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COMP108 Algorithmic Foundations Basics

COMP108 Algorithmic Foundations Basics. Prudence Wong http://www.csc.liv.ac.uk/~pwong/teaching/comp108/201314. Crossing Bridge @ Night. 1 min. each time, 2 persons share a torch they walk @ speed of slower person. Can you do it in 17 mins ?. 2 min. Target: all cross the bridge. 5 min.

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COMP108 Algorithmic Foundations Basics

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  1. COMP108Algorithmic FoundationsBasics Prudence Wong http://www.csc.liv.ac.uk/~pwong/teaching/comp108/201314

  2. Crossing Bridge @ Night 1 min each time, 2 persons share a torch they walk @ speed of slower person Can you do it in 17 mins? 2 min Target: all cross the bridge 5 min 10 min

  3. Module Information Dr Prudence Wong Rm 3.18 Ashton Building, pwong@liverpool.ac.uk office hours: Mon 3-5pm Demonstrators Mr David Hamilton, Miss Alison Liu, Mr Jude-Thaddeus Ojiaku References Main: Introduction to the Design and Analysis of Algorithms. A. V. Levitin. Addison Wesley. Reference: Introduction to Algorithms. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein. The MIT Press

  4. Module Information (2) Teaching, Assessments and Help 33 lectures, 11 tutorials 2 assessments (20%), 1 written exam (80%) Office hours, email Tutorials/Labs Location : Lecture/Seminar Rooms (theoretical) or Lab 1 (practical) Week 2: Theoretical – Lecture/Seminar Rooms

  5. Module Information (3) • Each assessment has two components • Tutorial participation (25%) • Class Test (75%) • Assessment 1 • Tutorials 1 – 5 (Weeks 2-6) • Class Test 1: Week 6, Thu 6th Mar • Assessment 2 • Tutorials 6 – 11 (Weeks 7-12) • Class Test 2: Week 12, Thu 8th May

  6. Why COMP108? Pre-requisite for: COMP202, COMP218(COMP202 is pre-requisite for COMP309) Algorithm design is a foundation for efficient and effective programs "Year 1 modules do not count towards honour classification…" (Career Services: Employers DO consider year 1 module results)

  7. Aims • To give an overview of the study of algorithms in terms of their efficiency. • To introduce the standard algorithmic design paradigms employed in the development of efficient algorithmic solutions. • To describe the analysis of algorithms in terms of the use of formal models of Time and Space. What do we mean by good? How to achieve? Can we prove?

  8. Ready to start … Learning outcomes Able to tell what an algorithm is & have some understanding why we study algorithms Able to use pseudo code to describe algorithm

  9. What is an algorithm? A sequence of precise and concise instructions that guide you (or a computer) to solve a specific problem Daily life examples: cooking recipe, furniture assembly manual(What are input / output in each case?) Algorithm Input Output

  10. 8 10 6 4 12 3 3 5 2 5 3 5 9 7 7 5 6 11 1 5 9 Why do we study algorithms? The obvious solution to a problem may not be efficient Given a map of n cities & traveling cost between them. What is the cheapest way to go from city A to city B? B Simple solution • Compute the cost of each path from A to B • Choose the cheapest one A

  11. Shortest path to go from A to B The obvious solution to a problem may not be efficient How many paths between A & B? involving 1 intermediate city? Simple solution • Compute the cost of each path from A to B • Choose the cheapest one B 2 A

  12. Shortest path to go from A to B The obvious solution to a problem may not be efficient How many paths between A & B? involving 3 intermediate cities? Simple solution • Compute the cost of each path from A to B • Choose the cheapest one B 2 Number of paths = 2 A

  13. Shortest path to go from A to B The obvious solution to a problem may not be efficient How many paths between A & B? involving 3 intermediate cities? Simple solution • Compute the cost of each path from A to B • Choose the cheapest one B 3 Number of paths = 2 Number of paths = 2 + 3 A

  14. Shortest path to go from A to B The obvious solution to a problem may not be efficient How many paths between A & B? involving 3 intermediate cities? Simple solution • Compute the cost of each path from A to B • Choose the cheapest one B Number of paths = 2 + 3 A

