(抽象データ型とデータ構造、スタック、キューなど)

http://www.sw.it.aoyama.ac.jp/2019/DA/lecture4.html

© 2009-19 Martin J. Dürst 青山学院大学

- Summary and homework of last lecture
- Polynomial vs. exponential time
- Finding the (asymptotic) time complexity of an algorithm
- Recurrence relations
- Abstract Data Types
- Stack
- Queue

The asymptotic growth (order of growth) of a function and the time (and
space) complexity of an algorithm can be expressed with the
Big-`O`/`Ω`/`Θ` notation:

`O`(`g`(`n`)): Set of functions with lower or same order of growth as`g`(`n`)

`Ω`(`g`(`n`)): Set of functions with larger or same order of growth as`g`(`n`)`Θ`(`g`(`n`)): Set of functions with same order of growth as`g`(`n`)

`f`(`n`)∈`O`(`g`(`n`)) ⇔
∃`c`>0: ∃`n`_{0}≥0:
∀`n`≥`n`_{0}:
`f`(`n`)≤`c`·`g`(`n`)

The order of growth of a function can be found by:

- Looking for appropriate
`c`and`n`_{0} - Calculating
lim
_{n→∞}(`f`(`n`)/`g`(`n`)) - Simplification (e.g.
`O`(5`n`^{2}+30`n`+200) →`O`(`n`^{2});`O`(17 log_{5}`n`) →`O`(log`n`))

When using Big-`O` notation, always try to simplify `g`()
as much as possible.

(no need to submit)

Review this lecture's material **every day**!

On the Web, find algorithms with time complexity `O`(1),
`O`(log `n`), `O`(`n`),
`O`(`n` log `n`),
`O`(`n`^{2}), `O`(`n`^{3}),
`O`(2^{n}), `O`(`n`!), and so
on.

`O`(1): Simple formulæ (e.g. interest calculation), initialization`O`(log`n`) (logarithmic order/time): binary search, other "divide and conquer" algorithms`O`(`n`) (linear order/time): proportional to size of data, checking all data items once (or a finite number of times)`O`(`n`log`n`): Many sorting algorithms, other "divide and conquer" algorithms`O`(`n`^{2}) (quadratic order/time),`O`(`n`^{3}) (cubic order/time): Considering all/most combinations of 2 or 3 data items`O`(2^{n}): Considering all/most subsets of data items`O`(`n`!): Considering all/most permutations of data items

Example of other order: `O`(`n`^{2.373}), fastest
known algorithm for matrix multiplication

Example:

1.1^{n} ≶ `n`^{20}

log(1.1)·`n` ≶ log(`n`)·20

`n`/log_{10}(`n`) ≶ 20/log_{10}(1.1)
≊483.2

`n`_{0} ≊ 1541

Conclusion: For any `a`, `b` > 1,
`a`^{n} will always eventually grow faster than
`n`^{b}
(`O`(`n`^{b})⊊`O`(`a`^{n}))

(`n`^{b} is polynominal,
`a`^{n} is exponential)

- What can be called a "realistic" time complexity depends on the problem
- In general:
- Polynomial time is realistic
- Exponential time is unrealistic

[We will discuss this in more detail in lecture 14]

- Find/define the variables that determine the size of the input (e.g.
`n`) - Find the basic operations (steps) in the algorithm that are most frequently executed
- Express the total number of basic operations (steps) using summation or a recurrence relation
- Determine the time complexity expressed with big-
`O`notation

Simplifications possible for big-`O` notation can be applied
early.

Example: Because constant factors are irrelevant in big-`O` notation,
they can be eliminated when counting steps.

- In many cases, the input size is the number of data items (examples: search, sort)
- For matrices, ..., often the number of rows or columns is used (matrices
of size
`n`×`n`or`n`×`m`) - In some cases, the size of individual data items has to be considered

Examples: Size in bits of integers with unlimited precision; length of variable-length strings, ... - Sometimes, there are two or more kinds of data, with different size

Example: String matching (text size`n`and pattern size`m`, lecture 11)

- Usually inside a loop (especially inside multiple loops)
- If there are several independent loops, check all of them
- If the number of operations depends on the values in the input, check the
worst case

Example: For search, time if value is not found - When methonds/functions are called, consider the content of the
function

Caution: Some methods/functions may hide complexity (e.g. Ruby
`sort`

, ...)

- Example program:
for (i=0; i<n; i++) for (j=i; j<n; j++) sum += i*j;

- Most frequent operation: Addition or multiplication in last line (inner loop)
- Expressing the number of operations as a sum:

∑_{i=0}^{n-1}∑_{j=i}^{n-1}1 - Evaluating the sum:

∑_{i=0}^{n-1}∑_{j=i}^{n-1}1 = ∑_{i=0}^{n-1}`n`-`i`=

=`n`+`n`-1 +`n`-2 + ... + 2 + 1 =`n`· (`n`+1) / 2 - Asymptotic time complexity:

`n`· (`n`+1) / 2 = 0.5`n`^{2}+ 0.5`n`∈`O`(`n`^{2})

- Example program (recursive version of binary search):

def binsearch(array, low, high, key) middle = (high+low)/2 if low>=high array[low]==key ? low : nil elsif key>array[middle] binsearch(array, middle+1, high, key) else binsearch(array, low, middle, key) end end

