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Several Graph Layout Problems for Grids
Vladimir LipetsBen-Gurion University of the Negev
Advisors:Prof. Daniel Berend
Prof. Ephraim Korach
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Graph layout problems
A large number of theoretical and practical problems in various areas may be formulated as graph layout problems.
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Graph layout problems
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Graph layout problems (MINCUT)
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Graph layout problems (Bisection)
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Graph layout problems (Bandwidth)
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Applications
Such problems arise in connection with:
• VLSI circuit design,
• graph drawing,
• embedding problems,
• numerical analysis,
• optimization of networks for parallel computer architectures.
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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]
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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
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Known Results (table)
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Known Results (table)
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Known Results (table)
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Grids
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Grids (example)
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Toroidal Grids (example)
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Location Matrix
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Location Matrix
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Double Monotonic
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Double Monotonic
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Main Results
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Main Lemma (non toroidal grids)
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Our Approach
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Our Approach (cont.)
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Prof of lower bound
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Prof of lower bound
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Constructions (upper bounds)
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Main Lemma (toroidal grids)
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Lower Boundsfor square toroidal grids
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Lower Boundsfor square toroidal grids
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Lower Boundsfor square toroidal grids
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Constructions (upper bounds) for square toroidal grids
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Constructions (upper bounds) for square toroidal grids
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Lower Boundsfor rectangular toroidal grids
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Constructions (upper bounds) for rectangular toroidal grids
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The End