1. 1 Objectives By the end of this lecture, students should: understand what an algorithms is appreciate the role of algorithm definitions in the problem solving process know the basic components of algorithms understand the difference between algorithms and programs be able to formulate algorithms for simple problems Reading: Walter Savitch. An Introduction to Problem Solving and Programming, Pearson, 2004. Sec. 1.1-1.3

2. 2 How do we solve problems? We "just do" Guesswork-and-luck Trial-and-error Experience (possibly someone else's) Systematically !

3. 3 sterilize(saw,alcohol); raise_hammer(); lower hammer(fast); start(saw); /* etc. etc. */ The Problem-solving Process Problem specification Algorithm Program Executable (solution) Analysis Design Implementation Compilation " Doctor, my head hurts" Patient has elevated pressure in anterior parietal lobe. 1. Sterilize cranial saw 2. Anaesthetize patient 3. Remove top of skull 4. Get the big spoon... 5. etc., etc. 01001110101100101010101010010 10101010100110010101010101001 011010011101010101010010010111 010011110101010111110101010001 10100001101...

4. 4 Problem Idea AlgorithmProgram Informal Description Refinement Coding Testing & Debugging...its not a linear process

5. 5 A sequence of instructions specifying the steps required to accomplish some task Named after: Muhammad ibn Musa al-Khwarizmi of Khowarezm (now Khiva in Uzbekistan) Circa 780-850 C.E. (Common Era) Algorithm

6. 6 Algorithm –History Muhammad ibn Musa Al-Khwarizmi http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Al-Khwarizmi.html Book on arithmetic: –Hindu numeration, decimal numbers, use of zero, method for finding square root –Latin translation (c.1120 CE): Algoritmi de numero Indorum Book on algebra –Hisab al-jabr wal-muqabala

7. 7 A sequence of instructions describing how to do a task Algorithm – Working Definition [ As opposed to actually executing the instructions]

8. 8 Algorithm -- Examples A cooking recipe Assembly instructions for a model The rules of how to play a game VCR instructions Description of a martial arts technique Directions for driving from A to B A knitting pattern A car repair manual

9. 9 Algorithm -- Examples A cooking recipe Assembly instructions for a model The rules of how to play a game VCR instructions Description of a martial arts technique Directions for driving from A to B A knitting pattern A car repair manual

10. 10 Algorithms are not Programs Algorithms are well-defined sequence of unambiguous instructions must terminate (to produce a result) Algorithm description relies on a well-defined instruction language Example: Manual Addition Describe the method! 123456 +789001 912457

11. 11 Algorithm – Examples (cont) Recipe for Almond and honey slice Recipe for Arroz con pollo

12. 12 Almond and Honey Slice 1/2 quantity Shortcrust Pastry 185 g unsalted butter 100 g castor sugar 5 tablespoons honey 50 ml cream 50 ml brandy or any other liqueur or spirit 300 g flaked almonds Preheat oven for 200° C Line a 30 cm 20 cm baking tray with baking paper, and then with pastry Bake blind for 20 minutes, then remove weights and foil Turn oven up to 220° C. Bring remaining ingredients to a boil, stirring. Spread evenly over pastry. Bake until topping is bubbling and has caramelised evenly, about 15 minutes. Cool before cutting into fingers or squares. From: Stephanie Alexander, The Cooks Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.

13. 13 Almond and Honey Slice 1/2 quantity Shortcrust Pastry 185 g unsalted butter 100 g castor sugar 5 tablespoons honey 50 ml cream 50 ml brandy or any other liqueur or spirit 300 g flaked almonds Preheat oven for 200° C Line a 30 cm 20 cm baking tray with baking paper, and then with pastry Bake blind for 20 minutes, then remove weights and foil Turn oven up to 220° C. Bring remaining ingredients to a boil, stirring. Spread evenly over pastry. Bake until topping is bubbling and has caramelised evenly, about 15 minutes. Cool before cutting into fingers or squares. From: Stephanie Alexander, The Cooks Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349. Instructions are given in the order in which they are performed (executed)

