Greedy Algorithms

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Greedy Algorithms
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Greedy Algorithms

• For some problems, dynamic programming is
  overkill
• Greedy algorithms make the choice thats best at
  that moment
• makes locally optimal choice in hopes for
  globally optimal solution
• will be useful when discussing MSTs, Dijkstras,
  etc...

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The Activity-Selection Problem

• Problem schedule several competing activities
  that require exclusive use of a common resource
• Could develop a solution similar to that of
  matrix-chain multiplication
• suppose the activity is included
• assume that times to the left/right are optimal

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Greedy Solution

• Obviously, there are 2n combinations
• Order activities by increasing finishing time
• Pick an activity if its non-conflicting
• Runs in linear time

5
a0
1 1 4
a1
a0
2 3 5
a2
a1
3 0 6
a3
a1
4 5 7
a4
a1
5 3 8
a5
a1
a4
6 5 9
a6
a1
a4
7 6 10
a7
a1
a4
8 8 11
a8
9 8 12
a9
a1
a4
a8
10 2 13
a10
a1
a4
a8
11 12 14
a11
12 ? -
a1
a4
a8
a11
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
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Elements of a Greedy Algorithm

• Determine the optimal substructure of the problem
• Develop a recursive solution
• Prove that at any stage of the recursion, the
  greedy choice is optimal
• Show that all but one of the subproblems induced
  by having made the greedy choice arae empty

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The Knapsack Problems

• Assume a thief is robbing a store with various
  valued and weighted objects. Her bag can hold
  only W pounds.
• The 0-1 knapsack problem must take all or
  nothing
• The fractional knapsack problem can take all or
  part of the item

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Where Greedy Fails
50
50
50
50
50
20
20 --- 30
30
30
30
20
20
knapsack
20
10
10
10
10
60
220
160
100
120
180
240
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Huffman Codes

• A widely used compression algorithm
• Savings of 20 - 90
• Uses the frequency of the characters
• Fixed codes versus variable-length codes
• Letter freq fixed code variable
• a 45 000 0
• b 13 001 101
• c 12 010 100
• d 16 011 111
• e 9 100 1101
• f 5 101 1101

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Constructing a Huffman Tree
f5
e9
c12
b13
d16
a45
11
Constructing a Huffman Tree
14
c12
b13
d16
a45
0
1
f5
e9
12
Constructing a Huffman Tree
14
25
d16
a45
0
1
0
1
f5
e9
c12
b13
13
Constructing a Huffman Tree
25
30
a45
1
0
0
1
14
c12
b13
d16
0
1
f5
e9
14
Constructing a Huffman Tree
55
a45
1
0
25
30
1
0
0
1
14
c12
b13
d16
0
1
f5
e9
15
Constructing a Huffman Tree
100
1
0
55
a45
1
0
25
30
1
0
0
1
14
c12
b13
d16
0
1
f5
e9
16
Summary

• For every greedy algorithm, there is a dynamic
  algorithm
• Greedy Algorithms make quick (local) decisions
• After a decision is made, problems of size n-1
  are left
• Greedy does not always yield an optimal solution