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

Martin J. Dürst


© 2009-22 Martin J. Dürst 青山学院大学

Today's Schedule

Moodle Registration,...


Summary of Last Lecture

Comparing Execution Times: From Concrete to Abstract

Very concrete

Very abstract


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

number of steps (counting additions and divisions)
n (number of data items) 1 8 64 512 4'096 32'768 262'144
linear search 1 8 64 512 4'096 32'768 262'144
binary search 1 10 19 28 37 46 55


Observations on Homework 1


How to Derive Steps from (Pseudo)Code


Why Worst Case


Thinking in Terms of Asymptotic Growth

⇒ 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)

n 1 10 100 1'000 10'000 100'000
5n 5 50 500 5'000 50'000 500'000
n1.2 1 15.8 251.2 3'981 63'096 1'000'000
n2 1 100 10'000 1'000'000 100'000'000 1e+10
n log2 n 0 33.2 664.4 9'966 132'877 1'660'964
1.01n 1.01 1.1046 2.7 20'959 1.636e+43 1.372e+432


Solution to Homework 3: Compare Function Growth

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

left right answer
100n n2
1.1n n20
5 log2 n 10 log4 n
20n n!
100·2n 2.1n


Using Ruby to Compare Function Growth

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


Classification of Functions by Asymptotic Growth

Various growth classes with example functions:


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

Set of functions that grow slower or as slow as n2:

Usage examples:
3n1.5O(n2), 15n2O(n2), 2.7n3O(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: ∀nn0:   f(n)≤c·g(n)  ⇔  f(n)∈O(g(n))


Example Algorithms


Comparing the Execution Time of Algorithms

(from last lecture)

Possible questions:

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


Confirming the Order of a Function


Method 1: Use The Definition

We want to check that 2n+15∈O(n)

The definition of Big-O is:

c>0: ∃n0≥0: ∀nn0:   f(n)≤c·g(n)  ⇔  f(n)∈O(g(n))

We have to find values c and n0 so that ∀nn0: 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 3n1.5, 15n2, and 2.7n3 are ∈ O(n2)

limn→∞(3n1.5/n2) = 0 ⇒
O(3n1.5)⊊O(n2), 3n1.5O(n2)

limn→∞(15n2/n2) = 15 ⇒

O(15n2)=O(n2), 15n2O(n2)

limn→∞(2.7n3/n2) = ∞ ⇒
O(n2)⊊O(2.7n3), 2.7n3O(n2)


Method 3: Simplification of Big-O Notation


Ignoring Lower Terms in Polynomials

Concrete Example:   500n2+30nO(n2)

Derivation for general case: f(n) = dna + enbO(na) [a > b > 0]

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

dna + enbcna [a > 0 ⇒ na>0]

d + enb/na = d + enb-ac [b-a < 0 ⇒ limn→∞enb-a = 0]

Some possible values for c and n0:

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

In general: a > b > 0 ⇒ O(na + nb) = O(na)


Ignoring Logarithm Base

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

(Hint: logb a = logc a / logc b = logc a · logb 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(loga n) = O(logb n) = O(log n)


Additional Notations: Ω and Θ

3n1.5O(n2), 15n2O(n2), 2.7n3O(n2)
3n1.5Ω(n2), 15n2Ω(n2), 2.7n3Ω(n2)
3n1.5Θ(n2), 15n2Θ(n2), 2.7n3Θ(n2)


Exact Definitions of Ω and Θ

Definition of Ω

c>0: ∃n0≥0: ∀nn0: c·g(n)≤f(n) ⇔ f(n)∈Ω(g(n))

Definition of Θ

c1>0: ∃c2>0: ∃n0≥0: ∀nn0:
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

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





(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.



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
constant factor
linear growth
quadratic growth
cubic growth
logarithmic growth
exponential growth
Omega (Ω)
オメガ (大文字)
capital letter
Theta (Θ)
シータ (大文字)
asymptotic upper bound
asymptotic lower bound
(式の) 項
(対数の) 底