  15. Shortest path to go from A to B The obvious solution to a problem may not be efficient How many paths between A & B? involving 3 intermediate cities? Simple solution • Compute the cost of each path from A to B • Choose the cheapest one B Number of paths = 2 + 3 Number of paths = 2 + 3 + 6 = 11 A 6

  16. Shortest path to go from A to B The obvious solution to a problem may not be efficient How many paths between A & B? involving 5 intermediate cities? TOO MANY!! Simple solution • Compute the cost of each path from A to B • Choose the cheapest one B For large n, it’s impossible to check all paths! We need more sophisticated solutions A

  17. Shortest path to go from A to B There is an algorithm, called Dijkstra's algorithm, that can compute this shortest path efficiently. B A

  18. 8 10 6 4 12 3 3 5 2 5 3 5 9 7 7 5 6 11 1 5 9 Idea of Dijkstra's algorithm Go one step from A, label the neighbouring cities with the cost. B 10 0 5 A 7

  19. 8 10 6 4 12 3 3 5 2 5 3 5 9 7 7 5 6 11 1 5 9 Idea of Dijkstra's algorithm Choose city with smallest cost and go one step from there. B 10 0 5 A 7

  20. 8 10 6 4 12 3 3 5 2 5 3 5 9 7 7 5 6 11 1 5 9 Idea of Dijkstra's algorithm This path must NOT be used because we find a cheaper alternative path If an alternative cheaper path is found, record this path and erase the more expensive one. B 10 8 Then repeat & repeat … 0 5 A 7

  21. Shortest path to go from A to B Lesson to learn: Brute force algorithm may run slowly. We need more sophisticated algorithms. B A

  22. How to represent algorithms … Able to tell what an algorithm is and have some understanding why we study algorithms Able to use pseudo code to describe algorithm

  23. Algorithm vs Program An algorithm is a sequence of precise and concise instructions that guide a person/computer to solve a specific problem Algorithms are free from grammatical rules • Content is more important than form • Acceptable as long as it tells people how to perform a task Programs must follow some syntax rules • Form is important • Even if the idea is correct, it is still not acceptable if there is syntax error

  24. Compute the n-th power Input: a number x & a non-negative integer n Output: the n-th power of x Algorithm: • Set a temporary variable p to 1. • Repeat the multiplication p = p * x for n times. • Output the result p.

  25. Pseudo Code C: p = 1; for (i=1; i<=n; i++) p = p * x; printf("%d\n", p); pseudo code: p = 1 for i = 1 to n do p = p * x output p C++: p = 1; for (i=1; i<=n; i++) p = p * x; cout << p << endl; Pascal: p := 1; for i := 1 to n do p := p * x; writeln(p); Java: p = 1; for (i=1; i<=n; i++) p = p * x; System.out.println(p);

  26. Pseudo Code suppose n=4, x=3 Another way to describe algorithm is by pseudo code p = 1 for i = 1 to n do p = p * x output p trace table similar to programming language Combination of both more like English

  27. Pseudo Code: conditional if a < 0 then a = -a b = a output b Conditional statement if condition then statement if condition then statement else statement if a > 0 then b = a else b = -a output b What is computed? absolute value

  28. Pseudo Code: iterative (loop) var automatically increased by 1 after each iteration Iterative statement for var = start_value to end_value do statement while condition do statement repeat statement until condition condition to CONTINUE the loop condition to STOP the loop condition for while loop is NEGATION of condition for repeat-until loop

  29. for loop for var = start_value to end_value do statement Sum of 1st n nos.: input: n sum = 0 for i = 1 to n do begin sum = sum + i end output sum sum=0 i=1 No i <= n? i=i+1 Yes sum = sum+i the loop is executed n times

  30. for loop for var = start_value to end_value do statement suppose n=4 Sum of 1st n nos.: input: n sum = 0 for i = 1 to n do begin sum = sum + i end output sum the loop is executed n times

