Becoming a better problem solver: a CS perspective
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This document discusses strategies and tactics for becoming a better problem solver from a computer science perspective. It begins by defining problem solving and outlining common steps. It then presents strategies for getting started, such as getting hands-on, restating the problem, and using wishful thinking to simplify issues. Tactics for making progress are also outlined, including considering extremes, exploiting symmetries, and trading space for time. Examples are provided to illustrate the approaches. The overall message is that varying attempts, recording progress, and finding easier related problems to solve can help overcome challenges.
![Example: Computing segment sums
Given an array A of integers, compute the sum of
any segment A[i, j] efficiently.
6 4 -3 0 5 1 8 7
For example,
A[1, 3] = 6 + 4 + −3 = 7
A[3, 7] = −3 + 0 + 5 + 1 + 8 = 11](https://image.slidesharecdn.com/print-120120025443-phpapp02/85/Becoming-a-better-problem-solver-a-CS-perspective-38-320.jpg)
![Example: Computing segment sums
Wishful thinking Computing for any segment A[i, j]
is difficult, what if we consider only
segments of the form A[1, j]?
Space-time tradeoff Sums for A[1, j] can be
precomputed and stored in a another
array P
A 6 4 -3 0 5 1 8 7
P 6 10 7 7 12 13 24 31](https://image.slidesharecdn.com/print-120120025443-phpapp02/85/Becoming-a-better-problem-solver-a-CS-perspective-39-320.jpg)
![Example: Computing segment sums
A 6 4 -3 0 5 1 8 7
P 6 10 7 7 12 13 24 31
Observation
The sum for A[i, j] can be computed as
P[j] − P[i] + A[i].](https://image.slidesharecdn.com/print-120120025443-phpapp02/85/Becoming-a-better-problem-solver-a-CS-perspective-40-320.jpg)
Becoming a better problem solver: a CS perspective
- 1. Becoming a Better Problem Solver: A CS Perspective Melvin Zhang melvinzhang.net http://www.slideshare.net/melvinzhang/ becoming-a-better-problem-solver-a-cs-perspective January 20, 2012
- 2. Becoming a Better Problem Solver: A CS Perspective
- 3. Outline What is problem solving? Strategies and tactics Getting started (Strategies) Making progress (Tactics) Summary What I’m working on
- 5. P´lya’s Mouse o A good problem solver doesn’t give up easily, but don’t keep banging your head on the same part of the wall. The key is to vary each attempt.
- 6. References
- 7. References
- 8. What is problem solving? Problem solving is the process of tackling problems in a systematic and rational way.
- 9. Steps in problem solving Understanding the problem Looking back Devising a plan Carrying out the plan
- 10. Outline What is problem solving? Strategies and tactics Getting started (Strategies) Making progress (Tactics) Summary What I’m working on
- 11. Strategies, tactics and tools Strategies General approaches and psychological hints for starting and pursuing problems. Tactics Diverse methods that work in many different settings. Tools Narrowly focused techniques and ”tricks” for specific situations.
- 12. Outline What is problem solving? Strategies and tactics Getting started (Strategies) Making progress (Tactics) Summary What I’m working on
- 13. Strategy 1. Get your hands dirty
- 14. Example: Generating Gray codes Named after Frank Gray, a researcher from Bell Labs. Refers to a special type of binary code in which adjacent codes different at only one position. 3-bit binary code 000 001 010 011 100 101 110 111
- 15. Example: Generating Gray codes 1-bit 2-bit 3-bit 0 00 000 1 01 001 11 011 10 010 110 111 101 100
- 16. Applications of Gray codes Figure: Rotary encoder for angle-measuring devices Used in position encoder (see figure). Labelling the axis of Karnaugh maps. Designing error correcting codes.
- 17. Strategy 2. Restate the problem The problem as it is stated may not have an obvious solution. Try to restate the problem in a different way. Find the Inverse Original Given a set of object, find an object satisfying some property P. Inverse Find all of the objects which does NOT satisfy P.
- 18. Example: Invitation You want to invite the largest group of friends, so that each person know at least k others at the party.
