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Graph Coloring

Graph Coloring. Vertex Coloring problem in VLSI routing. Share a track. Standard cells. channels. Share a track. Minimize channel width- assign horizontal Metal wires to tracks.. (min # of tracks = min channel width. Vertex Coloring problem in VLSI routing. Standard cells. 1. 2. 3.

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Graph Coloring

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  1. Graph Coloring

  2. Vertex Coloring problem in VLSI routing Share a track Standard cells channels 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 Standard cells 1 2 3 channels 4 5 Minimize channel width- assign horizontal Metal wires to tracks.. (min # of tracks = min channel width

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

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

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

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

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

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

  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 Consider a,b c e a d b c c e e d b a d b

  14. Consider a,b a c c e e d b d b a a c c e e d b d b a

  15. Consider a,b a c c e e d b d b a be a c a e c e c c e d b a d b SMALLEST Complete graph d be d be a

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

  17. B c R Ra ad G eb Ra Ge Bc Ge d b a c e Rd Gb d b

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