CSC 201 Analysis and Design of Algorithms Lecture 03: Introduction to a CSC 201 Analysis and Design of Algorithms Lecture 03: Introduction to a lgorithms analysis Asst. Dr.Surasak Mungsing Sep-151

17-Sep-15 2 CSC201 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

17-Sep-15 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 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.  CSC201 Analysis and Design of Algorithms

17-Sep , 90, 53, 23, 11, 34 INPUT OUTPUT instance 11 Algorithm m:= a[1]; for I:=2 to size of input if m > a[I] then m:=a[I]; return m Data-Structure Problem: Find the smallest integer from a given set of integers stored in an array Example CSC201 Analysis and Design of Algorithms

17-Sep-15 5 Algorithm A Problem Problem: Find the smallest integer from a given set of integers stored in an array CSC201 Analysis and Design of Algorithms

17-Sep copy the input a to array t1; assign n  size of input; 2.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; 3.Output t2[1]; Algorithm B (use two temporary arrays) CSC201 Analysis and Design of Algorithms

17-Sep Sort the input in increasing order. 2.Return the first element of the sorted data Sorting black box Algorithm C CSC201 Analysis and Design of Algorithms

17-Sep-15 8 Test each data whether it is the smallest one Algorithm D 1.i  0; flag  true; 2.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 CSC201 Analysis and Design of Algorithms

17-Sep-15 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 ระยะเวลาที่ใช้ในการทำงานของอัลกอริธึมจะ เพิ่มขึ้นเมื่อจำนวนข้อมูลนำเข้าเพิ่มขึ้น CSC201 Analysis and Design of Algorithms

17-Sep 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 Correctness, efficiency and measurement model CSC201 Analysis and Design of Algorithms

17-Sep Input Size Time vs. Size of Input 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. CSC201 Analysis and Design of Algorithms

17-Sep-1512 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 CSC201 Analysis and Design of Algorithms

17-Sep-1513 Time required for small size of inputs CSC201 Analysis and Design of Algorithms

17-Sep Time required for intermediate size of inputs CSC201 Analysis and Design of Algorithms

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 Sep-15 CSC201 Analysis and Design of Algorithms

17-Sep-1516 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. CSC201 Analysis and Design of Algorithms

17-Sep-1517Analysis 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 CSC201 Analysis and Design of Algorithms

17-Sep Growth rate of functions CSC201 Analysis and Design of Algorithms

17-Sep-1519 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  (g): functions with growth rates at least as fast as function g  (g): functions with growth rates as fast as function g O(g): functions with growth rates not faster than that of function g CSC201 Analysis and Design of Algorithms

17-Sep-1520 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. CSC201 Analysis and Design of Algorithms

17-Sep-1521 Example  Example: f(n) = n 2 + 5n 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 n 2 Therefore we may sat that f(n) = O(n 2 )  Question : What is the meaning of f(n) = O(g(n))?  Answer: This means f is the same order of magnitude as g CSC201 Analysis and Design of Algorithms

17-Sep-1522 The meaning of Big O  Q : What is the meaning of f 1 (n) = O(1)?  A : f 1 (n) = O(1) means that for all n> a certain value ( i.e. n 0 ), f 1 will be bounded by a constant value  Q : What is the meaning of f 2 (n) = O(n log n)?  A : f 2 (n) = O(n lg n) means that for all n> a certain value ( i.e. n 0 ) f 2 will be bounded by a constant number times n log n or f 2 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)) CSC201 Analysis and Design of Algorithms

17-Sep-1523 Exercise 1. What is the different between algorithms and programs? 2. What factors influence the performance of an algorithm? 3. How do we measure the performance of algorithms? 4. What are Big O, Big Theta, Big Omega? 5. Write Big-O of functions in ascending order CSC201 Analysis and Design of Algorithms

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