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UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Fall, 2008. Lecture 2 Tuesday, 9/16/08 Design Patterns for Optimization Problems Greedy Algorithms. Algorithmic Paradigm Context. Solve subproblem(s), then make choice. Make choice, then solve subproblem(s).

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Umass lowell computer science 91 503 analysis of algorithms prof karen daniels fall 2008

UMass Lowell Computer Science 91.503Analysis of AlgorithmsProf. Karen DanielsFall, 2008

Lecture 2

Tuesday, 9/16/08

Design Patterns for Optimization Problems

Greedy Algorithms


Algorithmic paradigm context
Algorithmic Paradigm Context

Solve subproblem(s), then make choice

Make choice, then solve subproblem(s)

Subproblem solution order



What is a greedy algorithm
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 presolving overlapping subproblems

    • Top-down algorithmic structure

      • With each step, reduce problem to a smaller problem

  • Greedy Choice Property:

    • “locally best” = globally best


Greedy strategy approach
Greedy Strategy Approach

  • Determine the optimal substructure of the problem.

  • Develop a recursive solution.

  • Prove that, at any stage of the recursion, one of the optimal choices is the greedy choice.

  • Show that all but one of the subproblems caused by making the greedy choice are empty.

  • Develop a recursive greedy algorithm.

  • Convert it to an iterative algorithm.

source: 91.503 textbook Cormen, et al.


Examples
Examples

  • Activity Selection

  • Minimum Spanning Tree

  • Dijkstra Shortest Path

  • Huffman Codes

  • Fractional Knapsack



Activity selection optimization problem
Activity Selection Optimization Problem

  • 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: 91.404 textbook Cormen, et al.


Activity selection1

1

2

4

5

9

3

12

16

6

10

14

15

11

7

13

8

1

2

3

4

5

6

7

8

Activity Selection

Activity Time Duration

Activity Number


Algorithmic progression
Algorithmic Progression

  • “Brute-Force”

    • (board work)

  • Dynamic Programming #1

    • Exponential number of subproblems

    • (board work)

  • Dynamic Programming #2

    • Quadratic number of subproblems

  • Greedy Algorithm


Activity selection2
Activity Selection

Solution to Sij including ak produces 2 subproblems:

1) Sik (start after ai finishes; finish before ak starts)

2) Skj (start after ak finishes; finish before aj starts)

c[i,j]=size of maximum-size subset of mutually compatible activities in Sij.

source: 91.404 textbook Cormen, et al.


Recursive Activity Selection

High-level call: RECURSIVE-ACTIVITY-SELECTOR(s,f,0,n)

Returns an optimal solution for

Errors from earlier printing are corrected in red.

source: web site accompanying 91.503 textbook Cormen, et al.



Greedy algorithm

Running time?

Greedy Algorithm

source: 91.503 textbook Cormen, et al.

  • Algorithm:

    • S’ = presort activities in Sby nondecreasing finish time

      • and renumber

    • GREEDY-ACTIVITY-SELECTOR(S’)

      • n length[S’]

      • A {1}

      • j 1

      • for i 2 to n

        • do if

        • then

        • ji

    • return A


Streamlined greedy strategy approach
Streamlined Greedy Strategy Approach

  • View optimization problem as one in which making choice leaves one subproblem to solve.

  • Prove there always exists an optimal solution that makes the greedy choice.

  • Show that greedy choice + optimal solution to subproblem optimal solution to problem.

Greedy Choice Property: “locally best” = globally best

source: 91.503 textbook Cormen, et al.



Minimum spanning tree1

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.



Single source shortest paths dijkstra s algorithm

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.



Huffman codes1
Huffman Codes

source: 91.503 textbook Cormen, et al.








Knapsack
Knapsack

Each item has value and weight.

Goal: maximize total value of items chosen, subject to weight limit.

50

fractional: can take part of an item

0-1: take all or none of an item

30

20

10

item1

item2

item3

“knapsack”

Value: $60 $100 $120



Additional examples
Additional Examples

On course web site under Miscellaneous Docs

  • Patriotic Tree

    • 404 review handout

  • Tree Vertex Cover

    • 91.503 midterm


Greedy heuristic
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


Homework
Homework

HW# Assigned DueContent

1 T 9/9 T 9/16 91.404 review & Chapter 15

2 T 9/16 T 9/23 Chapter 16


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