Steiner Tree

1. Steiner Tree Prof. Shiyan Hu shiyan@mtu.edu Office: EERC 518

2. Interconnect Topology Optimization • Problem: given a net consisting of a source and a set of sinks, build the best interconnect topology to minimize different design objectives • wire length and timing • In most cases, topology means tree • Because tree is the most compact structure to connect everything without redundancy • Delay analysis is easy

3. Terminology • For multi-terminal net, we can easily construct a tree (spanning tree) to connect the terminals together. • However, the wire length may be unnecessarily large. • Better use Steiner Tree: • A tree connecting all terminals as well as other added nodes (Steiner nodes). • Rectilinear Steiner Tree: • Steiner tree such that edges can only run horizontally and vertically. • Manhattan planes • Note: X (or Y)-architecture (non-Manhanttan) Steiner Node

4. Prim’s Algorithm for Minimum Spanning Tree • Grow a connected subtree from the source, one node at a time. • At each step, choose the closest un-connected node and add it to the subtree. Y X s

5. Conventional Algorithms Are Not Good Enough Minimum spanning tree may have very long source-sink path. Shortest path tree may have very large routing cost. Want to minimize path lengths and routing cost at the same time. Interconnect Topology Optimization Under Linear Delay Model

6. Timing-Driven Interconnect Topology Design • BPRIM algorithm • [Cong et al, ICCD’91, TCAD’92] • RSA algorithm (for Minimum Rectilinear Steiner Arborescences) • [Rao, Sadayappan, Hwang, Shor, Algorithmica’92] • Prim-Dijkstra tradeoff algorithm • [Alpert et al, TCAD’95] • SERT algorithm (Steiner Elmore Routing Tree) • [Boese, Kahng, McCoy, Robins, TCAD’95] • MVERT algorithm (Minimum Violation Elmore Routing Tree) • [Hou, Hu, Sapatnekar, TCAD’99]

7. Given Net N with source s and connected by tree T. Radius of net N: distance from the source to the furthest sink. Radius of a routing tree r(T): length of the longest path from the root to a leaf. Cost of an edge: distance between two. Cost of a routing tree cost(T): sum of the edge costs in T. minpathG(u,v): shortest path from u to v in G distG(u,v): cost of minpathG(u,v). Definitions r(T) radius of the net source s routing tree

8. Bounded Radius Minimum Spanning Tree • Basic Idea: Restrict the tree radius while minimizing the routing cost • Bounded radius minimum spanning tree problem (BRMST): Given a net N with radius R, find a minimum cost tree with radius r(T)(1+ )R source  =  radius = 4.03 cost = 4.03 source  = 1 radius = 1.77 cost = 4.26 source  = 0 radius = 1 cost = 4.95 trade-off between radius and the cost of routing trees • Parameter  controls the trade-off between radius and cost •  =  minimum spanning tree;  = 0 shortest path tree

9. x x y y s s x’ distT(s,x)+cost(x,y) (1+ )R distT(s,x’)+cost(x’,y) R BPRIM Algorithm forBounded-Radius Minimum Spanning Trees • Given net N with source s and radius R, and parameter . • Grow a connected subtree T from the source, one node at a time • At each step, choose the closest pair x T and y  N-T • If distT(s, x) + cost(x,y) (1+)R, add (x,y) • Else backtrack along minpathT(s,x) to find x’ such that distT(s, x’) + cost(x’, y)  R, then add (x’, y) • Slack R is introduced at each backtrace so we do not have to backtrace too often.

10. Experimental Results of BPRIM Algorithm

11. Prim-Dijkstra Algorithm Prim’s MST Dijkstra’s SPT Trade-off

12. Prim’s and Dijkstra’s Algorithms • d(i,j): length of the edge (i, j) • p(j): length of the path from source to j • Prim: d(i,j) Dijkstra: d(i,j) + p(j) p(j) d(i,j)

13. The Prim-Dijkstra Trade-off • Prim: add edge minimizing d(i,j) • Dijkstra: add edge minimizing p(i) + d(i,j) • Trade-off: c(p(i)) + d(i,j) for 0 <= c <= 1 • When c=0, trade-off = Prim • When c=1, trade-off = Dijkstra

14. Hanan Theorem • For rectilinear Steiner tree construction, there exists a routing tree with minimum total wire length on the grid formed by horizontal and vertical lines passing through source and sinks. Hanan nodes source Hanan Grid

