instance: a number of routing planes a set of nets, where each net is a set of pins (terminals) a set of shapes for each pin a set of blockage shapes rules that tell if two given shapes are connected, separated Task: Compute a feasible routing, i.e., a set of wire shapes for each net, connecting the pins, separate from blockages and shapes of other nets

10 Routing: task Instance: a number of routing planes a set of nets, where each net is a set of pins (terminals) a set of shapes for each pin a set of blockage shapes rules that tell if two given shapes are connected, separated timing constraints information on power consumption, yield,... Task: Compute a feasible routing, i.e., a set of wire shapes for each net, connecting the pins, separate from blockages and shapes of other nets such that all timing constraints are met and the (estimated) power consumption and/or manufacturing cost is minimized.

11 Routing: simplified view Find vertex-disjoint Steiner trees connecting given terminal sets in a 3-dimensional grid graph. NP-hard: no polynomial-time algorithm unless P = NP Order of magnitude: 10 million Steiner trees in a graph with 500 billion vertices Even linear-time algorithms are too slow!

12 Global and detailed routing Routing is usually performed in three phases: Global routing: performs global optimization, determines a routing area (corridor) for each net, but no detailed view Detailed routing: constructs wires connecting each net within this corridor, respecting all design rules necessary for the lithographic process in fabrication Postoptimization: fix remaining violated constraints, improve the wiring by spreading and do some postprocessing for more robust manufacturing Graphs are huge: vertices in detailed routing, still vertices in global routing.

13 Detailed routing in global routing corridor

14 Global routing: classical model In each routing plane: contract regions of approx. 70x70 tracks to a single vertex compute capacities of edges between adjacent regions pack Steiner trees with respect to these edge capacities global optimization of objective functions Steiner tree yields routing corridor for each net Detailed routing computes detailed wires in these corridors by fast goal-oriented variants of Dijkstra s algorithm and optimal pin access (Hetzel [1998], Müller [2009], Peyer, Rautenbach, Vygen [2009], Klewinghaus [2013], Gester et al. [2013], Ahrens et al. [2015],...)

15 Global routing: classical problem formulation Instance: a global routing (grid) graph with edge capacities a set of nets, each consisting of a set of vertices (terminals) Task: find a Steiner tree for each net such that the edge capacities are respected, and (weighted) netlength is minimum.

16 Simple example edge-disjoint paths problem 3 terminal pairs: blue, red, green each terminal pair has demand 1 each edge has capacity

17 Simple example edge-disjoint paths problem 3 terminal pairs: blue, red, green each terminal pair has demand 1 each edge has capacity 1 no solution exists edge-disjoint paths problem is NP-hard (even in planar grids) 01 01

18 Simple example edge-disjoint paths problem 3 terminal pairs: blue, red, green each terminal pair has demand 1 each edge has capacity 1 no solution exists fractional solution exists (route 1 2 along each colored path) edge-disjoint paths problem is NP-hard (even in planar grids) fractional relaxation can be solved in polynomial time by linear programming (but not fast enough) 01 01

19 Real example: global routing congestion map

20 Min-max resource sharing Instance finite sets R of resources and C of customers for each c C: a set B c R R 0 of feasible solutions Task Find a b c B c for each c C with minimum congestion (b c ) r. max r R c C

21 Min-max resource sharing Instance finite sets R of resources and C of customers for each c C: a set B c R R 0 of feasible solutions given by an oracle function f c : R R 0 B c with Task y r (f c (y)) r r R σ inf b B c r R y r b r for all price vectors y R R 0 and some fixed σ 1 (a σ-approximate oracle). Find a b c B c for each c C with minimum congestion (b c ) r. max r R c C

22 Min-max resource sharing Instance finite sets R of resources and C of customers for each c C: a set B c R R 0 of feasible solutions given by an oracle function f c : R R 0 B c with Task y r (f c (y)) r r R σ inf b B c r R y r b r for all price vectors y R R 0 and some fixed σ 1 (a σ-approximate oracle). Find a b c conv(b c ) for each c C with minimum congestion max r R (b c ) r. c C

23 Resource sharing for global routing Each net is a customer. Define a resource r (with a capacity cap(r)) for each edge of the global routing graph the total power consumption the critical area (for estimating the yield loss) each edge of the timing propagation graph For each net c, let B c contain, for each routing solution s for c, the vector ( usg(s,r) ) cap(r) where usg(s, r) tells how much s uses from r. r R We look for a solution with congestion at most 1. An objective function can be viewed as an additional resource (determine its capacity by binary search).

