greedy algorithms. greedy algorithm greedy algorithm - makes locally optimal choice at each stage....
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Greedy Algorithms
Greedy Algorithm
Greedy Algorithm
- Makes locally optimal choice at each stage.
- For optimization problems.
If the local optimum is a part of the global optimum, we get the global optimum.
Greedy Algorithm vs Dynamic Programming
DynamicProgramming
GreedyAlgorithm
Knapsack Problem
n items a thief’s knapsack of
size W
Knapsack Problem
0-1 knapsack problem
- Each item must be either taken or left behind.
Fractional knapsack problem
- The thief can take fractions of items.
Knapsack Problem
453
0201
0$60 $10
0$120
$135
knapsack($6/
unit)($5/unit)
($4/unit)
($3/unit)
n = 4,W = 50
50
Fractional Knapsack Problem
5045
30
20 1
0$60
$100
$120$13
5 knapsack
($6/unit)
($5/unit)
($4/unit)($3/
unit)
10
20
2030
Greedy algorithm:greatest value per unit
50
$240
0-1 Knapsack Problem
45
30
20
10
$60
$100
$120
$135 optima
l
($6/unit)
($5/unit)
($4/unit)
($3/unit)
50 50
value
50
valueper unit
50
10
20
45
30
20
$160
$135
$220
0-1 Knapsack Problem
optimal
50 50
value
50
valueper unit
50
10
20
45
30
20
$160
$135
$220
Difficult to get the optimal solution with a greedy strategy.
Dynamic Programming :
Optimal Substructure vsSubproblem Solution
48
5 7
4
1
3
2
T1,5 T1,6
T2,5 T2,6
Ti,j : the solution of a subproblem
A subproblem solution?
A local optimum?
Greedy Algorithm vs Dynamic Programming
Dynamic Programming
Greedy Algorithm
Computes all subproblems
Find a local optimum
Always finds the optimal solution
May not be able to find the optimal solution
Less efficient More efficient
Optimal Substructure vsSubproblem Solution
Subproblem solution
- From all subproblem solutions
Optimal substructure
- With only constant number of parameters
- Without subproblems or future choices
Usuallytop-down
UsuallyBottom-up
Huffman Codes
A lossless data compression algorithm.
It uses variable-length code
Variable-Length Code
Six characters : a, b, c, d, e, f
How can we represent them with binary strings?
a b c d e f
Fixed-length
000 001 010 011 100 101
Variable-Length Code
Six characters : a, b, c, d, e, f
How can we represent them with binary strings?
a b c d e f
Variable-length
0 1 00 01 10 11
What 0010 menas?
0 0 1 0 = a a b a0 01 0 = a d a00 10 = c e…
Prefix Code
Six characters : a, b, c, d, e, f
How can we represent them with binary strings?
a b c d e f
Variable-length
0 101 100 111 1101 1100
No codeword is a prefix of another codeword.
Prefix Code
Six characters : a, b, c, d, e, f
How can we represent them with binary strings?
a b c d e f
Variable-length
0 101 100 111 1101 1100
No codeword is a prefix of another codeword.
Variable-Length Code
a b c d e f Total
Frequency 45 13 12 16 9 5 100
Fixed-length 000 001 010 011 100 101 300
Variable-length
0 101 100 111 1101
1100
224
Is it the optimal way?
Variable-Length Code
a b c d e f Total
Frequency 45 13 12 16 9 5 100
Fixed-length 000 001 010 011 100 101 300
Variable-length
0 101 100 111 1101
1100
224
Alternative? 0 1100
1101
10 1110
1111
231
Huffman Tree
100cha
rcode
freq
a 000 45
b 001 13
c 010 12
d 011 16
e 100 9
f 101 5
Fixed-length code
86 14
58 1428
a: 45
b: 13
c: 12
d: 16
e: 9
f: 5
0 1
0 1 0
0 1 0 1 0 1
Huffman Tree
char
code
freq
a 0 45
b 101 13
c 100 12
d 111 16
e 1101
9
f 1100
5
Variable-length code
100
55
25
14
30
a: 45
b: 13c: 12 d: 16
f: 5 e: 9
0 1
0 1
0 1
10
0 1
Huffman’s Algorithm
A
B
0
0
Every non-leaf node has two children.
B
0
Observation 1The longest code : at least 2 characters
Huffman’s Algorithm
C H
I J
G10
A10
F10
Observation 1
Observation 2
The longest code : at least 2 charactersThe longest 2 codes : the least frequent two characters
D E
B10
Huffman’s Algorithm
Observation 1
Observation 2
Observation 3
The longest code : at least 2 charactersThe longest 2 codes : the least frequent two charactersA non-leaf node : handled like a leaf node
d: 1614
30 x: 30T T
’
Huffman’s Algorithm
d: 1614
30 x: 30
A non-leaf node : handled like a leaf node
Observation 3
e: 9 f: 5
10
T T ’
10
Total length in T = Total length in T ‘ + 16 * 1 + (9 + 5) * 2
Huffman’s Algorithm
Observation 1
Observation 2
Observation 3
The longest code : at least 2 charactersThe longest 2 codes : the least frequent two charactersA non-leaf node : handled like a leaf node
d: 1614
30 x: 30T T
’
Huffman’s Algorithm
a: 45 b: 13 c: 12 d: 16 e: 9 f: 5
Merging two least frequent nodes.
Huffman’s Algorithm
a: 45 b: 13 c: 12 d: 16
e: 9 f: 5
Merging two least frequent nodes.
1410
Huffman’s Algorithm
a: 45
c: 12 b: 13
d: 16
e: 9 f: 5
Merging two least frequent nodes.
1410
2510
Huffman’s Algorithm
a: 45
c: 12 b: 13 d: 16
e: 9 f: 5
Merging two least frequent nodes.
1410
2510
3010
Huffman’s Algorithm
a: 45
c: 12 b: 13 d: 16
e: 9 f: 5
Merging two least frequent nodes.
1410
2510
3010
550 1
Huffman’s Algorithm
a: 45
c: 12 b: 13 d: 16
e: 9 f: 5
Merging two least frequent nodes.
1410
2510
3010
550 1
1000 1 cha
rcode
a 0
b 101
c 100
d 111
e 1101
f 1100
Greedy Algorithm
Optimization Algorithms : finds a proper local optimum.
Not as powerful as Dynamic Programming, but simpler.
Greedy algorithms
- Knapsack Problem
- Huffman Code
S
Graphs
What is a graph?
1 2
3 4
5
V = { 1, 2, 3, 4, 5 }E = { {1,2}, {1,3}, {2,3}, {2,4}, {2,5}, {3,4} }
G = ( V, E )
Directed and Undirected
1 2
3 4
5
Directed graph
1 2
3 4
5
Undirected graph
Representations of Graphs
1 2
3 4
5
Directed graph
1
2
3
4
5
2 /
3
1
2
2 /
4 /
3 /
3 /
Adjacency-list
Representations of Graphs
1 2
3 4
5
Directed graph Adjacency-Matrix1 2 3 4
5
5
3
4
2
1
Representations of Graphs
Adjacency-Matrix
Adjacency-list
Adjacency List Adjacency Matrix
space |V| + |E| |V|2
Finding all edges
|V| + |E| |V|2
Finding one edge
num of edges 1
Adjacency Matrix of Graphs
1 2
3 4
5
Directed graph Adjacency-Matrix1 2 3 4
5
5
3
4
2
1
Adjacency Matrix of Graphs
A = AT =
Weighted Graphs
1 2
3 4
5
1.7
0.42.0
-0.3
3.1
-0.23.6-
2.1
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