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Invitation to Computer Science 6th Edition
Chapter 2
The Algorithmic Foundations of Computer Science
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Invitation to Computer Science, 6th Edition
Objectives
In this chapter, you will learn about:
• Representing algorithms
• Examples of algorithmic problem solving
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Invitation to Computer Science, 6th Edition
Introduction
• Chapter 1 – Introduced algorithms and algorithmic problem
solving
• This chapter– Develops more fully the notions of algorithm and
algorithmic problem solving
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Invitation to Computer Science, 6th Edition
Representing Algorithms
• Pseudocode– Natural language: used to express algorithms– Problems using natural language to represent
algorithms• Natural language can be extremely verbose
• Lack of structure makes it difficult to locate specific sections of the algorithm
• Natural language is too “rich” in interpretation and meaning
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Invitation to Computer Science, 6th Edition
Figure 2.1 The Addition Algorithm of Figure 1.2 Expressed in Natural Language
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Invitation to Computer Science, 6th Edition
Representing Algorithms(continued)
• High-level programming language– Examples: C++, Java– Problem with using a high-level programming
language for algorithms• During the initial phases of design, we are forced to
deal with detailed language issues
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Invitation to Computer Science, 6th Edition
Figure 2.2 The Beginning of the Addition Algorithm of Figure 1.2 Expressed in aHigh-Level Programming Language
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Invitation to Computer Science, 6th Edition
Pseudocode (continued)
• Natural languages– Not sufficiently precise to represent algorithms
• High-level programming language– During the initial phases of design, we are forced to
deal with detailed language issues
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Invitation to Computer Science, 6th Edition
Pseudocode
• Pseudocode– Used to design and represent algorithms– A compromise between the two extremes of natural
and formal languages
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Invitation to Computer Science, 6th Edition
Figure 1.2 Algorithm for Adding Two m-digit Numbers
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Invitation to Computer Science, 6th Edition
Pseudocode(continued)
• English language constructs modeled to look like statements available in most programming languages
• Steps presented in a structured manner (numbered, indented, and so on)
• No fixed syntax for most operations is required
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Invitation to Computer Science, 6th Edition
Pseudocode(continued)
• Less ambiguous and more readable than natural language
• Emphasis is on process, not notation
• Well-understood forms allow logical reasoning about algorithm behavior
• Can be easily translated into a programming language
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Invitation to Computer Science, 6th Edition
Sequential Operations
• Basic sequential operations – Computation, input, and output
• Instruction for performing a computation and saving the result– Set the value of “variable” to “arithmetic expression”
• Variable – Storage location that can hold a data value
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Invitation to Computer Science, 6th Edition
Sequential Operations (continued)
• Pseudocode – Not a precise set of notational rules to be memorized
and rigidly followed
• Input operations – Submit to the computing agent data values from the
outside world that it may then use in later instructions
• Output operations – Send results from the computing agent to the
outside world
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Invitation to Computer Science, 6th Edition
Figure 2.3 Algorithm for Computing Average Miles per Gallon
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Invitation to Computer Science, 6th Edition
Conditional and Iterative Operations
• Sequential algorithm– Sometimes called a straight-line algorithm– Executes its instructions in a straight line from top to
bottom and then stops
• Control operations– Conditional and iterative– Allow us to alter the normal sequential flow of control
in an algorithm
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Invitation to Computer Science, 6th Edition
Conditional and Iterative Operations (continued)
• Conditional statements– Ask questions and choose alternative actions based
on the answers– If then else– Example
• if x is greater than 25 then
print x
else
print x times 100
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Invitation to Computer Science, 6th Edition
Figure 2.4 The If/Then/Else Pseudocode Statement
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Invitation to Computer Science, 6th Edition
Figure 2.5 Second Version of the Average Miles per Gallon Algorithm
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Invitation to Computer Science, 6th Edition
Conditional and Iterative Operations (continued)
• Loop– The repetition of a block of instructions– While statement
• Continuation condition– Determines if statement is true or false
• Infinite loop– Continuation condition never becomes false
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Invitation to Computer Science, 6th Edition
Conditional and Iterative Operations (continued)
• Examples
– while j > 0 do
set s to s + aj
set j to j – 1
– do
print ak
set k to k + 1
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Invitation to Computer Science, 6th Edition
Figure 2.6 Execution of the While Loop23
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Invitation to Computer Science, 6th Edition
Conditional and Iterative Operations (continued)
• Loop example
Step Operation
1 Set the value of count to 1
2 While (count ≤ 100) do step 3 to step 5
3 Set square to (count x count)
4 Print the values of count and square
5 Add 1 to count
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Invitation to Computer Science, 6th Edition
Conditional and Iterative Operations (continued)
• Pretest loop – Continuation condition is tested at the beginning of
each pass through the loop
• Posttest loop– Continuation condition is tested at the end of the
loop body, not the beginning
• Primitives– Instructions that computing agent understands and is
capable of executing without further explanation
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Invitation to Computer Science, 6th Edition
Figure 2.7 Third Version of the Average Miles per Gallon Algorithm
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Invitation to Computer Science, 6th Edition
Figure 2.8 Execution of the Do/While Posttest Loop
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Invitation to Computer Science, 6th Edition
Figure 2.9 Summary of Pseudocode Language Instructions
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Examples of Algorithmic Problem Solving
• Go Forth and Multiply: Multiply two numbers using repeated addition
• Sequential search: Find a particular value in an unordered collection
• Find maximum: Find the largest value in a collection of data
• Pattern matching: Determine if and where a particular pattern occurs in a piece of text
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Invitation to Computer Science, 6th Edition
Examples of Algorithmic Problem Solving
• Example 1: Go Forth and MultiplyGiven 2 nonnegative integer values, a ≥ 0, b ≥ 0, compute and output the product (a 3 b) using the technique of repeated addition. That is, determine the value of the sum a + a + a + . . . + a (b times)
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Invitation to Computer Science, 6th Edition
Examples of Algorithmic Problem Solving((continued))
• Algorithm outline– Create a loop that executes exactly b times, with
each execution of the loop adding the value of a to a running total
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• Get values for a and b
• Set the values of count to 0
• Set the value of product to 0
• While (count < b) do– Set the value of product to (product + a)– Set the value of count to (count + 1)
• End of loop
• Print the value of product
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Invitation to Computer Science, 6th Edition
Figure 2.10 Algorithm for Multiplication of Nonnegative Values via Repeated Addition
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Modified version of Multiply
• Devise an algorithm that uses a loop to take in a positive integer n and a real number x as the input and computes and print xn . Assume that the computing agent knows how to multiply but do not know how to calculate exponential. Hint: x0 = 1; x1 = x; x2 = x*x; x3= x*x*x; etc. Make sure you stop the algorithm at some point.
