Graph_theory_Part2_students_2Pp

# Graph_theory_Part2_students_2Pp - Graphs and Related Theory...

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1 2/2/2006; rev. 9/24/06 M. A. Breuer and others 1 Graphs and Related Theory and Algorithms with applications to VLSI layout Part 2 of 3 Melvin A. Breuer EE581 Fall 2006 University of Southern California fill-in answer animated only 2/2/2006; rev. 9/24/06 M. A. Breuer and others 2 Graphs related to VLSI layout Graph type Relation to layouts Interval Segments of a line Permutation Straight line segments with end points on two parallel lines Co-comparability Curves between two parallel lines lying entirely in between these lines Chordal Subtrees in trees

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2 2/2/2006; rev. 9/24/06 M. A. Breuer and others 3 Review of complexity terminology Two classes of problems P - problems solvable by a known p olynomial time algorithm Example: Sorting - n log n. NP - such problems appear to take exponential time to solve NP-hard problems are decision problems, e.g., is graph G k-colorable? NP-complete problems are optimization problems, e.g., find the minimum k so that G is k-colorable 2/2/2006; rev. 9/24/06 M. A. Breuer and others 4 Layouts, routing and graphs Many layout problems can be modeled using graphs While most graph optimization problems are NP- complete, graphs associated with layouts and routing often have special properties, such as being perfect graphs, and as such often have polynomial time algorithms We next look at some problems and graphs and their associated special properties and algorithmic complexity Actual graph algorithms can be found elsewhere
3 2/2/2006; rev. 9/24/06 M. A. Breuer and others 5 Intersection graphs Two-terminal nets are often modeled as wires, lines or segments . Multi-terminal nets can be represented by multiple wires or polygons. Hence a layout can be modeled by lines and polygons. Most graphs related to VLSI routing and layout deal with the intersection of polygons and line segments. An intersection graph G is defined by the intersection of objects, such as lines and polygons, where each object is represented by a vertex, and an edge is defined between two objects iff the two objects intersect (overlap). 2/2/2006; rev. 9/24/06 M. A. Breuer and others 6 Example intersection graph D C B A A C B D

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4 2/2/2006; rev. 9/24/06 M. A. Breuer and others 7 Embedding An embedding of G (a drawing) divides the plane into a finite number of regions . The edges which bound a region define a (interior) face . An _____________ region is called the external or outside face . An interior
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Graph_theory_Part2_students_2Pp - Graphs and Related Theory...

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