What is Big 🅾?
Big O is fundamental in the world of computer science, yet many developers and engineers often struggle to understand it.
Most people know its important, and its commonly something software engineers to ruminate upon, exaggerating the complexity and difficulties in understanding and using Big O.
This fear can often results in avoidance of the topic.
Let's stop fearing it, and start getting excited about it. Big O can be a helpful tool to help developers and software engineers to improve the quality and efficiency of code.
Learning Objectives
 provide an introduction into Big O notation
 define "algorithms" in the context of Big O
 define time complexity/efficiency
 asymptotic behavior
 worstcase analysis
 explain how to calculate time complexity
 stepbystep breakdown
 code examples
 define the fundamental types of asymptotic complexity
 Differentiate between O, o, θ, Ω and ⍵
Here we will focus Big O are in terms of time complexity since this is what most people are referring to when discussing Big O.
Algorithms → Big 🅾
Algorithms are programmatic operations that performs a computation to calculate a value or values.
Big O is the standard notation computer scientists use to define, discuss, compare and quantify the performance of algorithms using mathematical terms.
 define how the algorithm responds to an increasing number of input values (quantify how algorithm scales)
 mathematically compare the efficiency of different algorithms
Complexity Analysis & Time Complexity
Complexity analysis is a tool for determining how an algorithm responds to increasing inputs. It can be defined in terms of both spacial efficiency as well as time complexity.
 time complexity refers to the efficiency of an algorithm in respect to the amount of time an algorithm requires to execute completely as its input increases.
 spacial complexity refers to the efficiency of an algorithm in respect to the amount of memory (RAM) it consumes as the input increases.
Note: this article, we will be focusing on time complexity
Lots of Factors
The speed at which a computer can execute (complete) and algorithm is impacted by a wide variety of factors:
 Hardware
 CPU Clock Speed
 CPU Cores (dualcore, quadcore, etc.)
 Cooling Ability (ability to dispense head from critical components under load)
 RAM read/write speed
 Harddrive read/write speed
 Available disc
 Network
 Network speed (up/down)
 Concurrent Network Operations
 Software
 Operating System (Mac, Windows, Linux)
 Concurrent applications
 Security & monitoring software
The key idea here is that finite execution speed is highly variable. When we calculate an algorithms time complexity we simplify the calculation to only include the specific instructions executed as a direct result of an algorithm, ignoring all the factors that may impact execution speed when they are outside the scope of the algorithm.
Calculating TimeComplexity
Calculating an algorithm's time complexity begins by breaking down the operation into individual steps.
What's a step?
In the context of big O and algorithmic complexity...
 a step is NOT → 🚫
 a function
 a line of code
 a statement in code
 a block of code
 a step is → ✅
 a single unit of work executed by the central processing unit (CPU).
This includes operations like:
 assigning a value to a variable
 looking up the value of an element in an array
 comparing two values
 incrementing/decrementing a value
 a single arithmetic operation (addition, subtraction, multiplication, division)
and many more.
The goal is to break down an algorithm into the smallest unit of work that a computer can perform in a single step.
Learn by Example
One of the best ways to illustrate calculating timecomplexity is to work through a real example.
Take the maximum
function below:
const arr = [45,23,67,32,11,17,94,16,47,42,23] // input
function maximum(arr) {
let max = arr[0]
for(let i = 1; i < arr.length; i++){
if(arr[i] > max) {
max = arr[i]
}
}
return max
}
The function maximum
defined above is an example of an algorithm. It is a strictly defined process for finding the largest value in a collection of numbers.
The maximum
function involves the following steps
 create
max
variable (allocate memory)  look up value of item at index
0
inarr
(e.g.arr[0]
)  assign the value returned as to
max
variable (e.g.max = arr[0]
)  create
i
variable (e.g.let i
)  assign
i
a value of1
(e.g.let i = 1
)  define loop termination condition (e.g.
i < arr.length
)  define loop increment (
i++
)  retrieve the value at position
1
inarr
(e.g.arr[1]
→23
)  compare the value of the element retrieved to
max
(e.g.arr[i] > max
)  since
arr[1]
is23
in the example above, the loop concludes and restarts  increment
i
(e.g.i++
)  lookup value at
arr[2]
(sincei = 2
)  compare
arr[2]
withmax
 since
arr[2] = 67
andmax = 45
,arr[i] > max
evaluates totrue
(e.g67 > 45
)  enter
if
block  reassign
max = arr[i]
(e.g.max = 67
)  increment
i
(e.g.i++
)
... and continuing the loop block until the loops terminating condition (i <arr.length
) is reached.
Now since we are focused on timecomplexity, we need to differentiate between setup steps, and the steps that are impacted by the size of the input. In other words, define which steps would increase as the input to the algorithm increases, from the steps that would execute regardless of the size of the input.
So given the maximum
algorithm:
 Independent Steps (required regardless of input size)
 create
max
variable  assign
max
to the first value of the input (arr[0]
)  creating loop statement
 the
i
variable  assigning the
i
variable a value of1
 the
 returning the
max
value
 create
 Dependent Operations (execute a variable number of times depending on input size)
 Number of times we have to increment
i
 Number of times we have to lookup a value from the input
arr[i]
 comparing the value of
arr[i]
tomax
 assigning
max
a value
 Number of times we have to increment
Remember: we are time complexity is quantifying how an algorithm responds to an increasing input
Given the above, we can state objectively that the maximum
function has
 4 independent operations
 4 dependent operations
Or as a mathematical function where n
represents the size of the input: f(n) = 4n + 4

