1. Greedy Algorithms

2. The two key components Optimal Sub-structure Greedy Property
You solve the problem by solving a sub-problem optimally
Greedy Property
Using the choice that seems best at the moment leads to the optimal result
This is tougher to show!
Greedy problems are the “easy” ones for finding optimal results
They are less common than the “tough” ones, unfortunately…

3. Classical problem: Making change
Given Coins: 1 cent, 5, 10, 25, 50, and 100 cents, how can you get a certain total in the fewest coins
Straightforward: choose most of the largest coin, then next largest, etc.
For these denominations, this is optimal, but this is NOT true for general coin values.
Example: 1, 10, 15, 20 cent denominations
25 cent goal
Greedy strategy would give
Optimal result would be 10+15
The point: you need to ensure that you have a situation where greedy applies!

4. Classical problem: Scheduling jobs
Every job has a start and a finish time
Can work on only one job at a time
Want to schedule as many jobs as possible

5. Possible greedy choices
Take jobs in order of earliest start time
Take jobs in order of earliest finish time
Take jobs in order of length of job (shortest job first)
Take jobs in order of fewest conflicts with other jobs

6. Earliest Start
Shortest Job
Fewest Conflicts

7. Possible greedy choices
Take jobs in order of earliest start time
Take jobs in order of earliest finish time
Take jobs in order of length of job (shortest job first)
Take jobs in order of fewest conflicts with other jobs
Only the earliest finish time leads to an optimal solution!
If there were any other optimal solution not including that, you could replace one of those choices with the one that ends earlier, and still have an optimal solution.
Need to think about what would happen if there WERE a better solution from the non-greedy choice – show there’s a conflict
Can extend this to figuring out how many rooms are needed to schedule events.

8. Example (last programming lab): Dragon of Loowater
N heads of dragons, M knights
Each dragon head and each knight has a value, only knights with greater value than the head can defeat it
Knights can defeat one head only
Match knights with dragon heads to minimize total knight height
Greedy solution:
Choose the shortest knight needed
Sort the heads and the knights
For smallest head, pick smallest knight possible, then repeat until done

9. More Greedy solutions
Will see several examples later on – e.g. in some graph applications
Balancing Stations (see book)
Grouping most w/least
Watering Grass (see book)
Minimizing Lateness
Optimal Caching