chapter 5 algorithms © 2007 pearson addison-wesley. all rights reserved

Chapter 5

Algorithms

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The central role of algorithms in computer science: examples

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Layering: physical communication

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The central role of algorithms in computer science

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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

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Definition of Algorithm

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

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In-Class Exercise

• In what sense do the steps described by the following list of instructions fail to constitute an algorithm?– Step1. Take a coin out of your pocket and put it on

the table;– Step2. Return to step 1Return to step 1Return to step 1 if there is a coin in the pocket.

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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

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Figure 5.2 Folding a bird from a square piece of paper

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Pseudocode Primitives

• Assignment name expression

• Conditional selection if condition then action

• Repeated execution while condition do activity

» repeat activity until condition

• Procedure procedure name (generic names)

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Figure 5.3 Origami primitives

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Figure 5.4 The procedure Greetings in pseudocode

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In-class Exercise

• 1. Does the following program represent an algorithm in the strict sense? Why or why not?

Count 0;

While (Count not 5) do

(Count Count + 2)

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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

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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.

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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)

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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?

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Figure 5.5

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Exercise

• Homework

1, 2, 4 in Page 213

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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

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Iterative Structures

• Pretest loop:

while (condition) do

(loop body)• Posttest loop:

repeat (loop body)

until(condition)

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Figure 5.6 The sequential search algorithm in pseudocode

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Figure 5.7 Components of repetitive control

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Figure 5.8 The while loop structure

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Figure 5.9 The repeat loop structure

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In-class Exercise

• Convert the pseudocode routinZ 0X 1while (X<6) do (ZZ+X; X X+1) to an equivalent routine using a

repeat statement.answer: Z 0X 1repeat (ZZ+X; X X+1) until (X=6)

• (Extra Point Question) Page 223-6 Selection sort

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

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Figure 5.11 The insertion sort algorithm expressed in pseudocode

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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.

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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

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Figure 5.14 The binary search algorithm in pseudocode

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Figure 5.15

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Figure 5.16

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Figure 5.17

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In-class Exercise

• What names are interrogated by the binary search when searching for the name Evelyn in the list Alice, Bill, Carol, David, Evelyn, Fred and George?

• Answer: David, Fred, and Evelyn

• What is the maximum number of entries that must be interrogated when applying the binary search to a list of 200 entries? What about a list of 100,000 entries?

• Answer:

100,000; n when 17

200;n when 8

log n2

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In-class Exercise

• What sequence of numbers would be printed by the following recursive procedure if we started it with N assigned the value 1?

Procedure Exercise(N)print the value of N;if (N<3) then (apply the procedure Exercise to

the value N+1)print the value of N;

Answer: 1 2 3 3 2 1

• What is the termination condition in the recursive procedure of the above question?

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Algorithm Efficiency

• Measured as number of instructions executed

• Big theta notation: Used to represent efficiency classes– Example: Insertion sort is in Θ(n2)

• Best, worst, and average case analysis

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Figure 5.18 Applying the insertion sort in a worst-case situation

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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

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Software Verification

• Proof of correctness– Assertions

• Preconditions

• Loop invariants

• Testing

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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?

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Figure 5.21 Separating the chain using only three cuts

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Figure 5.22 Solving the problem with only one cut

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Figure 5.23 The assertions associated with a typical while structure