4
referring to the operations that are not affected byn

4n
referring to the operations affected byn
Worstcase Scenario
After distinguishing the individual steps of an algorithm, we need to determine how those execution steps are affected by different types of input.
We start evaluating input types with what is called the worstcase scenario. In terms of timecomplexity, the worst case scenario refers to an input that would require the most operations to complete.
In the example of the maximum
function, the worstcase scenario would be the scenario in which the input array was sorted smallest to largest, which would require every iteration through the for
loop to evaluate the if (arr[i] > max)
as true
, and thus executing an additional steps defined inside the if
block.
if ( arr[i] > max ) {
max = arr[i]
}
So using the 4 independent steps, 4 dependent steps, and 2 conditional steps, we can alter the previous formula to add the conditionally dependent steps outlined as part of the worstcase scenario: f(n) = 4n + 4 + 2n
or simplified as f(n) = 6n + 4
Asymptotic behavior
In complexity analysis, we are primarily concerned with how the function (f(n)
) behaves as the input (n
) grows.
So the next step in calculating the timecomplexity is to eliminate (ignore) all the independent variables (the steps that occur regardless of the input size) from the equation.
In the case of the mathematical function defined earlier (f(n) = 6n + 4
), 4
is constant regardless of the input size, however the 6n
grows directly as the size of the input grows larger. We refer to these independent operations as 4
"initialization constants".
Initialization Constants
Initialization constants are the steps of an algorithm that will execute regardless of the size of the input. Including "behind the scene" initialization operations like the virtual machine required for Java.
When we remove our initialization constants, our mathematical function for maximum
is stated simply as f(n) = 6n
We remove initialization constants because these steps are not affected by the input size, or the input values. Instead they are a direct result of the syntax and rules of a highlevel programming language. Initialization constant can vary quite dramatically from one language to another, and do not help to clarify how an algorithm responds to increasing input sizes (time complexity).
Asymptotic behavior refers to an algorithms time complexity with intialization constants removed.
Common Types of Asymptotic Behavior
Given this behavior we can conclude:
 A algorithm that does not require any looping will have a complexity of
f(n) = 1
since the number of instructions is constant regardless of the size of the input  A algorithm with a single loop, which loops from
1
ton
will have a complexity off(n) = n
since it will perform the instructions inside the loop a constant number of times.  An algorithm with a nested loop (a loop inside another loop) will have a complexity of
f(n) = n²
since the execution time will increases by a factor ofn
as the size of the input (n
) increases in size.
Now to take everything that we've discussed an put it in the computer science language of Big O is pretty simple. Essentially we've already done it.

