1. Foundations of Algorithms, Fourth Edition
Richard Neapolitan, Kumarss Naimipour
Chapter 3
Dynamic Programming

2. Figure 3.1: The array B used used to compute the binomial coefficient.

3. Figure 3.2: A weighted, directed graph.

4. Figure 3. 3: W represents the graph in figure 3
Figure 3.3: W represents the graph in figure 3.2 and D contains the lengths of the shortest paths. Our algorithm for the Shortest Paths problem computes the values in D from those in W.

5. Figure 3.4: The shortest path uses vk

6. Figure 3. 5: The array P produced when Algorithm 3
Figure 3.5: The array P produced when Algorithm 3.4 is applied to the graph in Figure 3.2.

7. Figure 3.6: A weighted, directed graph with a cycle.

8. Figure 3.7: The number of columns in Ak-1 is the same as the number of rows in Ak.

9. Figure 3.8: The array M developed in Example 3.5.

10. Figure 3. 9: The array P produced when Algorithm 3
Figure 3.9: The array P produced when Algorithm 3.6 is applied to the dimensions in Example 3.5

11. Figure 3.10: Two binary search trees

12. Figure 3.11: The possible binary search trees when there are three keys.

13. Figure 3.12: The binary search trees composed of Key2 and Key3

14. Figure 3.13: Optimal binary search tree given that Keyk is at the root.

15. Figure 3. 14: The arrays A and R, produced when Algorithm 3
Figure 3.14: The arrays A and R, produced when Algorithm 3.9 is applied to the instance in Example 3.9

16. Figure 3. 15: The tree produced when Algorithms 3. 9 and 3
Figure 3.15: The tree produced when Algorithms 3.9 and 3.10 are applied to the instance in Example 3.9

17. Figure 3.16: The optimal tour is [v1, v3, v4, v2, v1]

18. Figure 3.17: The adjacency matrix representation W of the graph in Figure 3.16

19. Figure 3.18: A section of DNA

20. Figure 3.19: The array used to find the optimal alignment

21. Figure 3.20: The completed array used to find the optimal alignment