Lecture_Module3 Prof Saroj Kaushik, IIT Delhi

Transcription

1 Lecture_Module3

2 Problem definition Method (how to solve it) Algorithm Data structure Verification for correctness Analysis for efficiency Coding in given programming language Understanding of computer architecture Compilation, testing, de-bugging Documentation 2

3 Most difficult Requires interaction between programmer and user Specs include: Input data Type, accuracy, units, range, format, location, sequence Special symbols to signal end of data Output data (results) type, accuracy, units, range, format, location, headings How is output related to input Any special constraint Example: find the phone no. of a person Problems get revised often 3

4 It is a finite set of instructions which, if followed accomplish a particular task. It is basically used to describe a problem solving method suitable for implementation as a computer program. Algorithm is independent of the machine and language used for implementation. 4

5 Input Zero or more quantities are supplied externally Output At least one quantity is produced Definiteness Each instruction is clear & unambiguous Finiteness It terminates after finite steps Effectiveness Each instruction is simple to be carried out manually. 5

6 An unambiguous specification of a method Is characterized by: Ordered sequence of well-defined, effective operations that, when executed, will produce a result after terminating within a finite no of steps 6

7 Well-defined and effective No ambiguity, a method must exist Good Examples: Add 1 to x compute largest prime no. < 100 compute square root of x to 4 decimal places Bad examples: divide 10 by x compute largest prime compute square root of x 7

8 Always terminate, and be sure about it Produce correct results this may require some hard thinking testing helps, but is not adequate 8

9 Basic differences are: Program is written in programming language whereas algorithm is in English like pseudo language. Program may be non terminating (OS) whereas algorithm should terminate in finite steps. 9

10 Study of algorithm can be classified in four distinct areas namely how to devise express validate analyze algorithms 10

11 Devising good algorithm: Requires study of various design techniques Top down, bottom up and object oriented approaches Expressing an algorithm: Good algorithms are expressed using principle of structured programming Validation of algorithm: It should be validated for correctness of all possible legal inputs. (correct, incorrect, exceptions) Note that algorithm need not yet be expressed as a computer program. 11

12 Analysis of an algorithm: Study of behavior pattern or performance profile. It can be calculated in terms of computing time and space requirement in the machine. Time Complexity: Running time of the program as a function of the size of input. Space Complexity: Amount of computer memory required during the program execution. 12

13 In top-down model, an overview of the system is formulated, without going into detail for any part of it. Each part of the system is then refined in more details. Each new part may then be refined again, defining it in yet more details until the entire specification is detailed enough to validate the model. 13

14 This design model can also be applied while developing algorithm. It basically refers to successive refinement of the problem (task) into sub problems (subtasks). Refinement is applied until we reach to the stage where the subtasks can be directly carried out. 14

15 Main Task subtask1 subtask2 subtask3 15

16 In bottom-up design individual parts of the system are specified in details. The parts are then linked together to form larger components, which are in turn linked until a complete system is formed. Object-oriented languages such as C++ or JAVA use bottom-up approach where each object is identified first. 16

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18 It is a technique using which one can write algorithms (programs) in top down fashion. This technique should be used with every level of refinement. In SP, one entry and one exit principle is adopted in all the constructs. Basically there are three structures in SP 18

19 Sequential (sequence) entry T1 T2 exit 19

20 If cond then task1 Cond N Y task1 If cond then task1 else task2 task2 N Cond Y task1 20

21 While While (cond) do { } Repeat Repeat Until (cond) 21

22 for ordered sequence T1, while C then T2,T3 T1 N C T2 Y T3 22

23 for ordered sequence T1, Repeat T2 until C,T3 T1 T2 N C T3 Y 23

24 What is the difference between: S1 S1 S2 C N Y Y C N S2 S3 S3 S1, Repeat S2 until C, S3 S1, While C { S2 }, S3 24

25 Computation of income tax Given a tax table as below, compute the tax, T, on an income, X. INCOME TAX 0 <= INC <= <= INC <= *(INC ) <= INC <= *(INC ) <= INC *(INC ) Here is an algorithm: Step 1: Input INC; Step 2: Compute tax, T; Step 3: Output T 25

