1. Asst. Dr.Surasak Mungsing E-mail: Surasak.mu@spu.ac.th
CSE 221/ICT221 Analysis and Design of Algorithms Lecture 03: Introduction to algorithms analysis
Asst. Dr.Surasak Mungsing
Feb-19

2. CSE221/ICT221 Analysis and Design of Algorithms
Meaning of Algorithm
Algorithm
Recipe for getting things done successfully
"Recipe" – well defined steps of doing
"things" – computation problems which defined input/output
"done" – solved within definite time and steps
"successfully" – done correctly
Any special method of solving a certain kind of problem - Webster Dictionary
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CSE221/ICT221 Analysis and Design of Algorithms

3. CSE221/ICT221 Analysis and Design of Algorithms
Computer Program
A computer program (also a software program, or just a program) is a sequence of instructions written to perform a specified task for a computer.
A computer program in the form of a human-readable, computer programming language is called source code. Source code may be converted into an executable image by a compiler or executed immediately with the aid of an interpreter.
Algorithm is a step by step outline or flowchart how to solve a problem, but program is an implemented coding of a solution to a problem based on the algorithm.
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CSE221/ICT221 Analysis and Design of Algorithms

4. CSE221/ICT221 Analysis and Design of Algorithms
Example
Problem: Find the smallest integer from a given set of integers stored in an array
INPUT
Algorithm
instance
OUTPUT
m:= a[1];
for I:=2 to size of input
if m > a[I] then m:=a[I];
return m
25, 90, 53, 23, 11, 34 11
Data-Structure
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CSE221/ICT221 Analysis and Design of Algorithms

5. CSE221/ICT221 Analysis and Design of Algorithms
Problem
Problem: Find the smallest integer from a given set of integers stored in an array
Algorithm A
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CSE221/ICT221 Analysis and Design of Algorithms

6. CSE221/ICT221 Analysis and Design of Algorithms
Algorithm B (use two temporary arrays)
copy the input a to array t1;
assign n  size of input;
While n > 1
For i  1 to n /2
t2[ i ]  min (t1 [ 2*i ], t1[ 2*i + 1] );
copy array t2 to t1;
n n/2;
Output t2[1];
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CSE221/ICT221 Analysis and Design of Algorithms

7. CSE221/ICT221 Analysis and Design of Algorithms
Algorithm C
Sort the input in increasing order.
Return the first element of the sorted data. 8 9 5 6 11 34 7 20
black
box
Sorting 5 6 7 8 9 11 20 34
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CSE221/ICT221 Analysis and Design of Algorithms

8. CSE221/ICT221 Analysis and Design of Algorithms
Algorithm D
Test each data whether it is the smallest one
i  0;
flag  true;
while flag
i  i + 1;
min  a[ i ];
flag  false;
for j  1 to size of input
if min > a[ i ] then flag  true;
3. output min
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CSE221/ICT221 Analysis and Design of Algorithms

9. Which algorithm is better?
All algorithms can solve the problem correctly, but which one is better?
Consideration is based on running time (number of operations needed) and amount of memory used
หมายเหตุ
ระยะเวลาที่ใช้ในการทำงานของอัลกอริธึมจะเพิ่มขึ้นเมื่อจำนวนข้อมูลนำเข้าเพิ่มขึ้น
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CSE221/ICT221 Analysis and Design of Algorithms

10. CSE221/ICT221 Analysis and Design of Algorithms
Correctness, efficiency and measurement model
Correctness : ability to solve the problem correct ly in all cases
Efficiency : required resources for algorithm to work correctly
Time: number of execution
Space: memory space required
Measurement model : worst case
average case best case
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CSE221/ICT221 Analysis and Design of Algorithms

11. CSE221/ICT221 Analysis and Design of Algorithms
Time vs. Size of Input
Input Size
Measurement parameterized by the size of the input.
The algorithms A,B,C are implemented and run in a PC.
Algorithms D is implemented and run in a supercomputer.
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CSE221/ICT221 Analysis and Design of Algorithms

