Becoming a better problem solver: a CS perspective

Becoming a BetterProblem Solver:A CS Perspective

Melvin Zhangmelvinzhang.net

http://www.slideshare.net/melvinzhang/

becoming-a-better-problem-solver-a-cs-perspective

January 20, 2012

melvinzhang.net

http://www.slideshare.net/melvinzhang/becoming-a-better-problem-solver-a-cs-perspective

http://www.slideshare.net/melvinzhang/becoming-a-better-problem-solver-a-cs-perspective

Becoming a Better Problem Solver:A CS Perspective

Outline

What is problem solving?

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

Summary

What I’m working on

Polya’s Mouse

Polya’s Mouse

A good problem solverdoesn’t give up easily, butdon’t keep banging yourhead on the same part ofthe wall.

The key is to vary eachattempt.

References

References


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

Steps in problem solving

Understandingthe problem

Devising a plan

Carrying outthe plan

Looking back

Outline



Summary


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.

Outline



Summary


Strategy 1. Get your hands dirty

Example: Generating Gray codesNamed after Frank Gray, a researcher from BellLabs. Refers to a special type of binary code inwhich adjacent codes different at only one position.

3-bit binary code000001010011100101110111

Example: Generating Gray codes

1-bit01

2-bit00011110

3-bit000001011010110111101100

Applications of Gray codes

Figure: Rotary encoder for angle-measuring devices

I Used in position encoder (see figure).

I Labelling the axis of Karnaugh maps.

I Designing error correcting codes.

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 Inverse

Original Given a set of object, find an objectsatisfying some property P .

Inverse Find all of the objects which does NOTsatisfy P .

Example: Invitation

You want to invite thelargest group of friends,so that each person knowat least k others at theparty.

Invitation

Direct approach

1. For each subset of friends, check if everyoneknows at least k others.

2. Return the largest set of friends.

Looking backWorks but there are potentially 2n subsets to check,where n is the number of friends.

Invitation

Find the InverseInstead of finding the largest group to invite, findthe smallest group that is left out.

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

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.

Example: Largest rectangle

Find the largest white rectangle in an n × n grid.There is an easy solution which checks allrectangles. There are

(n2

)×(n2

)≈ n4 rectangles.

Example: Largest rectangle

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

There are(n2

)segments in a row, but we can find

the longest white segment using a single scan of therow. What is that so?

Example: Next Gray code

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

3-bit Gray code000001011010110111101100

Example: Next Gray code

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

Binary code

101

110

Gray code

111

101

Outline



Summary


Making progress

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

Sometimes, you mighthave to sleep on it.

Story of RSA

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

Tactic 1. Extremal principle

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

I Tallest/shortest

I Leftmost/rightmost

I Largest/smallest

Example: Activity selection

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


An intuitive approach is to repeatedly pick theleftmost activity.

Does this produce the largest set of activities?


This method may be fooled! Consider the following:


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

Tactic 2. Exploit symmetry

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

000 000001 001011 010010 011110 100111 101101 110100 111

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

(reflection).

Example: Gray code to binary code

3-bit Gray code000001011010110111101100

Order 0 1 01 0 1

Gray code 1 1 0Binary code 1 0 0

Tactic 3. Space-time tradeoff

Trading space for time: lookup tables, cachingTrading time for space: recalculation

Example: Computing segment sums

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,

I A[1, 3] = 6 + 4 + −3 = 7

I A[3, 7] = −3 + 0 + 5 + 1 + 8 = 11


Wishful thinking Computing for any segment A[i , j ]is difficult, what if we consider onlysegments of the form A[1, j ]?

Space-time tradeoff 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


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 ].


Looking back

I What is special about sum?

I Does this work with max/min? If not, can theidea be adapted?

Outline



Summary


Strategies and tactics

Strategies

1. Get your hands dirty

2. Restate the problem

3. Wishful thinking

Tactics

1. Extremal principle

2. Exploit symmetry

3. Space-time tradeoff

If there is a problem you can’t solve, thenthere is an easier problem you can solve:find it.

George Polya

Outline



Summary


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Becoming a better problem solver: a CS perspective