f(1)
 represents algorithms with a constant number of instructions
 alternatively, you could say the size of the input does not affect the execution time of the algorithm
 mathematically, we would represent this as
f(n) = 1
 is referred to as a "constant time"
 execution time is constant regardless of the size of the input

f(n)
 represents an algorithm that's execution time grows in direct proportion to the size of the input
 alternatively, the size of the input directly affects the algorithms execution time
 mathematically, we would represent this as
f(n) = n
 is referred to as a "linear time"
 execution time grows in direct proportion to the input (
n
)

f(n²)
 represents an algorithm that executes proportionally at a rate of n squared compared to the input siz
 mathematically, we would represent this as
f(n) = n²
 is referred to as quadratic time
 execution time grows in a quadratic fashion as the input increases.
... and this would continue as the exponents representing the function grows.
Remember the key idea here  we are trying to indicate how an algorithm scales as the input to the algorithm increases. Algorithms with a larger θ will execute more slowly than algorithms with a smaller θ
Asymptotic Complexity & Bounds
It is important to understand that O(n)
(pronounced "Oh of n") and θ(n)
(pronounced "theta of n") do not have the same meaning.
 Algorithms expressed in terms of
θ
("theta") are expressed in terms of "tight bounds".  With tightly bound complexity, an algorithm will consistently execute at the rate expressed by the function of theta.
 Algorithms with complexities expressed in
O
are defining the upper bounds or worse case scenario for execution steps.
The difference here is between O
and θ
really comes down to how much variability there is in the equation.
Take Array.find(x => condition)
function:
const input = [5, 12, 8, 130, 44];
const found = input.find(element => element > 10);
console.log(found); // 12
The .find
method iterates through every element in the array, and return the first element that matches the provided logical comparison.
Under the hood, the .find
function is performing the following:
function find(collection, condition){
for(let i = 0; i < collection.length; i++){
const item = collection[i]
if(condition(item)) {
return item
}
}
return null;
}
In summary .find
:
 loops through a collection
 With each iteration passing each element as a parameter to the function provided (
condition
)  Depending on the return value of
condition(item)

true
indicates a match has been found and the value is returned 
false
indicates a nonmatch, and the loop continues

 If no match has been found by the end of the loop through the collection, then a
null
value is returned
Assuming Array.find
works like the function defined above:
 If the first element in the collection matches the provided condition to the
Array.find
algorithm could complete with only a single iteration. This possibility indicates an algorithm that's worstcase scenario involves looping through the entire collection. This would be a lineartimecomplexity or
f(n) = n
 Since this is the upper bound, we could also express it as
o(n)
 This would be a lineartimecomplexity or
There are 3 primary types of bounds in asymptotic functions:

O(n)
denotes the mathematical function in the worst case scenario pronounced "big O of n"
 referred to as upper bound
 Worstcase scenario
 Most execution steps
 Longest execution time
 Greatest timecomplexity
 Least efficient

θ(n)
denotes the mathematical function of the exact timecomplexity of a an asymptotic function given any input. pronounced "theta of n"
 referred to as tightlybound
 Exact
 exact execution steps
 consistent response to increasing input size
 consistent efficiency regardless of input values
 consistent complexity regardless of input values

Ω(n)
denotes the mathematical function provided the best case scenario (fastest execution) pronounced "omega of n"
 referred to as lower bound
 Bestcase
 Least execution steps
 Bestcase scenario
 Lest execution steps
 Fastest execution time
 Least timecomplexity
 Most efficient
This can be broken down further between O
vs. o
and Ω
vs ꞷ
or "big O vs little O" and "big omega vs little omega":
Notation  Pronunciation  Bound  Numeric Operator  Meaning 

