Computational Problem Solving Chin-Sung Lin Eleanor Roosevelt High School.

Computational Problem Solving

Chin-Sung Lin

Eleanor Roosevelt High School

Computational Problem Solving

• Problems

• Problem Solving Approaches

• Computational Problems

• Solutions for Computational Problems

• Impacts of Computational Problem Solving

• What is an Algorithm?

• Representation of Algorithms – Flow Chart

• Representation of Algorithms – Pseudo Code

• Algorithms of Skyscrapers Project

Problems

Problems are

obstacles, difficulties, or challenges

which invite solutions

Problems are

EVERYWHERE

Problems

Healthcare Education

Transportation Internet Biology

Economic

Problem Solving Approaches

Problem Solving Approaches

3 Problem Solving Approaches in Science and Engineering

Analytical Experimental Computational

Problem Solving Example

How can we build a projectile intercepting system?

Problem Solving Approaches Example

3 Problem Solving Approaches (3 Types of Models)



Analytical (Mathematical) Model


Experimental Model


Computational (Simulation) Model


3 Problem Solving Approaches (3 Types of Models)


Computational Problems

Computational Problems

• Decision Problems

• Function Problems

• Search Problems

• Sort Problems

• Counting Problems

• Optimization Problems

Decision Problems

• A computational problem where the answer for every instance is

either yes or no.

• Example: The primality testing - "Given a positive integer n,

determine if n is prime." A decision problem is typically

represented as the set of all instances for which the answer is

yes.

Function Problems

• A computational problem where a single output of a function is

expected for every input, but the output is more complex than

that of a decision problem, that is, it isn't just YES or NO.

• Example: The integer factorization problem, which asks for the

list of factors.

Sort Problems

• A computational problem where elements of a list need to be put

in a certain order.

• The most-used orders are numerical order and lexicographical

order.

• Example: Order the students in a class according to their ages

from older to younger.

Search Problems

• A computational problem where it searches an element from a

given list of elements.

• Example: Find students with first name Peter in Elro classlist.

Counting Problems

• A computational problem where it counts the number of

occurrences of a type of elements in a set of elements.

• Example: A counting problem associated with factoring is "Given

a positive integer n, count the number of nontrivial prime factors

of n.”

Optimization Problems

• A computational problem where it finds the "best possible"

solution among the set of all possible solutions.

• Example: Use a Google Map to find the shortest path of driving.

Solutions for Computational Problems

Problem Solving Process

4-Step Problem Solving Process

Algorithm

Programming

Testing

Specification

Problem Solving Example: Counting Money

• Suppose you want to count out a certain amount of money, using the fewest

possible US bills and coins

• US monetary system: $100, $20, $10, $5, $1, 25¢, 10¢, 5¢, 1¢

• Example: To make $56.39, what should you do?






– At each step, take the largest possible bill or coin that does not overshoot






– At each step, take the largest possible bill or coin that does not overshoot• two $20 bills, to make $40• a $10 bill, to make $50• a $5 bill, to make $55• a $1 bill, to make $56• a 25¢ coin, to make $56.25• a 10¢ coin, to make $56.35• four 1¢ coins, to make $56.39


• Suppose you want to count out a certain amount of money, using the

fewest possible US bills and coins



– At each step, take the largest possible bill or coin that does not

overshoot• two $20 bills, to make $40• a $10 bill, to make $50• a $5 bill, to make $55• a $1 bill, to make $56• a 25¢ coin, to make $56.25• a 10¢ coin, to make $56.35• four 1¢ coins, to make $56.39

– For US money, this algorithm always gives the optimum solution



possible coins

• Fictional monetary system: $12, $8, $1 (coins)

• Example: To make $17, what should you do?



possible coins



– At each step, take the largest possible bill or coin that does not overshoot• a $12 coin, to make $12• five $1 coins, to make $17



possible coins




– This algorithm needs six coins and is NOT the optimum solution



possible coins





– The better way is to use



possible coins





– The better way is to use• two $8 coins, to make $16• a $1 coin, to make $17

– This way needs ONLY three coins and IS the optimum solution

Impacts ofComputational Problem Solving

Problem Solving Example: A Kidney Story

• Kidney disease affects 50,000 new Americans a year. Kidney Exchanges can

save lives!

• Kidney transplants are often their best option for saving their life.

• The demand for kidneys far outstrips the supply from deceased donors.

• For patients with kidney disease, the best option is to find a living donor —

a healthy person willing to donate one of their two kidneys.

• In a live donation, a potential donor and the intended recipient must be

compatible, which can be quite rare.

• In order for patients to obtain a compatible donor, they can swap donors.

• A pool of incompatible patient-donor pairs where one tries to find swaps is

called a kidney exchange.


2-Cycle Exchange 3-Cycle Exchange

• In the simplest exchange, you have two patients each with an available donor

kidney, and both patients are compatible with the other’s donor kidney.

• When compatibility doesn't match, looking for a third person can make the

exchange work —this is called a "3-cycle exchange”.

• Longer cycles can yield more matches, but are often difficult to manage.


• Consider the exchange below. A patient is connected to a donor if they are

biologically compatible. A donor will only donate a kidney if his or her friend

also receives a kidney. What is the optimal exchange for this situation? Why?

• What technique will you use to solve this problem? How would your

technique scale if there were ten donors and patients? 100? Thousands?


• 60 Lives, 30 Kidneys, All Linked (New York Time article: February 18, 2012)

• Prof. Tuomas Sandholm and his research group from CMU developed an

algorithm that can compute the optimal matching for a U.S.-wide kidney

exchange (~10,000 patients) in about an hour. The optimal matching is the

one that results in the largest number of exchanges.