24 Rounding A solution to the min-max resource sharing instance is a convex combination b B c p b b of solutions for each net c (p b 0 (b B c ), b B c p b = 1) However, we need a single solution for each net. Use randomized rounding: randomly choose b with probability p b, independently for each net. (Raghavan, Thompson [1987,1991], Raghavan [1988]) If no solution consumes very much of any capacity, this yields a solution that exceeds capacities only slightly. New rounding and correction algorithms even better?

25 Algorithms for min-max resource sharing oracle # oracle calls Grigoriadis, Khachiyan [1994] strong, bounded Õ(ω 2 C 2 ) Grigoriadis, Khachiyan [1994] strong, bounded Õ(ω 2 C ) random. Grigoriadis, Khachiyan [1996] strong, unbounded Õ(ω 2 C R ) Jansen, Zhang [2008] weak, unbounded Õ(ω 2 C R ) Müller, Radke, Vygen [2011] weak, unbounded Õ(ω 2 ( C + R )) Müller, Radke, Vygen [2011] weak, bounded Õ(ω 2 C ) All these algorithms and predecessors for special cases (Plotkin, Shmoys, Tardos [1995], Young [1995], Villavicencio, Grigoriadis [1996], Garg, Könemann [2007],...) compute a σ(1 + ω)-approximate solution for any given ω > 0. Running times dominated by number of oracle calls. Logarithmic terms and dependency on σ omitted. For a strong oracle, σ can be chosen arbitrarily close to 1. A bounded oracle respects a given upper bound on resource usage.

26 Algorithms for min-max resource sharing oracle # oracle calls Grigoriadis, Khachiyan [1994] strong, bounded Õ(ω 2 C 2 ) Grigoriadis, Khachiyan [1994] strong, bounded Õ(ω 2 C ) random. Grigoriadis, Khachiyan [1996] strong, unbounded Õ(ω 2 C R ) Jansen, Zhang [2008] weak, unbounded Õ(ω 2 C R ) Müller, Radke, Vygen [2011] weak, unbounded Õ(ω 2 ( C + R )) Müller, Radke, Vygen [2011] weak, bounded Õ(ω 2 C ) All these algorithms and predecessors for special cases (Plotkin, Shmoys, Tardos [1995], Young [1995], Villavicencio, Grigoriadis [1996], Garg, Könemann [2007],...) compute a σ(1 + ω)-approximate solution for any given ω > 0. Running times dominated by number of oracle calls. Logarithmic terms and dependency on σ omitted. For a strong oracle, σ can be chosen arbitrarily close to 1. A bounded oracle respects a given upper bound on resource usage.

27 Weak duality Let { λ := inf max r R (the optimum congestion ). } (b c ) r : b c B c (c C) c C Lemma (Weak duality) Let y R R 0 be some price vector, not all-zero, and opt c (y) := inf b Bc r R y r b r. Then c C opt c (y) r R y λ. r

28 Weak duality Let { λ := inf max r R (the optimum congestion ). } (b c ) r : b c B c (c C) c C Lemma (Weak duality) Let y R R 0 be some price vector, not all-zero, and opt c (y) := inf b Bc r R y r b r. Then c C opt c (y) r R y λ. r Proof Let (b c B c ) c C be a solution with congestion (1 + δ)λ. Then c C opt c (y) c C r R y r R y r (b c ) r r R r r R y = y r c C (b c) r r r R y r r R y r (1 + δ)λ r R y = (1 + δ)λ. r

29 Bounding λ Lemma (Weak duality) Let y R R 0 be some price vector, not all-zero. Then c C opt c (y) r R y λ. r

30 Bounding λ Lemma (Weak duality) Let y R R 0 be some price vector, not all-zero. Then c C opt c (y) r R y λ. r Corollary Let b c := f c (1) (c C) and λ ub := max r R c C (b c) r. Then λ ub R σ r R c C (b c) r R σ c C opt c (1) λ λ ub. R

31 Scaling We know λub R σ λ λ ub.

32 Scaling We know λub R σ λ λ ub. 1. Set j := Scale B (j) c := { 2j λ ub b : b B c }. Note that λ (j) 1.

33 Scaling We know λub R σ λ λ ub. 1. Set j := Scale B (j) c := { 2j λ ub b : b B c }. Note that λ (j) Find a solution with congestion λ (j) 5 4 σλ (j)

34 Scaling We know λub R σ λ λ ub. 1. Set j := Scale B (j) c := { 2j b : b B λ ub c }. Note that λ (j) Find a solution with congestion λ (j) 5 4 σλ (j) If λ (j) 1 2, then increment j and go to 2.

