Chapter 5 Algorithms © 2007 Pearson Addison-Wesley. All rights reserved

© 2007 Pearson Addison-Wesley. All rights reserved 0-2 The central role of algorithms in computer science: examples

© 2007 Pearson Addison-Wesley. All rights reserved 0-3 Layering: physical communication application transport network link application transport network link application transport network link application transport network link network link data

© 2007 Pearson Addison-Wesley. All rights reserved 0-4 The central role of algorithms in computer science

© 2007 Pearson Addison-Wesley. All rights reserved 0-5 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

© 2007 Pearson Addison-Wesley. All rights reserved 0-6 Definition of Algorithm An algorithm is an ordered set of unambiguous, executable steps that defines a terminating process.

© 2007 Pearson Addison-Wesley. All rights reserved 0-7 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 1 Return to step 1 Return to step 1 if there is a coin in the pocket.

© 2007 Pearson Addison-Wesley. All rights reserved 0-8 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

© 2007 Pearson Addison-Wesley. All rights reserved 0-9 Figure 5.2 Folding a bird from a square piece of paper

© 2007 Pearson Addison-Wesley. All rights reserved 0-10 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)

© 2007 Pearson Addison-Wesley. All rights reserved 0-11 Figure 5.3 Origami primitives

© 2007 Pearson Addison-Wesley. All rights reserved 0-12 Figure 5.4 The procedure Greetings in pseudocode

© 2007 Pearson Addison-Wesley. All rights reserved 0-13 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)

© 2007 Pearson Addison-Wesley. All rights reserved 0-14 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

© 2007 Pearson Addison-Wesley. All rights reserved 0-15 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.

© 2007 Pearson Addison-Wesley. All rights reserved 0-16 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)

© 2007 Pearson Addison-Wesley. All rights reserved 0-17 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?

© 2007 Pearson Addison-Wesley. All rights reserved 0-18 Figure 5.5

© 2007 Pearson Addison-Wesley. All rights reserved 0-19 Exercise Homework 1, 2, 4 in Page 213

© 2007 Pearson Addison-Wesley. All rights reserved 0-20 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

© 2007 Pearson Addison-Wesley. All rights reserved 0-21 Iterative Structures Pretest loop: while (condition) do (loop body) Posttest loop: repeat (loop body) until(condition)

© 2007 Pearson Addison-Wesley. All rights reserved 0-22 Figure 5.6 The sequential search algorithm in pseudocode

© 2007 Pearson Addison-Wesley. All rights reserved 0-23 Figure 5.7 Components of repetitive control

© 2007 Pearson Addison-Wesley. All rights reserved 0-24 Figure 5.8 The while loop structure

© 2007 Pearson Addison-Wesley. All rights reserved 0-25 Figure 5.9 The repeat loop structure

© 2007 Pearson Addison-Wesley. All rights reserved 0-26 In-class Exercise Convert the pseudocode routin Z Z 0 X X 1 while (X<6) do (Z  Z+X; X X X+1) to an equivalent routine using a repeat statement. answer: Z Z 0 X X 1 repeat (Z  Z+X; X X X+1) until (X=6) (Extra Point Question) Page Selection sort

© 2007 Pearson Addison-Wesley. All rights reserved 0-27 Figure 5.10 Sorting the list Fred, Alex, Diana, Byron, and Carol alphabetically

© 2007 Pearson Addison-Wesley. All rights reserved 0-28 Figure 5.11 The insertion sort algorithm expressed in pseudocode

© 2007 Pearson Addison-Wesley. All rights reserved 0-29 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.

© 2007 Pearson Addison-Wesley. All rights reserved 0-30 Figure 5.12 Applying our strategy to search a list for the entry John

© 2007 Pearson Addison-Wesley. All rights reserved 0-31 Figure 5.13 A first draft of the binary search technique

© 2007 Pearson Addison-Wesley. All rights reserved 0-32 Figure 5.14 The binary search algorithm in pseudocode

© 2007 Pearson Addison-Wesley. All rights reserved 0-33 Figure 5.15

© 2007 Pearson Addison-Wesley. All rights reserved 0-34 Figure 5.16

© 2007 Pearson Addison-Wesley. All rights reserved 0-35 Figure 5.17

© 2007 Pearson Addison-Wesley. All rights reserved 0-36 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:

© 2007 Pearson Addison-Wesley. All rights reserved 0-37 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: What is the termination condition in the recursive procedure of the above question?

© 2007 Pearson Addison-Wesley. All rights reserved 0-38 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

© 2007 Pearson Addison-Wesley. All rights reserved 0-39 Figure 5.18 Applying the insertion sort in a worst-case situation

© 2007 Pearson Addison-Wesley. All rights reserved 0-40 Figure 5.19 Graph of the worst-case analysis of the insertion sort algorithm

© 2007 Pearson Addison-Wesley. All rights reserved 0-41 Figure 5.20 Graph of the worst-case analysis of the binary search algorithm

© 2007 Pearson Addison-Wesley. All rights reserved 0-42 Software Verification Proof of correctness –Assertions Preconditions Loop invariants Testing

© 2007 Pearson Addison-Wesley. All rights reserved 0-43 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?

© 2007 Pearson Addison-Wesley. All rights reserved 0-44 Figure 5.21 Separating the chain using only three cuts

© 2007 Pearson Addison-Wesley. All rights reserved 0-45 Figure 5.22 Solving the problem with only one cut

© 2007 Pearson Addison-Wesley. All rights reserved 0-46 Figure 5.23 The assertions associated with a typical while structure