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