2. algorithms 1.ppt

8/14/2019 2. Algorithms 1.ppt

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Book 3: Internal Design and

Programming

Chapter 2: Algorithms


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Introduction

This chapter describes the basics of

algorithms used for module logic design

and the algorithms used to solve typicalproblems.

Technology Engineers No.3 Internal Design

and Programming Chapter 2


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The basis of programming is algorithm design. In

designing the logical structure of a program, it is

important to use the most appropriate algorithm.

In choosing a program from a library, what

algorithm to use is one of the most important

criteria for choosing the most appropriate

program.

Overview


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Objectives

Understand the basics of algorithms

Algorithm definition, design, relationships with

data structures, representation methods, etc.

Understand the characteristics and meaning

of representative search, sort, character

string processing, and file processingalgorithms


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What is an Algorithm?

Definition of an algorithm

Defined in the Japanese Industrial Standard as:

A set of a limited number of clearly defined rulesthat are applied a limited number of times to solve

problems

To be an algorithm, a set of rules must be

unambiguous (to solve a problem effectively

and efficiently) and have a clear stopping point


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Algorithm and Programming

Programming describes data and algorithms in

a programming language to execute its giventasks

Algorithm vs Program

A program consists of algorithms and data An algorithm consists of logic and control


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What is an Algorithm?4 different types of programming:

Procedural programming

Most commonly used type of programming

E.g. FORTRAN, COBOL, Pascal, C, etc.

Characteristics:

Divided into modules (coding is done for each module) Structured theorems are introduced

Limitations (as it is procedure oriented)

Beside declaring procedures, variables must be declared

Instructions are executed one by one in sequential manner

(parallel processing cannot be done)

Comparisons and calculations must be made to solve

problems


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Functional programming

A type of function-oriented programming

E.g. used in field of Artificial Intelligence (AI), basic

calculation theories, other research tasks, etc.

Characteristics:

Expressions are built by nesting and they are replaced

with results of calculations to execute a program

Recursive calls are described more easily

Level of similarity with parallel processing is high

Calculation process or procedure does not need to be

considered


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Logic programming

Predicate based on facts and inferences

E.g. Prolog

Characteristics:

Facts are described based on the prejudice logic,

making the process of programming easier

Inferences and logical calculations can be made easily

Object-oriented programming System is considered a group of objects

E.g. Smalltalk, C++, Java, etc.


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Algorithm and the Data Structure 1. Data Structure

A procedure that a program follows to store data andperform given tasks

Basic data structure

Most basic data structure unit Data is stored in main memory unit

Data type must be declared

Problem-oriented data structure

Built by combining basic data structures containing one or more ofthe following:

List

Stack

Queue

Tree Structure


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Algorithm and the Data Structure 2. Relationships between algorithm and the data

structure

Array processing

Linear search algorithm and the data structure

Most frequently used data search method

Data is searched while subscripts in an array are beingincremented

Maximum number of times the search is repeated is the size ofan array

File processing Process of reading, editing and printing a file is repeated until

there is no data in a file


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Algorithm and the Data Structure

List processing

Data arranged in terms of logic

Data does not need to be shifted during insertion ordeletion

Data can be inserted or deleted by simply

manipulating a pointer


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Various Algorithms

Describes the representative algorithms that

have until now been developed in relation to

each type of problem


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Search Algorithm 1. Linear (or sequential) search method

Exhaustive search method

A method of combining a table from the top to the end

Data retrieved is collated with each data in a table Search is successful only if data matches data to retrieve

Search ends when first successful match occurs

Inappropriate for numerous data retrieval

Comparisons are made (N x 2) times

15 3 2 9 6 1 7 8

1 2 3 4 5 6 7 8

TBL

6

(Match)

Searching from the top to the end one by one


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Search Algorithm Sentinel search method

Same data (sentinel) as the data to retrieve is placed at the end

of a table

Used when authenticity of data and data matches data to

retrieve

Comparisons are made (N + 1) times

Loop does not need to carry out the end-of-array test,

making search more efficient

H K A I S D E A

1 2 3 4 5 6 7 8

TBL

N pieces

The same data as data to retrieve is stored

N + 1 (sentinel) pieces


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Search Algorithm

2. Binary search method

A method that narrows down target data while successivelydividing a search area into two parts

Requires elements to be arranged in ascending ordescending order

Data smaller than reference data does not need to besearched if data being searched is larger than reference data

Average number of times = [log2N] Maximum number of times = -[log2N] + 1

** Decimal numbers of the value in [] are truncated


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Binary Search Method Step1: A total of the value of a subscript representing the top of a table and

that of a subscript representing the end of a table is a divided by 2. Step 2: The elements having the value obtained in step 1 as a subscript are

compared with a target element.

Step 3: If there is an element that matches as target element the search issuccessful.

Step 4: If the value of a target element is smaller than that of an element ina table,1 is subtracted from the current subscript and the the end of table.Ifthe value of a target element is larger than that of an element in a table,1 isadded to the current subscript and the value is used as a subscript forrepresenting the top of a table.