15. Rectilinear Steiner Arborescence Algorithm • Given n nodes lying in the first quadrant • Purpose is to maintain shortest paths from source to sink and minimize total wire length • RSA algorithm • Start with a forest of n single-node A-trees. • Iteratively substituting min(p,q) for pair of nodes p, q where min(p,q) = (min{xp, xq}, min{yp, yq}). • The pair p, q are chosen to maximize ||min(p,q)|| over all current nodes. p q min(p,q)

16. r r r r r r Example of RSA Algorithm

17. Performance of RSA Algorithm • Time Complexity O(n log n) when implemented using a plane-sweep technique. • Wirelength of the tree by RSA algorithm  2 x Optimal solution (i.e., 2 x wirelength of minimumRectilinear Steiner Arborescence

18. Rectilinear Steiner Tree • Chris Chu and Yiu-Chung Wong, FLUTE: Fast Lookup Table Based Rectilinear Steiner Minimal Tree Algorithm for VLSI Design. IEEE Transactions on Computer-Aided Design, vol. 27, no. 1, pages 70-83, January 2008 • Obstacle-avoiding Rectilinear Steiner Minimum Tree Construction: An Optimal Approach, Tao Huang and Evangeline F.Y. Young, Proceedings IEEE International Conference on Computer-Aided Design, 2010

19. Steiner Elmore Routing Tree (SERT) Heuristic • Use Elmore Delay Model directly in construction of routing tree T. • Add nodes to T one-by-one like Prim’s MST algorithm. • Two versions: • SERT Algorithm: • At each step, choose v T and uT s.t. the maximum Elmore-delay to any sink has minimum increase. • SERT-C Algorithm: • SERT with critical sink • First connect the critical sink to the source by a shortest path, • then continues as in SERT, except that we minimize the Elmore delay of the critical sink rather than the maximum delay.

20. Steps of SERT Algorithm 7 7 7 8 8 8 3 3 3 6 6 6 4 4 4 1 1 1 5 5 5 source source source 2 2 2 9 9 9 7 7 7 8 8 8 3 3 3 6 6 6 5 5 5 4 4 4 1 1 1 source source source 2 2 2 9 9 9

21. Examples of SERT-C Construction 7 7 7 8 8 8 6 6 3 6 3 3 5 5 4 4 4 1 1 1 5 source source 2 2 2 9 9 9 source c) Node 5 critical a) Node 2 or 4 critical b) Node 3 or 7 critical 7 7 7 8 8 8 3 6 3 6 6 3 5 4 4 4 1 1 1 5 5 source source 2 2 2 9 9 9 source d) Node 6 critical f) Node 9 critical e) Node 8 critical (also SERT)

22. Maze Routing • http://foghorn.cadlab.lafayette.edu/MazeRouter.html

23. Global Routing Prof. Shiyan Hu shiyan@mtu.edu Office: EERC 518

24. Global Routing • 2D Routing • Route all the nets • Minimize total wirelength • Satisfy the congestion constraint or minimize the overflow • 3D Layer Assignment

25. Global Routing Approaches • Sequential Approach • Net ordering based approach • Concurrent Approach • Integer Programming

26. Sequential Routing • In sequential approach, we need some net ordering. • A bad net ordering will increase the total wire length, and may even prevent completion of routing for some circuits which are indeed routable. A B A B B B A B first (Good order) A first (Bad order) A

27. Criteria for Net Ordering • Criticality of net - critical nets first. • Estimated wire length - short nets first since they are less flexible. • Consider bounding rectangles (BR): A B B A B is in A’s BR

28. Net Ordering

29. Nets Set of possible routing trees net 1 T11, T12, ...... , T1k1 : : : : net n Tn1, Tn2, ... , Tnkn Concurrent Routing • Consider all the nets simultaneously. • Formulate as an integer program. • Given: Lij = Total wire length of Tij Ce= Capacity of edge e • Determine variable xij s.t. xij = 1 if Tij is used xij = 0 otherwise.

30. Integer Program Formulation

31. Solution 1 1 2 3 What are the constraints for edge capacity? 3 1 2 2 Concurrent Approach: Example Possible trees: 1 1 2 3 net 1: 2 3 3 3 1 2 2 net 2: 2 3 3 net 3: 2 2

32. Integer Programming Approach • Standard techniques to solve integer programming. • No net ordering. Give global optimum. • Can be extremely slow, especially for large problems. • To make it faster, a fewer choices of routing trees for each net can be used.