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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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Greedy Algorithms

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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

Sij

aa

ac

ak

ax

az

ab

aw

ay

Jeff Chastine

### Visualization

Skj

Sik

Sij

• Optimal solution includes

aa

ac

ak

ax

az

ab

aw

ay

Jeff Chastine

### Visualization

Skj

Sik

Sij

• We know

• How to choose activity ?

aa

ac

ak

ax

az

ab

aw

ay

Jeff Chastine

### Visualization

Skj

Sik

Sij

• Iterate through all (like Matrix Mults)!

aa

ac

ak

ax

az

ab

aw

ay

Jeff Chastine

### Recursive Solution

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

1m ← i + 1

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

3dom ← m + 1

4ifm ≤ n

5then return {am}

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

6else return

Jeff Chastine

Code Trace

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Jeff Chastine

First Call

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Jeff Chastine

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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

50

item 3

30

item 2

20

item 1

10

\$100

\$60

\$120

Jeff Chastine

50

item 3

30

item 2

20

item 1

10

\$100

\$60

\$120

Jeff Chastine

50

item 3

30

item 2

20

item 1

10

10

\$100

\$60

\$120

Bad Idea…

Jeff Chastine

50

item 3

30

20

item 2

20

item 1

10

10

\$100

\$60

\$120

Total = \$160

Jeff Chastine

### Greedy Fail

50

30

item 3

30

item 2

20

item 1

10

10

\$100

\$60

\$120

Try again! Total = \$180

Jeff Chastine

50

30

item 3

30

item 2

20

20

item 1

10

\$100

\$60

\$120

Total = \$220

Jeff Chastine

50

item 3

30

item 2

20

item 1

10

\$100

\$60

\$120

Jeff Chastine

50

item 3

30

item 2

20

item 1

10

\$100

\$60

\$120

Jeff Chastine

50

item 3

30

item 2

20

item 1

10

10

\$100

\$60

\$120

Jeff Chastine

50

item 3

30

20

item 2

20

item 1

10

10

\$100

\$60

\$120

Jeff Chastine

Take only part

50

2030

item 3

30

20

item 2

20

item 1

10

10

\$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?

50

50

30

item 3

30

20

item 2

20

item 1

10

10

10

\$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

abcdef

Frequency (in thousands)45131216 9 5

Fixed-length codeword000001010011100101

Variable-length codeword010110011111011100

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

a:45

f:5

Jeff Chastine

### How it works

• Combine and re-sort

14

c:12

b:13

d:16

a:45

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

• Re-sort

14

25

d:16

a:45

1

1

0

0

e:9

f:5

c:12

b:13

Jeff Chastine

14

25

d:16

a:45

1

1

0

0

e:9

f:5

c:12

b:13

Jeff Chastine

### How it works

25

30

a:45

0

1

1

0

14

c:12

b:13

d:16

1

0

e:9

f:5

Notice what’s happening to the first grouping we did

Jeff Chastine

25

30

a:45

0

1

1

0

14

c:12

b:13

d:16

1

0

e:9

f:5

Jeff Chastine

55

a:45

0

1

25

30

0

1

1

0

14

c:12

b:13

d:16

1

0

e:9

f:5

Jeff Chastine

100

1

0

55

a:45

0

1

25

30

0

1

1

0

14

c:12

b:13

d:16

1

0

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