Step 5: Step 1 through step 4 is repeated. If an element matching a target

element cannot be found at the point where the value of a subscriptrepresenting the top of a table becomes larger than that of a subscriptrepresenting the end of a table, the search is unsuccessful. This completesthe search.


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Binary Search Method

1002 1005 1010 1024 1028 1052 1211 1322 1866 2132

TBL(1) TBL(2) TBL(3) TBL(4) TBL(5) TBL(6) TBL(7) TBL(8) TBL(9) TBL(10)

1052 1211 1322 1866 2132

1866 2132

1866

1866

1866

TBL

The first comparison

The second comparison

The third comparison

x

x

x

X>TBL(i)

X>TBL(i)

X=TBL(i)


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Sort Algorithm Reorganizing data in a specific order

Internal sorting

Sorting data in a main memory unit

External sorting Sorting data stored on a magnetic disk and other auxiliary

storage unit

Selection of sorting method

Speed of sort How data is sorted

Computational complexity


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Sort Algorithm

Sorting

Internal sorting

External sorting

Shaker sort method

Basic selection method

Shell sort method

Basic exchange method

Merge sort method

Quick sort method

Basic insertion method


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Exchange Sort

1. Basic Exchange method (bubble sort) One of the simplest sort methods

Used to compare a pair of data sequentially from the

head of an array Efficiency is low as data items are unconditionally

compared even if they have been sorted correctly

Process is time-consuming when data volume is large


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Exchange Sort Steps of Basic Exchange (Bubble) method

1st & 2nd elements in a table are compared

If 1st element is larger than 2nd, 1st is exchanged with 2nd

If 2nd element is larger than 1st, no exchange occurs

2nd and 3rd elements are compared and steps 2 & 3 are repeated

This routine operation is repeated to the last element in the table.

Maximum value is stored in the last element in that table as it reaches

the last element

Steps 1 through 4 and 5 are executed until the operation reaches the

last but one

Steps 1 through 6 are repeated until only the 1st & 2nd elements in a

table remain


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Selection Sort

2. Basic selection method

Like bubble sort, it is one of the simplest sort method

Smallest/largest value is first selected from all data items and it is

exchanged with the data item at the head of an array, then sameroutine repeats its execution on all remaining data items

Efficiency is low as data items are unconditionally compared even

if they have been sorted correctly


Maximum computational complexity: 0 (n2)

Average computational complexity: 0 (n2)


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Selection Sort Steps of basic selection method

Data item with the smallest value is detected in data itemsstored in a table

Data item with the smallest detected is exchanged with the firstdata item in a table (space into which the data item with the

smallest value can be temporarily in a table)

Data item with the smallest value is detected in the 2nd item ina table

Data item with the smallest value is detected is exchanged withthe 2nd

The same operation is repeatedly performed until the last dataitem but one is reached. When the last data item but one isreached, data sorting is completed


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Selection Sort


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Insertion Sort

3. Basic insertion method

Like basic exchange method, this is one of the simplest

sort method

While data items are being sorted, an unsorted data item

is inserted into a proper position in the sequence of

sorted data items

Comparison and insertion speeds are fast as preceding

data items are sorted



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Insertion Sort

Steps of the basic insertion method 1st & 2nd elements in a table are compared

If 1st element < 2nd element, nothing is done

If 1st element > 2nd element, 1st element is exchanged with2nd, i.e. 1st & 2nd elements are in correct order

2nd & 3rd elements are compared

If 2nd element < 3rd element, nothing is done

If 2nd element > 3rd element, 2nd element is exchanged with3rd element. Then this element is compared with the

preceding element according to steps 2 and 3. These steps arerepeated until it is placed in the right position

Steps 4, 5 and 6 are repeated until the last element in the tableis inserted into the correct position


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Insertion Sort


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Shaker Sort

Shaker sort method

Algorithm is the same as that of basic exchange method(bubble sort)

Data items are first compared from left to right, such thatmaximum (minimum) value is set at rightmost position

Data items are then compared from right to left, such thatminimum (maximum) value is set at leftmost position

Operation is repeated to perform data sorting



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Shell Sort

Shell sort method

Extended version of basic insertion method

Execution speed is faster than basic insertion method

Two items located away from each other a certain interval are

picked out of data sequence and sorting is executed while picked

data items are compared with each other

Interval is called gap, such that gap = number of data (n) 2, which

gradually made smaller and finally set to 1

Pieces of data that are away from each other and located in different

positions are quickly exchanged


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Shell Sort

Data items that are sorted in wrong positions

can be resorted to correct positions in earliest

stages of the operationIf part of the data is already sorted, sorting can

be completed very quickly


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Shell Sort


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Quick Sort

Quick sort method Fastest sort method using the recursive call method

A method of dividing a large problem into small problems

and solving each small problem individually

Basic algorithm of quick sort method: Choose a pivot (a key to partition the set of numbers into equal

halves)

Divide all numbers into two partitions

- All numbers in first partition are less than the pivot

- All numbers in second partition are greater than or equal to the pivot

Recursively sort numbers in the two partitions.

