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Several Graph Layout
Problems for Grids
Vladimir Lipets
Ben-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 Bounds
for square toroidal grids
Lower Bounds
for square toroidal grids
Lower Bounds
for square toroidal grids
Constructions (upper bounds)
for square toroidal grids
Constructions (upper bounds)
for square toroidal grids
Lower Bounds
for rectangular toroidal grids
Constructions (upper bounds)
for rectangular toroidal grids
The End