1 / 29

Greedy Algorithms:

Greedy Algorithms:. A greedy algorithm always makes the choice that looks best at the moment. It makes a local optimal choice in the hope that this choice will lead to a globally optimal solution. Greedy algorithms yield optimal solutions for many (but not all) problems. .

fineen
Download Presentation

Greedy Algorithms:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Greedy Algorithms: • A greedy algorithm always makes the choice that looks best at the moment. • It makes a local optimal choice in the hope that this choice will lead to a globally optimal solution. • Greedy algorithms yield optimal solutions for many (but not all) problems. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  2. The 0-1 Knapsack problem: • The 0-1 knapsack problem: • N items, where the i-th item is worth vi dollars and weight wi pounds. 11 p 3 p 4p 58 p 8p 88p • vi and wi are integers. 3$ 6 $ 35$ 8$ 28$ 66$ • We can carry at most W (integer) pounds. • How to take as valuable a load as possible. • An item cannot be divided into pieces. • The fractional knapsack problem: • The same setting, but the thief can take fractions of items. • W may not be integer. W CS4335 Design and Analysis of Algorithms/WANG Lusheng

  3. Solve the fractional Knapsack problem: • Greedy on the value per pound vi/wi. • Each time, take the item with maximum vi/wi . • If exceeds W, take fractions of the item. • Example:(1, 5$), (2, 9$), (3, 12$), (3, 11$) and w=4. • vi/wi : 5 4.5 4.0 3.667 • First: (1, 5$), Second: (2, 9$), Third: 1/3 (3, 12$) • Total W: 1, 3, 4. Can only take part of item CS4335 Design and Analysis of Algorithms/WANG Lusheng

  4. Proof of correctness: (The hard part) X i1 w1 i2 w2wj . . . ip wp . . . ikwk Let X = i1, i2, …ik be the optimal items taken. • Consider the item j : (vj, wj)with the highest v/w. • if j is not used in X (the optimal solution), get rid of some items with total weight wj (possibly fractional items) and add item j. (since fractional items are allowed, we can do it.) • Total value is increased. Why? • One more item selected by greedy is added to X • Repeat the process, X is changed to contain all items selected by greedy WITHOUT decreasing the total value taken by the thief. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  5. The 0-1 knapsack problem cannot be solved optimally by greedy • Counter example: (moderate part) • W=10 21.8 • Items found (6pounds, 12dollars), (5pounds, 9 dollar), 1.8 1. 1 (5pounds, 9 dollars), (3pounds, 3 dollars), (3 pounds, 3 dollars) • If we first take (6, 12) according to greedy algorithm, then solution is (6, 12), (3, 3) (total value is 12+3=15). • However, a better solution is (5, 9), (5, 9) with total value 18. To showthata statement does not hold, we only have to give an example. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  6. A subset of mutually compatibles jobs: {c, f} CS4335 Design and Analysis of Algorithms/WANG Lusheng

  7. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  8. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  9. Sorting the n jobs based on fi needs O(nlog n) time CS4335 Design and Analysis of Algorithms/WANG Lusheng

  10. Example: • Jobs (s, f): (0, 10), (3, 4), (2, 8), (1, 5), (4, 5), (4, 8), (5, 6) (7,9). Sorting based on fi: (3, 4) (1, 5), (4, 5) (5, 6) (4,8) (2,8) (7, 9)(0,10). Selecting jobs: (3,4) (4, 5) (5,6) (7, 9) CS4335 Design and Analysis of Algorithms/WANG Lusheng

  11. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  12. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  13. Sort ob finish time: b, c, a, e, d, f, g, h. Greedy algorithm Selects: b, e, h. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  14. Depth: The maximum No. of jobs required at any time. Depth:3 CS4335 Design and Analysis of Algorithms/WANG Lusheng

  15. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  16. Depth: The maximum No. of jobs required at any time. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  17. Greedy on start time CS4335 Design and Analysis of Algorithms/WANG Lusheng

  18. Greedy Algorithm: c d g j b f i Depth: The maximum No. of jobs required at any time. a e h Depth:3 CS4335 Design and Analysis of Algorithms/WANG Lusheng

  19. ddepth CS4335 Design and Analysis of Algorithms/WANG Lusheng

  20. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  21. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  22. l1=0, l2=1 l2=0, l1=0 0 11 10 l1=9, l2=0 l1=0, l2=1 1 0 11 CS4335 Design and Analysis of Algorithms/WANG Lusheng

  23. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  24. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  25. di<dj We check every pair of consecutive numbers, if there is not inversion, then the whole sequenc has no inversion. n1n2 … nk Example: 1, 2, 3, 4, 5, 6, 7, 8, 9 1, 2, 6, 7, 3, 4, 5, 8, 9 CS4335 Design and Analysis of Algorithms/WANG Lusheng

  26. di<dj CS4335 Design and Analysis of Algorithms/WANG Lusheng

  27. CS4335 Design and Analysis of Algorithms/WANG Lusheng

  28. Example: Job ti di • 2 2 • 3 4 • 4 6 • 4 8 • 6 10 j1 j2 5 9 13 j5 19 2 j3 j4 CS4335 Design and Analysis of Algorithms/WANG Lusheng

  29. CS4335 Design and Analysis of Algorithms/WANG Lusheng

More Related