1. Lecture 6 Greedy Algorithms

2. Basic Algorithm Design Techniques
Divide and conquer
Dynamic Programming
Greedy
Common Theme: To solve a large, complicated problem, break it into many smaller sub-problems.

3. Greedy Algorithm
If a problem requires to make a sequence of decisions, for the first decision, make the “best” choice given the current situation.
(This automatically reduces the problem to a smaller sub-problem which requires making one fewer decisions)

4. Warm-up: Walking in Manhattan
“Walk in a direction that reduces distance to the destination.”

5. Greedy does not always work
Driving in New York: one-way streets, traffic…

6. Design and Analysis Designing a Greedy Algorithm:
1. Break the problem into a sequence of decisions.
2. Identify a rule for the “best” option.
Analyzing a Greedy Algorithm:
Important! Often fails if you cannot find a proof.
Technique: Proof by contradiction. Assume there is a better solution, show that it is actually not better than what the algorithm did.

7. Fractional Knapsack Problem
There is a knapsack that can hold items of total weight at most W. There are now n items with weights w1,w2,…, wn. Each item also has a value v1,v2,…,vn.
The items are infinitely divisible: can put ½ (or any fraction) of an item into the knapsack.
Goal: Select fractions p1,p2,…,pn such that
Capacity constraint: p1w1+p2w2+…+pnwn <= W
Maximum Value: p1v1+p2v2+…+pnvn maximized.

8. Example
Capacity W = 10,
3 items with (weight, value) = (6, 20), (5, 15), (4, 10)
Solution: Item Item 2.
Weight = 10, Value = 32

9. Interval Scheduling
There are n meeting requests, meeting i takes time (si, ti)
Cannot schedule two meeting together if their intervals overlap.
Goal: Schedule as many meetings as possible.
Example: Meetings (1,3), (2, 4), (4, 5), (4, 6), (6, 8)
Solution: 3 meetings ((1, 3), (4, 5), (6, 8))