Representation and Evaluation of Algorithms

(アルゴリズムの表現と評価)

Data Structures and Algorithms

2nd lecture, October 6, 2016

http://www.sw.it.aoyama.ac.jp/2016/DA/lecture2.html

Martin J. Dürst

© 2009-16 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Summary of last lecture・Last week's homework
• Representation of algorithms
• The programming language Ruby
• Evaluation of algorithms

Summary of Last Lecture

• Course overview
• The fascination of algorithms and data structures
• Data structures: concept (example: linked list)
• Algorithms: concept (example: linear search, binary search)

前回の宿題 1: 膨大なデータ

1. 東京証券取引所の第一部の取引で、一つの株式会社の株が営業時間内に平均で 30秒で一回売買されていると想定して、合計で年間に (一売買行為を一つの項目と考えるとき) 何項目のデータが集まるかを、計算しなさい。
2. 問題 1 の結果よりもデータ項目数がもっと多くて、実際に計算機で扱えそうな問題を考え、説明しなさい (他人と同じものの場合には減点対象)。

Methods for Representing Algorithms

• Algorithms are abstract ideas
• But we need some way to represent them
• The main representation methods are:
  • Text(ual description)
  • Diagrams
  • Pseudocode
  • Programming languages
• Each method has advantages and disadvantages

Text

Principle: Describe the algorithm in a natural language

Advantages: Possible to understand/write for non-experts

Disadvantages:

• Ambiguity of natural language
• Difficulty to be precise
• Structure is unclear
• Dependent on natural language choice (e.g. Japanese vs. English)

Diagrams

Examples: Flowchart, ...

Advantages: Visual expression

Disadvantages:

• Creation is time-consuming
• Problem of data exchange (many different formats)
• Gap to structured programming

(structured programming: replacing goto (jump to an arbitrary location in a program) with structured branches (if/switch/...) and loops (for/while/...) only)

What is Pseudocode

• Notation that is close to actual program code
• Commonalities (not required):
  • Simple notation (e.g.: no semicolon at end of line)
  • No variable declarations
  • Use of special symbols (e.g. ←, ∨, ∧, ≦, ...)
  • Partial use of natural language (incl. comments)
• Important: You can create your own pseudocode!

Evaluation of Pseudocode

Advantages:

• Middle ground between (natural language) text and program
• Adjustable to author's preferences
• Adjustable to the characteristics of the algorithm
• Strong connection to structured programming
• Possible to concentrate on the essence of the algorithm

Disadvantages:

• Many different variations
• Obfuscation is possible
• Not executable

Programing Languages

Advantages:

• Precision
• Executable

Disadvantages:

• Many different languages
• Dependent on execution environment
• More precise than necessary
• Different from essence of algorithm

The Programming Language Ruby

• Created in Japan by Yukihiro Matsumoto (松本 行弘; nickname: Matz) starting in 1993
• Gaining popularity outside Japan since 2000
• Increased attention after publication of Ruby on Rails in 2004

• Completely object-oriented
• Easy-to-use scripting language
• Easy for beginners, satisfying for experts
• Contributions to Ruby implementation by Master/Bachelor students (internationalization,...)
• Labowork (see last week's slides)

Ruby for Algorithm Representation

• Simple syntax
• No need to declare variables
• Data structures (e.g. array) are orthogonal to data types (e.g. int)
  (i.e. no need for "array of int", "array of float", ...)
• Ruby is often called "executable pseudocode"
• Excellent environment (profiler, ...)

Important: In this course, you have to learn how to read Ruby programs. But you do not need to write Ruby programs.

First Ruby Example

Linear search and binary search: 2search.rb

How to execute:

• Open a Cygwin Terminal or Start Command Prompt with Ruby
• Use the cd command to move to a directory of your choice
• Download the file into the choosen directory
• Execute ruby 2search.rb

Basics of Ruby Syntax

• # starts a comment (up to end of line)
  (we use # for comments about the algorithms, ## for comments about Ruby itself)
• No need for ; (semicolon) at end of line
• No need for () for conditions (if/while, ...) and most method calls
• Methods (functions) are defined using def
• class/def/if/while/do all end with end (no {}!)

Overview of Algorithm Evaluation

Main evaluation criteria:

• Execution time (computational (time) complexity)
  (this is the main focus of this lecture, and of research on algorithms in general)
• Amout of memory used (space complexity)
  (in general, there is not that much difference between different algorithms or data structures)
• Ease of implementation
  (this is highly subjective, but in general, faster algorithms may be more difficult to implement)

Contextual information used for evaluation:

• How often will the algorithm be used?
• With how much data will the algorithm be used?

Comparing the Execution Time of Algorithms

Example: Comparing linear search and binary search

Possible questions:

• How many seconds faster is binary search when compared to linear search?
• How many times faster is binary search when compared to linear search?

Problem: These questions do not have a single, clear answer.

When we compare algorithms, we want a simple answer.

Comparing Execution Times: From Concrete to Abstract

Concrete ↑

• Measure actual execution time
  
• Count operation steps
  
• Estimate worst case number of steps
  
• Think about asymtotic behavior

Abstract ↓

Measuring Execution Time

• Advantages:

  exact results (if conditions match)

• Problems:
  • Results depend on hardware used
  • Results depend on implementation details
  • Results vary slightly with each execution
  • Impossible without implementation

⇒ We need a better method to compare algorithms, not implemenations or hardware

Counting Operation Steps

(Step: An operation that can be calculated in constant time; examples: arithmetic operations (addition, multiplication, ...), comparisons, memory access, ...)

• Advantages:

  Independent of hardware

• Problems:
  • The number of steps depends on implementation details
  • There are many ways to count steps
    (e.g. is an assignment one step or two steps (two memory accesses))
  • The number of steps changes depending on the number of data items in the input and on the details of the data

⇒ We need a more abstract way to compare algorithms

Summary of Today's Lecture

• There are four main ways of describing algorithms:
  text, diagrams, pseudocode, and programs
• Each description has its advantages and disadvantages
• In this course, we will use Ruby as "executable pseudocode"
• The main criterion to evaluate algorithms is time complexity as a function of the number of (input) data items

This Week's Homework

(提出は不要不要だが、表など記入したまま資料を持ってくること)

1. Ruby を使って探索アルゴリズムのステップ数を調べ、テーブルを完成
   (散布図を作成してもよい)
   
2. n の増加の場合、複数の関数の値を計算しなさい
3. 関数の増加の比較で、どちらの関数が「最終的に勝つ」か、そしてその理由を調べなさい
4. 高校の教科書やウェブなどで対数 (ln, log₁₀, log₂ など) と極限 (lim_n→∞など) について調査・再確認

Glossary

representation

表現

evaluation

評価

pseudocode

疑似コード

non-expert

素人

ambiguity

曖昧さ

natural language

自然言語 (すなわち、人間が話す言語)

flowchart

流れ図

structured progamming

構造化プログラミング

obfuscation

ごまかし

object-oriented (programming language)

オブジェクト指向 ((プログラム) 言語)

scripting language

スクリプト言語

criterion (複数 criteria)

基準

execution time

実行時間

computational complexity

計算量

worst case

最悪の場合

asymptotic

漸近的

round

四捨五入する、概数で表す

logarithm

対数

limit

極限