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ABSTRACT

On the Skew-Bounded Minimum Buffer Routing Tree Problem C. Albrecht (Synopsys), A.B. Kahng, B. Liu, I. Mandoiu (UCSD), A. Zelikovsky (GSU). ABSTRACT

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ABSTRACT

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  1. On the Skew-Bounded Minimum Buffer Routing Tree Problem C. Albrecht (Synopsys), A.B. Kahng, B. Liu, I. Mandoiu (UCSD), A. Zelikovsky (GSU) • ABSTRACT • We consider the problem of buffering a given tree with the minimum number of buffers under load cap and buffer skew constraints. Our contributions include: • A proof that the greedy algorithm proposed by Tellez and Sarrafzadeh (TCAD’97) is suboptimal for all non-zero skew bounds • Anoptimaldynamic programming algorithmfor the problem • Experimental results on test cases extracted from recent industrial designs showingthat the dynamic programming algorithm has practical run time and saves up to 20% of the buffers inserted by the algorithm of Tellez and Sarrafzadeh

  2. Motivation • In order to initiate meaningful placement and timing optimizations, every design flow requires early elimination of all electrical violations (e.g., load cap and slew violations), even for non-critical nets. Bounds on load caps • - Serve as proxies for signal slew rate bound • - Improve coupling noise immunity • - Reduce delay uncertainty due to coupling noise • - Improve reliability with respect to hot-carrier and AC self-heating effects • - Facilitate technology migration since designs are more balanced • - Guarantee bounded input rise/fall times at buffers and sinks • For clock and test distribution an additional design requirement is bounding the buffer skew, i.e., the difference between the maximum and the minimum number of buffers over all source-to-sink paths in a routing tree, since buffer skew is one of the main factors affecting the actual delay skew • To make progress with any methodology, it is crucial to have a fast and resource efficient method for fixing load cap and buffer skew violations. Of particular interest are practical methods for buffering non-critical nets that have up to tens of thousands of sinks (e.g., scan enable)

  3. Minimum-Buffered Routing Problem 0.75CU Cw=Cb=0 CU 0.75CU Tree with bounded buffer load cap 0.75CU Cw=Cb=0 D=0 CU 0.75CU Tree with bounded buffer load cap andzero buffer-skew Given: • Net N with source r and set of sinks S • Binary routing tree T = (r, V, E) for N • Input capacitance csfor each sinks S • Buffer input capacitance Cb • Unit-length wire capacitance Cw • Capacitive load upper-bound CU • Buffer-skew bound D Find: bufferingof the routing tree T such that • The load cap of each buffer and of the source r is at most CU • The buffer skew is at most D • The number of inserted buffers is minimized

  4. The Greedy Algorithm • Proposed by Tellez and Sarrafzadeh(IEEETrans. on CAD, vol. 16, 1997, pp. 333-342) packNode(u) w/ buffer skew bound D • A.0Ifl(Tv)<l(Tw) (longest path of v is less than longest path of w) then swap v and w. • A.1 If l(Tv) - l(Tw)> D then insertl(Tv) - l(Tw) -D buffers at the topmost position of (u,w); exit if cap(Tu)<Cu • A.2 Perform packNode(u) excluding child branches with maximum longest path; exit if cap(Tu)<Cu • A.3 Insert buffers at topmost position of child branches with shortest path equal to l(u) – D • A.4 Perform packNode(u) considering only child branches with maximum longest path Bounded load capw/o buffer skew bound For each u  V,in bottom-uporder, do • A.packNode(u): Let v and w be the two children of u. If cap(Tv) + cap(Tw) > Cu add a buffer at the topmost position of the child branch with the largest cap (the greedy choice) then remove the subtree driven by the buffer • B.packEdge(u): While cap(Tu)>Cuadd a buffer on edge (u,parent(u))at the highest possible position still meeting the load cap bound Cu

  5. The Greedy Algorithm is Suboptimal The greedy algorithm of Tellez and Sarrafzadeh finds the optimum buffering whenD = 0 However, the algorithmissuboptimalfor any buffer skewD > 0 Buffer skew D = 1, sink input cap Cu=CU, Cv=Cx=0.75CU Interconnect and buffer have zero cap 0.75CU 0.75CU CU 0.75CU CU 0.75CU Greedy buffering Optimum buffering Counterexample 1.

