Computational Thinking

Transcription

1 Computational Thinking Giri Narasimhan School of Computing and Information Sciences College of Engineering and Computing

2 2 What is Computational Thinking? u [J. Wing, CACM 2006] Thought processes involved in: Problem solving Decomposition/Induction; Reduction/Transformation; Pattern Recognition; Abstraction; Algorithm Design [ edu/computational-thinking/] Designing Systems Metacomputing u A basic skill for all modern human beings u A concept that grew out of teaching CS u How to teach it?

3 3 Luciano and others said u Girls love (math) puzzles. u Girls prefer to collaborate than to work alone.... u Then girls should love (algorithmic) puzzles! u With a bit of work, puzzles can be made collaborative. u It might even get them excited about programming the algorithm.

4 4 1. Twenty Questions? u I have thought of a number X between 1 and a million. u Your job is to infer this number by asking questions u If you guess a number X, I will respond with either: LOWER or HIGHER or YES u How many guesses do you need to correctly infer? u Observation: You will never need more than a million guesses!

5 5 Twenty Questions for X = Guess: 500,000 LOWER 2. Guess: 250,000 LOWER 3. Guess: 125,000 LOWER 4. Guess: 62,500 LOWER 5. Guess: 31,250 LOWER 6. Guess: 15,625 LOWER 7. Guess: 7,812 LOWER 8. Guess: 3,906 LOWER 9. Guess: 1,953 LOWER 10. Guess: 976 LOWER 11. Guess: 488 HIGHER 12. Guess: 732 LOWER 13. Guess: 610 LOWER 14. Guess: 549 HIGHER 15. Guess: 579 HIGHER 16. Guess: 594 LOWER 17. Guess: 586 HIGHER 18. Guess: 590 LOWER 19. Guess: 588 YES DONE! u At most log 2 (N)+1 questions

6 6 Twenty Questions Analysis u If X is between 1 and N, then the initial range for X is of width N u After first question, range narrows to width N/2 u After each question, range width is halved u Stops when the width is at most 1 u Hence at most log 2 (N)+1 questions needed. u But then, log 2 (1,000,000) < 20

7 7 Binary Search Algorithm ALGORITHM BinarySearch (Min, Max, X) Lo = Min Hi = Max repeat while (Lo <= Hi) COMMENT: X is between Lo and Hi Guess: Mid = (Lo + Hi) / 2 If answer is LOWER then Hi = Mid -1 else If answer is HIGHER then Lo = Mid + 1 else return Mid

8 8 Modified Twenty Questions u I have thought of a positive number X. u Your job is to infer this number by asking questions u If you guess a number X, I will respond with either: LOWER or HIGHER or YES u How many guesses do you need to correctly infer?

9 9 Solution Strategy u Guess an upper bound B u Then perform Binary Search in range [1, B] u How to guess an upper bound? Doubling Search

10 10 Modified 20 Questions: X = Guess: 2? HIGHER 2. Guess: 4? HIGHER 3. Guess: 8? HIGHER 4. Guess: 16? HIGHER 5. Guess: 32? HIGHER 6. Guess: 64? HIGHER 7. Guess: 128? HIGHER 8. Guess: 256? HIGHER 9. Guess: 512? HIGHER 10. Guess: 1024? LOWER 11. Guess: 768? LOWER 12. Guess: 640? LOWER 13. Guess: 576? HIGHER 14. Guess: 608? LOWER 15. Guess: 594? LOWER 16. Guess: 585? HIGHER 17. Guess: 589? LOWER 18. Guess: 587? HIGHER 19. Guess: 588? Yes DONE! u At most 2log 2 (X)+1 questions

11 11 What did we do here? u Found a fun way to think about a standard computer science algorithm. u Forced students to be creative with a standard algorithm. u Decomposition or Divide-and-Conquer After every question we narrowed down the search range by a factor of 2.

12 12 Algorithms are recipes!

13 13 2. Celebrity Problem u A Celebrity is one that knows nobody and that everybody knows. u Observation: There can be at most one celebrity in a room. u Observation: There may be no celebrity in the room. u Goal: To identify the celebrity if one exists with the least number of questions. u Allowable YES/NO Questions: Ask X if he/she knows Y

14 14 Celebrity Problem Cont d u A Celebrity is one that knows nobody and that everybody knows. u Goal: To identify the celebrity if one exists with the least number of questions. u Allowable YES/NO Questions: Ask X if he/she knows Y u You need at most n 2 questions if there are n people.

