1. STEINER TREE BASED ALGORITHMS

2. INTRODUCTION Maze routing and line probing algorithms cannot be used for multiterminal nets. There are several approaches to extend these algorithms 1.Decomposition of multi terminal nets to two terminal nets. Two terminal nets routed by usint the above two algorithms. Quality of routing depends on decomposition No interaction between decomposition and routing

3. 2.Exploration can be carried out from several terminals in the net at the same time. No predetermined net decomposition. RECTILINEAR STEINER TREES(RST) Used to route mutiterminal nets. RST is a steiner tree with only rectilinear edges. sum of all edges of the tree length or cost of the tree

4. Finding minimun cost RST is NP hard,several heuristic algorithms have been developrd. Ratio of cost of MST to that of a RST is not greater than 1.5. RECTILINEARIZATION Underlying Grid of a set of points S is the grid obtained by drawing horizontal and vertical lines Through each point of S.

5. Staircase Layout of an edge (i,j) of an MST T is a rectilinear shortest path between the points i and j on the underlying grid. MSP (length=11) Steiner point Steiner tree (len=13

6. Z-Shaped Layout: A staircase layout having at most two turns. L-Shaped Layout: A staircase layout having at most one turn S-RST: an RST that is obtained from MST T by using staircase layouts for edges of the MST. L-RST: an S-RST of T in which the layouts of each MST edge is a L-shaped layout.

8. SEPARABLE MST SMST (Separable MST): a rectilinearMST where arbitrary staircase layouts of any two non- adjacent edges of the MST do not intersect or overlap.

9. SEPARABILITY BASED ALGORITHMS Works in 2 steps: 1)using modified Prims algorithm SMST is constructed. 2)Optimal ZRST is constructed from this SMST. Z-SUFFICIENCY THEOREM Given a separable MST T of a point set S, there exists a Z-RST derivable from T whose cost is equal to the minimum cost S-RST derivable from T. à the min Z-RST is the min S-RST too.

10. ALGORITHM SMST

11. ALGORITHM ZRST Select a leaf edge (r) and hang the tree from it. (àTree: Tr) Let Te be the subtree with root e and Mz[e] be the optimal Z-RST of Te Mz[e] can be found recursively. For each ei (children of e) and every Z-shape layout ( Zij) of ei, calculate Mzij[ei] of Tei. Merge these Mzij[ei]’s with the Z-shape layout of edge e.

12. Since T is separable, the overlaps can occur only among e, e1, …, ed which have common node. total overlaps in Z-RST of Te= ∑ overlaps + ∑ overlaps e, e1.. ed Tei Try all combinations of Mzij[ei]’s for each Tei and find the Z-RST of Te with minimum cost. For Tr, calculate Mz[r] recursively.

13. ALGORITHM ZRST

14. LEAST COST FUNCTION

15. NON RECTILINEAR STEINER TREE BASED ALGORITHMS δ-geometry introduced:to improve layout and enhance performance. Δ is always taken a positive integer(>=2) Here,edges with (iΠ⁄δ) are allowed. δ = 2 :-rectilinear geometry δ = 4:-45 degree geometry δ = ά :-eucleadian geometry

17. STEINER MIN MAX TREE BASED ALGORITHMS Maximum weight edge is removed Weight of any edge is a function of current density,capacity and crowdness of boarder

18.  The nets are arranged according to their priority,length and multiplicity numbers.  Global routing is performed in 2 steps: 1.SMMT phase: Nets are routed one by one,at the jth step if the length of routing of the net is within a constant factor cj,the minimum length is accepted.or else the routinfg is rejected.

19. 2.SP phase The nets are routed one by one using shortest path algorithm and the results of SMMT phase.At the jth step we accept the routing only if it is better than the previous routing.  The number of steps are the parameters that depend on the density and length minimization in the problem

20. WEIGHTED STEINER TEE BASED ALGORITHM Weighted steiner tree:-steiner tree with weigthted lengths.(length=l,weight=w,weighted length=lw) Steps for finding minimum cost WRST Find an MST (T)using Prims algorithm. Rectilinearize each edge of T.

21. e1,e2….en:edges of T There are more than one possible staircase layouts for each edge. (P 1 (e j ),P 2 (e j )….,P k (e j )):subset of all possible staricase layouts of edge ej. Let L (j-1) :-staircase layouts of edges e1,e2.....e(j-1). Merge P i (e j ) and L (j-1) to get Q ij