1. Java Coding 6-extra David Davenport Computer Eng. Dept.,
Computational complexity
David Davenport
Computer Eng. Dept.,
Bilkent University Ankara - Turkey.

2. IMPORTANT… Students… Instructors…
This presentation is designed to be used in class as part of a guided discovery sequence. It is not self-explanatory! Please use it only for revision purposes after having taken the class. Simply flicking through the slides will teach you nothing. You must be actively thinking, doing and questioning to learn!
Instructors…
You are free to use this presentation in your classes and to make any modifications to it that you wish. All I ask is an saying where and when it is/was used. I would also appreciate any suggestions you may have for improving it.
thank you, David.

3. The Story so far… Searching Sorting O(N) – on unordered list
O(log2N) – on ordered list!
Sorting
O(N2)
O(N.log2N)
For arrays
(storing data in first N elements)
Insertion & deletion
O(1) - unordered
O(N) - ordered
Different implementations offer different complexity measures.
Slide shows values for the common simple method of storing valid data in first N elements of array
Note that, in practice, may also need to be concerned with initialisation. What is its complexity?
Consider alternative implementations, eg. using –1 values to mark invalid array elements.
Next semester we will look at other more complex data structures that offer different performance characteristics.

4. Powers of 2… log2N N N2 N times N2 times 10 103 106 20 1012 30 109
1,024
103
106
1/1000 sec
1 second 20
1,048,576
1012
~12 days 30
1,073,741,824
109
1018
16 minutes
~30,000 years 40
1,099,511,627,776
1024 50
1,125,899,906,842,624
1015
1030 60
1,152,921,504,606,846,976
1036 70
1,180,591,620,717,411,303,424
1021
1042 80
1,208,925,819,614,629,174,706,176
1048
~32 E 12 microseconds in a year!
@ 1 E 6 operations / sec

5. A “simple” Decision Problem
Does a 5 by 5 arrangement exist?
Yes/No?
Worst case: Need to check all possible arrangements
No rotation. 25 possible pieces for first square, 24 left for next, then 23 for next, and so on. Gives 25 factorial.
Big bang was ~15 billion years ago! Oops!
How many arrangements? 25x24x23x … x3x2x1 = 25!
How 1 million/second? ~490 billion years

6. Values for various functions10 50
100
300
1000
logN
>4 6 7 9 N
N.logN 33
282
665
2469
9966 N2
2500
10000
90000
1 million (7 digits) 2N
1024
16-digit number
31-digit number
91-digit number
302-digit number N!
3.6 million (7 digits)
65-digit number
161-digit number
623-digit number
Unimaginably large NN
10 billion (11 digits)
85-digit number
201-digit number
744-digit number
Polynomial
For comparison:
number of protons in known universe has 79 digits
number of microseconds since big bang has 24 digits
Polynomial time algorithms are said to be reasonable or tractable
Exponential time algorithms are said to be unreasonable or intractable!
Source: Algorithmics, David Harel.
Exponential

7. Times (at one million instructions per second)10 50
100
300
1000
logN
1/10000 second N
1/1000 second
N.logN
1/100 second N2
1 second 2N
35.7 years
400 trillion centuries
75-digit # centuries N!
3.6 seconds
49-digit # centuries NN
2.8
hours
70-digit # centuries
185-digit # centuries
728-digit # centuries
Polynomial
For comparison:
big bang was about 15 billion years ago.
Tower of Hanoi problem
2 to the power N
64 1 million per second = 0.5 million years!
Tiling puzzles
see next slide…
Adapted from: Algorithmics, David Harel.
Exponential

8. Worse still… Tractable (polynomial) Intractable (exponential)
Uncomputable
e.g. Halting problem!
Should we be concerned?