- 19. Invitation Direct approach 1. For each subset of friends, check if everyone knows at least k others. 2. Return the largest set of friends. Looking back Works but there are potentially 2n subsets to check, where n is the number of friends.
- 20. Invitation Find the Inverse Instead of finding the largest group to invite, find the smallest group that is left out. Observation A person with less than k friends must be left out.
- 21. Strategy 3. Wishful thinking Make the problem simpler by removing the source of difficulty! 1. Identify what makes the problem difficult. 2. Remove or reduce the difficulty.
- 22. Example: Largest rectangle Find the largest white rectangle in an n × n grid. There is an easy solution which checks all rectangles. There are n × n ≈ n4 rectangles. 2 2
- 23. Example: Largest rectangle 2D seems to be difficult, how about 1D? There are n segments in a row, but we can find 2 the longest white segment using a single scan of the row. What is that so?
- 24. Example: Next Gray code Given an n-bit Gray code, find the next code. 3-bit Gray code 000 001 011 010 110 111 101 100
- 25. Example: Next Gray code Gray code is tough! What if we worked in binary? Gray code Binary code 111 101 101 110
- 26. Outline What is problem solving? Strategies and tactics Getting started (Strategies) Making progress (Tactics) Summary What I’m working on
- 27. Making progress Record your progress Any form of progress is good, record your workings and keep track of interesting ideas/observations. Sometimes, you might have to sleep on it.
- 28. Story of RSA Figure: Left to right: Adi Shamir, Ron Rivest, Len Adleman
- 29. Tactic 1. Extremal principle Given a choice, it is useful to consider items which are extreme. Tallest/shortest Leftmost/rightmost Largest/smallest
- 30. Example: Activity selection Each bar represents an activity with a particular start and end time. Find the largest set of activities with no overlap.
- 31. Example: Activity selection An intuitive approach is to repeatedly pick the leftmost activity. Does this produce the largest set of activities?
- 32. Example: Activity selection This method may be fooled! Consider the following:
- 33. Example: Activity selection How would you normally pick among a set of tasks? Do the one with the earliest deadline first!
- 34. Tactic 2. Exploit symmetry
- 35. Example: Gray code to binary code 3-bit Gray code 3-bit binary code 000 000 001 001 011 010 010 011 110 100 111 101 101 110 100 111 Some observations: The leftmost column is always the same. After a column of ones, the order flips (reflection).
- 36. Example: Gray code to binary code 3-bit Gray code 000 001 Order 0 1 0 011 1 0 1 010 Gray code 1 1 0 110 Binary code 1 0 0 111 101 100
- 37. Tactic 3. Space-time tradeoff Trading space for time: lookup tables, caching Trading time for space: recalculation
- 38. Example: Computing segment sums Given an array A of integers, compute the sum of any segment A[i, j] efficiently. 6 4 -3 0 5 1 8 7 For example, A[1, 3] = 6 + 4 + −3 = 7 A[3, 7] = −3 + 0 + 5 + 1 + 8 = 11
- 39. Example: Computing segment sums Wishful thinking Computing for any segment A[i, j] is difficult, what if we consider only segments of the form A[1, j]? Space-time tradeoff Sums for A[1, j] can be precomputed and stored in a another array P A 6 4 -3 0 5 1 8 7 P 6 10 7 7 12 13 24 31
- 40. Example: Computing segment sums A 6 4 -3 0 5 1 8 7 P 6 10 7 7 12 13 24 31 Observation The sum for A[i, j] can be computed as P[j] − P[i] + A[i].
- 41. Example: Computing segment sums Looking back What is special about sum? Does this work with max/min? If not, can the idea be adapted?
- 42. Outline What is problem solving? Strategies and tactics Getting started (Strategies) Making progress (Tactics) Summary What I’m working on
- 43. Strategies and tactics Strategies Tactics 1. Get your hands dirty 1. Extremal principle 2. Restate the problem 2. Exploit symmetry 3. Wishful thinking 3. Space-time tradeoff
- 44. If there is a problem you can’t solve, then there is an easier problem you can solve: find it. George P´lya o
- 45. Outline What is problem solving? Strategies and tactics Getting started (Strategies) Making progress (Tactics) Summary What I’m working on