O 
"big O"  tightupper  < 
maximum possible number of execution steps (tightly bound) 
o 
"little O"  upper  ≤ 
highest possible number of execution steps 
θ 
"theta"  tight  = 
exact number of execution steps (tightly bound) 
Ω 
"big omega"  lower  ≥ 
least number of execution steps 
ꞷ 
"little omega"  tightlower  > 
least number of execution steps (tightly bound) 
Tight refers to how accurate the mathematical function representing the asymptotic complexity is for a given algorithm.
Learn by example
The best way to learn is to do it, so join me in taking apart a sorting algorithm and examining its asymptotic timecomplexity.
Sorting Algorithms
Sorting Algorithms are algorithms that sort a collection of items by a specific attribute. These algorithms are prevalent in nearly every modern application.
Examples of sorting algorithms:
 Sorting files in Finder/Windows explorer (Alphabetically, by size, by file type, etc.)
 Sorting shopping results (by price, by rating, by sales, etc.)
 Sorting contacts (by first name, by last name, by date of birth, by workplace, etc.)
 Sorting job posting (by post date, by industry, by company, etc.)
There are numerous realistic and practical application for sorting algorithms, and like many algorithms, there is multiple ways to tackle the same problem.
Selection Sort
A "selection sort" algorithm takes an input, and sorts it according to some arbitrary value. Take a number selection sorting algorithm like the one below:
function selectionSort(arr){
const inputArr = [...arr]
const output = []
while (inputArr.length > 0) {
let min = inputArr[0]
let minIndex = 0
for(let i = 1; i < inputArr.length; i++) {
if (inputArr[i] < min) {
min = inputArr[i]
minIndex = i
}
}
output.push(inputArr.splice(minIndex, 1))
}
return output
}
The basic sorting function (selectionSort
) defined above sorts values from smallest to largest by:
 copying the input (to avoid modifying by reference)
 iterating through the copied array (
inputArr
) and removing the smallest element (min
) from the input array amd adding it to theoutput
array.  repeating step 2 until the
inputArr
has no elements left
Challenge Go ahead and see if you can calculate the complexity (in big O notation and θ notation) yourself.
Calculating Selection Sort Time Complexity
Recall, we mathematically determine complexity by:
 determining the individual steps executed by the CPU
 create
inputArr
 copy
arr
intoinputArr
 create
output
 initialize
output = []
 create loop
 create
i
 assign
i = 1
 lookup
inputArr[i]
 compare
inputArr[i] < min
 optionally, assign
min = inputArr[i]
 optionally, assign
minIndex = i
 increment
i
 check condition (
i < inputArr.length
)  repeat loop while
i < inputArr.length
 create
 remove
min
element frominputArr
 add it to minimum value to
output
array  check
inputArr.length > 0
 optionally, loop
 create
 determine which steps are dependent steps
 dependent
 lookup
inputArr[i]
 inside loop:
 compare
inputArr[i] < min
 optionally, assign
min = inputArr[i]
 optionally, assign
minIndex = i
 increment
i
 repeat loop while
i < inputArr.length
 compare
 remove
min
element frominputArr
 add it to minimum value to
output
array  check
inputArr.length > 0
 optionally, loop
 lookup
 independent
 create
inputArr
 copy
arr
intoinputArr
 create
output
 initialize
output = []
 create loop
 create
i
 assign
i = 1
 create
 create
 dependent
 determine the worstcase scenario
 the execution of both loops (
for
andwhile
) are both directly impacted by the size of the input, and unaffected (in terms of execution time) by the values of the input.
 the execution of both loops (
 remove initialization constants
Walking through the Selection Sort
The while
loop that states we continue looping until every element from inputArr
has been removed. Then with each iteration through the while
loop, a nested for
loop through the same collection of elements is executed to find the smallest value.
 the first pass
inputArr
, will determine11
is the smallest element in theinputArr
and remove it, placing it inoutput
 The second pass through
inputArr
will determine12
is the smallest value, and remove it frominputArr
, placing the value inoutput
 The third pass through
inputArr
, will determine22
is the smallest value, remove it frominputArr
, placing the value inoutput
 The fourth pass through
inputArr
, will determine25
is the smallest value, remove it frominputArr
, placing the value inoutput
 The fifth pass through
inputArr
, will determine64
is the smallest value, remove it frominputArr
, placing the value inoutput
 The sixth pass through
inputArr
, will determine that there are no more elements withininputArr
, and exit the loop.  The
output
will be returned with the elements sorted from smallest to largest[11, 12, 22, 25, 64]
Overall we can summarize the behavior of an input [11, 25, 12, 22, 65]
with the following table:
loop  output 
inputArr 
min 
inputArr.length 
output.length 