15 15 Celebrity Problem Cont d u Question: Does X know Y? NO: Eliminate Y from being a celebrity YES:?? Eliminate X from being a celebrity u Every query eliminates one candidate u Need n-1 questions to eliminate n-1 candidates u Verification is necessary 3(n-1) questions are sufficient!

16 16 What did we do here? u Induction Inductive Hypothesis: We know how to find k-1 noncelebrities among a set of k people, i.e., we know how to find at most one person among a set of k people that could potentially be a celebrity.

17 Algorithms can be simple 17

18 18 Matching Residents to Hospitals 2012 Nobel Prize in Economics u American Mathematical Monthly, 69(1):9-15, Joanne says: Know your audience u Prom Date problem Each boy and girl has a ordered list of preferred dates. Boy asks favorite girl to accompany him as his Prom Date. If she has a better prior offer, then she says NO. What if she is expecting another offer, then what?

19 19 3. Prom Date Problem Cont d u Assume equal number of boys and girls. u Two major issues: What if a boy or girl is unmatched for the PROM? Is there a stable matching one that does not lead to a scene at the PROM? An undesirable scene is when Alice and Bob discover at the PROM that they like each other over their PROM dates!

20 Prom Dates Agony and Ecstasy 20

21 21 Prom Date Preference Lists Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

22 22 Gale-Shapley Algorithm: Round 1 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

23 23 Gale-Shapley Algorithm: Round 1 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

24 24 Gale-Shapley Algorithm: Round 2 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

25 25 Gale-Shapley Algorithm: Round 2 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

26 26 Gale-Shapley Algorithm: Round 3 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

27 27 Gale-Shapley Algorithm: Round 3 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

28 28 Gale-Shapley Algorithm: Round 4 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

29 29 Gale-Shapley Algorithm: Round 5 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

30 30 Gale-Shapley Algorithm: Round 5 Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

31 31 Gale-Shapley Algorithm u In each round, each unmatched girl proposes to the boy on top of her preference list. u If boy is unmatched and gets only one offer in a round, he tentatively accepts the offer. u If boy is already matched or gets more than one offer in a round, he rejects all but the best offer.

32 32 Gale-Shapley Algorithm Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G Each Boy and Girl will have a Prom Date. There will be no ugly scene at the Prom!

33 33 What did we do here? u Algorithm Design Why this algorithm? Designed with correctness in mind.

34 34 Switching the roles of boys & girls Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G

35 35 Boy-optimal vs Girl-optimal Ami Cat Eva Gia Ivy Kia Bob Dev Feo Hugo Jon Leo F D F F B J A C E G I K D J D H D F C G K C A A J B J J L L E K G E K E L L L L H H G E C I C C B H B B F D I I I A E I H F H D J B K A A K G G Four Boy-Girl pairs were same in BO & GO.

36 If you like Algorithms, not to worry! 36

37 37 Reflection u How to turn mathematical puzzles into computing challenges? u How to balance mathematical rigor and fun? Collaborative assignments versus single-person projects? Stronger students and weaker students? Male and female students?

38 Thanks! 38

39 39 4. Billiards shot problem: EASY! u What is the distance traveled by the cue ball in order to strike the red ball head on without touching any of the walls of the table? (3,3) (7,2)

40 40 4. Billiards shot Cont d u What is the minimum distance traveled by the cue ball in order to strike the red ball head on after one bounce off the walls of the table?

41 41 4. Billiards shot u What is the minimum distance traveled by the cue ball in order to strike the red ball head on after one bounce off the NORTH wall of the table?

42 42 4. Billiards shot u What is the minimum distance traveled by the cue ball in order to strike the red ball head on after one bounce off the NORTH walls of the table? Simple trigonometry: Similar triangles It s messy!

43 43 4. Billiards shot u What is the minimum distance traveled by the cue ball in order to strike the red ball head on after one bounce off the NORTH walls of the table? Use reflections It s now easy!