Greedy Algorithms

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# Greedy Algorithms - PowerPoint PPT Presentation

Greedy Algorithms. Greedy Algorithms. Compared to Dynamic Programming Both used in optimization problems More efficient Not always optimal Top-down instead of bottom up Make a locally optimal (quick) choice Still have optimal substructure No need to dive into sub-problems.

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## PowerPoint Slideshow about 'Greedy Algorithms' - bailey

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### Greedy Algorithms

Jeff Chastine

Greedy Algorithms
• Compared to Dynamic Programming
• Both used in optimization problems
• More efficient
• Not always optimal
• Top-down instead of bottom up
• Make a locally optimal (quick) choice
• Still have optimal substructure
• No need to dive into sub-problems

Jeff Chastine

Activity Selection
• Given a set of activities , find the maximum # you can schedule
• NOT optimal use of time of a room
• One activity is active at one time
• Activity has start time of and finish of
• Order activities by finish time

isifi

1 2 3 4 5 6 7 8 9 10 11

1 3 0 5 3 5 6 8 8 2 12

4 5 6 7 8 9 10 11 12 13 14

Jeff Chastine

Using Dynamic Programming
• Defining the subproblem:
• Set of activities that “fit” in this time chunk
• Picking any activity creates two sub-problems

Jeff Chastine

Visualization

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Jeff Chastine

Visualization

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• Optimal solution includes

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Jeff Chastine

Visualization

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• We know
• How to choose activity ?

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Jeff Chastine

Visualization

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• Iterate through all (like Matrix Mults)!

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Jeff Chastine

Recursive Solution

RECURSIVE-ACT-SELECT (s, f, i, n)

1 m ← i + 1

2 whilem ≤ n and sm < fi //find 1st activity

3 dom ← m + 1

4 ifm ≤ n

5 then return {am}

RECURSIVE-ACT-SELECT(s, f, m, n)

6 else return

Jeff Chastine

Code Trace

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Jeff Chastine

First Call

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Jeff Chastine

Second Call

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Jeff Chastine

Second Call

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Jeff Chastine

Second Call

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Jeff Chastine

Third Call

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Jeff Chastine

Third Call

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Jeff Chastine

Third Call

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Jeff Chastine

Third Call

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Jeff Chastine

Third Call

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Jeff Chastine

Fourth Call

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Jeff Chastine

Fourth Call

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Jeff Chastine

Fourth Call

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Jeff Chastine

Fourth Call

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Jeff Chastine

Result

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Jeff Chastine

Where Greedy Fails
• Knapsack Problem
• Steal items of different weight and value
• Sack has weight limit
• Goal: steal as much as possible
• Two flavors of problem
• 0-1 version (take all of something)
• Fractional (can take only part)

Jeff Chastine

0-1 Example

50

item 3

30

item 2

20

item 1

10

\$100

\$60

\$120

Jeff Chastine

Greedy – Go for the Gold!

50

item 3

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item 2

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item 1

10

\$100

\$60

\$120

Jeff Chastine

Greedy – Go for the Gold!

50

item 3

30

item 2

20

item 1

10

10

\$100

\$60

\$120

Jeff Chastine

Greedy – Pick the next best

50

item 3

30

20

item 2

20

item 1

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10

\$100

\$60

\$120

Total = \$160

Jeff Chastine

Greedy Fail

50

30

item 3

30

item 2

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item 1

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10

\$100

\$60

\$120

Try again! Total = \$180

Jeff Chastine

Optimal for 0-1 Problem

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30

item 3

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item 2

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item 1

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\$100

\$60

\$120

Total = \$220

Jeff Chastine

Fractional Example

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item 3

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item 2

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item 1

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\$100

\$60

\$120

Jeff Chastine

Greedy – Go for the Gold!

50

item 3

30

item 2

20

item 1

10

\$100

\$60

\$120

Jeff Chastine

Greedy – Go for the Gold!

50

item 3

30

item 2

20

item 1

10

10

\$100

\$60

\$120

Jeff Chastine

Greedy – Get next best…

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item 3

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item 2

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item 1

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\$100

\$60

\$120

Jeff Chastine

Greedy – AHA!

Take only part

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item 3

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item 2

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item 1

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\$100

\$60

\$120

Jeff Chastine

Greedy – Go for the Gold!

50

2030

\$80

item 3

30

20

item 2

\$100

20

item 1

10

10

\$60

\$100

\$60

\$120

Total = \$240

Note: optimal for 0-1 was \$220

Jeff Chastine

Why the Greedy Fail?

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item 3

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item 2

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item 1

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\$100

\$60

\$120

Notice the waste of space!

Jeff Chastine

Huffman Codes
• Objective: compress text data
• Raw ASCII text file uses 8 bits/char (fixed)
• 1000 characters = 8K bits
• Book uses 6 characters for 3 bits/char
• Trick
• Analyze: not all characters are used!
• Analyze: frequency of characters
• Prefix codes: no binary code is prefix of another

Jeff Chastine

Analysis

a b c d e f

Frequency (in thousands) 45 13 12 16 9 5

Fixed-length codeword 000 001 010 011 100 101

Variable-length codeword 0 101 100 111 1101 1100

110001001101 = face //12 bits

01011010 = abba // 8 bits

Jeff Chastine

How it works
• Sort by least frequent

e:9

c:12

b:13

d:16

a:45

f:5

Jeff Chastine

How it works
• Group lowest two

e:9

c:12

b:13

d:16

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f:5

Jeff Chastine

How it works
• Combine and re-sort

14

c:12

b:13

d:16

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1

0

e:9

f:5

Watch how least frequent characters are pushed down the tree!

Jeff Chastine

How it works
• Group lowest two

14

c:12

b:13

d:16

a:45

1

0

e:9

f:5

Jeff Chastine

How it works
• Re-sort

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25

d:16

a:45

1

1

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0

e:9

f:5

c:12

b:13

Jeff Chastine

How it works

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25

d:16

a:45

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e:9

f:5

c:12

b:13

Jeff Chastine

How it works

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c:12

b:13

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f:5

Notice what’s happening to the first grouping we did

Jeff Chastine

How it works

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30

a:45

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c:12

b:13

d:16

1

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e:9

f:5

Jeff Chastine

How it works

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b:13

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Jeff Chastine

How it works

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b:13

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e:9

f:5

Jeff Chastine

Summary
• Greedy algorithms have two properties
• Greedy choice property: a globally optimal solution can be created by the greedy choice
• Optimal substructure: the optimal solution is contains optimal solutions to subproblems
• Greedy is top-down
• Doesn’t always yield optimal solution

Jeff Chastine