2. Several Graph Layout Problems for Grids Vladimir Lipets Ben-Gurion University of the Negev Advisors: Prof. Daniel Berend Prof. Ephraim Korach

3. Graph layout problems A large number of theoretical and practical problems in various areas may be formulated as graph layout problems.

4. Graph layout problems

5. Graph layout problems (MINCUT)

6. Graph layout problems (Bisection)

7. Graph layout problems (Bandwidth)

8. Applications Such problems arise in connection with: VLSI circuit design, graph drawing, embedding problems, numerical analysis, optimization of networks for parallel computer architectures.

9. History Historically, bandwidth was the the first layout problem, as a means to speed up several computations on sparse matrices during the fifties. The bandwidth problem for graphs was first posed as an open problem during a graph theoretical meeting in 1967 by Harary. For more detailed survey of graph layout problems see. The Minimal Cutwidth Linear Arrangement problem (MINCUT) was first used in the seventies as a theoretical model for the number of channels in an optimal layout of a circuit [Adolphson and Hu 1973]

10. Known Results All above problems are NP-hard in general, MINCUT remains NP-hard even when restricted, for example, to polynomially (edge-) weighted trees planar graphs with maximum degree 3. MINCUT remains NP-hard even when restricted, for example, to trees, with maximum degree 3

11. Known Results (table)

14. Grids

15. Grids (example)

16. Toroidal Grids (example)

17. Location Matrix

19. Double Monotonic

21. Main Results

22. Main Lemma (non toroidal grids)

23. Our Approach

24. Our Approach (cont.)

25. Prof of lower bound

27. Constructions (upper bounds)

28. Main Lemma (toroidal grids)

29. Lower Bounds for square toroidal grids

32. Constructions (upper bounds) for square toroidal grids

34. Lower Bounds for rectangular toroidal grids

35. Constructions (upper bounds) for rectangular toroidal grids

36. The End