1. CS 312: Algorithm Design & Analysis Lecture #24: Optimality, Gene Sequence Alignment This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick

2. Announcements  Homework #15 due now  Project #5: Gene Sequence Alignment  Kick-off: today  Read directions now  Whiteboard experience: due Monday  Early: Monday after mid-term exam  Due: Wednesday after mid-term exam  Mid-term Exam  Start preparing your one page of notes  Must be prepared by you. No cutting and pasting.

3. Objectives  Revisit the main ideas behind Dynamic Programming  Define the optimality property for DP  Develop the algorithm for gene sequence alignment (or at least begin)  Prepare for Project #5

4. Dynamic Programming The six steps: 1.Ask: am I solving an optimization problem? 2.Devise a minimal description (address) for any problem instance and sub-problem 3.Divide problems into sub-problems: define the recurrence to specify the relationship of problems to sub-problems 4.Check that the optimality property holds: An optimal solution to a problem is built from optimal solutions to sub- problems. 5.Store results – typically in a table – and re-use the solutions to sub-problems in the table as you build up to the overall solution. 6.Back-trace / analyze the table to extract the composition of the final solution.

5. Optimality Property An optimal solution to a problem is built from optimal solutions to sub-problems.  The optimality property is a necessary condition for solving an optimization problem by DP!  It allows us to store and re-use optimal results to sub-problems.

6. Optimality A BC EFG HI D JK

7. Shortest Path American Fork Orem Provo Sundance Geneva 20 10 12 3 15 18 10 12 Goal: the shortest path from AF to Provo. Does this problem exhibit the optimality property? Pair up. Discuss

8. Shortest Path American Fork Orem Provo Sundance Geneva 20 10 12 3 15 18 10 12 Goal: the shortest path from AF to Provo. Does this problem exhibit the optimality property? Pair up. Discuss

9. Questions  Q. In general, do you know which sub-problem solutions to use in advance?  A. No. So a very greedy algorithm is not an option. (But Dijkstra’s is.)  Q: How does having a table of intermediate shortest path results help find the shortest path from AF to Provo?  A: Reuse those results for intermediate destinations as you try different routes.  Q. Do you have to reconsider alternative sub-optimal solutions for the intermediate destinations?  A. No  Thus,, the Optimality Property holds  Therefore, the shortest path problem can be solved by DP. American Fork Orem Provo Sundance Geneva 20 10 12 3 15 18 10 12

10. Optimality in Driving The shortest route from American Fork to Provo passes through Orem. Assume we have found this route. Then what can we say about the shortest route from AF to Orem? It follows that optimal route from AF to Provo. Could it be otherwise?

11. A related problem Now suppose you drive from AF to Orem as fast as you can on your way to Provo, But you are limited by the gas in your tank.

12. Does the Optimality Property Hold? AFOremProvo Goal: get to Provo in as little time as possible. No refueling. Does this problem (formulation) satisfy the optimality property or not? Why? 5/9 10/5 20/1 5/9 10/5 20/1 “takes 20 minutes using 1 gallon of gas” Start with 10 gallons

13. Problem Solving Advice  Start by asking: which sub-problems should be solved?  If you know how to choose in advance using local information only, then  greedy might work.  Else if sub-problems don’t overlap, then  divide and conquer would be a good choice.  Else if the optimality property holds, then  DP is a good choice.  Else the optimality property does NOT hold, so  apply another strategy. (Stay tuned for more guidance) Important!

14. x=ACGCTGA y=ACTGT Gene Sequence Alignment

15. Virtually Identical Problems  Edit Distance  aka Levenshtein Distance  Sequence Alignment  E.g., Gene Sequence Alignment  Fundamentally the same thing!  We’re focusing on gene sequence alignment.

16. Edit Distance / Sequence Alignment Problem Contrast the 2 perspectives.

17. Edit Distance / Sequence Alignment Problem x: ACGCT-C y: A--CTGT Alignment Example: The ‘-’ is a “gap”

18. Edit Distance / Sequence Alignment Problem x: ACGCT-C y: A--CTGT Divide into Pairs

19. Edit Distance / Sequence Alignment Problem x: ACGCT-C y: A--CTGT Each Pair has a type and a cost

20. x: ACGCT-C y: A--CTGT Edit Distance / Sequence Alignment Problem

21. x: ACGCT-C y: A--CTGT Edit Distance / Sequence Alignment Problem

22. x: ACGCT-C y: A--CTGT

23. Edit Distance / Sequence Alignment Problem x: ACGCT-C y: A--CTGT

24. Edit Distance / Sequence Alignment Problem x: ACGCT-C y: A--CTGT How would you solve this problem?

25. Solution Ideas  Enumerate all and score  Pro: Easy to code  Pro: Optimal  Con: exponential  Greedy: work from left to right, gobbling up matches and inserting gaps or allowing substitutions as necessary  Pro: Easy  Pro: Linear = fast / efficient  Con: not optimal  DP  Pre-req: optimality property  Pre-req: define addressable sub-problems  Pre-req: determine relationship between problem and sub-problems  Pro: Optimal  Con: ?  Divide and Conquer?

26. Designing the DP Algorithm for Gene Sequence Alignment

27. DP?

28. Example: Sub-problems x=ACGCTGA y=ACTGT

29. Example: Sub-problems x=ACGCTGA y=ACTGT

30. Example: Sub-problems x=ACGCTGA y=ACTGT

31.  To be continued in Lecture #25

32. Assignment  HW #16  Read Section 6.3, if you haven’t done so already.  Thursday: Screencast & Quiz