CSE 101. Algorithm Design and Analysis Miles Jones Office 4208 CSE Building Lecture 9: Greedy

Transcription

1 CSE 101 Algorithm Design and Analysis Miles Jones Office 4208 CSE Building Lecture 9: Greedy

2 GENERAL PROBLEM SOLVING In general, when you try to solve a problem, you are trying to find a solution from among a large space of possibilities. You usually do this by making a series of decisions and continuing based on what the new state of the problem has become.

3 GENERAL PROBLEM SOLVING In general, when you try to solve a problem, you are trying to find a solution from among a large space of possibilities. You usually do this by making a series of decisions and continuing based on what the new state of the problem has become. If you have no information about which choice is best, you may have to use exhaustive enumeration or Brute Force to try out all possibilities and find the optimal one. This might take a long time to do. What are some other ideas in general?

4 THE GREEDY METHOD In some cases (not all!!!!!), there is sufficient structure that allows you to reach the correct solution by just picking the straightforward best decision at each stage. This is called the Greedy Method.

5 THE GREEDY METHOD In some cases (not all!!!!!), there is sufficient structure that allows you to reach the correct solution by just picking the straightforward best decision at each stage. This is called the Greedy Method. It doesn t always work. Just as in life, acting in one s immediate best interest is not always the best longer-term strategy.

6 COOKIES Suppose you are the cookie monster and you have a 6x6 sheet of freshly baked cookies in front of you. The cookies are all chocolate chip cookies but they may have different sizes If you are only allowed to take six cookies. Devise an algorithm to do this.

7 COOKIES What is an algorithm you could use to select the best option? (The best option means that the sum of all the cookie s sizes is the highest possible.)

8 COOKIES What is an algorithm you could use to select the best option? (The best option means that the sum of all the cookie s sizes is the highest possible.) =555

9 COOKIES What is an algorithm you could use to select the best option if you can only select one cookie from each row?

10 COOKIES What is an algorithm you could use to select the best option if you can only select one cookie from each row? =

11 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column?

12 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column?

13 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column?

14 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column?

15 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column?

16 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column? =404

17 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column? = =414

18 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column? = = =458!!!!!!!!

19 COOKIES What is an algorithm you could use to select the best option if you can t select 2 cookies from the same row or column? = = =458!!!!!!!!

20 THE GREEDY METHOD As you have seen, the Greedy Method does not always work. Because of this, in order to use the Greedy Method, we must prove the correctness of the algorithm. Or else, we must present a counterexample to show that a particular greedy method will not work.

21 THE GREEDY METHOD (WARNING!!) As you have seen, the Greedy Method does not always work. Because of this, in order to use the Greedy Method, we must prove the correctness of the algorithm. Furthermore, for a single problem, there may be more than one potential greedy strategy i.e. more than one way to choose the best possible choice at each step. The problem may be solved in one way but not the other.

22 GREEDY DIJKSTRA S We have already seen a greedy algorithm. Instance: a graph with positive edge weights and a vertex s Solution form: dist(v) is the length of a path from s to v Constraints: No constraints that are not obvious Objective: dist(v) is set to the minimum distance from s to v

23 GREEDY DIJKSTRA S What is the greedy property? Pick the next vertex u such that the distance from s to u is smallest (and u has not been picked yet.)

24 EVENT SCHEDULING Suppose you are running a conference and you have a collection of events (or talks) that each have a start time and an end time. Oh no!!! You only have one conference room!!!! Your goal is to schedule the most events possible that day such that no two events overlap.

25 EVENT SCHEDULING Your goal is to schedule the most events possible that day such that no two events overlap. Brute Force: Say that T is the set of events and T =n. Let s check all possibilities. How would we do that?

26 EVENT SCHEDULING Your goal is to schedule the most events possible that day such that no two events overlap. Brute Force: Say that T is the set of events and T =n. Let s check all possibilities. How would we do that? Go through all subsets of T. Check if it is a valid schedule, i.e., no conflicts, and count the number of events. Take the maximum out of all valid schedules. (How many subsets of T are there?)

27 EVENT SCHEDULING

28 EVENT SCHEDULING Your goal is to schedule the most events possible that day such that no two events overlap. Exponential is too slow. Let s try some greedy strategies:

29 EVENT SCHEDULING Your goal is to schedule the most events possible that day such that no two events overlap. Exponential is too slow. Let s try some greedy strategies: Shortest duration Earliest start time Fewest conflicts Earliest end time

30 SHORTEST DURATION

31 EARLIEST START TIME

32 FEWEST CONFLICTS

33 COUNTEREXAMPLE FOR FEWEST CONFLICTS

34 EARLIEST FINISH TIME

35 EVENT SCHEDULING Your goal is to schedule the most events possible that day such that no two events overlap. Exponential is too slow. Let s try some greedy strategies: Shortest duration Earliest start time Fewest conflicts Earliest end time (We can t find a counterexample!!) Let s try to prove it works!!!!!

36 EVENT SCHEDULING

37 MODIFY-THE-SOLUTION There are a few proof techniques to prove greedy strategies work. We will use modify-the-solution to prove the correctness of this greedy strategy. General structure of modify-the-solution: 1. Let d be a decision point, and let g be the greedy choice at d. 2. Let S be a solution achieved by not choosing g. 3. Show how to transform S into some solution S that chooses g, and that is at least as good as S Conclude (by induction) that any S with a series of non-greedy decisions at d 1,, d n can be transformed into an equal-or-better greedy solution, and that therefore the greedy algorithm is optimal.