Lecture 13: Shortest Path and Graph Coloring

1. Lecture 13: Shortest Path and Graph Coloring • Shortest Path Problems • Graph Coloring MACM101 Discrete Mathematics I

2. 13.1. Shortest Path Problems • A weighted graph: • Is a graph in which a number called the weight is assigned to each edge. • The weight of a path: • Is the sum of weights of the edges in the path. • The shortest path from u to v: • Is the path of the smallest weight between the two vertices. • The weight of that path is called the distance between them. MACM101 Discrete Mathematics I

3. 13.1. Shortest Path Problems • Examples: • Weight of the edge on A and B is 3. • Weight of the path A, C, D, F is 4 + 2 + 5 = 11. • The shortest path between A and D is A, B, D. • The distance between A and D is 4. MACM101 Discrete Mathematics I

4. 13.1. Shortest Path Problems • Examples: • The weight can be mileage, fares, response time, etc. MACM101 Discrete Mathematics I

5. 13.1. Shortest Path Problems • Rationale • Given shortest path Pa-b from vertex a to b, and Pa-b passes vertex c • Path Pa-c (a subpath of Pa-b) is a shortest path from vertex a to c • Length of Pa-c <= Length of Pa-b MACM101 Discrete Mathematics I

6. 13.1. Shortest Path Problems • Dijkstra’s Algorithm: Dijkstra (1930-2002) • Is a popular algorithm for finding shortest path. • Applications: Internet routing. MACM101 Discrete Mathematics I

7. 13.1. Shortest Path Problems • Examples: • Use Dijkstra’s algorithm to find the length of the shortest path between the vertices a and z in the following weighted graph. • Solution: • At each iteration of the algorithm, the vertices of the set S is circled. The algorithm terminates when z is circled. MACM101 Discrete Mathematics I

8. 13 (a, c, b, d, e) 13.1. Shortest Path Problems MACM101 Discrete Mathematics I

9. 13.1. Shortest Path Problems • Question: • What about longest path ? MACM101 Discrete Mathematics I

10. 13.1. Shortest Path Problems • Question: • What about negative weight ? MACM101 Discrete Mathematics I

11. 13.1. Shortest Path Problems • Question: • In the algorithm, weight is additive, i.e., min w1+w2+w3… • How about multiplicative, i.e., min w1*w2*w3… MACM101 Discrete Mathematics I

12. 13.1. Shortest Path Problems • Question: • In the algorithm, weight is additive, i.e., min w1+w2+w3… • How about multiplicative, i.e., min w1*w2*w3… • Solution: • Min w1*w2*w3… is equivalent to min (log w1*w2*w3…) MACM101 Discrete Mathematics I

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15. (E) (D) (A) (C) (B) 13.1. Shortest Path Problems • Important shortest path problem: • The traveling salesman problem. • A traveling salesman wants to visit each of n cities exactly once and return to his starting point. • In which order should he visit these cities to travel the minimum total distance? MACM101 Discrete Mathematics I

16. 13.1. Shortest Path Problems • Solution: • Find all possible Hamilton circuits. • Assume that the salesman starts in Detroit. • Then we need to examine (n-1)!/2 different circuits. • Calculate the length of each circuit. • Select one with minimum total length. • A-B-C-D-E-A (or its reverse): 458 km. MACM101 Discrete Mathematics I

17. 13.2. Graph Coloring • A coloring of a graph: • Is the assignment of a color to each vertex of the graph so that adjacent vertices have different colors. • For most graphs, we can use fewer colors than the number of vertices in the graph • The chromatic number of a graph: • Is the least number of colors needed for coloring the graph. MACM101 Discrete Mathematics I

18. 13.2. Graph Coloring • Examples: • When a cycle Cn has an even number of vertices, it can be colored using 2 colors. • When a cycle Cn has an odd number of vertices, it can be colored using 3 colors. MACM101 Discrete Mathematics I

19. 13.2. Graph Coloring • Examples: • Coloring for the complete graph Kn – • How many ? MACM101 Discrete Mathematics I

20. 13.2. Graph Coloring • Examples: • The complete graph Kn can be colored with n colors. • No smaller number of colors is possible since every two vertices of this graph are adjacent. MACM101 Discrete Mathematics I

21. 13.2. Graph Coloring • Examples: • The complete bipartite graph Km,n can be colored with 2 colors. • i.e. color the set of m vertices with one color and the set of n vertices with a second color. • e.g. A coloring of K3,4. MACM101 Discrete Mathematics I

22. 13.2. Graph Coloring • The 4-color problem Chromatic number of a map (planar graph) ≤ 4 • 1850 – asked by a De Morgan’s student • 1976 – proved by K. Appel and W. Haken using a computer MACM101 Discrete Mathematics I

23. 13.2. Graph Coloring • Applications of graph coloring: • Scheduling final exams. • How can the final exams can be scheduled so that no student has two exams at the same time? • Solution: • Use a graph model with vertices representing courses and with an edge between two vertices if there is a student taking both courses. • Each time slot of a final exam is represented by a different color. MACM101 Discrete Mathematics I

24. 13.2. Graph Coloring • Solution (continued): • Suppose there are 7 finals with common students in courses {(1,2), (1,3), … } as represented by the graph. So, we need four color or time slots. MACM101 Discrete Mathematics I

25. 13.3. Further Readings • Shortest Path Problems : Section 8.6. • Graph Coloring : Section 8.8. MACM101 Discrete Mathematics I