Incremental and On-demand Random Walk for Iterative Power Distribution Network Analysis Yiyu Shi*, Wei Yao*, Jinjun Xiong+ and Lei He* *Electrical Engineering Dept., University of California, Los Angeles +IBM T. J. Watson Research Center, Yorktown Heights, NY This paper is supported in part by an NSF CAREER award CCR0306682 and a UC MICRO grant sponsored by Actel and Fujitsu.

Outline PDN analysis and random walk algorithm Incremental random walk Walk Ordering Optimization Experimental Results Conclusions

PDN Design Challenges Designing reliable supply network is essential VDD decreases Wire width decreases Load current increases Efficient power grid analysis tool is needed To handle large circuit size with millions of nodes On-demand analysis is necessary Only calculate the response at the nodes of interest Random walk based solver is a natural solution To handle various design changes in the design loop Incremental analysis is necessary Update analysis results instead of calculating from scratch

Random Walk Based Solver The voltage at an internal node x can be solved from It can be rewritten as At pad node, we have x 3 2 1 4 g1 g2 g3 g4 Ix ∑ 1

Random Walk Based Solver Alternatively, the above equations can be viewed as 3 1 x px,1 px,3 px,2 px,4 2 4 M walks from i-th node Take average This yields xi

Outline PDN analysis and random walk algorithm Incremental random walk Walk Ordering Optimization Experimental Results Conclusions

Incremental Analysis of Power Network Incremental analysis exists in literature for full-solvers Fictious Domain Method [Zhao:ICCAD’07] Large Change Sensitivities However, they both require a full solve for every node of the original circuits Cannot be applied to the on-demand solutions obtained from random walk based solver. Q: How to perform incremental analysis for random walk based solver? 7

Basic Idea for Incremental Random Walk When design changes 25 20 (-5) 0.1 0.1 0.2 0.1 0.1 20 (-5) 0.1 25 100 40 (+15) 25 100 0.1 0.1 25 20 (-5) Increase or decrease the number of visits from the node of change Propagate the change throughout the network.

Incremental Random Walk Suppose the probability of walking to node i from an adjacent node j is After design changes it becomes Then, the number of visits of node i needs to be changed to is positive => Extra random walks are needed is negative => Remove existing random walks

Negative Random Walk It is easy to start extra random walks from a node But to remove existing walks is difficult Bookkeeping of all walks is infeasible due to the huge memory requirement Instead of removing ΔN number of random walks starting from a node directly, we perform ΔN number of extra random walks from that node. Whenever a node is visited, the number of visits at that node is decreased by one opposed to the normal random walk which should increase the number of visits by one. It can be proved that the statistical effect of such a strategy is equal to the removal of ΔN random walks from that node.

Outline PDN analysis and random walk algorithm Incremental random walk Walk Ordering Optimization Experimental Results Conclusions

Merging of Positive and Negative Walks During positive (negative) random walk, when arriving at a node that requires additional negative (positive) random walks, The walk terminates. The number of required additional walks at current node is decreased by one. Order of the incremental random walks matters Assume ΔN positive walks from A and ΔN negative walks from B A first, then B => ~2ΔN walks are needed B first, then A => only ~ΔN walks are needed Intuitively, when the walks from a node has a higher probability of hitting the nodes that require opposite walks, that node should have an earlier start. ΔN -ΔN

Runtime for Different Ordering: an Example Runtime distribution on a circuit with 1027 nodes 500 Monte Carlo runs with different node ordering 6X runtime difference can be observed with different ordering scheme

Walk Ordering Optimization We order the nodes according to P(A→i) is the probability of a walk staring from node A to hit node i can can be computed as where Φc is a set of all paths from node A to i. In practice, we only need to consider K paths with largest probability. Can be solved by K shortest paths algorithm. where

Outline PDN analysis and random walk algorithm Incremental random walk Walk Ordering Optimization Experimental Results Conclusions

Impact of Ordering Algorithm PDN design with 102,471 nodes and 215 ports Consider topology changes and magnitude changes Our ordering algorithm can reduce the runtime close to the minimum time in 10000 Monte Carlo simulations. 8X speedup compared with the worst ordering case.

Linear Complexity for On-demand Analysis Random walk from scratch has a warm-up cost Becomes linear after the observation node is more than 40% of the total nodes However, in reality on-demand analysis hardly falls in this region Incremental random walk algorithm has a linear runtime complexity w.r.t. number of observation nodes Very suitable for on-demand incremental analysis Up to 100X speedup over the random walk from scratch

Iterative PDN Sizing Design information Runtime and accuracy comparison 18X speedup compared with random walk from scratch with similar accuracy.

Iterative Package Ball Assignment with Substrate Rerouting To reduce crosstalk noise Iteratively optimize ball assignment Different ball assignment leads to different substrate rerouting Topology changes in RLC network Incremental random walk can be used to speedup the iterative design process 13X speedup for total analysis time during iterative design Compared to random walk from scratch

Conclusion We develop an incremental random walk algorithm, which is capable of both incremental and on-demand analysis. We propose an efficient algorithm to optimize the walk ordering problem Achieve 8× runtime reduction compared to the worst order Compared with random walk from scratch, runtime is reduced by up to 18× for on-chip iterative PDN sizing, and by up to 13× for iterative package ball assignment with substrate rerouting. On-demand analysis has a runtime linear with respect to the number of observed nodes for incremental random walk. 2× difference compared to non-incremental random walk.