Chapter 5: Algorithms

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Chapter 5:

Algorithms

Computer Science: An Overview Eleventh Edition J. Glenn Brookshear by

Dennis Brylow

Chapter 5: Algorithms

• 5.1 The Concept of an Algorithm

• 5.2 Algorithm Representation

• 5.3 Algorithm Discovery

• 5.4 Iterative Structures

• 5.5 Recursive Structures

• 5.6 Efficiency and Correctness

Definition of Algorithm

An algorithm is an ordered set of unambiguous, executable steps that defines a terminating process.

Algorithm Representation

• Requires well-defined primitives

• A collection of primitives constitutes a

programming language.

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Figure 5.2 Folding a bird from a square piece of paper

Figure 5.3 Origami primitives

Pseudocode Primitives

• Assignment

name = expression

• Example

RemainingFunds = CheckingBalance + SavingsBalance

Pseudocode Primitives (continued)

• Conditional selection

if (condition):

activity

• Example

if (sales have decreased):

lower the price by 5%

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Pseudocode Primitives (continued)

• Conditional selection

if (condition):

activity else:

activity

• Example

if (year is leap year):

daily total = total / 366 else:

daily total = total / 365

Pseudocode Primitives (continued)

• Repeated execution

while (condition):

body

• Example

while (tickets remain to be sold):

sell a ticket

Pseudocode Primitives (continued)

• Indentation shows nested conditions

if (not raining):

if (temperature == hot):

go swimming else:

play golf else:

watch television

Pseudocode Primitives (continued)

• Define a function

def name():

• Example

def ProcessLoan():

• Executing a function if (. . .):

ProcessLoan() else:

RejectApplication()

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Figure 5.4 The procedure Greetings in pseudocode

def Greetings():

Count = 3

while (Count > 0):

print('Hello') Count = Count - 1

Pseudocode Primitives (continued)

• Using parameters

def Sort(List):

. .

• Executing Sort on different lists Sort(the membership list) Sort(the wedding guest list)

Polya’s Problem Solving Steps

• 1. Understand the problem.

• 2. Devise a plan for solving the problem.

• 3. Carry out the plan.

• 4. Evaluate the solution for accuracy and its potential as a tool for solving other problems.

Polya’s Steps in the Context of Program Development

• 1. Understand the problem.

• 2. Get an idea of how an algorithmic function might solve the problem.

• 3. Formulate the algorithm and represent it as a program.

• 4. Evaluate the solution for accuracy and

its potential as a tool for solving other

problems.

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Getting a Foot in the Door

• Try working the problem backwards

• Solve an easier related problem

– Relax some of the problem constraints – Solve pieces of the problem first (bottom up

methodology)

• Stepwise refinement: Divide the problem into smaller problems (top-down methodology)

Ages of Children Problem

• Person A is charged with the task of determining the ages of B’s three children.

– B tells A that the product of the children’s ages is 36.

– A replies that another clue is required.

– B tells A the sum of the children’s ages.

– A replies that another clue is needed.

– B tells A that the oldest child plays the piano.

– A tells B the ages of the three children.

• How old are the three children?

Figure 5.5 Figure 5.6 The sequential search

algorithm in pseudocode

def Search (List, TargetValue):

if (List is empty):

Declare search a failure else:

Select the first entry in List to be TestEntry while (TargetValue > TestEntry and entries remain):

Select the next entry in List as TestEntry if (TargetValue == TestEntry):

Declare search a success else:

Declare search a failure

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Figure 5.7 Components of repetitive control

Iterative Structures

• Pretest loop:

while (condition):

body

• Posttest loop:

repeat:

body until(condition)

Figure 5.8 The while loop structure Figure 5.9 The repeat loop structure

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Figure 5.10 Sorting the list Fred, Alex, Diana, Byron, and Carol alphabetically

Figure 5.11 The insertion sort

algorithm expressed in pseudocode

def Sort(List):

N = 2

while (N <= length of List):

Pivot = Nth entry in List

Remove Nth entry leaving a hole in List while (there is an Entry above the

hole and Entry > Pivot):

Move Entry down into the hole leaving a hole in the list above the Entry Move Pivot into the hole

N = N + 1

Recursion

• The execution of a procedure leads to another execution of the procedure.

• Multiple activations of the procedure are formed, all but one of which are waiting for other activations to complete.

Figure 5.12 Applying our strategy to

search a list for the entry John

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Figure 5.13 A first draft of the binary search technique

if (List is empty):

Report that the search failed else:

TestEntry = middle entry in the List if (TargetValue == TestEntry):

Report that the search succeeded if (TargetValue < TestEntry):

Search the portion of List preceding TestEntry for TargetValue, and report the result of that search if (TargetValue > TestEntry):

Search the portion of List following TestEntry for TargetValue, and report the result of that search

Figure 5.14 The binary search algorithm in pseudocode

def Search(List, TargetValue):

if (List is empty):

Report that the search failed else:

TestEntry = middle entry in the List if (TargetValue == TestEntry):

Report that the search succeeded if (TargetValue < TestEntry):

Sublist = portion of List preceding TestEntry Search(Sublist, TargetValue)

if (TargetValue < TestEntry):

Sublist = portion of List following TestEntry Search(Sublist, TargetValue)

Figure 5.15 Figure 5.16

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Figure 5.17

Algorithm Efficiency

• Measured as number of instructions executed

• Big theta notation: Used to represent efficiency classes

– Example: Insertion sort is in Θ(n ² )

• Best, worst, and average case analysis

Figure 5.18 Applying the insertion sort in

a worst-case situation Figure 5.19 Graph of the worst-case

analysis of the insertion sort algorithm

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Figure 5.20 Graph of the worst-case analysis of the binary search algorithm

Software Verification

• Proof of correctness – Assertions

• Preconditions

• Loop invariants

• Testing

Chain Separating Problem

• A traveler has a gold chain of seven links.

• He must stay at an isolated hotel for seven nights.

• The rent each night consists of one link from the chain.

• What is the fewest number of links that must be cut so that the traveler can pay the hotel one link of the chain each morning without paying for lodging in advance?