Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Lateral computing wikipedia , lookup
Theoretical computer science wikipedia , lookup
Clique problem wikipedia , lookup
Multiple-criteria decision analysis wikipedia , lookup
Computational complexity theory wikipedia , lookup
Inverse problem wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Mathematical optimization wikipedia , lookup
Travelling salesman problem wikipedia , lookup
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