The sorted list of numbers is obtained by putting the pivot at the

beginning of the second partition and then appending the numbers in

the second partition after the first partition


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Quick Sort


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Quick Sort Example : Sort the list of numbers [15, 17, 8, 13, 9, 24, 18,

2]. Let us choose 15 as the pivot(First element of the set)

The first partition is the list [8, 13, 9, 2] - all smaller than 15.The second partition is the list [17, 24, 18] - all larger than 15.

The sorted first partition is [2,8,9,13]The sorted second partition is [17, 18, 24]

The sorted list for the input numbers is obtained as follows :

put the pivot 15 at the beginning of the sorted second partition -giving the list [15, 17, 18, 24]

append this list at the end of the first partition giving [2,8,9,13, 15, 17,18, 24]

http://cs.oregonstate.edu/~minoura/cs162/javaProgs/sort/QuickSort.html


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Sort Algorithm

Merge sort method

The recursive call and divide-and-conquer methods are used, as in the

case of the quick sort method

All data items are divided into sections and sorting is executed in eachdivided section. This is repeated until there is only one remaining

element in a data sequence

After a data sequence is divided, divided sections are sequentially

merged

The process overall is thus: Split the original list into two halves

sort each half (using merge sort)

merge the two sorted halves together into a single sorted list W


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Merge Sort

Example

34 56 78 12 45 3 99 23

34 56 78 12 45 3 99 23

34 56 78 12 45 3 99 23

34 56 78 12 45 3 99 23

34 56 12 78 3 45 23 99

12 34 56 78 3 23 45 99

3 12 23 34 45 56 78 99


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sorting 100 elements:

bubble sort : 13.639008 sec/ 30.08%

insertion sort: 13.368042 sec/ 29.48% (fast for sortedlists)

shell sort : 13.478973 sec/ 29.73% (poorimplementation)

merge sort : 2.232971 sec/ 4.92%

quick sort : 2.433960 sec/ 5.37%

Source: http://www.faqs.org/faqs/CAD/autolisp-faq/part1/section-

9.html

Sort Algorithm


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Recursive Algorithm

A recursive algorithm is one which calls

itself to solve smaller versions of an input

problem Recursive computer programs require more

memory and computation compared with

iterative algorithms, but they are simplerand for many cases a natural way of

thinking about the problem.


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Example:Algorithm for finding the k-th even

Algorithm -> Even(positive integer k)

Input: k, a positive integerOutput: k-th even natural number (the first even being 0)

Algorithm:

if k= 1,thenreturn 0;elsereturn Even(k-1) + 2.

Recursive Algorithm


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Now, the same problem can be solved by aniterative algorithm.

Algorithm -> Even(positive integer k)

Input:k

, a positive integerOutput: k-th even natural number (the first even being 0)

Algorithm:int i, even;i:= 1;

even:= 0;while( i < k) {even:= even+ 2;i:= i+ 1;

}return even.

Recursive Algorithm


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Example:Algorithm for testing whether or not a

number xis a natural number

Algorithm -> Natural(a number x)Input:A number x

Output:"Yes" ifxis a natural number, else "No"

Algorithm:if x< 0, thenreturn "No"else

if x= 0, thenreturn "Yes"elsereturn Natural( x - 1)

http://www.cs.queensu.ca/home/cisc121/2002f/lecturenotes/malamb/CISC121A2002fset11-4up.pdf

Recursive Algorithm


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Character String Processing

Character string search

Simple string search

Text is compared character by character sequentiallyfrom the head

Boyer-Moore method (BM method)

The basis of the fastest known ways to find one string

of characters in another

Data is collated while characters in the text are skipped


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Simple collation Text is compared character by character sequentially from the

head. When a character string on which a search should be performed

is detected, the position where it is located is set as a returnvalue.

Principle:

comparison is repeatedly executed until thefirst character of a characterstring to search matches a character in character strings in text.

If there is a match, each remaining character of the matched characterstring is compared with each of a character string to search.


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BM Method Data is collated while characters in the text are

skipped.

Case 1: If there are no character string to search a character at the tail, and all other characters in the first text portion do

not match any character of a character string to be searched. Therefore,

a search point is moved by the length of a character stringto besearched to allow the next search to start from that point.


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BM Method Case 2: If there is a match with a character at the tail of a

character string to be searched Because a character at the tail of a character string matches a character in the

text, characters that precede the matched character must be compared. If all

characters match, the value of a subscript of the first character of thatmatched text pattern is returned.


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BM Method Case 3: If there is a match with one character

somewhere in a character string but unmatchedwith a character at the tail of a character string Because a character at the tail of a character string matches a

character in the text, characters that precede the matched character

must be compared. If all characters match, the value of a subscript of

the first character of that matched text pattern is returned.

two


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BM Method

Distance of movement


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