Algorithms and Data Structures
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1 Algorithms and Data Structures Course Introduction Antonio Carzaniga Faculty of Informatics University of Lugano February 21, 2012
2 General Information On-line course information on Moodle: INFO.B018 and on my web page: last edition also on-line:
3 General Information On-line course information on Moodle: INFO.B018 and on my web page: last edition also on-line: Announcements you are responsible for reading the announcements page or the messages sent through Moodle
4 General Information On-line course information on Moodle: INFO.B018 and on my web page: last edition also on-line: Announcements you are responsible for reading the announcements page or the messages sent through Moodle Office hours Antonio Carzaniga: by appointment Koorosh Khazaei: by appointment Elena Khramtcova: by appointment
5 Textbook Introduction to Algorithms Third Edition Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein The MIT Press
6 Additional Readings (Textbook 2) Algorithms Sanjoy Dasgupta Christos H. Papadimitriou Umesh V. Vazirani Mc Graw-Hill
7 +30% projects Evaluation 3 5 assignments grades added together, thus resulting in a weighted average +30% midterm exam +40% final exam ±10% instructor s discretionary evaluation participation extra credits trajectory...
8 +30% projects Evaluation 3 5 assignments grades added together, thus resulting in a weighted average +30% midterm exam +40% final exam ±10% instructor s discretionary evaluation participation extra credits trajectory % plagiarism penalties
9 Plagiarism Antonio Carzaniga
10 Plagiarism A student should never take someone else s material and present it as his or her own. Doing so means committing plagiarism.
11 Plagiarism A student should never take someone else s material and present it as his or her own. Doing so means committing plagiarism. material means ideas, words, code, suggestions, corrections on one s work, etc. Using someone else s material may be appropriate e.g., software libraries always clearly identify the external material, and acknowledge its source. Failing to do so means committing plagiarism. the work will be evaluated based on its added value
12 Plagiarism A student should never take someone else s material and present it as his or her own. Doing so means committing plagiarism. material means ideas, words, code, suggestions, corrections on one s work, etc. Using someone else s material may be appropriate e.g., software libraries always clearly identify the external material, and acknowledge its source. Failing to do so means committing plagiarism. the work will be evaluated based on its added value Plagiarism on an assignment or an exam will result in failing that assignment or that exam loosing one or more points in the final note! Penalties may be escalated in accordance with the regulations of the Faculty of Informatics
13 Deadlines Antonio Carzaniga
14 Deadlines Deadlines are firm.
15 Deadlines Deadlines are firm. Exceptions may be granted at the instructor s discretion only for documented medical conditions or other documented emergencies
16 Deadlines Deadlines are firm. Exceptions may be granted at the instructor s discretion only for documented medical conditions or other documented emergencies Each late day will reduce the assignment s grade by one third of the total value of that assignment
17 Deadlines Deadlines are firm. Exceptions may be granted at the instructor s discretion only for documented medical conditions or other documented emergencies Each late day will reduce the assignment s grade by one third of the total value of that assignment Corollary 1. The grade of an assignment turned in more than two days late is 0
18 Deadlines Deadlines are firm. Exceptions may be granted at the instructor s discretion only for documented medical conditions or other documented emergencies Each late day will reduce the assignment s grade by one third of the total value of that assignment Corollary 1. The grade of an assignment turned in more than two days late is 0 The proof of Corollary 1 is left as an exercise
19 Now let s move on to the real interesting and fun stuff Antonio Carzaniga
20 Fundamental Ideas Antonio Carzaniga
21 Fundamental Ideas Johannes Gutenberg invents movable type and the printing press in Mainz, circa 1450 AD (already known in China, circa 1200 AD)
22 Maybe More Fundamental Ideas Antonio Carzaniga
23 Maybe More Fundamental Ideas The decimal numbering system (India, circa 600 AD)
24 Maybe More Fundamental Ideas The decimal numbering system (India, circa 600 AD) Persian mathematician Khwārizmī writes a book (Baghdad, circa 830 AD) Muhammad ibn Musa al-khwārizmī
25 Maybe More Fundamental Ideas The decimal numbering system (India, circa 600 AD) Persian mathematician Khwārizmī writes a book (Baghdad, circa 830 AD) methods for adding, multiplying, and dividing numbers (and more) Muhammad ibn Musa al-khwārizmī
26 Maybe More Fundamental Ideas The decimal numbering system (India, circa 600 AD) Persian mathematician Khwārizmī writes a book (Baghdad, circa 830 AD) methods for adding, multiplying, and dividing numbers (and more) these procedures were precise, unambiguous, mechanical, efficient, and correct Muhammad ibn Musa al-khwārizmī
27 Maybe More Fundamental Ideas The decimal numbering system (India, circa 600 AD) Persian mathematician Khwārizmī writes a book (Baghdad, circa 830 AD) methods for adding, multiplying, and dividing numbers (and more) these procedures were precise, unambiguous, mechanical, efficient, and correct they were algorithms! Muhammad ibn Musa al-khwārizmī
