1. Global Routing

2. 2 B (2, 6) A (2, 1) C (6, 4) B (2, 6) A (2, 1) C (6, 4) S (2, 4) Rectilinear Steiner minimum tree (RSMT) Rectilinear minimum spanning tree (RMST) Single Net Routing Single net routing:  Basis for full routing  Rectilinear routing

3. Rectilinear Routing  RMST by Prim’s Algorithm  O(p 2 )  p: number of terminals using methods such as  Starts with a single terminal and greedily adds least-cost edges to the partially-constructed tree  Advanced computational-geometric techniques reduce the runtime to O(p log p) 3

4. 4 Characteristics of an RSMT  No. of Steiner points: 0 ≤ s ≤ p – 2 −p: No. of pins  The degree of any terminal pin is 1, 2, 3, or 4 −The degree of a Steiner point: 3 or 4  Always enclosed in the minimum bounding box (MBB) of the net  L RSMT  L MBB / 2 Rectilinear Routing 1 2 3 4 56 7

5. 5 Transforming an initial RMST into a low-cost RSMT p1p1 p2p2 p3p3 p1p1 p3p3 p2p2 S1S1 p1p1 p3p3 p2p2 Construct L-shapes between points with (most) overlap of net segments p1p1 p3p3 S p2p2 Final tree (RSMT) Rectilinear Routing

6. 6 Hanan grid:  Consists of the lines x = x p, y = y p that pass through the location (x p,y p ) of each terminal pin p  Maurice Hanan proved: −For finding Steiner points, it suffices to consider only intersections  At most (n 2 -n) candidates for Steiner points Rectilinear Routing

7. 7 Hanan points ( ) RSMTIntersection lines Terminal pins Hanan Grid

8. 8 Rectilinear Steiner Tree

9. 9 Sequential Steiner Tree Heuristic Rectilinear Steiner Tree

10. 10 A Sequential Steiner Tree Heuristic 1.Find the closest (in terms of rectilinear distance) pin pair, construct their minimum bounding box (MBB) 2.Find the closest point pair (p MBB,p C ) between any point p MBB on the MBB and p C from the set of pins to consider 3.Construct the MBB of p MBB and p C 4.Add the L-shape that p MBB lies on to T (deleting the other L-shape). If p MBB is a pin, then add any L-shape of the MBB to T. 5.Goto step 2 until the set of pins to consider is empty Steiner Tree Algorithm Finds RSMT for p ≤ 4

11. 11

12. 12 1 Sequential Steiner Tree Heuristic

13. 13 1 2 1 Sequential Steiner Tree Heuristic

14. 14 1 2 3 1 Sequential Steiner Tree Heuristic MBB pcpc

15. 15 1 2 3 1 2 3 12 4 Sequential Steiner Tree Heuristic p MBB

16. 16 1 2 3 1 2 3 4 5 1 2 3 4 12 3 Sequential Steiner Tree Heuristic

17. 17 1 2 3 1 2 3 4 5 1 2 3 4 1 2 3 4 56 12 3 4 Sequential Steiner Tree Heuristic

18. 18 1 2 3 4 56 7 1 2 3 1 2 3 4 5 1 2 3 4 1 2 3 4 56 12 3 4 5 Sequential Steiner Tree Heuristic

19. 19 1 2 3 4 56 7 1 2 3 1 2 3 4 5 1 2 3 4 56 7 1 2 3 4 1 2 3 4 56 12 3 4 5 6 Sequential Steiner Tree Heuristic

20. 20 1 2 3 4 56 7 1 2 3 1 2 3 4 5 1 2 3 4 56 7 1 2 3 4 1 2 3 4 56 12 3 4 5 6 Sequential Steiner Tree Heuristic