14. 14 Correct Algorithm? Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil. Remove the chicken and to the juices in the casserole add garlic, onions and green peppers, and sauté until onion is golden. Add bay leaf, whole tomatoes, and chicken broth. When the broth boils add salt, saffron and rice. Arrange chicken on rice, cover casserole and bake in a moderate oven (350°F) for 20 minutes or until the rice is tender. Add beans and artichokes during last 10 minutes of cooking. From: Arroz Con Pollo in The Margaret Fulton Cookbook, Hamlyn, Sydney, 1968.

15. 15 Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil. Remove the chicken and to the juices in the casserole add garlic, onions and green peppers, and sauté until onion is golden. Add bay leaf, whole tomatoes, and chicken broth. When the broth boils add salt, saffron and rice. Arrange chicken on rice, cover casserole and bake in a moderate oven (350°F) for 20 minutes or until the rice is tender. Add beans and artichokes during last 10 minutes of cooking. From: Arroz Con Pollo in The Margaret Fulton Cookbook, Hamlyn, Sydney, 1968. Correct Algorithm?

16. 16 Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil. Remove the chicken and to the juices in the casserole add garlic, onions and green peppers, and sauté until onion is golden. Add bay leaf, whole tomatoes, and chicken broth. When the broth boils add salt, saffron and rice. Arrange chicken on rice, cover casserole and bake in a moderate oven (350°F) for 10 minutes. Add beans and artichokes. Cover, and bake for another 10 minutes or until rice is tender. Correct Algorithm?

17. 17 From Algorithms to Programs Problem Program Program Algorithm Algorithm: A sequence of instructions describing how to do a task (or process)

18. 18 Algorithms are not Programs always design the algorithm before you start to program, the idea comes before the coding.

19. 19 Example How to get out?

20. 20 Example repeat if no wall left then turn left, step ahead else if no wall ahead then step ahead else turn right until at exit run through the maze always sticking to the wall on your left

21. 21 Algorithms and Languages the very same Algorithm may look very different when described (or implemented) in different (programming-) languages, e.g. Sum up the numbers between 1 and 10 by adding each of them to a total starting with 0 s +/( 10) APL-style int s:=0; for i = 0 to 10 do s:= s+i; end Pascal-style

22. 22 Components of an Algorithm Input and Output Specification Variables and values Instructions –Sequences –Selections –Repetitions Abstraction Mechanisms –Objects –Methods Also required: Documentation

23. 23 Values Represent quantities, amounts or measurements May be numerical or alphabetical (or of some other defined type) Example: –Recipe ingredients

24. 24 Almond and Honey Slice 1/2 quantity Shortcrust Pastry 185 g unsalted butter 100 g castor sugar 5 tablespoons honey 50 ml cream 50 ml brandy or any other liqueur or spirit 300 g flaked almonds Preheat oven for 200° C Line a 30 cm 20 cm baking tray with baking paper, and then with pastry Bake blind for 20 minutes, then remove weights and foil Turn oven up to 220° C. Bring remaining ingredients to a boil, stirring. Spread evenly over pastry. Bake until topping is bubbling and has caramelised evenly, about 15 minutes. Cool before cutting into fingers or squares. From: Stephanie Alexander, The Cooks Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.

25. 25 Almond and Honey Slice 1/2 quantity Shotcrust Pastry 185 g unsalted butter 100 g castor sugar 5 tablespoons honey 50 ml cream 50 ml brandy or any other liqueur or spirit 300 g flaked almonds Preheat oven for 200° C Line a 30 cm 20 cm baking tray with baking paper, and then with pastry Bake blind for 20 minutes, then remove weights and foil Turn oven up to 220° C. Bring remaining ingredients to a boil, stirring. Spread evenly over pastry. Bake until topping is bubbling and has caramelised evenly, about 15 minutes. Cool before cutting into fingers or squares. From: Stephanie Alexander, The Cooks Companion, Viking/Penguin, Ringwood, Victoria, 1996, p. 349.

