1. Graph Coloring

2. Vertex Coloring problem in VLSI routing channels Standard cells Share a track Minimize channel width- assign horizontal Metal wires to tracks.. (min # of tracks = min channel width

3. Vertex Coloring problem in VLSI routing channels Standard cells 3 4 12 5 Minimize channel width- assign horizontal Metal wires to tracks.. (min # of tracks = min channel width

4. Vertex=wire edge=overlapped wires color = track 1 2 3 4 5 R B R G B CHORDAL GRAPH

5. Vertex Coloring problem in register allocation channels Share a register Minimize # of registers: assign variable (lifetimes) to registers (min # of registers ) TIME increasing

6. Clique paritioning: edges are connected if there is no conflict (no overlapping wires, no overlapping lifetimes) 1 2 3 4 5 R B R G B COMPLEMENT OF CHORDAL GRAPH IS COMPARABILITY GRAPH

7. Clique paritioning: example 1 2 3 4 5 R B R G B

8. Garey & Johnson Text Instance: graph G=(V,E), positive integer K<=|V|. Question: is G K-colorable ? Solvable in polynomial time for K=2, NP- complete for K>=3. General problem solvable in polynomial time for comparability graphs, chordal graphs, and others.

9. Also same for clique partitioning Graph G=(V,E), K<=|V| Question : can vertices of G be partitioned into k<=K disjoint sets V1, V2,…Vk such that for 1<=i<=k the subgraph induced by Vi is a complete graph?

10. abab cdcd a b d c ? In our application, our graphs are always chordal ! (in channel routing problem)

11. Register allocation in loops coloring of a circular arc graph which is NP-complete a b d c ? LOOP- variable c is defined in loop iteration i and used in the next loop iteration i+1 Time increasing c ? LOOP

12. Channel routing is still a hard problem due to the vertical constraints Which we cannot accommodate in our graph theory formulation (which Only looks at horizontal constraints i.e. horizontal intervals) top down view

13. Berge’s Algorithm(contract- connect) for Vertex Coloring a b c e d Consider a,b a b c e d a b c e d

15. ab c e d b c e d a b c e d a b c e d a

17. b c e d a b c e d a b c e d a be c e d a b c e d a c d a SMALLEST Complete graph

18. Consider a,b b c e d a be c e d a

19. adG eb BcBc R b Ge d a Ra Gb Bc Ge Rd b c e d Ra