Net Criticality Revisited: An Effective Method to Improve Timing in Physical Design H. Chang 1, E. Shragowitz 1, J. Liu 1, H. Youssef 2, B. Lu 3, S. Sutanthavibul 4 1 University of Minnesota, USA 2 Universite du Centre, Tunisia 3 Cadence Design Systems Inc., USA 4 Intel Corp., USA

Basic Ideas and Goals Perform timing-driven placement and routing based on IMP (Iterative Minimax Pert), a zero slack distribution algorithm for net delay bound calculation Propose new criticality metrics for placement and routing Achieve better timing results in one-pass physical design

Why IMP Algorithm? Review of related works on delay budgeting problem –First ZSA algorithm in O(np), 1989 –IMP algorithm based on Minimax formulation with linear time complexity O(n+p), 1990 –Application of IMP to placement problem in O(n+p), 1992 –PWL-GBS algorithm solving the problem in linear programming formulation in O(m 2 logH), 1997 –MISA algorithm based on the maximal independent set (MIS) of a transitive slack equalization graph in O(kn 3 ), 2000

Calculation of Net Delay Bounds by IMP Algorithm Path slack U  for any path , U  = T cr - T  T cr : longest path delay, T  : delay of the path  U  >0 for any noncritical path, U  =0 for any critical path. Delays on non-critical paths can be increased without increase in a clock cycle

Continued.. The IMP algorithm is based on two ideas: a) The net delay slacks on each of the paths can be distributed among constituent nets, according to the relative weights of the nets along the path.

Continued.. b) Each net may belong to the multiple paths. Therefore, the propagation delay on the net should not exceed the minimal value among all maximal delays defined on this net for each path separately Reference: H.Youssef, R-B. Lin and E. Shragowitz in IEEE TCAS, 1992 This problem is NP-hard.

Asymptotically Converging Approximation Algorithm where, both minimal path slack and maximal path weight for all the paths traversing edge e, can be computed in linear time by a pert-like algorithm. Repetitive application of step 1 results in convergence to the optimal solution of the initial Minimax problem, On each step of the algorithm, a lower bound on the value of is found by a linear algorithm,

Criticality Metrics Net delay bounds provide new opportunities for identification of timing-critical nets Net Criticality Metric =

Probabilistic Interpretation of Criticality Metrics Projected Net Delays could be considered as random values. Assuming Gaussian distribution N(m x,  x ) of net delay x, x min = m x - 3  x, x max = m x +3  x, x max  b x, where b x is a bound on net delay if x min  0  x max = 2m x  b x Net Criticality Metric can be rewritten as, Net Criticality Metric = 2m x /b x The probability for the net delay to be below the bound b x is decreasing when a ratio m x /b x is increasing.

Statistical Approximation Formulas for Net Criticality Metrics The ranking of nets according to criticality metrics is preserved when a mathematical expectation of a net delay is replaced by a net parameter. Criticality metrics for placement (CMP) Criticality metrics for routing (CMR)

One-Pass Flow v.s. Multiple-Pass Flow in Physical Design One-pass criticality-based layout flowTraditional iterative layout flow

Experiments Environment: –Cadence Silicon Ensemble DSM Automation Layout System –4-layer, 0.18 micron technology standard cell library Integration of the proposed criticality metrics in layout flow of Cadence Silicon Ensemble –Application of criticality metrics to placement (CMP) –Application of criticality metrics to routing (CMR) –Application of CMP and CMR to placement and routing in one pass of a layout process

Placement Results WLM: layout by Cadence in Wire Length Minimization mode; TDP: layout by Cadence in Timing-Driven Placement mode; CPF: layout by Cadence in WLM mode with nets on Critical Paths given higher weights during placement (2-pass solution); CMP: layout by Cadence in WLM mode with weights derived from new Criticality Metrics for Placement (1-pass solution). 26.4% improvement versus Cadence Wire Length Minimization mode and 13.8% improvement versus Cadence timing driven placement mode

Routing Results WLM: layout by Cadence in Wire Length Minimization mode; TDR: layout by Cadence in Timing-Driven Routing mode; CPF: layout by Cadence in WLM mode with nets on Critical Paths given higher weights during routing (2-pass solution); CMR: layout by Cadence in WLM mode with weights derived from new criticality Metrics for Routing (1-pass solution). 9.2% improvement versus Cadence Wire Length Minimization mode and 3.7% improvement versus Cadence timing driven routing mode

Placement and Routing Results WLM: layout by Cadence in Wire Length Minimization mode; TDPR: layout by Cadence in Timing-Driven Placement and Routing mode; CPF: layout by Cadence in WLM mode with nets on Critical Paths given higher weights during placement and routing (2-pass solution); CMP+CMR: layout by Cadence in WLM mode with weights derived from new Criticality Metrics for Placement and Criticality Metrics for Routing (1-pass solution). 29.5% improvement versus Cadence Wire Length Minimization mode and 12.4% improvement versus Cadence timing driven placement and routing mode

Conclusion The proposed criticality metrics: –achieve substantially better timing results in one pass of physical design. –could be integrated with any layout system that allows weights for nets in the design. –can be applied to timing optimization of placement alone or routing alone