26. 26 Variables This jar can contain 10 cookies 50 grams of sugar 3 slices of cake etc. ValuesVariable Are containers for values – places to store values Example:

27. 27 Restrictions on Variables Variables may be restricted to contain a specific type of value

28. 28 Components of an Algorithm Values and Variables Instruction (a.k.a. primitive) Sequence (of instructions) Selection (between instructions) Repetition (of instructions) Abstractions (with Objects and Methods) Documentation (beside instructions)

29. 29 Instructions (Primitives) Some action that is –simple –unambiguous –that the system knows about... –...and should be able to actually do

30. 30 Instructions – Examples Take off your shoes Count to 10 Cut along dotted line Knit 1 Purl 2 Pull rip-cord firmly Sift 10 grams of arsenic Directions to perform specific actions on values and variables.

31. 31 Instructions -- Application Some instructions can only be applied to a specific type of values or variables Examples:

32. 32 Instructions (Primitives) -- Recommendations When writing an algorithm, make each instruction simple and unambiguous Example: Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil. Cut chicken into pieces. Heat olive oil in a casserole dish. Brown the chicken pieces in the casserole dish.

33. 33 Components of an Algorithm Values and Variables Instruction (a.k.a. primitive) Sequence (of instructions) Selection (between instructions) Repetition (of instructions) Abstractions (with Objects and Methods) Documentation (beside instructions)

34. 34 Instruction (Primitives) When writing an algorithm, make the instructions simple and unambiguous. Example: Cut chicken into pieces and brown the pieces on all sides in a casserole dish in hot olive oil. Cut chicken into pieces. Heat olive oil in a casserole dish. Brown the chicken pieces in the casserole dish. A sequence of simple instructions

35. 35 Sequence A series of instructions...to be carried out one after the other......without hesitation or question Example: –How to cook a Gourmet Meal TM

36. 36 Sequence -- Example 1. Open freezer door 2. Take out Gourmet Meal 3. Close freezer door 4. Open microwave door 5. Put Gourmet Meal on carousel 6. Shut microwave door 7. Set microwave on high for 5 minutes 8. Start microwave 9. Wait 5 minutes 10. Open microwave door 11. Remove Gourmet Meal 12. Close microwave door

37. 37 Components of an Algorithm Values and Variables Instruction (a.k.a. primitive) Sequence (of instructions) Selection (between instructions) Repetition (of instructions) Abstractions (with Objects and Methods) Documentation (beside instructions)

38. 38 Selection An instruction that decides which of two possible sequences is executed The decision is based on a single true/false condition Examples: –Car repair –Reciprocals

39. 39 Selection Example -- Car Repair if (motor turns) then { CheckFuel CheckSparkPlugs CheckCarburator } else { CheckStarterMotor CheckEngineBlock }

40. 40 Selection Example – Car Repair (cont) Should be a true or false condition. if (motor turns) then { CheckFuel CheckSparkPlugs CheckCarburator } else { CheckStarterMotor CheckEngineBlock }

41. 41 Selection Example -- Car Repair (cont) Sequence if the condition is true. if (motor turns) then { CheckFuel CheckSparkPlugs CheckCarburator } else { CheckStarterMotor CheckEngineBlock }

42. 42 Selection Example -- Car Repair (cont) Sequence if the condition is false. if (motor turns) then { CheckFuel CheckSparkPlugs CheckCarburator } else { CheckStarterMotor CheckEngineBlock }

43. 43 Selection Example -- Reciprocals Q. Give an algorithm for computing the reciprocal of a number. Examples: Reciprocal of 2: 1/2 Reciprocal of -3/4: 1/(-3/4) = -4/3 Reciprocal of 0: undefined

44. 44 Selection Example – Reciprocals (cont) Q. Give an algorithm for computing the reciprocal of a number. Algorithm: input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity" }

45. 45 Selection Example-- Reciprocals input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity" } Algorithm: Num is a variable whose value depends on the actual number the user provides.

