ch23

Advanced Engineering Mathematics

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
CHAPTER 23 Graphs. Combinatorial Optimization SECTION 23.1. Graphs and Digraphs, page 954 Purpose. To explain the concepts of a graph and a digraph (directed graph) and related concepts, as well as their computer representations. Main Content, Important Concepts Graph, vertices, edges Incidence of a vertex v with an edge, degree of v Digraph Adjacency matrix Incidence matrix Vertex incidence list, edge incidence list Comment on Content Graphs and digraphs have become more and more important, due to an increase of supply and demand—a supply of more and more powerful methods of handling graphs and digraphs, and a demand for those methods in more and more problems and fields of application. Our chapter, devoted to the modern central area of combinatorial optimization, will give us a chance to get a feel for the usefulness of graphs and digraphs in general. SOLUTIONS TO PROBLEM SET 23.1, page 958 6. The adjacency matrix is WX . Adding the edge between 3 and 4, we would have a complete graph. The only zeros of the matrix outside the main diagonal correspond to that edge. 8. ST 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 J 3 J 2 J 1 W 3 W 2 W 1 J 4 2. 362 im23.qxd 9/21/05 2:01 PM Page 362
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
10. ST 12. UV 20. If and only if G is complete. In this case the adjacency matrix of G has n 2 2 n 5 n ( n 2 1) ones, and since every edge contributes two ones, the number of edges is n ( n 2 1)/2. This gives another proof of Prob. 19. 22. The matrix is Edge e 1 e 2 e 3 e 4 e 5 e 6 e 7 1 1010000 2 1101000 3 0100100 4 0011111 5 0000010 6 0000001 1 4 2 3 18. 1 4 2 3 16. 1 2 3 4 14. 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 Instructor’s Manual 363 S T Vertex im23.qxd 9/21/05 2:01 PM Page 363
Background image of page 2
24. The matrix is Edge e 1 e 2 e 3 e 4 1 2 1 2 11 2 1 2 1000 30 1 2 10 4 0001 SECTION 23.2. Shortest Path Problems. Complexity, page 959 Purpose. To explain a method (by Moore) of determining a shortest path from a given vertex s to a given vertex t in a graph, all of whose edges have length 1. Main Content, Important Concepts Moore’s algorithm (Table 23.1) BFS (Breadth First Search), DFS (Depth First Search) Complexity of an algorithm Efficient, polynomially bounded Comment on Content The basic idea of Moore’s algorithm is quite simple. A few related ideas and problems are illustrated in the problem set. SOLUTIONS TO PROBLEM SET 23.2, page 962 2. There are three shortest paths, of length 4 each: Which one we obtain in backtracking depends on the numbering (not labeling!) of the vertices and on the backtracking rule. For the rule in Example 1 and the numbering shown in the following figure we get (B). If we change the rule and let the computer look for largest (instead of smallest) numbers, we get (A). 4 8 4 3 1 0 s t 1 3 3 4 2 3 3 2 6 2 1 5 10 11 3 7 9 12 t s t t s s ( a )( b c ) 364 Instructor’s Manual Vertex W X im23.qxd 9/21/05 2:01 PM Page 364
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4. The length of a shortest path is 5. No uniqueness.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 09/28/2008.

Page1 / 12

ch23 - im23.qxd 9/21/05 2:01 PM Page 362 CHAPTER 23 Graphs....

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online