0  [] 
[11, 25, 12, 22, 64] 
11 
5  0 
1  [11] 
[25, 12, 22, 64] 
12 
4  1 
2  [11, 12] 
[25, 22, 64] 
22 
3  2 
3  [11, 12, 22] 
[25, 64] 
25 
2  3 
4  [11, 12, 22, 25] 
[64] 
64 
1  4 
5  [11, 12, 22, 25, 64] 
[] 
0  5 
 Overall the
selectionSort
function Is comprised of two (2) loops
 The outermost loop executes
n
times  the inner loop executes
n
times
In the worse case scenario both loops are executed 5
times for an input with 5
elements. In terms of a mathematical function, we could represent this as f(n) = n * n
or f(n) = n²
.
Since the there is no scenario in which this algorithm will execute faster than f(n) = n²
Cheers 🥂 to you if you got that 👍
The sorting algorithm outlined above is a highly inefficient method of sorting 🧠 If you are up for a challenge, see if you can think of a more efficient algorithm for sorting numbers 🧠
Recursion
Recursion refers to the process in which a function calls itself to determine the end result. Recursion occurs very commonly, you'll find recursive functions in:
 calculating the number of
.jpeg
files in a folder every folder inside the root folder must also run the same function
 calculating the size of a folder
 every folder inside the folder must also be calculated
 factorials
 to find the factorial of
5
, you first find the factorials of1
,2
,3
and4
 to find the factorial of
... and many more.
Recursive Complexity
One of the simplest examples of a recursive algorithm is factorials. A factorial is a defined as the product of every integer between a number and 0. So determining the factorials of 1
through 5
would look like:
Input  Math Equation 

1 
f(1) = 1 
2 
f(2) = 2 * 1 
3 
f(3) = 3 * 2 * 1 
4 
f(4) = 4 * 3 * 2 * 1 
Functionally, we would calculate a factorial using recursion.
function factorial(number) {
if (number = 1) return 1
return number * factorial(number  1)
}
Given the above function, can you determine the asymptotic complexity?
The factorial(number)
function above does not have any loops. Previously algorithms without any loops had a constant runtime, however with the factorial(number)
function, this is not the case. As you can imagine the greater the input value, the greater the number of execution steps, and the simple recursive function above would have a linear complexity or complexity of θ(n)
.
This begs the question: Do recursive functions always have linear complexities?
Another recursive function referred to as "binary search" is slightly more complicated. In binary search algorithms, the function receives a collection of sorted values. Since the input to the function is sorted, some optimizations can be made to more quickly find the desired value.
Imagine the function:
function findIndexFromSorted(array, target) {
let start = 0
let end = array.length  1
while(start <= end){
let middle = Math.ceil((start + end) / 2)
// make sure we always have a valid middle value
if (middle > array.length  1) middle
// see if middle element is target
if (array[middle] === target) return middle
// check right side
if (target > array[middle]) {
start = middle + 1
continue
}
// check left side
if (target < array[middle]){
end = middle  1
continue
}
}
}
Here we begin by checking the value at the middle of the array
to see if it is the target
. If it is, the function has found the target
and should return the index. If not, we see if the value of the middle of the array
is less than or greater than the target
. Then we change our search parameters (start
and end
) to inspect the rightside (higher numbers) or leftside (lower number) of array
. With each iteration through the while
loop, n
or the number elements (relevant input) shrinks.
So with each iteration through the recursive algorithm, the number of elements inspected shrinks by a factor of 2. Or as a mathematical function: f(n) = n / (2 ^ i)
/ f(n) = n / 2ⁱ
where i
represents the iteration.
Let's represent the iteration with i
:
Iteration  Number of relevant elements 
i function 

