Lecture 6 Greedy Algorithms

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.

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)

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

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

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.

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.

Example Capacity W = 10, 3 items with (weight, value) = (6, 20), (5, 15), (4, 10) Solution: Item 1 + 0.8 Item 2. Weight = 10, Value = 32

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))