1. A General Framework for Track Assignment in Multilayer Channel Routing (Multi layer routing) -VLSI Layout Algorithm KAZY NOOR –E- ALAM SIDDIQUEE 04080541

2. Abstract We need Routing solution Using minimum number of tracks Feasible routing solutions were minimum clique cover (MCC1, MCC2) Channels with cyclic vertical constraints cannot be routed in VH and HVH routing models Now we need heuristic algorithm –(TAH) Track Assignment Heuristic »Nets don’t induce cycles, are assigned to extreme horizontal layer(s), track by track just done in two layer algorithm. »Remaining sets of nets are assigned to other horizontal layers flanked by vertical layers using either MCC1 or MCC2. –Used for two layer routing solution, Three layer HVH CRP etc. Extension of TAH framework (Algorithm) –For two and three layer no-dogleg routing To multilayer V i H i, (2 ≤ i < d max ) routing model and V i H i+1,(2 ≤ i < d max -1 ) routing model

3. –Efficient Polynomial time heuristic algorithm for CRP (Channel Routing Problem) –Runs in O(k(n+e)) time, k= no. of tracks, n= no. of nets, e= size of HNCG –Polynomial time Heuristic algorithm- »produce an acceptable solution to a problem in many practical scenarios »in the fashion of a general heuristic »there is no formal proof of its correctness. CRP »is a problem of computing a feasible route for nets »The number of tracks required (channel area) is minimized –Horizontal non Constraint Graph (HCNG) is used »Complement of HCG, HC = (V,E), by HNC = (V, E) »Horizontal wire segments of a net vertices in HNCG are non overlapping – so, may be safely assigned to same track. TAH (Track Assignment Heuristic)

4. TAH (Cont.) If t is 1 (t = no. of tracks) TAH terminates by assigning nets to horizontal layers following MCC1 or MCC2 If t > 1 The algorithm assigns nets to different tracks in –Stage 1: –Stage 2: –Stage 3:

5. For channel TOP: 3 1 3 0 0 5 6 0 3 0 0 BOTTOM: 1 2 4 2 4 1 5 7 0 7 6 VCG Stage 1 If VCG is acyclic then nets in t(i-1) tracks of first (i-1) layers are assigned using MCC1 or MCC2 1 2 3 4 5 6 7

6. For channel TOP: 3 1 3 2 0 5 6 0 3 0 0 BOTTOM: 1 2 4 3 4 1 5 7 0 7 6 1 2 3 4 5 6 7 VCG Stage 1 (cont.) Let us make the VCG cyclic Then a cycle is detected and a vertex with maximum adjacency number is deleted, here 3 (or 1), and that vertex is added to a set D. This process continues until the graph becomes acyclic.

7. For channel TOP: 3 1 3 2 0 5 6 0 3 0 0 BOTTOM: 1 2 4 3 4 1 5 7 0 7 6 1 2 3 4 5 6 7 VC’ D = { } the nets in set D are assigned to t(i-1) tracks with higher priority according to TAH framework, while the rest nets are assigned normal weights. Stage 1 (cont.) When D is exhausted the rest of the VC’ are assigned according to MCC1 or MCC2.

8. -If all the nets are not assigned in stage 1 TAH framework is used for assignment of nets to the t tracks of the extreme layers. -Nets outside D are assigned to extreme layers in this stage. And if D is not empty that is dealt in next stage. Stage 2 2 4 5 7 VC” 2 4 5 7 HNCG

9. If after stage 1 and 2, tracks are already filled up then this stage occurs. -Then unassigned nets are assigned iteratively track by track. -For example, there are unassigned nets from the set D. These nets are assigned to the flanked layers using TAH frame work as in stage 1. -If already all nets in D are assigned then the only remaining nets (or a subset of nets) are those in VC”; MCC1 or MCC2 are applied for assignment of nets to flanked layers and use the TAH framework for assignment of nets to the extreme layer(s). Stage 3

10. Criticism The novel feature is that –The realization of a class heuristics for track assignment into a graph theoretic optimization problem. Weights to vertices are assigned corresponding to intervals and used the maximum weighted clique algorithm. These weights are tuples that encode the class of heuristics in a logical sequence of priority. It is possible to easily incorporate new heuristics for track assignment by defining appropriate sets of tuple weights. –Backtracking is not used here, rather it computes optimal or near optimal routing solutions for most of the benchmark channels.

11. Thank you