1  n 
i = n / 2 
2  n / 2 
i = n / 2² 
3  n / 4 
i = n / 2³ 
4  n / 8 
i = n / 2⁴ 
5  n / 16 
i = n / 2⁵ 
But what is the value of i
for a given input n
?
n 
i 
n function 
n value 

1  i = n / 2¹ 
f(1) = 1/2¹ = 1/2 
0.5 
2  i = n / 2² 
f(2) = 2/2² = 2/4 
0.25 
3  i = n / 2³ 
f(3) = 3/2³ = 3/8 
0.375 
4  i = n / 2⁴ 
f(4) = 4/2⁴ = 4/16 
0.25 
5  i = n / 2⁵ 
f(5) = 5/2⁵ = 5/32 
0.15625 
6  i = n / 2⁶ 
f(6) = 6/64 = 3/32 
0.09375 
That's may seem kinda messy, but given this information we can solve for n
using 1 = n/2ⁱ
. Multiply both side of the equation by 2
, we get 2ⁱ = n
or n = 2ⁱ
. Recall back to algebra, how do you define a variable in terms of the exponent it was raised to?
Answer: Logarithms
Logarithms and Complexity
A logarithm answers the question "How many of this number do we multiply to get that number?"
 the base of the logarithm refers to the number being raised to an exponent
 the number in parenthesis specifies the desired value of the base raised to that number
For Example:
Sentence  Logarithm 
::
 What exponent does 2
need to be raised to, in order to get the value of 8
?  log₂(8)

 What exponent does 2
need to be raised to, in order to get the value of 256
?  log₂(256)

Using logarithms, we could define the binary search algorithm above as: f(n) = n / 2 * log₂(n)
This introduces the last type of asymptotic complexity, logarithmic complexity. In Logarithmic asymptotic algorithms which the algorithms, as the input n
increases, the efficiency of the algorithm increases.
We can summarize these various algorithms with a chart to demonstrate how the algorithm responds to increasing inputs:
All Together
In short, here are the general rules regarding Big O and calculating algorithmic complexity
 Algorithms that execute faster with a larger input, will often execute faster will smaller inputs as well.
 Many simple algorithms can be analyzed by counting the loops
 An algorithm with a single loop, will generally have a linear complexity.
 Every nested loop will increase the complexity beyond
f(n) = n
by a factor of n a single loop will have a complexity of
f(n) = n
 a nested loop with will have a complexity of
f(n) = n²
 a nested loop nested inside another nested loop (3 loops) will have a complexity of
f(n) = n³
 a single loop will have a complexity of
 Given an algorithm with a series of loops (not nested loops), the slowest loop (e.g. the loop with the most computational steps) will determine the asymptotic behavior of the algorithm.
 Two nested loops followed by a single loop is asymptotically the same as the nested loops since the nested loops will have a greater impact on execution time as the input grows.
 While all the symbols
O
,o
,Ω
,ω
andΘ
are useful at times, O is the one used more commonly, as it's easier to determine than Θ and more practically useful thanΩ
. It is easier to determine
O
complexity thanθ
complexity orΩ
complexity 
O
complexity provides the best insight into performance, since it assumes the worstcase scenario
 It is easier to determine
 Logarithmic complexity generally occurs in recursive function, and represents an algorithm that gets more efficient as the input increases.
References
 Dionysis Zindros  A Gentle Introduction to Algorithm Complexity Analysis
 Big O Cheat Sheet  Know Thy Complexities!
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