- Expressing the number of operations as a recurrence:

`B`(1) = 1

`B`(`n`) =`B`(⌈`n`/2⌉) + 1

(⌈⌉: ceiling function)

- A
*recurrence (relation)*is a recursive definition of a mathematical function - There are several ways to solve recurrences
- One way to solve a recurrence is to discover a pattern by repeated
substitution:

`B`(`n`) =`B`(⌈`n`/2⌉) + 1 =`B`(⌈⌈`n`/2⌉/2⌉) + 1 + 1 =`B`(⌈`n`/2^{2}⌉) + 2 =

=`B`(⌈`n`/2^{3}⌉) + 3 =`B`(⌈`n`/2^{k}⌉) +`k` - Using
`B`(1) = 1:

`⌈``n`/2^{k}⌉ = 1 ⇒ 1 ≥`n`/2^{k}(>1/2) ⇒ 2^{k}≥`n`(> 2^{k-1}) ⇒`k`≥ log_{2}`n`(>`k`-1) ⇒`k`= ⌈log_{2}`n`⌉ `B`(`n`) = 1 + ⌈log_{2}`n`⌉ ∈`O`(log`n`)- The asymptotic time complexity of binary search is
`O`(log`n`)

(from previous lectures)

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?- What is the order [of growth of the execution time] of linear search and
binary search?

Linear search is`O`(`n`), binary search is`O`(log`n`).

Conclusion: Expressing time complexity as `O`() allows to evaluate
the essence of an algorithm, ignoring hardware and implementation
differences.

- Combination of data with functions operating on data
- The data can only be accessed/changed using the functions (encapsulation)
- Goals
- Data integrity (examples: birthday and age; bank account)
- Modularization of big software projects

- Related to type theory
- Often implemented by objects in object-oriended programming languages
- Type → class
- Function → member function/method

- Stack
- Queue
- Linear list
- Dictionary

Caution: A dictionary ADT is**not**the same as a book dictionary - Priority queue

- Principle:
- last-in-first-out (LIFO)
- General example:
- Stack of trays in cafeteria
- Example from IT:
- Function stack (local variables, return address, ...)
- Main methods:
- new, add/push, delete/pop
- Other methods:
- empty? (check whether the stack is empty or not)

top (return the topmost element without removing it from the stack)

It is possible to define a stack using the following four axioms:

- Stack.new.empty? ↔ true
- s.push(e).empty? ↔ false
- s.push(e).top ↔ e
- s.push(e).pop ↔ s (here, pop returns the new stack, not the top element)

(s is any arbitrary stack, e is any arbitrary data item)

Axioms can define a contract between implementation and users

- Principle:
- first-in-first-out (FIFO)
- General example:
- Queue in cafeteria waiting for food
- Example from IT:
- Queue of processes waiting for execution
- Main methods:
- add/enqueue, remove/delete/dequeue
- Explain the meaning of GIGO: Garbage in, garbage out.

Implementation: 4ADTs.rb; some complexities can be improved by using additional variables

ADT | stack | queue | ||
---|---|---|---|---|

Implemented as | `Array` |
`LinearList` |
`Array` |
`LinearList` |

create | O(n) |
O(1) |
O(n) |
O(1) |

add | O(1) |
O(1) or O(n) |
O(n) or O(1) |
O(n) or O(1) |

delete | O(1) |
O(n) or O(1) |
O(1) or O(n) |
O(1) or O(n) |

`empty?` |
O(1) |
O(1) |
O(1) |
O(1) |

length | O(1) |
O(n) |
O(1) |
O(n) |

- The order (of growth)/(asymptotic) time complexity of an algorithm can be calculated from the number of the most frequent basic operations
- Calculation can use a summation or a recurrence (relation)
- The big-
`O`notation compactly express the inherent efficiency of an algorithm - An
*abstract data type*(ADT) combines data and the operations on this data *Stack*and*queue*are typical examples of ADTs

(no need to submit)

- Order the following orders of growth, and explain the reason for your
order:
`O`(`n`^{2}),`O`(`n`!),`O`(`n`log log`n`),`O`(`n`log`n`),`O`(20)^{n} - Write a simple program that uses the classes in 4ADTs.rb.

Use this program to compare the implementations.

Hint: Use the second part of 2search.rb as an example. - Implement the
*priority queue*ADT (use Ruby or any other programming language)A priority queue keeps a priority (e.g. integer) for each data item.

In the simplest case, the only data is the priority.

The items with the highest priority leave the queue first.

Your implementation can use an array or a linked list or any other data structure.

- polynomial growth
- 多項式増加
- exponential growth
- 指数的増加
- integers with unlimited precision
- 非固定長整数
- summation
- 総和
- recurrence (relation)
- 漸化式
- ceiling function
- 天井関数
- substitution
- 置換
- abstract data type
- 抽象データ型
- encapsulation
- カプセル化
- data integrity
- データの完全性
- modularization
- モジュール化
- type theory
- 型理論
- object-oriended
- オブジェクト指向 (形容詞)
- type
- 型
- class
- クラス
- member function
- メンバ関数
- method
- メソッド
- stack
- スタック
- cafeteria
- 食堂
- axiom
- 公理
- queue
- 待ち行列、キュー
- ring buffer
- リングバッファ
- priority queue
- 順位キュー、優先順位キュー、優先順位付き待ち行列