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 or that the greedy choice is just as good

8. Room Scheduling
Assume classes with start and end times (like tasks, above)
How many rooms are needed to teach all of them?
Same as the depth – the number of tasks overlapping at once
Can use a greedy approach:
Just assign to first available room.
Sort start and end times together. Then linearly process
Each time a start is passed, increase by 1, each time an end is passed, decrease by 1
Keep track of how many rooms max were needed
This is also the basis for the sweep-and-prune approach for collision detection in graphics!

9. Minimizing maximum lateness
Tasks of particular length, and deadline time. Want to minimize the maximum lateness
Can always do task with earliest deadline first
Ensures that there is no “idle” time – the best solution must also have no idle time.
Reversing any two from that order will lead to a worse result
So, that order can’t change.

10. More Greedy solutions
Will see several examples later on – e.g. in some graph applications
Balancing Stations (see book)
Grouping most w/least
See book for some more specific examples