28 Algorithms are the essence of computer programs
29 Algorithms are the essence of computer programs
30 Algorithms are the essence of computer programs
31 Algorithms are the essence of computer programs
32 Algorithms are the essence of computer programs
33 Example A sequence of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...
34 Example A sequence of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... The well-known Fibonacci sequence Leonardo da Pisa (ca ca. 1250) son of Guglielmo Bonaccio a.k.a. Leonardo Fibonacci
35 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1
36 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1 Implementation on a computer: Scheme (define (F n) (cond ((= n 0) 0) ((= n 1) 1) (else (+ (F (- n 1)) (F (- n 2))))))
37 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1 Implementation on a computer: Java public class Fibonacci { public static int F(int n) { if (n == 0) { return 0; } else if (n == 1) { return 1; } else { return F(n-1) + F(n-2); } } }
38 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1 Implementation on a computer: C or C++ int F(int n) { if (n == 0) { return 0; } else if (n == 1) { return 1; } else { return F(n-1) + F(n-2); } }
39 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1 Implementation on a computer: Ruby def F(n) case n when 0 return 0 when 1 return 1 else return F(n-1) + F(n-2) end end
40 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1 Implementation on a computer: Python def F(n): if n == 0: return 0 elif n == 1: return 1 else: return F(n-1) + F(n-2)
41 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1 Implementation on a computer: very concise C/C++ (or Java) int F(int n) { return (n<2)?n:f(n-1)+f(n-2); }
42 The Fibonacci Sequence 0 if n = 0 Mathematical definition: F n = 1 if n = 1 F n 1 + F n 2 if n > 1 Implementation on a computer: pseudo-code Fibonacci(n) 1 if n = = 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2)
43 Questions on Our First Algorithm Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2)
44 Questions on Our First Algorithm Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) 1. Is the algorithm correct? for every valid input, does it terminate? if so, does it do the right thing?
45 Questions on Our First Algorithm Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) 1. Is the algorithm correct? for every valid input, does it terminate? if so, does it do the right thing? 2. How much time does it take to complete?
46 Questions on Our First Algorithm Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) 1. Is the algorithm correct? for every valid input, does it terminate? if so, does it do the right thing? 2. How much time does it take to complete? 3. Can we do better?
47 Correctness Antonio Carzaniga
48 Correctness Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) 0 if n = 0 F n = 1 if n = 1 F n 1 + F n 2 if n > 1
49 Correctness Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) 0 if n = 0 F n = 1 if n = 1 F n 1 + F n 2 if n > 1 The algorithm is clearly correct assuming n 0
50 Performance How long does it take?
51 Performance How long does it take? Let s try it out...
52 Results running time (seconds) Ruby Scheme Python C C-wiz Java C-gcc n
53 Comments Antonio Carzaniga
54 Different implementations perform differently Comments it is better to let the compiler do the optimization simple language tricks don t seem to pay off
55 Different implementations perform differently Comments it is better to let the compiler do the optimization simple language tricks don t seem to pay off However, the differences do not seem to be substantial all implementations sooner or later seem to hit a wall...