35 Scaling We know λub R σ λ λ ub. 1. Set j := Scale B (j) c := { 2j b : b B λ ub c }. Note that λ (j) Find a solution with congestion λ (j) 5 4 σλ (j) If λ (j) 1 2, then increment j and go to Now 1 5σ λ (j) Find a solution with congestion λ (j) σ(1 + ω 2 )λ (j) + ω 10 (hence at most σ(1 + ω) times the optimum).

36 Scaling We know λub R σ λ λ ub. 1. Set j := Scale B (j) c := { 2j b : b B λ ub c }. Note that λ (j) Find a solution with congestion λ (j) 5 4 σλ (j) If λ (j) 1 2, then increment j and go to Now 1 5σ λ (j) Find a solution with congestion λ (j) σ(1 + ω 2 )λ (j) + ω 10 (hence at most σ(1 + ω) times the optimum). Lemma (Main Lemma) Let δ, δ > 0. Suppose that λ 1. Then we can compute a solution with congestion at most σ(1 + δ)λ + δ in ( ) O (δδ ) 1 ( C + R ) θ log R time, where θ is the time for an oracle call.

37 Core algorithm ( multiplicative weights method ) Input: An instance of the min-max resource sharing problem. Output: A convex combination of vectors in B c for each c C. t := 3σ ln R δδ, ɛ := δ 3σ. α r := 0, y r := 1 for each r R. x c,b := 0 for each c C and b B c. For p := 1 to t do: For c C do: b := f c (y). (perform t phases) (call oracle) x c,b := x c,b + 1. (record solution) α := α + b. (update resource consumption) For each r R with b r 0 do: y r := e ɛαr. (update prices) x c,b := 1 t x c,b for each c C and b B c. (normalize)

38 Core algorithm ( multiplicative weights method ) Input: An instance of the min-max resource sharing problem. Output: A convex combination of vectors in B c for each c C. t := 3σ ln R δδ, ɛ := δ 3σ. α r := 0, y r := 1 for each r R. x c,b := 0, X c := 0 for each c C and b B c. For p := 1 to t do: (perform t phases) While there exists c C with X c < p do: b := f c (y). (call oracle) ξ := min{p X c, 1/ max{b r : r R}}. x c,b := x c,b + ξ, X c := X c + ξ. (record solution) α := α + ξb. (update resource consumption) For each r R with b r 0 do: y r := e ɛαr. (update prices) x c,b := 1 t x c,b for each c C and b B c. (normalize)

39 Proof of performance guarantee (sketch) Lemma Let (x, y) be the output of the algorithm with congestion λ := max r R λ r, where λ r := ( ) x c,b b r c C b B c Then λ 1 ( ɛt ln y r ). r R

40 Proof of performance guarantee (sketch) Lemma Let (x, y) be the output of the algorithm with congestion λ := max r R λ r, where λ r := ( ) x c,b b r c C b B c Then λ 1 ( ɛt ln y r ). r R Proof: For r R: λ r = x c,b b r = α r t c C b B c = 1 ɛt ln (eɛαr ) = 1 ɛt ln y r 1 ( ɛt ln y r ). r R

41 Proof of performance guarantee (sketch) Lemma (Main Lemma) Let δ, δ > 0. Suppose that ɛ λ < 1, where ɛ := (e ɛ 1)σ. Then the algorithm computes a solution with congestion at most σ(1 + δ)λ + δ.

42 Proof of performance guarantee (sketch) Lemma (Main Lemma) Let δ, δ > 0. Suppose that ɛ λ < 1, where ɛ := (e ɛ 1)σ. Then the algorithm computes a solution with congestion at most Sketch of proof: σ(1 + δ)λ + δ. Congestion is at most 1 ɛt ln ( r R y (t) r ). where y (i) is the price vector at the end of the i-th phase.

