3621511 (6 cp)
Pasi Fränti
16.9.2013

DAA course at Autumn 2013
• Web: http://cs.uef.fi/pages/franti/asa/
• ETCS 6 (=richer content)
• Lectures (34h):
- Mon 14-16, 16-18
- Tue 14-16
• Exercises (16h):
- Fri 10-12
• Exam dates: 1.11. 18.12. x.5.
2

Motivation
• Design
- 90% cases simple algorithms found from
Bachelor level courses
- Focus on the 10% tough ones
• Analysis
– Why not just measure processing time?
– Time vs. space complexity analyses
– Upper and lower limits
3

Key concepts
• Problem
- Can the problem be solved?
- How difficult is the problem?
- Real-life problems to algorithmic problems
• Algorithm
– How to find suitable algorithm?
– How to make it efficient?
• Instance
– Upper and lower limits
4

199
117
110
216
246
182
121
315
170
231
79
142
191
242
148
136
78
191
178 149
126
116
89
51 112
90
163
59
143
116
79
170
58
73
72
63
105
135
131 109
53
86
27
120
234
Graph problems
• Coloring problem
• Shortest path
• Traveling salesman

Clustering problem
Input Output
6

What is algorithm?
• We need a well-specified problem that tells what needs to be achieved
• Algorithm solves the problem
• It consists of a sequence of commands that takes an input and gives output
Input Algorithm Output
7

Non-solvable problems
• Give list of all rational numbers in [0..1]
• Longest sequence of  without the 0 digit
• Stopping problem
Does algorithm A stop always?
Any algorithm
A
Solve
Stopping
B
8

Non-solvable problems
• Give list of all rational numbers in [0..1]
• Longest sequence of  without the 0 digit
• Stopping problem
Algorithm C NOT stop always!
Any algorithm
A
Solve
Stopping
B
Algorithm
“Vice Versa”
Eternal loop
C
Stop
9

Algorithm principles
• Sequence
- One command at a time
- Parallel and distributed computing
• Condition
- IF
- CASE
• Loops
- FOR
- WHILE
- REPEAT

Complexity
Time complexity:
– How much time it takes to compute
– Measured by a function T ( N )
Space complexity:
– How much memory it takes to compute
– Measured by a function S ( N )
11

Time complexity
•
•
N
T ( N
= Size of the input
) = Time complexity function
• Order of magnitude:
– How rapidly T ( N ) grows when N grows
– For example: O( N ) O(log N ) O( N ²) O(2 N )
12

Asymptotic analysis
Examples: Bubble sort ( N ²), Quicksort ( N∙ log N )
13

Problem size vs. processing time
f(N)
100 N
5 N ²
½ N ³
2 N log N
Processing power increases →
T(N) = 1,000 T(N)=10,000 Growth rate
N = 10 N =100 10 x
14
12
9
2 1,000
44
27
13
2 10,000
3.1 x
2.3 x
1.4 x
Very high!!
14

Exponential time complexity
1 2 4 8 16 32 64 128
256 512
1k 2k
…
1M
1G
Halfway…?
1T
1P
1E
Large than Everest
2 64 = 18.4 ∙ 10 18
15

Graph algorithms

Graph algorithms