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Invitation to Computer Science, 6th Edition
Example 2: Looking, Looking, Looking
• Algorithm discovery– Finding a solution to a given problem
• Sequential search– Standard algorithm for searching an unordered list of
values
• Binary Search (extended example)– In computer science, a binary search or half-interval
search algorithm finds the position of a specified value (the input “key”) within a sorted array.
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Example 2: Looking, Looking, Looking (Continued)
• Task– Find a particular person’s name from an unordered
list of telephone subscribers
• Algorithm outline– Start with the first entry and check its name, then
repeat the process for all entries
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Invitation to Computer Science, 6th Edition
Figure 2.11 First Attempt at Designing a Sequential Search Algorithm
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Invitation to Computer Science, 6th Edition
Example 2: Looking, Looking, Looking (continued))
• For each entry, write a separate section of the algorithm that checks for a match
• Problems of naïve sequential search algorithm– Only works for collections of exactly one size– Duplicates the same operations over and over
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Example 2: Looking, Looking, Looking (continued)
• Correct sequential search algorithm– Uses iteration to simplify the task– Refers to a value in the list using an index (or
pointer)– Handles special cases (such as a name not found in
the collection)– Uses the variable Found to exit the iteration as soon
as a match is found
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Invitation to Computer Science, 6th Edition
Figure 2.12 Second Attempt at Designing a Sequential Search Algorithm
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Invitation to Computer Science, 6th Edition
Figure 2.13 The Sequential Search Algorithm
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Example 2: Looking, Looking, Looking (continued)
• The selection of an algorithm to solve a problem is greatly influenced by the way the data for that problem is organized
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Invitation to Computer Science, 6th Edition
Example 3: Big, Bigger, Biggest
• Library– Collection of useful algorithms
• ProblemGiven a value n ≥ 1 and a list containing exactly n unique numbers called A1, A2, . . . , An, find and print out both the largest value in the list and the position in the list where that largest value occurred
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Invitation to Computer Science, 6th Edition
Figure 2.14 Algorithm to Find the Largest Value in a List
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Invitation to Computer Science, 6th Edition
Example 4: Meeting Your Match
• Pattern matching– Process of searching for a special pattern of symbols
within a larger collection of information– Is assisting microbiologists and geneticists studying
and mapping the human genome
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Invitation to Computer Science, 6th Edition
Example 4: Meeting Your Match (continued)
• Pattern-matching problemYou will be given some text composed of n characters that will be referred to as T1 T2 . . . Tn. You will also be given a pattern of m characters, m ≤ n, that will be represented as P1 P2 . . . Pm. The algorithm must locate every occurrence of the pattern within the text. The output of the algorithm is the location in the text where each match occurred. For this problem, the location of a match is defined to be the index position in the text where the match begins
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Invitation to Computer Science, 6th Edition
Example 4: Meeting Your Match (continued)
• STEP 1– The matching process: T1 T2 T3 T4 T5 … P1 P2 P3
• STEP 2– The slide forward: T1 T2 T3 T4 T5 …– 1-character slide -> P1 P2 P3
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Invitation to Computer Science, 6th Edition
Figure 2.15 First Draft of the Pattern-Matching Algorithm
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Invitation to Computer Science, 6th Edition
Figure 2.16 Final Draft of the Pattern-Matching Algorithm
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Invitation to Computer Science, 6th Edition
Example 4: Meeting Your Match (continued)
• Examples of higher-level constructs– Sort the entire list into ascending order– Attempt to match the entire pattern against the text– Find a root of the equation
• Abstraction– Use of high-level instructions during the design
process
• Top-down design– Viewing an operation at a high level of abstraction
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Invitation to Computer Science, 6th Edition
Summary
• Algorithm design – A first step in developing an algorithm
• Algorithm design must: – Ensure the algorithm is correct– Ensure the algorithm is sufficiently efficient
• Pseudocode – Used to design and represent algorithms
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Invitation to Computer Science, 6th Edition
Summary (continued)
• Pseudocode – Readable, unambiguous, and able to be analyzed
• Algorithm design – Uses multiple drafts and top-down design to develop
the best solution
• Abstraction – A key tool for good design
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Test 1
• 4 types questions (11)– Concept: computer science, algorithm, Von
Neumann architecture– Design: using Pseudocode code– Correction: verify the given algorithms– Discussion: be creative & comprehensive
• Close notes & Close books
• Panther Card
• Feb 4th: 3:00-4:15pm
• Assignment 1 due on Feb 4th before 4:15pmInvitation to Computer Science, 6th Edition 60