アルゴリズムの設計と解析

黄 潤和、佐藤 温（TA） 2015.4～

Contents (L1 - Introduction)

What is an Algorithm? How to design? How to analyze? About this course Text book Teaching plan Evaluation

What is a Tree? 2

Why study algorithms? Algorithms play the central role both in the science practice

From a practical standpoint - you have to know a standard set of important algorithms - you should be able to design new algorithms

From theoretical standard - the study of algorithms is the core of computer science related to many other fields useful in developing analytical skills

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Of computing

Analysis of Algorithms 4

Introduction What is an Algorithm? An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time.

A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 1 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved.

合理的な

ambiguous あいまいな

Two main approaches

1. from typical problem types (a number of algorithms to a problem type)

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Two main approaches

2. from algorithm design techniques (a design techniques to solve different problems)

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e.g. Euclid’s Algorithm

Problem: Find gcd(m,n), the greatest common divisor of two nonnegative, not both zero integers m and n e.g.: gcd(60,24) = 12, gcd(60,0) = 60, … Euclid’s algorithm: it is based on repeated application of equality gcd(m,n) = gcd(n, m mod n) until the second number becomes 0, which makes the problem trivial. e.g.: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12

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最大公約数

Other methods (to the same problem)

(2) Step 1 Assign the value of min{m,n} to t Step 2 Divide m by t. If the remainder is 0, go to Step 3;

otherwise, go to Step 4 Step 3 Divide n by t. If the remainder is 0, return t and stop;

otherwise, go to Step 4 Step 4 Decrease t by 1 and go to Step 2 (3) Step 1 Find the prime factorization of m Step 2 Find the prime factorization of n Step 3 Find all the common prime factors Step 4 Compute the product of all the common prime factors

and return it as gcd(m,n)

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Analysis of Algorithms 10

Analysis of Algorithms 11

Two algorithm descriptions (1) Readable description Step 1 If n = 0, return m and stop; otherwise go to Step 2 Step 2 Divide m by n and assign the value fo the remainder to r Step 3 Assign the value of n to m and the value of r to n. Go to Step 1. (2) Concise description (簡潔な, pseudo 擬似) while n ≠ 0 do r ← m mod n m← n n ← r return m

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Which is better?

Two main issues: (1) How to design algorithms? (solve the problem) (2) How to analyze algorithms? (evaluate/optimize the algorithms)

Algorithm design techniques

Brute force Divide and conquer Decrease and conquer Transform and conquer Space and time tradeoffs Greedy approach Dynamic programming Iterative improvement Backtracking Branch and bound ……

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Analysis of algorithms How good is the algorithm? time efficiency space efficiency

Does there exist a better algorithm? lower bounds optimality

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Often used data structures list

array

linked list

string

stack

queue

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graph

tree

set and dictionary

About this course

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About this course (1) Teaching plan

syllabus

Evaluation - In class: attendance, exercises, self-correction/memo

- Exams: midterm, final

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Exercise 1-1

What is the output of Test1(200) ? Test1（200）の出力結果は何ですか？ Test1は次のアルゴリズムで

す。 Algorithm Test1(n) b ← 0 for i 1 to n do if i mod 6 = 0 then b ← b + 1 else if i mod 9 = 0 then b ← b +10 return b

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Exercise 1-2 What are the output of Test2(100)? Test2（100）の出力結果は何ですか？ Test2は次のアルゴリズムです。 Algorithm Test2(n) b ← 0 for i 1 to n do for j 1 to i do b ← b +1 return b

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Exercise 1-3 What are the output of Test3(1000) ? Test3（1000 ）の出力結果は何ですか？ Test3は次のアルゴリズムです。 Algorithm Test3(n) i ← 1 b ← 0 while i < n do b ← b + 1 i ← 2i return b

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