1. 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
CCR and a UC MICRO grant sponsored by Actel and Fujitsu.

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

3. 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

4. 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 ∑

5. 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

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

7. 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

8. 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.

9. 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

10. 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.

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

12. 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

13. 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

14. 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

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

16. 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 Monte Carlo simulations.
8X speedup compared with the worst ordering case.

17. 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

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

19. 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

20. 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.