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Online Job Scheduling

Online Job Scheduling. Marek Chrobak University of California, Riverside. Outline: Online optimization problems Competitive analysis Example: caching 2. List scheduling for minimum makesnan Graham’s algorithm is 2-competitive 3. List scheduling for related machines

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Online Job Scheduling

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  1. Online Job Scheduling Marek Chrobak University of California, Riverside 1

  2. Outline: • Online optimization problems • Competitive analysis • Example: caching 2. List scheduling for minimum makesnan • Graham’s algorithm is 2-competitive 3. List scheduling for related machines 4. Minimizing total completion time • Optimal algorithm for the preemptive case • 2-competitive algorithm for the non-preemptive case 5. Scheduling of unit jobs for maximum throughput • 2-competitive greedy algorithm • 1.618-competitive algorithm for the 2-bounded case • Randomized 1.58-competitive algorithm 6. Scheduling equal-length jobs for maximum throughput 2

  3. RAM disk k CPU • Paging: • 2-Level memory system with large disk and k pages in RAM • Sequence of requests to pages • If the requested page is not in RAM (miss), it must be fetched from disk (another page may need to be evicted) • Objective: minimize the number of misses 3

  4. 4 4 4 5 5 5 3 3 3 3 3 3 2 2 2 2 2 1 4 6 6 7 7 7 7 4 1 3 9 9 9 6 6 3 5 2 6 6 1 1 • Paging example: • k= 5 • Initial RAM state: 4 3 5 6 1 6 1 2 3 9 7 5 Requests: • Questions: • Can we reduce the number of misses? Yes No • Can we reduce it to zero? 4

  5. 4 2 4 2 2 2 5 3 3 5 3 5 1 1 5 1 1 5 7 6 6 7 6 6 • Main question: What page to evict on a miss? • Common replacement strategies: • LRU = least recently used • FIFO = first-in first-out LRU Example: Requests: …4 5 3 6 2 1 5 7 4 5

  6. online offline usually b=0 Performance measure? Algorithm A is R-competitive if for any sequence S of page requests (# of misses of A on S) ≤ R·(minimum # of misses on S) + b where C is a constant independent of S Competitive ratio of A = smallest such R 6

  7. 4 4 4 4 4 4 4 4 1 1 3 1 2 1 1 1 5 5 5 5 9 9 7 9 6 6 6 6 6 6 6 6 How to determine optimum? Belady Algorithm: at each step evict the page whose next request is farthest in the future. …4 5 3 6 2 1 5 7 9 1 4 6 5 3 7 7

  8. LRU: 3 3 2 2 2 1 1 1 2 3 1 3 3 2 1 3 1 2 3 2 2 2 1 1 3 1 3 1 1 1 1 3 2 3 3 2 1 2 3 1 2 3 1 3 1 2 2 Optimum: How bad is LRU? Try k=2 each request is a miss every second request is a miss So competitive ratio of LRU is ≥ 2 8

  9. misses of LRU Requests: … x y … r z r… p r p … s p s … r’ … p’ … s’ optimum must have r in cache phase = request until the 2nd miss In this phase: at least 2 different requests other than r, so optimum must miss at least once How good is LRU? Try k=2 again We have p  r and s  p Case 1: s  r. Then r,p,s are different Case 2: s = r. So on p we evicted r. There must be another request z  r,p between r, p for otherwise LRU would not evict r So competitive ratio of LRU is ≤ 2 Exercise: Show that the competitive ratio of LRU for k pages is ≤ k. Hint: Similar idea. If all faults in a phase different, you are done. Else, consider two equal faults… 9

  10. k requests Q B = Belady’s pages in memory So at most one fault here Is it possible to improve ratio k? Theorem: No online paging algorithm has competitive ratio smaller than k. Proof: A = some online algorithm. Adversary strategy: fix a set of pages 1,2,…,k+1. At each step request the page that A does not have in memory. • Then: • Algorithm A faults at each time step • Belady faults at most once in each k consecutive steps (why?) Then B and Q differ by at most 1 Therefore the competitive ratio of A is at least k 10

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