43 Proof of performance guarantee (sketch) Lemma (Main Lemma) Let δ, δ > 0. Suppose that ɛ λ < 1, where ɛ := (e ɛ 1)σ. Then the algorithm computes a solution with congestion at most σ(1 + δ)λ + δ. Sketch of proof: ( ) Congestion is at most 1 ɛt ln r R y (t) r. ( ) Initially, we have r R y (0) r = R. where y (i) is the price vector at the end of the i-th phase.

44 Proof of performance guarantee (sketch) Lemma (Main Lemma) Let δ, δ > 0. Suppose that ɛ λ < 1, where ɛ := (e ɛ 1)σ. Then the algorithm computes a solution with congestion at most σ(1 + δ)λ + δ. Sketch of proof: ( ) Congestion is at most 1 ɛt ln r R y (t) r. ( ) Initially, we have r R y (0) r = R. Price y r multiplied by e ɛξbr Short calculation yields y (p) r y (p 1) r + ɛ opt c (y (p) ), r R c C r R (1 + (e ɛ 1)ξb r ) in each iteration. where y (i) is the price vector at the end of the i-th phase.

45 Proof of performance guarantee (sketch) We had r R y (p) r r R y (p 1) r + ɛ c C opt c (y (p) ).

46 Proof of performance guarantee (sketch) We had r R r R By weak duality, ɛ c C opt c (y (p) ) r R y (p) We get y (p) r r R y (p) r y (p 1) r + ɛ c C opt c (y (p) ). r ɛ λ < ɛ λ r R y (p 1) r

47 Proof of performance guarantee (sketch) We had r R r R By weak duality, ɛ c C opt c (y (p) ) r R y (p) We get and thus r R y (t) r y (p) r r R y (p) r y (p 1) r + ɛ c C opt c (y (p) ). r ɛ λ < ɛ λ r R y (p 1) r R (1 ɛ λ ) t = R (1 + ɛ λ ) t 1 ɛ λ R e tɛ λ /(1 ɛ λ ).

48 Proof of performance guarantee (sketch) We had r R r R By weak duality, ɛ c C opt c (y (p) ) r R y (p) We get and thus r R y (t) r y (p) r r R y (p) r y (p 1) r + ɛ c C opt c (y (p) ). r ɛ λ < ɛ λ r R y (p 1) r R (1 ɛ λ ) t = R (1 + ɛ λ ) t 1 ɛ λ R e tɛ λ /(1 ɛ λ ). Together with λ 1 ɛt ln( r R y (t) ) r, this proves the claim.

49 Number of oracle calls After each oracle call either X c = p (happens only t C times) or ξb r = 1 for some r R (increases the price of r by e ɛ ). Hence the number of oracle calls is ( ) O (δδ ) 1 ( C + R ) log R.

50 Number of oracle calls After each oracle call either X c = p (happens only t C times) or ξb r = 1 for some r R (increases the price of r by e ɛ ). Hence the number of oracle calls is ( ) O (δδ ) 1 ( C + R ) log R. The above scaling algorithm computes a σ(1 + ω)-approximate solution in O((ω 2 + log R )( C + R ) θ log R ) time, where θ is the time for an oracle call.

51 Main result The above scaling algorithm computes a σ(1 + ω)-approximate solution in O((ω 2 + log R )( C + R ) θ log R ) time.

52 Main result The above scaling algorithm computes a σ(1 + ω)-approximate solution in O((ω 2 + log R )( C + R ) θ log R ) time. Using binary search instead of simple scaling, and a variant of the Main Lemma in which λ 1 is not guaranteed, we obtain: Theorem We can compute a σ(1 + ω)-approximate solution in O((ω 2 + log log R )( C + R ) θ log R ) time. Faster and more general than all previous algorithms!

53 Main result The above scaling algorithm computes a σ(1 + ω)-approximate solution in O((ω 2 + log R )( C + R ) θ log R ) time. Using binary search instead of simple scaling, and a variant of the Main Lemma in which λ 1 is not guaranteed, we obtain: Theorem We can compute a σ(1 + ω)-approximate solution in O((ω 2 + log log R )( C + R ) θ log R ) time. Faster and more general than all previous algorithms! Extensions for practical application: Most oracle calls not necessary; reuse previous result if still good enough. Use lower bounds to decide Speed-up heuristics, fast approximate oracles Parallelization (without loss of theoretical guarantees!)

