Asymptotic Time Complexity and Big-O Notation

(漸近的計算量と O 記法)

Data Structures and Algorithms

3rd lecture, October 6, 2022

https://www.sw.it.aoyama.ac.jp/2022/DA/lecture3.html

Martin J. Dürst

© 2009-22 Martin J. Dürst 青山学院大学

Today's Schedule

• Summary from last lecture, last week's homework
• Comparing execution times: From concrete to abstract
• Classification of Functions by Asymptotic Growth
• Big-O notation

Moodle Registration,...

• If your name on your student card is in Kanji, change it to Kanji in Moodle
• If your name on your student card is in wide Latin letters (全角英字), change it to wide Latin letters in Moodle
• Make sure your name in Moodle uses the same Kanji as on your student card
• If you are interested in ラボワーク, please contact me after this lecture

Summary of Last Lecture

• There are four main ways to describe algorithms: natural language text, diagrams, pseudocode, programs
• Each description has advantages and disadvantages
• Pseudocode is close to structured programming, but ignores unnecessary details
• In this course, we will use Ruby as "executable pseudocode"
• The main criterion to evaluate and compare algorithms is time complexity as a function of the number of (input) data items

Comparing Execution Times: From Concrete to Abstract

Very concrete

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

Very abstract

Last Lecture's Homework 1: Example for Asymptotic Growth of Number of Steps

[都合により削除]

Observations on Homework 1

• RANGE_TOP = 1_000_000_000 makes sure that we (almost) never find an item.
• Approximating the number of steps by a formula:
  • Linear search: n
  • Binary search: 3 log₂ n + 1
• Most important term for increasing n:
  • Linear search: n
  • Binary search: 3 log₂ n

How to Derive Steps from (Pseudo)Code

• Identify basic operations (arithmetic operations, assignments, comparisons,...)
• Count or calculate number of times each operation is executed
• For branches, count the longest (worst) branch
• For loops, include the loop logic, and multiply by highest (worst) number of times the loop is executed
• For functions, include some steps for function overhead and multiply by highest (worst) number of times the function is called

Why Worst Case

• For the same input size, some algorithms always take the same number of steps.
  Example: Sum of an array of numbers
• For other algorithms, the number of steps depends on the input values.
  Example: Linear search: Finding 'Aarau' is very fast, finding 'Zürich' is much slower.
• An algorithm that is sometimes fast, but often slow is not very good.
• It is safest to consider the worst case behavior.
  Example for linear search: Search the whole dictionary without finding the target word.

  (We will see exceptions later in this course.)

Thinking in Terms of Asymptotic Growth

• The execution time of an algorithm and the number of executed steps depend on the size of the input
  (the number of data items in the input)
• We can express this dependency as a function f(n)
  (where n is the size of the input)
• Rules for comparing functions:
  • Concentrate on what happens when n increases (gets really big)
    → Ignore special cases for small n
    → Ignore constant(-time) differences (example: initialization time)
  • Concentrate on the essence of the algorithm
    → Ignore hardware differences and implementation differences
    → Ignore constant factors

⇒ Independent of hardware, implementation details, step counting details

⇒ Simple expression of essential differences between algorithms

Last Lecture's Homework 2: Example for Asymptotic Growth of Number of Steps

Fill in the following table
(use engineering notation (e.g. 1.5E+20) if the numbers get very big;
round liberally, the magnitude of the number is more important than the exact value)

[都合により削除]

Solution to Homework 3: Compare Function Growth

Which function of each pair (left/right column) grows larger if n increases?

Using Ruby to Compare Function Growth

• Start irb (Interactive Ruby, a 'command prompt' for Ruby)
• Write a loop: (start..end).each { |n| comparison }
• Example of comparison: puts n, 1.1**n, n**20
• Change the start and end values until appropriate
• If necessary, convert integers to floating point numbers for easier comparison
• Define the factulty function: def fac(n) n<2 ? 1 : n*fac(n-1) end

Caution: Use only when you understand which function will eventually grow larger

Classification of Functions by Asymptotic Growth

Various growth classes with example functions:

• Linear growth: n, 2n+15, 100n-40, 0.001n,...
• Quadratic growth: n², 500n²+30n+3000,...
• Cubic growth: n³, 5n³+7n²+80,...
• Logarithmic growth: ln n, log₂n, 5 log₁₀n²+30,...
• Exponential growth: 1.1ⁿ, 2ⁿ, 2^0.5n+1000n¹⁵,...
• ...

