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Algorithms, Part 1 of 3

Topics

• Definition of an Algorithm• Example: The Euclidean Algorithm• Syntax versus Semantics

Reading

• Sections 1.4 – 1.5

The First step in the programming process

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Problem Solving

• Problem solving is the process of transforming the description of a problem into the solution of that problem.

• We use our knowledge of the problem domain.

• We rely on our ability to select and use appropriate problem-solving strategies, techniques, and tools.

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Algorithms

• An algorithm is a step by step solution to a problem.

• Why bother writing an algorithm ?o For your own use in the future. You won’t have

to rethink the problem.o So others can use it, even if they know very

little about the principles behind how the solution was derived.

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Examples of Algorithms

• Washing machine instructions• Instructions for a ready-to-assemble piece

of furniture• A classic: finding the greatest common

divisor (GCD)o The Euclidean Algorithm

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Washing Machine Instructions

• Separate clothes into white clothes and colored clothes.

• For white clothes:o Set water temperature knob to HOT.o Place white laundry in tub.

• For colored clothes:o Set water temperature knob to COLD.o Place colored laundry in tub.

• Add 1 cup of powdered laundry detergent to tub.• Close lid and press the start button.

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Start

Separate Clothes into Light / Darkand Select Which one to wash

YES NO

Set Temp. to HOT Are Cloths Light Set Temp. to Cold?

Place Cloths in Tub

Add 1 cup detergent to Tub

Close Lid and Press Start Button

Stop

Flow Chart for Algorithm

If / else StructurePage 159 in

book.

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Observations About the Washing Machine Instructions

• There are a finite number of steps.• We are capable of doing each of the

instructions.• When we have followed all of the steps,

the washing machine will wash the clothes and then will stop.

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Refinement of the Definition

• Our old definition: o An algorithm is a step by step solution to a

problem.

• Adding our observations: o An algorithm is a finite set of executable

instructions that directs a terminating activity.

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Instructions for a Ready-to-Assemble Piece of Furniture

• "Align the marks on side A with the grooves on Part F.“

• How could these instructions be hard to follow?o Which side is A? A & B look alike -- both

line up with Part F. This instruction is ambiguous.

o The steps are not sequential, do not follow any particular order.

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Poor Example of an Algorithm

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Poor Example of Algorithm

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Poor example of an algorithm.

No Start, No Stop,No Steps

CarburetorRebuildingDiagram.

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Final Version of the Definition

• Our old definition: o An algorithm is a finite set of executable

instructions that directs a terminating activity.

• Final version:o An algorithm is a finite set of unambiguous,

executable instructions that directs a terminating activity.

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History of Algorithms

• The study of algorithms began as a subject in mathematics.

• The search for algorithms was a significant activity of early mathematicians.

• Goal: To find a single set of instructions that can be used to solve any problem of a particular type (a general solution).

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The Euclidean Algorithm

Problem: Find the largest positive integer that divides evenly into two given positive integers (i.e., the greatest common divisor).

Algorithm: Assign M and N the values of the larger and smaller of

the two positive integers, respectively. Divide M by N and call the remainder R. If R is not 0, then assign M the value of N, assign N the

value of R, and return to Step 2. Otherwise, the greatest common divisor is the value currently assigned to N.

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Finding the GCD of 24 and 9

M N R

24 9 6

9 6 3

6 3 0

So, 3 is the GCD of 24 and 9.

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Euclidean Algorithm (con’t)

• Do we need to know the theory that Euclid used to come up with this algorithm in order to use it ?

• What intelligence is required to find the GCD using this algorithm ?

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Greatest Com. Divisor Flow Chart Start

Assign M = LargerN = Smaller

Get Remainder R of M / N

Get Remainder R of M / N

Assign M value of N N Does R = 0Assign R value of N ?

Yes

N is GCD

Stop

While loop StructurePage 214 in book.

Assign M the value of NAssign N the value of R

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The Idea Behind Algorithms

• Once an algorithm behind a task has been discovered

o We don't need to understand the principles.

o The task is reduced to following the instructions.

o The intelligence is "encoded into the algorithm."

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Algorithm Representation

• Syntax and Semantics

o Syntax refers to the representation itself.

o Semantics refers to the concept represented. (i.e., the logic).

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Contrasting Syntax and Semantics

• In the English language, we have both syntax and semantics.

• Syntax is the grammar of the language.• Semantics is the meaning.• Given the following sentence,

I walked to the corner grocery store.

o Is this sentence syntactically correct?o Is it semantically correct?

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Contrasting Syntax and Semantics (con’t)

• Given the following sentence,

I talked to the circular grocery store.

o Is this sentence syntactically correct?o Is it semantically correct?

• How about

I grocery store walked corner the to.

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Contrasting Syntax and Semantics (con’t)

• Conclusion: An English sentence may be syntactically correct, yet semantically (logically) incorrect.

• This is also true of algorithms.

• And it is also true of computer code.

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Flowcharting

• Flowcharting is another technique used in designing and representing algorithms.

• A flowchart is a graph consisting of geometric shapes that are connected by flow lines.

• From the flowchart one can write the program code.

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Symbols (I)

Terminal symbol

Process symbol

Input-Output symbol

or

Decision symbol

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Symbols (II)

Flow Lines indicating the logical sequence of statements

One must follow the flow of thearrows direction, one cannot goin the opposite direction of thearrows flow.