better problem solver

8/9/2019 Better Problem Solver

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Becoming a Better

Problem Solver:

A CS Perspective

Zhang Zhiyong Melvinhttp://www.comp.nus.edu.sg/~melvin

School of Computing,National University of Singapore

May 27, 2009
http://www.comp.nus.edu.sg/~melvinhttp://www.comp.nus.edu.sg/~melvinhttp://www.comp.nus.edu.sg/~melvinhttp://www.comp.nus.edu.sg/~melvin


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Becoming a Better Problem Solver:

A CS Perspective


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Outline

What is problem solving?

Strategies and tacticsGetting started (Strategies)Making progress (Tactics)

Summary


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Polyas Mouse


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Polyas Mouse

A good problem solverdoesnt give up easily, butdont keep banging your

head on the same part ofthe wall.

The key is to vary eachattempt.


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References


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References


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Problem solving is the process of tacklingproblems in a systematic and rational way.


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Steps in problem solving

Understandingthe problem

Devising a plan

Carrying outthe plan

Looking back


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Outline



Summary


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Strategies, tactics and tools

StrategiesGeneral approaches and psychological hints forstarting and pursuing problems.

TacticsDiverse methods that work in many differentsettings.

ToolsNarrowly focused techniques and tricks forspecific situations.


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Outline



Summary


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Strategy 1. Get your hands dirty


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Example: Generating Gray codesNamed after Frank Gray, a researcher from Bell

Labs. Refers to a special type of binary code inwhich adjacent codes different at only one position.

3-bit binary code

000001010011

100101110111


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Example: Generating Gray codes

1-bit01

2-bit0001

1110

3-bit000001

011010110111101100


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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.


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Strategy 2. Restate the problem

The problem as it is stated may not have an obvioussolution. Try to restate the problem in a differentway.

Find the InverseOriginal Given a set of object, find an object

satisfying some property P.

Inverse Find all of the objects which does NOTsatisfy P.


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Example: Invitation

You want to invite the

largest group of friends,so that each person knowat least k others at theparty.


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Invitation

Direct approach

1. For each subset of friends, check if everyoneknows 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.

I


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Invitation

Find the InverseInstead of finding the largest group to invite, find

the smallest group that is left out.

ObservationA person with less than k friends must be left out.

S 3 Wi hf l hi ki


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Strategy 3. Wishful thinking

Make the problem simpler by removing the sourceof difficulty!

1. Identify what makes the problem difficult.

2. Remove or reduce the difficulty.

E l L l


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Example: Largest rectangle

Find the largest white rectangle in an n n grid.

There is an easy solution which checks allrectangles. There are

n

2

n

2

n4 rectangles.

E l L l


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Example: Largest rectangle

2D seems to be difficult, how about 1D?

There aren

2

segments in a row, but we can find

the longest white segment using a single scan of the

row. What is that so?

E l N t G d


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Example: Next Gray code

Given an n-bit Gray code, find the next code.

3-bit Gray code000001011010110

111101100

E l N t G d


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Example: Next Gray code

Gray code is tough! What if we worked in binary?

Binary code

101

110

Gray code

111

101

O tli


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Outline



Summary

Maki g og ess


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Making progress

Record your progressAny form of progress is good, record your workingsand keep track of interesting ideas/observations.

Sometimes, you might

have to sleep on it.

Story of RSA


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Story of RSA

Figure: Left to right: Adi Shamir, Ron Rivest, Len Adleman

Tactic 1 Extremal principle


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Tactic 1. Extremal principle

Given a choice, it is useful to consider items whichare extreme.

Tallest/shortest Leftmost/rightmost

Largest/smallest

Example: Activity selection


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Each bar represents an activity with a particularstart and end time. Find the largest set of activitieswith no overlap.



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An intuitive approach is to repeatedly pick theleftmost activity.

Does this produce the largest set of activities?



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This method may be fooled! Consider the following:



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How would you normally pick among a set of tasks?Do the one with the earliest deadline first!

Tactic 2 Symmetry


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Tactic 2. Symmetry

Example: Gray code to binary code


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Example: Gray code to binary code3-bit Gray code 3-bit binary code

000 000001 001011 010010 011

110 100111 101101 110100 111

Some observations: The leftmost column is always the same. After a column of ones, the order flips

(reflection).



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3-bit Gray code000001

011010110111

101100

Order 0 1 0

1 0 1Gray code 1 1 0

Binary code 1 0 0

Tactic 3 Space/time trade-off


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Tactic 3. Space/time trade off

Instead of always computing an answer from

scratch, it may be worthwhile to precompute somepartial results.

Example: Computing segment sums


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Given an array A of integers, compute the sum ofany 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



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Wishful thinking Computing for any segment A[i,

j]is difficult, what if we consider onlysegments of the form A[1,j]?

Space/time trade-off Sums for A[1,j] can beprecomputed and stored in a anotherarray P

A 6 4 -3 0 5 1 8 7P 6 10 7 7 12 13 24 31



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A 6 4 -3 0 5 1 8 7P 6 10 7 7 12 13 24 31

ObservationThe sum for A[i,j] can be computed asP

[j

] P

[i] +

A[i].



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p p g g

Looking back

What is special about sum? Does this work with max/min? If not, can the

idea be adapted?

Outline


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Summary

Strategies and tactics


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g

Strategies

1. Get your hands dirty2. Restate the problem

3. Wishful thinking

Tactics

1. Extremal principle2. Symmetry

3. Space/time trade-off

Duality of Problem and Solution


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y

If we really understand the problem, the

answer will come out of it, because the

answer is not separate from the problem.

J. Krishnamurti

better problem solver

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