Big-O Notation: Set of Functions

Big-O notation is a notation for expressing the order of growth of a function (e.g. time complexity of an algorithm).

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

Example:
Set of functions that grow slower or as slow as n²:
O(n²)

Usage examples:
3n^1.5 ∈ O(n²), 15n² ∈ O(n²), 2.7n³ ∉ O(n²)

Exact Definition of O

Iff we can find values c and n₀ greater 0 so that for all n greater n₀,   f(n)≤c·g(n), then   f(n)∈O(g(n)).

(Iff: If and only if)

∃c>0: ∃n₀≥0: ∀n≥n₀:   f(n)≤c·g(n)  ⇔  f(n)∈O(g(n))

• g(n) is an asymptotic upper bound of f(n)
• In some references (books, ...):
  • f(n)∈O(g(n)) is written f(n)＝O(g(n))
    
  • In this case, O(g(n)) is always on the rigth side
  • However, f(n)∈O(g(n)) is more precise and easier to understand
• Role of c: Ignore constant-factor differences (e.g. one computer or programming language being twice as fast as another)
• Role of n₀: Ignore initialization costs and behavior for small values of n

Example Algorithms

• The number of steps in linear search is: an + b
  ⇒ Linear search has time complexity O(n)
  (linear search is O(n), linear search has linear time complexity)
• The number of steps in binary search is:
  d log₂ n + e
  ⇒ Binary search has time complexity O(log n)
• Because O(log n) ⊊ O(n), binary search is faster

Comparing the Execution Time of Algorithms

(from last lecture)

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 answer.

When we compare algorithms, we want a simple answer.

The simple and general answer is using big-O notation:
Linear search is O(n), binary search is O(log n).

Binary search is faster than linear search (for inputs of significant size)

Additional Examples for O

• Linear growth:
  n∈O(n); 2n+15∈O(n);  100n-40∈O(n);
  5 log₁₀n+30∈O(n), ...
  O(1)⊂O(n);  O(log n)⊂O(n);  O(20 n)=O(4n + 13), ...
• Quadratic growth:
  n²∈O(n²);  500n²+30n+3000∈O(n²), ...
  O(n)⊂O(n²);  O(n³)⊄O(n²), ...
• Cubic Growth:
  n³∈O(n³);  5n³+7n²+80∈O(n³), ...
• Logarithmic growth:
  ln n∈O(log n);  log₂n∈O(log n);
  5 log₁₀n²+30∈O(log n), ...

Confirming the Order of a Function

• Method 1: Use the definition
  Find appropriatie values for n₀ and c, and check the definition
• Method 2: Use the limit of a function
  lim_n→∞(f(n)/g(n)):
  • If the limit is 0:   O(f(n))⊊O(g(n)), f(n)∈O(g(n))
  • If the limit is 0 < d < ∞:   O(f(n))=O(g(n)), f(n)∈O(g(n))
  • If the limit is ∞:   O(g(n))⊊O(f(n)), f(n)∉O(g(n))
• Method 3: Simplification

Method 1: Use The Definition

We want to check that 2n+15∈O(n)

The definition of Big-O is:

∃c>0: ∃n₀≥0: ∀n≥n₀:   f(n)≤c·g(n)  ⇔  f(n)∈O(g(n))

We have to find values c and n₀ so that ∀n≥n₀: f(n)≤c·g(n)

Example 1: n₀: = 5, c=3

∀n≥5: 2n+15≤3n ⇒ false, either n₀ or c (or both) are not big enough

Example 2: n₀: = 10, c=4

∀n≥10: 2n+15≤4n ⇒ true, therefore 2n+15∈O(n)

Method 2: Use the Limit of a Function

We want to check which of 3n^1.5, 15n², and 2.7n³ are ∈ O(n²)

lim_n→∞(3n^1.5/n²) = 0 ⇒
O(3n^1.5)⊊O(n²), 3n^1.5∈O(n²)

lim_n→∞(15n²/n²) = 15 ⇒

O(15n²)=O(n²), 15n²∈O(n²)

lim_n→∞(2.7n³/n²) = ∞ ⇒
O(n²)⊊O(2.7n³), 2.7n³∉O(n²)

Method 3: Simplification of Big-O Notation

• Big-O notation should be as simple as possible
• Examples (for all functions except constant functions, we assume increasing):
  • Constant functions: O(1)
  • Linear functions: O(n)
  • Quadratic functions: O(n²)
  • Cubic functions: O(n³)
  • Logarithmic functions: O(log n)
• For polynomials, all terms except the term with the biggest exponent can be ignored
• For logarithms, the base is left out (irrelevant)
• Simplification can be applied early (e.g. when counting steps)

