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Algorithm Design and Analysis

Explore greedy algorithms in minimizing lateness, scheduling jobs effectively to reduce lateness, analysis of the greedy algorithm, and strategies for achieving optimal solutions. Understand the concepts through practical examples and algorithm design techniques.

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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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