  6. Why No Greedy Algorithm Will Work • To guarantee optimality, solutions w/D different longest path lengthsmay be required for a subtree in any bottom-up algorithm • Counterexample 3:Cw=Cb=0, • D‘u’leaves, each with cu = CU – e, one ‘v’ leaf withcv = e • Optimum: depending on upstream tree topology, each of the following Dbufferings may be the only way to complete the optimum solution • To guarantee optimality, arbitrarily manysolutionsmaybe need for asubtree in any bottom-up algorithm • Counterexample 2:D=1, Cw=Cb=0, cu=CUand cvsatisfies cv2d-2<CU and cv(2d-2+1)>CUwhered is depth of Ta • Greedy buffers one of the two branches into node a,this triggers the insertion of arbitrarily many buffers upstream due to the skew constraint • Optimum: buffersas many of the ‘v’ nodes as needed in one of the two subtrees of node a v u u u a v u u u v u u u u v v u u v v u

  7. Dynamic Programming Algorithm • Initialize solution set L(u) =  ,  u  V • For each u  V, in bottom-up order, do (1) Let v and w be the children of u (2) For each buffering X  L(v) and Y  L(w), withl(X) ≥ l(Y), do (a) Let Z be XY with max{0,l(X)-s(Y)} buffers added at the top (b) For each i = 0, …, min{max{0, s(X) – s(Y)}, l(X) – l(Y)} do • Let Zi be Z with ibuffers added at the top of edge(w,u) • EdgeBuffering(Zi,u) (3) Remove from L(u)all bufferings with more than NB buffers (4) For each buffering with (nb, l, s) buffers in total, on longest path, and on shortest path, respectively, remove from L(u)all bufferings with parameters (nb+k, l+k, s+k) where k ≥2 • Return the bufferingX  L(v)with minimum number of buffers Procedure EdgeBuffering(X,u): • While cap(X) > CU, add a buffer on edge (u, parent(u)) at the highest position meeting the load cap bound Cu • L(u) L(u) + X • If cap(X) > CbthenL(u) L(u) + X’ where X’ is X with an additional buffer just below parent(u)

  8. Analysis Corectness: • By induction: for each buffering X of the branch driven by (u,parent(u)) there exists k>0 and a buffering Y L(u) such that X is dominated by Y with k buffers added at the top • The dynamic programming algorithm returns an optimum feasible buffering Runtime: • For each node u  T, the solution set L(u) computed by the dynamic programming algorithm contains at most 2(D+1)NBbufferings • The running time of the algorithm isO(n(D+1)3NB2) time, where n, D and NB are the number of sinks, the given skew bound and a given upper-bound on the optimum number of buffers, respectively • The bound is not known to be tight, in practice the runtime is much better

  9. Experimental Results • DP has practical runtime (less than 1 second for the above 2676-sink test) • DP saves up to 20% of the buffers inserted by Tellez-Sarrafzadeh algorithm • Compared to zero-skew buffering, DP achieves a significant reduction in the number of inserted buffers evenwith a very small buffer skew (D=1 or 2)

  10. On the Skew-Bounded Minimum Buffer Routing Tree Problem C. Albrecht (Synopsys Inc.) A.B. Kahng, B. Liu, I. Mandoiu (UC San Diego) A. Zelikovsky (Georgia State U.)

  11. Minimum-Buffered Routing • Early elimination of load cap and slew violations is neededfor all nets, even for non-critical ones. Bounds on load caps • - Serve as proxies for signal slew rate bound • - Improve coupling noise immunity • - Reduce delay uncertainty due to coupling noise • - Improve reliability with respect to hot-carrier and AC self-heating effects • - Facilitate technology migration since designs are more balanced • Guarantee bounded input rise/fall times at buffers and sinks • For clock and test distribution an additional design requirement is bounding the buffer skew, i.e., the difference between the maximum and the minimum number of buffers over all source-to-sink paths in a routing tree • Minimum-Buffered Routing Problem:Given a routed net, sink/buffer input caps, and unit-wire cap, insert the minumum number of buffers to satisfy given load cap and buffer skew constraints • Introduced by Tellez and Sarrafzadeh (IEEE TCAD’97) who gave a greedy algorithm

  12. Our Contributions • A proof that the greedy algorithm of Tellez and Sarrafzadeh is suboptimal for all non-zero skew bounds • - We give examples showing that no greedy algorithm can achieve optimality • Anoptimaldynamic programming algorithmfor the problem • The algorithm computes lists of undominated feasible solutions for all subtrees, in bottom-up order • Worst-case runtime isO(n(D+1)3NB2) time, where n, D and NB are the number of sinks, the skew bound, and a given upper-bound on the optimum number of buffers, respectively • Runtime is much better in practice • Experimental study of buffering algorithms on test cases extracted from recent industrial designs • - The dynamic programming algorithm uses significantly fewer buffers than the algorithm of Tellez and Sarrafzadeh

  13. Results on a 2676-sink testcase • DP has practical runtime (less than 1 second per run) • DP saves up to 20% of the buffers inserted by Tellez-Sarrafzadeh algorithm • Compared to zero-skew buffering, DP achieves a significant reduction in the number of inserted buffers evenwith a very small buffer skew (D=1 or 2)

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