1 / 19

UMass Lowell Computer Science 91.404 Analysis of Algorithms Prof. Karen Daniels Fall, 2002

UMass Lowell Computer Science 91.404 Analysis of Algorithms Prof. Karen Daniels Fall, 2002. Monday, 12/2/02 Design Patterns for Optimization Problems Greedy Algorithms. Algorithmic Paradigm Context. Greedy Algorithms. What is a Greedy Algorithm?. Solves an optimization problem

jyi
Download Presentation

UMass Lowell Computer Science 91.404 Analysis of Algorithms Prof. Karen Daniels Fall, 2002

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. UMass Lowell Computer Science 91.404Analysis of AlgorithmsProf. Karen DanielsFall, 2002 Monday, 12/2/02 Design Patterns for Optimization Problems Greedy Algorithms

  2. Algorithmic Paradigm Context

  3. Greedy Algorithms

  4. What is a Greedy Algorithm? • Solves an optimization problem • Optimal Substructure: • optimal solution contains in it optimal solutions to subproblems • Greedy Strategy: • At each decision point, do what looks best “locally” • Choice does not depend on evaluating potential future choices or solving subproblems • Top-down algorithmic structure • With each step, reduce problem to a smaller problem • Greedy Choice Property: • “locally best” = globally best

  5. Examples • Minimum Spanning Tree • Dijkstra Shortest Path • Huffman Codes • Fractional Knapsack • Activity Selection

  6. Minimum Spanning Tree

  7. Invariant: Minimum weight spanning forest Becomes single tree at end Invariant: Minimum weight tree 2 A B 4 3 G 6 Spans all vertices at end 5 1 1 E 7 6 F 8 4 D C 2 Minimum Spanning Tree Time: O(|E|lg|E|) given fast FIND-SET, UNION • Produces minimum weight tree of edges that includes every vertex. Time: O(|E|lg|V|) = O(|E|lg|E|) slightly faster with fast priority queue • forUndirected, Connected, Weighted Graph • G=(V,E) source: 91.503 textbook Cormen et al.

  8. Dijkstra Shortest Path

  9. 2 A 4 3 6 5 1 1 7 B 6 F 8 G 4 C 2 E D Single Source Shortest Paths: Dijkstra’s Algorithm for (nonnegative)weighted,directed graph G=(V,E) source: 91.503 textbook Cormen et al.

  10. Huffman Codes

  11. Huffman Codes

  12. Fractional Knapsack

  13. Fractional Knapsack 50 Value: $60 $100 $120 30 20 10 “knapsack” item1 item2 item3

  14. Activity Selection

  15. Example Optimization Problem: Activity Selection • Problem Instance: • Set S = {1,2,...,n} of n activities • Each activity i has: • start time: si • finish time : fi • Activities i, j are compatible iff non-overlapping: • Objective: • select a maximum-sized set of mutually compatible activities source: 91.503 textbook Cormen, et al.

  16. source: 91.503 textbook Cormen, et al.

  17. Algorithmic Progression • “Brute-Force” • Dynamic Programming #1 • Exponential number of subproblems • Dynamic Programming #2 • Quadratic number of subproblems • Greedy Algorithm

  18. Running time? Greedy Algorithm source: 91.503 textbook Cormen, et al. • Algorithm: • S’ = presort activities in S by nondecreasing finish time • and renumber • GREEDY-ACTIVITY-SELECTOR(S’) • n length[S’] • A {1} • j 1 • for i 2 to n • do if • then • j i • return A

  19. Greedy Heuristic • If optimization problem does not have “greedy choice property”, greedy approach may still be useful as a heuristic in bounding the optimal solution • Example: minimization problem Upper Bound (heuristic) Solution Values Optimal (unknown value) Lower Bound

More Related