Ignoring Lower Terms in Polynomials

Concrete Example:   500n²+30n ∈ O(n²)

Derivation for general case: f(n) = dn^a + en^b ∈ O(n^a) [a > b > 0]

Definition of O: f (n) ≤ cg(n) [n > n₀; n₀, c > 0]

dn^a + en^b ≤ cn^a [a > 0 ⇒ n^a>0]

d + en^b/n^a = d + en^b-a ≤ c [b-a < 0 ⇒ lim_n→∞en^b-a = 0]

Some possible values for c and n₀:

• n₀ = 1, c ≥ d+e
• n₀ = 2, c≥ d+2^b-ae
• n₀ = 10, c≥ d+10^b-ae

Some possible values for concrete example (500n²+30n):

• n₀ = 1, c ≥ 530 → 500n²+30n ≤ 530n² [n≥1]
• n₀ = 2, c ≥ 515 → 500n²+30n ≤ 515n² [n≥2]
• n₀ = 10, c ≥ 503 → 500n²+30n ≤ 503n² [n≥10]

In general: a > b > 0 ⇒ O(n^a + n^b) = O(n^a)

Ignoring Logarithm Base

How do O(log₂ n) and O(log₁₀ n) differ?

(Hint: log_b a = log_c a / log_c b = log_c a · log_b c)

log₁₀ n = log₂ n · log₁₀ 2 ≅ 0.301 · log₂ n

O(log₁₀ n) = O(0.301... · log₂ n) = O(log₂ n)

∀ a>1, b>1:   O(log_a n) = O(log_b n) = O(log n)

Additional Notations: Ω and Θ

• 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)

Examples:
3n^1.5 ∈ O(n²), 15n² ∈ O(n²), 2.7n³ ∉ O(n²)
3n^1.5 ∉ Ω(n²), 15n² ∈ Ω(n²), 2.7n³ ∈ Ω(n²)
3n^1.5 ∉ Θ(n²), 15n² ∈ Θ(n²), 2.7n³ ∉ Θ(n²)

Exact Definitions of Ω and Θ

Definition of Ω

∃c>0: ∃n₀≥0: ∀n≥n₀: c·g(n)≤f(n) ⇔ f(n)∈Ω(g(n))

Definition of Θ

∃c₁>0: ∃c₂>0: ∃n₀≥0: ∀n≥n₀:
c₁·g(n)≤f(n)≤c₂·g(n)   ⇔   f(n)∈Θ(g(n))

Relationships between Ω and Θ

f(n)∈Θ(g(n)) ⇔f(n)∈O(g(n)) ∧ f(n)∈Ω(g(n))  

Θ(g(n)) = O(g(n)) ∩ Ω(g(n))

Use of Order Notation

• O: Maximum (worst-case) time complexity of algorithms
• Ω: Minimally needed time complexity to solve a problem
• Θ: Used when expressing the fact that a time complexity is not only possible, but actually reached

In general as well as in this course, mainly O will be used.

Summary

• To compare the time complexity of algorithms:
  • Ignore constant terms (initialization,...)
  • Ignore constant factors (differences due to hardware or implementation)
  • Count basic steps executed in the worst case
  • Look at asymptotic growth when input size increases
• Asymptotic growth can be expressed with big-O notation
• The time complexity of algorithms can be expressed as O(log n), O(n), O(n²), O(2ⁿ), ...

Homework

(no need to submit)

Review this lecture's material and the additional handout (Section 2.2, pp 52-59 of The Design & Analysis of Algorithms by Anany Levitin) every day!

On the Web, find algorithms with time complexities
O(1), O(log n), O(n), O(n log n), O(n²), O(n³), O(2ⁿ), O(n!), and so on.

Glossary

big-O notation

O 記法 (O そのものは漸近記号ともいう)

asymptotic growth

漸近的 (な) 増加

approximate

近似する

essence

本質

constant factor

一定の係数、定倍数

eventually

最終的に

linear growth

線形増加

quadratic growth

二次増加

cubic growth

三次増加

logarithmic growth

対数増加

exponential growth

指数増加

Omega (Ω)

オメガ (大文字)

capital letter

大文字

Theta (Θ)

シータ (大文字)

asymptotic upper bound

漸近的上界

asymptotic lower bound

漸近的下界

appropriate

適切

limit

極限

polynomial

多項式

term

(式の) 項

logarithm

対数

base

(対数の) 底