Solving Hard Instances of FPGA Routing with a Congestion-Optimal Restrained-Norm Path Search Space

1. Solving Hard Instances of FPGA Routing with a Congestion-Optimal Restrained-Norm Path Search Space Keith So School of Computer Science and Engineering University of New South Wales, Australia ISPD2007

2. Presentation Outline • Introduction to FPGA Routing • Brief Review of Prior Work • Observed Problems with Standard Negotiated Congestion Formulations • New Lexicographical Path Search Space and Associated Properties • Application to Wirelength-Driven Routing • Future Work • Conclusions

3. Research Goals • Develop a range of FPGA routers for different designer constraints • Identify potential for improvements through the study of known best state-of-the-art routers • More robust FPGA routers can save unit costs and allow area improvement in future FPGA architectures

4. FPGA Routing Problem • FPGA Components • Logic blocks • Wire segments • Wire switches • Problem: To find an assignment of resources to nets that • Satisfies required connectivity • Satisfies electrical design rules • (Plus optional designer constraints)

5. Routing Resource Graph • Converts the FPGA architecture into a digraph • Pins, Logic Blocks, Wire Segments  Vertices • Switches  Edges • Problem translates to finding mutually vertex-disjoint trees that implement the connectivity of nets

6. A Simple Example • Example Netlist • Net 1: s1->t1 • Net 2: s2->t2 • Net 3: s3->t2 and t3 • In practice, 1000s+ of nets and more complex graph

7. Some Prior Approaches to FPGA Routing • Rip and reroute [Frankle92] • Global routing then detailed routing [Brown96,Lemieux97] • Transforms to SAT [Wood97,Nam99,Xu03] • Min-cost flow based[Lee03] • Negotiated Congestion based [Mcmurchie95,Betz97,Betz00,Fung03]

8. Negotiated Congestion • Iterative route nets and gradually increase sharing penalty of overused resources • A* routing used to connect each net, with a cost function with congestion and secondary cost components • Sharing penalty cost is based on current congestion and congestion in previous iteration rounds and is monotonically increasing • Iterations continue until no electrical design rules are violated or until a maximum number of iterations reached (declares unroutability)

9. NC on the Simple Example

10. Typical Form for Negotiated Congestion Cost Functions • For vertex v: • occ(v): guard for congested vertex • p(v): present congestion cost, exponentially increasing • h(v): historical congestion cost • s(v): designer issued secondary cost • Path cost = sum of vertex costs

11. Problems with Scalar Path Evaluation • In all previous implementations a weighted scalar projection is used for path evaluation • Optimality to the weighted functions has limited meaning to conditions in the terrain • Sometimes lead to suboptimal choices for congestion avoidance • Magnitude of congestion cost is exponentially increasing • Needed to ensure convergence • Will lead to numerical instability with floating point representations

12. Suboptimal Choice with Respect to Congestion Cost • Suppose x is chosen by the A* router • Any path y on the Pareto line might have been chosen • Path z2 which is congestion free is not chosen • because it is not on the lower bound of candidate paths in the scalar function

13. Phases in the Negotiated Congestion Cycle Phase 1 Phase 2

14. Phases in the Negotiated Congestion Cycle Phase 3 Phase 4 Precision Lost!

15. Solution: Lexicographical Ordered Pairs • Use a ordered pair to store congestion and secondary cost components • Pairwise addition: (c1,s1) + (c2,s2) = (c1+c2, s1+s2) • Define a dictionary order on elements: (c1,s1) < (c2,s2) if c1<s1 or [c1=s1 and c2<s2]

16. Proof: Preservation of Admissibility • The OP transformation preserves admissiblity of scalar secondary heuristics • Proof: • Define f’(x)=(0,f(x)) for admissible f in scalar form • In OP space, the actual cost will be of the form (d,f(x)+e) • (0,f(x)) < (d,f(x)+e) for any nonnegative d and e so f’(x) is admissible in OP space • Shows that A* search will be optimal when using the OP comparator

17. Properties of Searches in the Ordered Pair Space • Property 1: Path chosen will have minimum congestion cost • (w,x) < (y,z) if w < y • Property 2: If multiple paths of congestion cost x are available, path chosen will be the one with minimum secondary cost • (x,y) < (x,z) if y < z • Both properties handy for ensuring an “optimum choice” is made for each pin, assuming congestion cost measure “problem size’’

18. Application: CornNC Wirelength Driven Router • Secondary cost is total wirelength • CornNC cost is an OP of congestion cost and expected wirelength to target: f(v)=(occ(v)c(v),WL(v)) • Congestion cost increased linearly per iteration, allows many iterations to be run (representation limit ~millions!)

19. Experimental Evaluation • CornNC and VPR4.30 on the FPGA Challenge architecture and placements [Betz97] • Number of tracks parameterised to determine robustness • Runtimes and utilisation measured between mutually routable instances • Max iterations for CornNC = 1600

25. Further Aspects for Investigation • Runtime Improvements • Congestion cost schedule refinement • Inclusion of multiple objectives • e.g. simultaneous power and timing/mintime-maxtime • Application to other domains • where negotiated congestion has been successfully applied

26. Conclusions • Lexicographical ordered pair evaluation of candidate paths has useful properties and is numerically stable • Effective for solving difficult cases of wirelength FPGA routing with a significant (10%) improvement • Analytically promising for many other objectives because of provably desirable path choices