Homework

• Chapter 1: Problem 6, 7
• Chapter 2: Problem 6, 7
• Read Chapter 3

Scheduling

• Jobs have a duration and deadlines
• Example: Jobs 1, 2, 3, each 1 hour long, but only have 2 hours total
• Goal: Minimize maximum lateness of any job
• Possible greedy algorithms:
  • Do the shortest job first
    • Counterexample:
    • Job 1: Takes 1 day, deadline 100 days from now
    • Job 2: Takes 10 days, deadline 10 days
    • Doing the shortest job first would make the second job late by one day
      • However, you could do the longest job first and since the one day has 100 days deadline, could easily do that last
  • Define the flexibility of a job as the deadline subtracted from the duration
    • Order by flexibility
    • Counterexample:
    • Job 1: Takes 1 day, deadline 2 days
    • Job 2: Takes 10 days, deadline 10 days
    • We would do job 2 first (10-10 = 0), then job 1 second. That would make job 1 nine days late
      • Doing job 1 first would only make job 2 late by 1 day, so this isn’t optimal
  • Do earliest deadline job first
    • Works

Claim: There exists an optimum solution (minimizes lateness) without any idle time

How do we prove that the greedy solution is as good as the optimum solution?

<aside> <img src="/icons/document_gray.svg" alt="/icons/document_gray.svg" width="40px" /> If the optimum solution has the earliest deadline first, it is the same anyway. However, either 1) some jobs that have the same deadline were swapped, or 2) a job with an earlier deadline was scheduled later. For option 1, if any two jobs that have the same deadline were swapped, let the lateness of job 1 be A and the lateness of job 2 be B. Swapping these two jobs would simply swap the latenesses of both jobs, hence the solution is still optimal. For option 2, if any two jobs that don’t have the same deadline were swapped, we can create a permutation of these jobs based on adjacent inversions. Any adjacent inversion won’t change the lateness of either job. Perform a bubble sort on this permutation. This will arrive at the greedy solution. Hence, the greedy solution is optimal.

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Eviction Strategies (for landlords)

Belady’s optimal eviction strategy: evict page which is requested furthest into the future.

Least recently used page eviction strategy.