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UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Fall, 2008

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

Lecture 2

Tuesday, 9/16/08

Design Patterns for Optimization Problems

Greedy Algorithms

Solve subproblem(s), then make choice

Make choice, then solve subproblem(s)

Subproblem solution order

Greedy Algorithms

- 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

- 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.

- Activity Selection
- Minimum Spanning Tree
- Dijkstra Shortest Path
- Huffman Codes
- Fractional Knapsack

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: 91.404 textbook Cormen, et al.

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 Time Duration

Activity Number

- “Brute-Force”
- (board work)

- Dynamic Programming #1
- Exponential number of subproblems
- (board work)

- Dynamic Programming #2
- Quadratic number of subproblems

- Greedy Algorithm

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.

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

Running time?

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}
- j1
- for i 2 to n
- do if
- then
- ji

- return A

- S’ = presort activities in Sby nondecreasing finish time

- 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 Tree

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

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.

Dijkstra Shortest Path

2

A

4

3

6

5

1

1

7

B

6

F

8

G

4

C

2

E

D

for (nonnegative)weighted,directed graph G=(V,E)

source: 91.503 textbook Cormen et al.

Huffman Codes

source: 91.503 textbook Cormen, et al.

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

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

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

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

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

Fractional 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

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

On course web site under Miscellaneous Docs

- Patriotic Tree
- 404 review handout

- Tree Vertex Cover
- 91.503 midterm

- 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

HW#AssignedDueContent

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

2 T 9/16 T 9/23 Chapter 16