Fast Global Routing with Forbidden-Region Rerouting

1. Fast Global Routing with Forbidden-Region Rerouting Chin-Hsiung Hsu, Huang-Yu Chen, Yao-Wen Chang

2. Outline • Introduction • Problem Formulation • Routing Methodology • Experimental Results • Conclusion

3. Introduction • Although many routing techniques have been well studied and developed, such as • maze routing [14] • A*-search routing [7] • pattern routing [13] • monotonic routing [18] • multicommodity flow [1] • integer linear programming (ILP) [10] • It is not clear whether or not these traditional methods have great magic power to face the upcoming design challenges in the next generations.

4. Introduction • The iterative negotiation-based rip-up/rerouting (INR) has evolved as the main stream for the state-of-the-art global routers. • However, INR may get stuck at the local optima as the iterations increased. An enhanced iterative forbidden-region rip-up/rerouting (IFR), additional schemes to resolve this problem.

5. Problem Formulation • The routing region is partitioned into tiles and a routing graph composed of tile nodes and edges. • The tile node represents a tile, and the global edge models the relationship between adjacent tiles.

6. Problem Formulation • Each edge is associated with a capacity such as the number of available detailed routing tracks on the tile boundary or the maximum allowable via count between adjacent layers.

7. Problem Formulation • The main objective of global routing is to minimize the total overflows, which is calculated by the total amount of routing demand that exceeds the capacity for all edges.

8. Problem Formulation • The ISPD’08 metrics evaluate the global routing solution by the prioritized order: • the total overflow • the maximum overflow • the weighted(cpu time) total wirelength

9. Routing Methodology • This paper adopt the paradigm of 3D-to-2D capacity mapping followed by the planar (2D) routing and the 2D-to-3D layer assignment, such as previous routers [9, 19], instead of the time-consuming 3D routing approach.

10. Routing Methodology • The global routing flow mainly contains two parts: initial routing, and enhanced INR. • And this paper’s planar routing features the iterative forbidden-region rip-up/rerouting (IFR).

11. Routing Methodology • Iterative Forbidden-Region Rip-up/Rerouting (IFR) • Rerouting • Multiple Forbidden Regions Expansion • Look-ahead Historical Cost Increment

12. Routing Methodology • The forbidden regions mean that overflows in these regions is almost forbidden, or it would get huge cost penalty. • BoxRouter [6] and NTHU-Route [9] both apply the concept of the congested regions.

13. Rerouting • When searching a path by the node propagation from the source node s to the target node t, IFR sets the cost function of a current node x as where g(x) is the Manhattan distance from s to x, h(x) is the Manhattan distance from x to t. t x s

14. The costRe(e) compute by • where costp(e) is called the present-congestion cost defined as • where Pn denotes an overflow penalty constant and is set as 1000. Andthe costh(e) is called the historical-congestion cost defined as • where he represents the historical cost of e, and the base cost pe is usually set as pe=de/ce for e with the capacity ce and the demand de. Costp(e)=1000*2/3 2 3 t x s

15. However, the overflows between iterations would decline faster if we enlarge the gap between the overflow-free global edges and the over-congestion ones. • Therefore, we apply the modifies pe in IFR as follows: • the A*-search algorithm can attain the optimality properties. In this paper’s implementation, set ζ = 3.0.

16. Routing Methodology • Iterative Forbidden-Region Rip-up/Rerouting (IFR) • Rerouting • Multiple Forbidden Regions Expansion • Look-ahead Historical Cost Increment

17. Multiple Forbidden Regions Expansion • There are three phases for the multiple forbidden regions construction in IFR. • Construct forbidden regions and rip-up/rerouting • Region propagation levelling (RPL) • Final expansion

18. Phase 1: • Find the global edges that have the maximal overflow and are not covered by any forbidden region. • Forbidden regions are constructed by performing box expansion. • Expands until the stop-expanding criteria is satisfied. • Rip-up/rerouting • The stop-expanding criteria is • where β is a bound that decreases as the number of iterations in IFR increases. In this paper, β ranges from 1.1 to 1.4. • For a forbidden region Rf , define the four boundaries of Rf as: • where d ϵ {L, R, F, B} denotes the left-, right-, front-, or back- direction boundary of Rf.

19. Phase 2: Region propagation levelling The second phase is invoked when the number of overflows in the first phase stops declining and gets stuck at the local optimal solution. • Although the overflows can be dispersed out of the forbidden regions, they may stay around these forbidden regions and will go back inside the forbidden regions after the next iteration. • RPL inherits all forbidden regions at the previous iterations and then expands these forbidden regions simultaneously.

20. Phase 3: final expansion The third phase starts when the number of overflows is less than 0.5% of the initial overflows. IFR expands the forbidden region to the whole routing graph to quickly reduce the remaining overflows. • Because that INR loses it magnetic power to reduce the overflows when the total overflows are smaller.

21. Routing Methodology • Iterative Forbidden-Region Rip-up/Rerouting (IFR) • Rerouting • Multiple Forbidden Regions Expansion • Look-ahead Historical Cost Increment

22. Look-ahead Historical Cost Increment • The most negotiation-based global routers use the following equation as the updating scheme for the historical cost he: where K is a constant, and ce and de represent the capacity and demand of e, respectively.

23. Look-ahead Historical Cost Increment • Although the Equation (7) performs well at the beginning few iterations, but as the iteration increases it cannot minimize the overflows and just exchanges the positions of overflows. • Thus, it tends to get stuck in local optimal solutions.

24. Look-ahead Historical Cost Increment • Look-ahead historical cost is defined as: where N > 0 represents a positive integer. • The key idea is not only to increase the historical cost on the global edges with overflows, but also on those near-overflow global edges. In this paper,N and K are both set to be 1.

25. Experimental Results • Note that we do not compare with the ISPD’08 contestants since their binaries are not available to the public. • “OF” and “Max OF” give the numbers of the total overflows and the maximum overflows. • “WL” give the total wirelengths, which is calculated by the metrics of the ISPD’07 and ISPD’08 global routing contests, respectively. • “Quality Metric”, which is the weighted wirelength, is the metric of the ISPD’08 contest.

28. Conclusion • This paper develop enhanced iterative forbidden-region rip-up/rerouting (IFR). • It features novel techniques such as the region propagation levelling (RPL) and look-ahead historical cost increment to improve the traditional iterative negotiation-based rip-up/rerouting (INR).