Non-Hierarchical Sequencing Graphs

1. Non-Hierarchical Sequencing Graphs

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3. example Algorithmic Graph Theory

4. Example Algorithmic Graph Theory

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32. Algorithmic Graph Theory and its Applications Martin Charles Golumbic Algorithmic Graph Theory

33. Algorithmic Graph Theory and its Applications Martin Charles Golumbic Algorithmic Graph Theory

34. Algorithmic Graph Theory and its Applications Martin Charles Golumbic Algorithmic Graph Theory

35. Introduction • Intersection Graphs • Interval Graphs • Greedy Coloring • The Berge Mystery Story • Other Structure Families of Graphs • Graph Sandwich Problems • Probe Graphs and Tolerance Graphs Algorithmic Graph Theory

36. Theconcept of an intersection graph • applications in computation • operations research • molecular biology • scheduling • designing circuits • rich mathematical problems Algorithmic Graph Theory

37. Defining some terms • graph: a collection of vertices and edges • coloring a graph: assigning a color to every vertex, such that adjacent vertices have different colors Algorithmic Graph Theory

38. independent set: a collection of vertices NO two of which are connected Example: { d, e, f } or the green set • clique (or complete set): EVERY two of which are connected Example: { a, b, d } or { c, e } Algorithmic Graph Theory

39. complement of a graph: interchanging the edges and the non-edges __ The complement G The original graph G Algorithmic Graph Theory

40. directed graph: edges have directions (possibly both directions) • orientation: exactly ONE direction per edge cyclic orientation acyclic orientation Algorithmic Graph Theory

41. Phase 1 Phase 2 Phase 3 Jan Feb Mar Apr May Jun July Sep Oct Nov Dec Interval Graphs The intersection graphs of intervals on a line: - create a vertex for each interval - connect vertices when their intervals intersect Task 5 Task 4 1 2 3 The interval graph G 4 5

42. History of Interval Graphs • Hajos 1957: Combinatorics (scheduling) • Benzer 1959: Biology (genetics) • Gilmore & Hoffman 1964: Characterization • Booth & Lueker 1976: First linear time recognition algorithm • Many other applications: mobile radio frequency assignment VLSI design temporal reasoning in AI computer storage allocation Algorithmic Graph Theory

43. Scheduling Example • Lectures need to be assigned classrooms at the University. • Lecture #a: 9:00-10:15 • Lecture #b: 10:00-12:00 • etc. • Conflicting lectures  Different rooms • How many rooms?

44. Scheduling Example (cont.)

45. Scheduling Example (graphs) • The interval graph • Its complement (disjointness)

46. Coloring Interval Graphs • interval graphs have special properties • used to color them efficiently • coloring algorithm sweeps across from left to right assigning colors • in a ``greedy manner” • This is optimal ! Algorithmic Graph Theory

47. Coloring Interval Graphs Algorithmic Graph Theory

48. Coloring Intervals (greedy) Algorithmic Graph Theory

49. Is greedy the best we can do? • Can we prove optimality? • Yes: It uses the smallest # colors. Proof: Let k be the number of colors used. Look at the point P, when color k was used first. At P all the colors 1 to k-1 were busy! We are forced to use k colors at P. And, they form a clique of size k in the interval graph. Algorithmic Graph Theory

50. Coloring Intervals (greedy) P(needs 4 colors) Algorithmic Graph Theory