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Algorithm Design and Analysis

L ECTURE 6 Greedy Algorithms Cont’d Minimizing lateness. Algorithm Design and Analysis. CSE 565. Adam Smith. Greedy Algorithms. Build up a solution to an optimization problem at each step shortsightedly choosing the option that currently seems the best. Sometimes good

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Algorithm Design and Analysis

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  1. LECTURE 6 • Greedy Algorithms Cont’d • Minimizing lateness Algorithm Design and Analysis CSE 565 Adam Smith A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  2. Greedy Algorithms • Build up a solution to an optimization problem at each step shortsightedly choosing the option that currently seems the best. • Sometimes good • Often does not work • Proving correctness can be tricky, because algorithm is unstructured A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  3. Last lecture (KT, Chapter 4.1) Two types of correctness arguments • “Greedy stays ahead” • Interval scheduling problem • “Structural argument” • Interval partitioning • Today: “exchange” a b c d e j 4 e g c d 3 b h 2 f a f i 1 g 3 3:30 4 4:30 9:30 10 10:30 11 11:30 12 12:30 1 1:30 9 2 2:30 h Time 0 1 2 3 4 5 6 7 8 9 10 11 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  4. Scheduling to minimize lateness A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

  5. 1 2 3 4 5 6 tj 3 2 1 4 3 2 dj 6 8 9 9 14 15 Scheduling to Minimizing Lateness • Minimizing lateness problem. • Single resource processes one job at a time. • Job j requires tj units of processing time and is due at time dj. • If j starts at time sj, it finishes at time fj = sj + tj. • Lateness: j = max { 0, fj - dj }. • Goal: schedule all jobs to minimize maximumlateness L = max j. • Ex: lateness = 2 lateness = 0 max lateness = 6 d3 = 9 d2 = 8 d6 = 15 d1 = 6 d5 = 14 d4 = 9 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11

  6. Greedy strategies?

  7. Minimizing Lateness: Greedy Algorithms • Greedy template. Consider jobs in some order. • [Shortest processing time first] Consider jobs in ascending order of processing time tj. • [Earliest deadline first] Consider jobs in ascending order of deadline dj. • [Smallest slack] Consider jobs in ascending order of slack dj - tj.

  8. 1 1 2 2 1 1 10 10 2 100 10 10 Minimizing Lateness: Greedy Algorithms • Greedy template. Consider jobs in some order. • [Shortest processing time first] Consider jobs in ascending order of processing time tj. • [Smallest slack] Consider jobs in ascending order of slack dj - tj. tj counterexample dj tj counterexample dj

  9. d1 = 6 d2 = 8 d3 = 9 d4 = 9 d5 = 14 d6 = 15 Minimizing Lateness: Greedy Algorithm • Greedy algorithm. Earliest deadline first. Sort n jobs by deadline so that d1 d2 …  dn t  0 for j = 1 to n Assign job j to interval [t, t + tj] sj t, fj  t + tj t  t + tj output intervals [sj, fj] max lateness = 1 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11

  10. Minimizing Lateness: No Idle Time • Observation. There exists an optimal schedule with noidle time. • Observation. The greedy schedule has no idle time. d = 4 d = 6 d = 12 0 1 2 3 4 5 6 7 8 9 10 11 d = 4 d = 6 d = 12 0 1 2 3 4 5 6 7 8 9 10 11

  11. Minimizing Lateness: Inversions • Def. An inversion in schedule S is a pair of jobs i and j such that:i < j but j scheduled before i. • Observation. Greedy schedule has no inversions. • Observation. If a schedule (with no idle time) has an inversion, it has one with a pair of inverted jobs scheduled consecutively. inversion before swap j i

  12. Minimizing Lateness: Inversions • Def. An inversion in schedule S is a pair of jobs i and j such that:i < j but j scheduled before i. • Claim. Swapping two adjacent, inverted jobs reduces the number of inversions by one and does not increase the max lateness. • Pf. Let  be the lateness before the swap, and let  ' be it afterwards. • 'k = k for all k  i, j • 'ii • If job j is late: inversion fi before swap j i after swap i j f'j

  13. Minimizing Lateness: Analysis of Greedy Algorithm • Theorem. Greedy schedule S is optimal. • Pf. Define S* to be an optimal schedule that has the fewest number of inversions, and let's see what happens. • Can assume S* has no idle time. • If S* has no inversions, then S = S*. • If S* has an inversion, let i-j be an adjacent inversion. • swapping i and j does not increase the maximum lateness and strictly decreases the number of inversions • this contradicts definition of S* ▪

  14. Greedy Analysis Strategies • Greedy algorithm stays ahead. Show that after each step of the greedy algorithm, its solution is at least as good as any other algorithm's. • Exchange argument. Gradually transform any solution to the one found by the greedy algorithm without hurting its quality. • Structural. Discover a simple "structural" bound asserting that every possible solution must have a certain value. Then show that your algorithm always achieves this bound.

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