Introduction to Algorithm Analysis - Understanding Algorithms and Their Efficiency

1. CSE 221/ICT221 Analysis and Design of AlgorithmsLecture 03:Introduction to algorithms analysis Asst. Dr.SurasakMungsing E-mail: Surasak.mu@spu.ac.th

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 ofproblem - Webster Dictionary

3. 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 programin 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. • http://wiki.answers.com/Q/What_is_the_differences_between_algorithm_and_program#ixzz1GXvHCUbb CSE221/ICT221 Analysis and Design of Algorithms

4. 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 11 25, 90, 53, 23, 11, 34 Data-Structure CSE221/ICT221 Analysis and Design of Algorithms

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

6. 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]; CSE221/ICT221 Analysis and Design of Algorithms

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

8. 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 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 CSE221/ICT221 Analysis and Design of Algorithms

10. 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 CSE221/ICT221 Analysis and Design of Algorithms

11. 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. 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 CSE221/ICT221 Analysis and Design of Algorithms

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

14. Time required for intermediate size of inputs 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

16. Asymptotic Performance • 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. CSE221/ICT221 Analysis and Design of Algorithms

17. Analysis 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 CSE221/ICT221 Analysis and Design of Algorithms

18. Growth rate of functions 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):functions with growth rates as fast as function g g O(g):functions with growth rates not faster than that of function g CSE221/ICT221 Analysis and Design of Algorithms

20. Classifying functions by theirAsymptotic 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. CSE221/ICT221 Analysis and Design of Algorithms

21. Example • Example: f(n) = n2 + 5n + 13. The constant 13 is not change, when nis larger so there is no significant for considering the lower order terms , which is +5n, when in comparison with the term in the order ofn2 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 CSE221/ICT221 Analysis and Design of Algorithms

22. 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)) CSE221/ICT221 Analysis and Design of Algorithms

23. 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 CSE221/ICT221 Analysis and Design of Algorithms