• http://www.nsf.gov/cise/csbytes/newsletter/vol1/vol1i6.html


• Matching algorithms can be applied to a variety of other problems that

require similar barter exchanges.

• The National Odd Shoe Exchange lets people with differently sized feet

agree to swap shoes to avoid having to buy two pairs each

(http://www.oddshoe.org),

• Read It Swap It lets people swap books

(http://www.readitswapit.co.uk//TheLibrary.aspxx),

• Intervac allows people to swap vacation houses (http://www.intervac-

homeexchange.com), and

• Netcycler allows people to swap, give away, and get items for free

(www.netcycler.com).

What is an Algorithm?

An Algorithm

is a finite set of well-defined instructions for

effectively carrying out a procedure or solving a problem.

Examples of Algorithms

Representations of Algorithms


• Show the logic of how the problem is solved - not how it is

implemented.

• Readily reveal the flow of the algorithm.

• Be expandable and collapsible.

• Lend itself to implementation of the algorithm.


• Flowcharts

– Graphical representation of control flow

• Pseudocode

– A sequence of statements, more precise notation

– Not a programming language, no formal syntax

Representation of Algorithms

– Flow Chart

Start or stop

Process

Input or output

Connector

Decision

Flow line

Off-page connector

Representation - Flowcharts

Elements of flowcharts

Examples of Flowcharts

An algorithm for finding the area of a circle of radius r.

Review of

Flowchart Elements

Start or stopConnector


Process Input or output


Decision Off-page connector


Flow lines



ProcessDecision

Off-page connector


Connector

Input or output


Start or stop

Representation of Algorithms

– Pseudocode

Representation - Pseudocode

Conventions of pseudocode

1. Give a valid name for the pseudo-code procedure

2. Use the line numbers for each line of code.

3. Use proper Indentation for every statement in a block structure.

4. For a flow control statements use if-else. Always end an if statement with an end-if. Both if, else and end-if should be aligned vertically in same line.

Ex: If (conditional expression)

statements

else

statements

end-if

Representation - PseudocodeConventions of pseudocode (Continued)

5. Use “=” or “” operator for assignment statements.

Ex: i = j or I j

n = 2 to length[A] or n 2 to length[A]

6. Array elements can be represented by specifying the array name followed by the index in square brackets. For example, A[i] indicates the ith element of the array A.

7. For looping or iteration use for or while statements. Always end a for loop with end-for and a while with end-while.

8. The conditional expression of for or while can be written as shown in rule (4). You can separate two or more conditions with an “and” or an “or”.

Examples of PseudocodeAn algorithm for calculating the class average of test scores

Examples of PseudocodeAn algorithm for calculating the class average of test scores

1. Prompt “Enter the number of tests” and store in testCount 2. Let counter = 1 3. Let totalScore = 04. While counter is less than or equal to testCount 5. Prompt “enter test score” and store into testScore 6. If testScore is numeric then 7. totalScore = testScore + totalScores 8. counter = counter + 1 9. else 10. display “error: test score must be numeric” 11. end-if 12. end-while 13. Average = totalScore / testCount 14. display “Average is :” average

64

Building Blocks of Algorithms

Three building blocks of algorithms:

• Sequences

• Conditionals

• Loops

A sequence of instructions that are executed in the precise order they are written in:

statement block 1statement block 2statement block 3

Building Blocks - Sequences

statement block 1

statement block 2

statement block 3

Select between alternate courses of action depending upon the evaluation of a condition

If ( condition = true )statement block 1

Elsestatement block 2

End if

Building Blocks - Conditionals

statementblock 1

conditionTrue False

statementblock 2

Loop through a set of statements as long as a condition is true

Loop while ( condition = true )statement block

End Loop

Building Blocks - Loops

conditionTrue

False

statementblock

Examples of Flowchart

Make a robot trace the boundary of a square.

Examples of Flowchart

Calculate 12 + 22 + ... + n2

Examples of Pseudocode

Calculate 12 + 22 + ... + n2


Calculate 12 + 22 + ... + n2

1. Input n

2. Sum = 0

3. i = 1

4. While i <= n

5. sq = I * i

6. Sum = Sum + sq

7. i = I + 1

8. End-While

9. Output Sum


Fibonacci numbers or Fibonacci series are the numbers in the following integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, ......

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

Write an algorithm to find the Nth Fibonacci number.

Examples of PseudocodeAn algorithm for finding the Fibonacci Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ……)

Fibonacci ()1. Input N 2. Let result0 = 0 3. Let result1 = 1 4. If N = 1 or N = 25. result = N - 16. else7. for loop counter = 2 to N – 1 8. result = result1 + result09. result0 = result110. result1 = result11. end-for loop12. end-if13. output result

Algorithms of Skyscrapers Project


Rules:

• Using prefabricated-modular blocks of 100 meters long, 100 meters

wide, and 5 meters tall.

• The blocks interlock on top and bottom (like Legos), and they cannot be

stacked sideways.

• Using special lifters, putting stacks of blocks at the same height on

ground or on top of another set of equal-height stacks takes one week

regardless of how tall the stacks are or how many stacks are lifted.

• The prefabrication time of the blocks doesn’t count since they are

already in stock.

• No resource/budget limitations (i.e., you can have as many stacks as

possible at the same time, however, you don’t waste materials.)


Problem:

Part 1:

• If a client wants to build a 100-meter long, 100-meter wide, and

1250-meter high tower as quickly as possible, what is the shortest

amount of time that it will take to build the tower?

• Show your algorithm in both flowchart and pseudocode forms.

Part 2:

• Develop a general algorithm for skyscrapers of 100-meter long,

100-meter wide, and N-meter high (where N is a multiple of 5) in

both flowchart and pseudocode forms.

Computational Problem Solving Chin-Sung Lin Eleanor Roosevelt High School.

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