12. What is Algorithm Analysis?
Measurement of time complexity of algorithms
Techniques that drastically reduce the running time of an algorithm
A mathematical framework that more rigorously describes the running time of an algorithm
ดูจากเทคนิคที่สามารถลดเวลาการทำงานของขั้นตอนวิธีได้อย่างมาก
ดูจากกรอบการทำงานทางคณิตศาสตร์ที่บ่งชี้ระยะเวลาที่ต้องใช้ในทำงานของขั้นตอนวิธี
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CSE221/ICT221 Analysis and Design of Algorithms

13. Time required for small size of inputs
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CSE221/ICT221 Analysis and Design of Algorithms

14. Time required for intermediate size of inputs
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CSE221/ICT221 Analysis and Design of Algorithms

15. Asymtotic behavior
A line whose distance to a given curve tends to zero. An asymptote may or may not intersect its associated curve.
Asymptote
The x and y axes are asymptotes of the hyperbola xy = 1.
Asymptotic
1. (Mathematics) of or referring to an asymptote
2. (Mathematics) (of a function, series, formula, etc.) approaching a given value or condition, as a variable or an expression containing a variable approaches a limit, usually infinity
Feb-19

16. Asymptotic Performance
In mathematics, computer science, and related fields, big-O notation (along with the closely related big-Omega notation, big-Theta notation, and little o notation) describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.
Efficiency of an algorithm
Running time
Memory/storage requirements
Bandwidth/power requirements/logic gates/etc.
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CSE221/ICT221 Analysis and Design of Algorithms

17. CSE221/ICT221 Analysis and Design of Algorithms
In computer science, best, worst and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. Usually the resource being considered is running time, but it could also be memory or other resources.
Best case
The term best-case performance is used in computer science to describe the way an algorithm behaves under optimal conditions.
Worst case
the worst-case execution time is often of particular concern since it is important to know how much time might be needed in the worst case to guarantee that the algorithm will always finish on time.
Average case
Random (equally likely) inputs
Real-life inputs
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CSE221/ICT221 Analysis and Design of Algorithms

18. Growth rate of functions
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CSE221/ICT221 Analysis and Design of Algorithms

19. Classification of functions based on growth rate
asymptotic growth rate, asymptotic order, or order of functions
Comparison of functions by ignoring constant factors and small input
big oh O(g), big theta (g) and big omega (g)
(g): functions with growth rates at least as fast as function g g
(g): functions with growth rates as fast as function g
O(g): functions with growth rates not faster than that of function g
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CSE221/ICT221 Analysis and Design of Algorithms

20. Classifying functions by their Asymptotic Growth Rates
O(g(n)), Big-Oh of g of n, the Asymptotic Upper Bound;
(g(n)), Theta of g of n, the Asymptotic Tight Bound; and
(g(n)), Omega of g of n, the Asymptotic Lower Bound.
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CSE221/ICT221 Analysis and Design of Algorithms

21. CSE221/ICT221 Analysis and Design of Algorithms
Example
Example: f(n) = n2 + 5n + 13.
The constant 13 is not change, when n is larger so there is no significant for considering the lower order terms , which is +5n, when in comparison with the term in the order of n2
Therefore we may sat that f(n) = O(n2)
Question : What is the meaning of f(n) = O(g(n))?
Answer: This means f is the same order of magnitude as g
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CSE221/ICT221 Analysis and Design of Algorithms

22. CSE221/ICT221 Analysis and Design of Algorithms
The meaning of Big O
Q : What is the meaning of f1(n) = O(1)?
A : f1(n) = O(1) means that for all n> a certain value ( i.e. n0 ), f1 will be bounded by a constant value
Q : What is the meaning of f2(n) = O(n log n)?
A : f2(n) = O(n lg n) means that for all n> a certain value ( i.e. n0 ) f2 will be bounded by a constant number times n log n or f2 is in the same order of magnitude as f(n log n).
In general, f(n) = O(g(n)) means f(n) and g(n) are in the same order of magnitude (i.e. O(g(n))
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CSE221/ICT221 Analysis and Design of Algorithms

23. CSE221/ICT221 Analysis and Design of Algorithms
Exercise
What is the different between algorithms and programs?
What factors influence the performance of an algorithm?
How do we measure the performance of algorithms?
What are Big O, Big Theta, Big Omega?
Write Big-O of functions in ascending order
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CSE221/ICT221 Analysis and Design of Algorithms

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