56 Different implementations perform differently Comments it is better to let the compiler do the optimization simple language tricks don t seem to pay off However, the differences do not seem to be substantial all implementations sooner or later seem to hit a wall... Conclusion: the problem is with the algorithm
57 Complexity of Our First Algorithm We need a mathematical characterization of the performance of the algorithm We ll call it the algorithm s computational complexity
58 Complexity of Our First Algorithm We need a mathematical characterization of the performance of the algorithm We ll call it the algorithm s computational complexity Let T (n) be the number of basic steps needed to compute Fibonacci(n)
59 Complexity of Our First Algorithm We need a mathematical characterization of the performance of the algorithm We ll call it the algorithm s computational complexity Let T (n) be the number of basic steps needed to compute Fibonacci(n) Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2)
60 Complexity of Our First Algorithm We need a mathematical characterization of the performance of the algorithm We ll call it the algorithm s computational complexity Let T (n) be the number of basic steps needed to compute Fibonacci(n) Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) T (0) = 2; T (1) = 3
61 Complexity of Our First Algorithm We need a mathematical characterization of the performance of the algorithm We ll call it the algorithm s computational complexity Let T (n) be the number of basic steps needed to compute Fibonacci(n) Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) T (0) = 2; T (1) = 3 T (n) = T (n 1) + T (n 2) + 3
62 Complexity of Our First Algorithm We need a mathematical characterization of the performance of the algorithm We ll call it the algorithm s computational complexity Let T (n) be the number of basic steps needed to compute Fibonacci(n) Fibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else return Fibonacci(n 1) + Fibonacci(n 2) T (0) = 2; T (1) = 3 T (n) = T (n 1) + T (n 2) + 3 T (n) F n Antonio Carzaniga
63 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2
64 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2
65 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2 2(2F n 4 )
66 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2 2(2F n 4 ) 2(2(2F n 6 ))
67 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2 2(2F n 4 ) 2(2(2F n 6 ))...
68 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2 2(2F n 4 ) 2(2(2F n 6 ))... 2 n 2
69 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2 2(2F n 4 ) 2(2(2F n 6 ))... 2 n 2 This means that T (n) ( 2) n (1.4) n
70 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2 2(2F n 4 ) 2(2(2F n 6 ))... 2 n 2 This means that T (n) ( 2) n (1.4) n T (n) grows exponentially with n
71 Complexity of Our First Algorithm (2) So, let s try to understand how F n grows with n T (n) F n = F n 1 + F n 2 Now, since F n F n 1 F n 2 F n 3... F n 2F n 2 2(2F n 4 ) 2(2(2F n 6 ))... 2 n 2 This means that T (n) ( 2) n (1.4) n T (n) grows exponentially with n Can we do better?
72 A Better Algorithm Again, the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...
73 A Better Algorithm Again, the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... Idea: we can build F n from the ground up!
74 A Better Algorithm Again, the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... Idea: we can build F n from the ground up! SmartFibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else pprev = 0 6 prev = 1 7 for i = 2 to n 8 f = prev + pprev 9 pprev = prev 10 prev = f 11 return f
75 Results running time (seconds) Ruby Scheme Python C C-wiz Java C-gcc (Python) SmartFibonacci n
76 Complexity of SmartFibonacci SmartFibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else prev = 0 6 pprev = 1 7 for i = 2 to n 8 f = prev + pprev 9 pprev = prev 10 prev = f 11 return f
77 Complexity of SmartFibonacci SmartFibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else prev = 0 6 pprev = 1 7 for i = 2 to n 8 f = prev + pprev 9 pprev = prev 10 prev = f 11 return f T (n) =
78 Complexity of SmartFibonacci SmartFibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else prev = 0 6 pprev = 1 7 for i = 2 to n 8 f = prev + pprev 9 pprev = prev 10 prev = f 11 return f T (n) = 6 + 6(n 1)
79 Complexity of SmartFibonacci SmartFibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else prev = 0 6 pprev = 1 7 for i = 2 to n 8 f = prev + pprev 9 pprev = prev 10 prev = f 11 return f T (n) = 6 + 6(n 1) = 6n
80 Complexity of SmartFibonacci SmartFibonacci(n) 1 if n == 0 2 return 0 3 elseif n == 1 4 return 1 5 else prev = 0 6 pprev = 1 7 for i = 2 to n 8 f = prev + pprev 9 pprev = prev 10 prev = f 11 return f T (n) = 6 + 6(n 1) = 6n The complexity of SmartFibonacci(n) is linear in n Antonio Carzaniga
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