46. 46 Selection Example – Reciprocals (cont) input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity" } Algorithm: Condition depends on the value of Num

47. 47 Selection Example – Reciprocals (cont) input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity" } Algorithm: For a given value of Num, only one of these two sequences can be executed

48. 48 Selection Example – Reciprocals (cont) input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity" } Algorithm: Executed if Num is not equal to 0

49. 49 Selection Example – Reciprocals (cont) input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity" } Algorithm: Executed if Num is equal to 0

50. 50 Selection -- Exercise input Num if (Num is not equal 0) then { output 1/Num } else { output "infinity" } Will the following algorithms produce the same output? Algorithm 1: input Num if (Num is not equal 0) then { output 1/Num } output "infinity" Algorithm 2:

51. 51 Selection – Several Conditions What if several conditions need to be satisfied? if ( today is Wednesday and the time is 10.00am ) then { Go to CSE1301 Lecture } else { Go to Library } Solution 1

52. 52 Selection – Several Conditions (cont) Solution 2 Often called anested selection if ( today is Wednesday ) then { if ( the time is 10.00am ) then { Go to CSE1301 Lecture } else...etc...etc...etc...

53. 53 Selection – At Least One of Several Conditions What if at least one of several conditions needs to be satisfied? if ( I feel hungry or the time is 1.00pm or my mate has his eye on my lunch ) then { Eat my lunch now }

54. 54 Components of an Algorithm Values and Variables Instruction (a.k.a. primitive) Sequence (of instructions) Selection (between instructions) Repetition (of instructions) Abstractions (with Objects and Methods) Documentation (beside instructions)

55. 55 Repetition Repeat an instruction... –...while (or maybe until) some condition occurs –Test the condition each time before repeating the instruction Also known as iteration or loop Example: –Algorithm for leaving the maze

56. 56 Example repeat if no wall left then turn left, step ahead else if no wall ahead then step ahead else turn right until at exit run through the maze always sticking to the wall on your left Danger: endless loop if termination test not correct

57. 57 Components of an Algorithm Values and Variables Instruction (a.k.a. primitive) Sequence (of instructions) Selection (between instructions) Repetition (of instructions) Abstractions (with Objects and Methods) => Object-orientation (next lectures) Documentation (beside instructions)

58. 58 Documentation Records what the algorithm does Describes how it does it Explains the purpose of each component of the algorithm Notes restrictions or expectations Example: –Getting a date

59. 59 Documentation -- Example Think of something romantic to do decide on time and location Work through address book to look for a person initialise booking to unsuccessful until (successfully booked) { get next Name in little black book AskOnDate(Name, Time, Location) DetermineBookingSuccess } Assumes that I will find someone in the book before it runs out SighWithRelief

60. 60 Components of an Algorithm Values and Variables Instruction (a.k.a. primitive) Sequence (of instructions) Selection (between instructions) Repetition (of instructions) Abstractions (with Objects and Methods) => Object-orientation (next week) Documentation (beside instructions)

61. 61 Analysis of Algorithms we are interested in... Well-definedness (otherwise its not an algorithm) Correctness (does it fulfill the specification?) Termination (does it always end?) Resource consumption (time and memory)

62. 62 Example picking the lightest item... 1.put first item on scale 2.memorize weight 3.take item from scale in hand 4.repeat 5.put next item on scale 6.compare to memorized weight 7.take lighter item in hand; put other item away 8.until no more items left...is this well-defined?...can we be sure that this always terminates with the correct item in hand?

63. 63 The Software Development Process Define the problem clearly Analyse the problem thoroughly Design an algorithm carefully Code the algorithm efficiently Test the code thoroughly Document the system lucidly

64. 64 Summary Problem Solving Process Algorithms Components of Algorithms –Values and Variables –Instructions –to be continued…(with Abstractions) Documentation Analysis of Algorithms