26 Step 1: Input INC; Step 2a: if INC > then T *(INC ); Step 2b: if INC > and INC then T *(INC ); Step 2c: if INC > and INC then T *(INC ); Step 2d: if INC then T 0; Step 3: Output T 26

27 Table look-up: Consider: special key K, length of list, 5 or N, more generally unsorted L = [(x1, y1), (x2, y2), (x3, y3), (x4, y4), (x5, y5)], where xi is key and yi is corresponding output value Problem is to search whether key K is in the list and output relevant pair if it exist Here is an algorithm: Step 1: Input all data, K, and list L; Step 2: Search for K in L; Step 3: Output results 27

28 Here is refined version of the algorithm: Input K; Input (x1, y1); Input (x2, y2); Input (x3, y3); Input (x4, y4); Input (x5, y5); If K = x1 then output (x1, y1); If K = x2 then output (x2, y2); If K = x3 then output (x3, y3); If K = x4 then output (x4, y4); If K = x5 then output (x5, y5) 28

29 This algorithm does the same thing, except that it is compact: Input K; Repeat these operations 5 times: { Input (x, y); If K = x then output (x, y) } 29

30 An even better algorithm: Input n; If n > 0 then { Input K; Repeat these operations n times: { Input (x, y); If K = x then output (x, y) } } 30

31 Formula: x1 = [(-b) + (b 2 4ac)]/2a x2 = [(-b) - (b 2 4ac)]/2a Algorithm: input a, b, c; d = b*b 4*a*c; if (d 0) then { e = sqrt(d); r1 = (-b + e) / (2*a); r2 = (-b - e) / (2*a); i1=0; i2 =0 } else { e = sqrt(-d); r1 = (-b) / (2*a); r2 = r1; i1= e / (2*a); i2 = -e/(2*a); } output r1,r2,i1,i2

32 Formula: fact = n * n-1 * * 2 * 1 Algorithm: input n; i = 0; initialization fact = 1; looping while (i < n) { i = i + 1; fact = i * fact; } output fact Execution Let n = 4; i = 0; fact = 1; while loop 0 < 4; i = i +1=1; fact = 1 * 1; 1 < 4; i = i +1=2; fact = 1 * 2; 2 < 4; i = i +1=3; fact = 2 * 3; 3 < 4; i = i +1=4; fact = 6 * 4 4 < 4 is false so exit the while loop; output fact = 24 32

33 Algorithm: input num; rev = 0; while (num > 0) { rev = rev*10 + num mod 10; num = num div 10; } output rev Execution num = 245; rev = 0; while loop rev = 0 * = 5; num = 24; rev = 5 * = 54; num = 2; rev = 54 * = 542; num = 0; exit while output rev as

34 Algorithm: Here number is read in the loop. At the end of loop, read elements are not available. max = 0; i=0; while (i < 100) { i = i+1; input num; if (num > max) then max = num; } output max 34

35 Algorithm: input num(i), i = 1,100; max = 0; i = 0; while (i < 100) { i = i+1; if (num(i) > max) then max = num(i); } output max, num(i), i = 1,100 Execution Let numbers are: 4, 2,7,1 max = 0; i = 0; while loop i = 1; 4 > max so max = 4; i = 2; 2 < max so no change i = 3; 7 > max so max = 7; i = 4; 1 < max so no change exit of while output max as 7 and numbers as 4, 2, 7, 1 35

36 Series:f=0!+1!+2!+ +n! (n 0) Algorithm: Here we are given the value of n. Series sum is calculated by finding sum after adding previous sum with a term at each step. input n; f=0;term=1;i=0; while (i < n) { f=f+term; i=i+1; term = term * i; } output f