54 The algorithm in practice In practice, results are much better than theoretical performance guarantees. Usually iterations suffice. Only few constraints are violated (even after rounding); these are corrected easily by ripup-and-reroute. Detailed routing can realize the solution well, due to good capacity estimations. Small integrality gap and approximate dual solution infeasibility proof can be found for most infeasible instances New constraints can be added easily Open problems: What if one resource is very bad? Guarantee for the rest? Warmstart with good prices?

55 Near optimality chip ɛ = 100 λ t fract λ lb λ rounded λ final % gap % gap running (# nets) (fract.) (final) time Rose :01:45 ( ) :01: :00:28 Georg :02:38 ( ) :01: :00:49 Camilla :51:09 ( ) :33: :11:23 Tomoko :23:43 ( ) :15: :07:09 Andre :18:54 ( ) :50: :17:32

56 Global routing result for the telecommunication chip

57 Overall result: comparison to an industrial router Time (h:mm:ss) Wires Vias Scenic nets Errors BonnRoute Total (m) 25% 50% 100% Industrial 9:18: Bonn 1:17:35 3:55: % -10.3% -22.6% -93.9% -98.4% -98.9% -61.1% 14 nm testbed (14 instances with a total of nets) both routers with 20 threads BonnRoute followed by industrial cleanup tool

58 Congestion map of a difficult instance 110% 100% 94% 87% 76% 60% 30% 0% CRB_PCL RESEARCH INSTITUTE FOR DISCRETE MATHEMATICS, UNIVERSITY OF BONN

59 Modeling timing constraints via dynamic delay budgets Timing constraints are modeled by an acyclic digraph Arrival times at sources and latest allowed arrival times at sinks are given Delays on arcs depend on routing solution Add a resource for each of these arcs Add new customers for determining arrival times at intermediate vertices Results in dynamic delay budgets v 1 v 2 v 3 v 4 N1 N 2 N 3 usg(n 1 ) resource capacity usg(n 3 ) usg(v 2 ) usg(n 2 ) usg(v 2 ) a(v 2 ) a(v 3 ) a(v 1 ) a min (v 2 ) a max(v 2 ) a min (v 3 ) a max(v 3 ) a(v 4 )

60 Theorem (Held, Müller, Rotter, Traub, Vygen [2015]) Let ω > 0. (a) Given a σ-approximate routing oracle, we can compute, with Õ(ω 2 (C + R )) oracle calls, a solution that minimizes the congestion up to a factor σ(1 + ω). (b) For nets with a bounded number of pins, we can obtain σ = 1 in polynomial time. Then, if there is a solution that satisfies all constraints, we get a solution that overloads edges by at most a factor 1 + ω and violates timing constraints by at most ω h, where h is a constant that depends on the instance only. Sketch of proof: Oracle for arrival time customers is trivial Resource sharing algorithm yields (a) Routing oracle for routing a net based on Dijkstra-Steiner algorithm (Hougardy, Silvanus, Vygen [2017]) Increase capacities of timing resources so that all source-sink paths have the same capacity h.

61 Experimental results Bounds Industrial Our algorithm after... timing- routing- tool iteration iteration iteration rounding unaware unaware (heuristic) (final)

62 Conclusion: mathematics leads to better chips! D. Müller, K. Radke, J. Vygen: Faster min-max resource sharing in theory and practice. Mathematical Programming Computation 3 (2011), 1 35 M. Gester, D. Müller, T. Nieberg, C. Panten, C. Schulte, J. Vygen: BonnRoute: Algorithms and data structures for fast and good VLSI routing. ACM Transactions on Design Automation of Electronic Systems 18 (2013), Article 32 S. Held, D. Müller, D. Rotter, V. Traub, J. Vygen: Global routing with inherent static timing constraints. Proceedings of the IEEE/ACM International Conference on Computer-Aided Design (ICCAD, 2015), S. Hougardy, J. Silvanus, J. Vygen: Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm. Mathematical Programming Computation 9 (2017), to appear