  31. while loop condition to CONTINUE the loop while condition do statement Sum of 1st n numbers: input: n sum = 0 i = 1 while i <= n do begin sum = sum + i i = i + 1 end output sum • Do the same as for-loop in previous slides • It requires to increment i explicitly

  32. while loop – example 2 execute undetermined number of times Sum of all input numbers: sum = 0 while (user wants to continue) do begin ask for a number sum = sum + number end output sum No continue? Yes ask for number sum = sum+number

  33. repeat-until repeat statement until condition ask for number sum = sum+number Stop? No condition to STOP the loop Yes Sum of all input numbers: sum = 0 repeat ask for a number sum = sum + number until (user wants to stop) output sum • also execute undetermined number of times • How it differs from while-loop? execute at least once

  34. More Example 1 suppose x=14, y=4 input: x, y r = x q = 0 while r = y do begin r = r − y q = q + 1 end output r and q remainder & quotient suppose x=14, y=5 suppose x=14, y=7 What is computed?

  35. suppose x=12, y=4 More Example 2 all common factors input: x, y if x < y then swap x & y i = y while i = 1 do begin if x%i==0 && y%i==0 then output i i = i-1 end suppose x=15, y=6 a%b remainder ofa divided b What values are output?

  36. More Example 3 Highest Common Factor (HCF) Greatest Common Divisor (GCD) input: x, y if x < y then swap x & y i = y found = false while i = 1 && !found do begin if x%i==0 && y%i==0 then found = true else i = i-1 end output i What value is output? • Questions: • what value of found makes the loop stop? • when does found change to such value?

  37. Pseudo Code: Exercise assuming x and y are both integers Write a while-loop to • Find the product of all integers in interval [x, y] • e.g., if x is 2 & y is 5, then output is 2*3*4*5 = 120 product = ?? i = ?? while ?? do begin ?? i = ?? end output ??

  38. Pseudo Code: Exercise assuming x and y are both integers Write a while-loop to • Find the product of all integers in interval [x, y] product = 1 i = x while i <= y do begin product = product * i i = i+1 end output product

  39. Pseudo Code: Exercise 2 Write a while-loop for this: • Given two positive integers x and y, list all factors of x which are not factors of y • if x is 30 & y is 9, output is 2, 5, 6, 10, 15, 30 (not 1 or 3) i = ?? while ?? do begin if ?? then output ?? i = ?? end

  40. Pseudo Code: Exercise 2 Write a while-loop for this: • Given two positive integers x and y, list all factors of x which are not factors of y i = 1 while i <= x do begin if x%i == 0 && y%i != 0 then output i i = i+1 end

  41. Challenges … Convert while-loops to for-loops & repeat-loop

  42. Convert to for loops Find the product of all integers in interval [x, y] assuming x and y are both integers product = ?? for i = ?? to ?? do begin ?? end output ??

  43. Convert to for loops Find the product of all integers in interval [x, y] assuming x and y are both integers product = 1 for i = x to y do begin product = product * i end output product

  44. Convert to repeat loops Find the product of all integers in interval [x, y] assuming x and y are both integers product = 1 i = x repeat product = product * i i = i+1 until i > y output product What’s wrong with this? If given x is larger than y?

  45. Convert to repeat loops Find the product of all integers in interval [x, y] assuming x and y are both integers product = 1 if x <= y then begin i = x repeat product = product * i i = i+1 until i > y end output product

  46. Convert to for loops (2) Given two positive integers x and y, list all factors of x which are not factors of y for i = ?? to ?? do begin if ?? then ?? end

  47. Convert to for loops (2) Given two positive integers x and y, list all factors of x which are not factors of y for i = 1 to x do begin if x%i == 0 && y%i != 0 then output i end

  48. Convert to repeat loops (2) Given two positive integers x and y, list all factors of x which are not factors of y i = ?? repeat if ?? then ?? i = ?? until ??

  49. Convert to repeat loops (2) Given two positive integers x and y, list all factors of x which are not factors of y i = 1 repeat if x%i==0 && y%i!=0 then output i i = i+1 until i > x

  50. Searching …

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