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