Algorithm Analysis:
Asymptotic Complexity
Agenda
 Prerequisites
 Why is the Analysis of Algorithms important?
 What's the goal of Asymptotic Complexity?
 What are we really measuring?
 RAM computational model
 Machine Operations as a Function
 Asymptotic Complexity
 Asymptotic Notation (Θ, O, Ω)
Prerequisites
 Basic Programming
 Basic Algebra
 Mathematical Functions
 Graphing
 Logarithms
 Set Notation (helpful)
Why is the Analysis of Algorithms Important?
 Foundational concept for computation
 Design performant algorithms
 Understand language primitives
 Develop an intuition for scale
 Identify intractable problems
 Learn patterns and techniques from the "greatest hits" of programming
 Poorly taught in academia
 Impetus for my book (https://hideoushumpbackfreak.com/algorithms/)
Litmus Test
If you think you don't need it, or you know it and never use it, you most likely need to study it.
What's the Goal of Asymptotic Complexity?
 Performancecentric analysis (sort of...)
 Why not measure clock time?
 Machine variation
 Input size must be a factor
 Concisely express how an algorithm will scale without regard to a particular physical machine
 Simple means for comparing algorithms
 "[Asymptotic Complexity] significantly simplifies calculations because it allows us to be sloppy – but in a satisfactorily controlled way"  Donald Knuth
 Theoretical framework for advanced algorithm analysis
What are we measuring?
 Approximate number of Machine Operations
 It's not practical to measure the actual number of machine operations
 Programming language variations
 Machine variations
 Parallelism
 Radom Access Machine (RAM) Model
 Not to be confused with Random Access Memory
Random Access Machine (RAM)
 Theoretical computing machine
 Algorithm Comparisons independent of a physical machine
 Runs pseudo code
 Assumptions
 Sequential execution of instructions
 Fixed word sizes
 Standard integer and floatingpoint numbers

Operations (loosely defined)
 Arithmetic (add, subtract, multiply, divide, remainder, floor, ceiling)
 Data Movement (load, store, copy)
 Control (conditional branch, unconditional branch, subroutine call, return)
RAM Operations
1 data operation
1 control operation X 3 iterations
1 arithmetic and 1 data operation X 3 iterations
1 control operation
Total = 12 operations
If we are being pedantic, this is how to count operations...
1 control operation
j ← 0
for i∈{1,2,3} do
j←i × 2
end for
return j
RAM Operations
 An operation is roughly equivalent to an executed line of pseudo code
 It's acceptable to take reasonable liberties
\( d \gets \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} \)
1 operation
1 operation X 3 iterations
1 operation X 3 iterations
1 operation
ignore
1 operation
How do machine operations relate to scale?
j ← 0
for i∈{1,2,3} do
j←i × 2
end for
return j
Total = 8 operations
Machine Operations as a \( f(x) \)
 When considering scale, growth rate in relation to input size is what's important
 Mathematical function describing the relationship between operations and input size
 Input Size
 Could be anything: number of bits, a scalar value, etc...
 Typically number of items: elements in array, nodes in graph, etc...
 Conventions
 Input size = \( n \)
 Growth rate function = \( T(n) \)
\( T(n) \) Example
function square(A)
for i ← 0 to A −1 do
A[i] = A[i] x A[i]
endfor
end function
1 operation per item
1 operation per item
Total = 2 operations per item
\( T(n) = 2n \)
More Complex \( T(n) \)
 Not all algorithms can be expressed by a single \( T(n) \)
 Upper Bound
 Lower Bound
function findDuplicates(A)
D←∅
for i←0 to A −1 do
for j←0 to A −1 do
if i≠j and A[i] = A[j] then
D←D ∪ {A[i]}
end if
end for
end for
return D
end function
1 operation
\( n \) operations
\( n^2 \) operations (\( n \) per item)
\( n^2 \) operations (\( n \) per item)
Between 0 and \( n^2 \) operations
1 operation
Upper Bound
Lower Bound
Comparing \( T(n) \)
Under reasonable assumptions, square scales much better than findDuplicates
Asymptotic Complexity
Review: The goal is to concisely express how an algorithm will scale (growth rate)
Loosely consider the asymptotic complexity to be the "shape" of \( T(n) \)
Deriving Asymptotic Complexity
Remove all constants and nonleading terms
Constants
NonLeading Term
The "shape" does not change
6 Basic Complexity Classes
Name  Growth Rate  Example 

Constant  Hash Table (Look up)  
Logarithmic  Binary Tree (Search)  
Linear  Linked List (Search)  
Quadratic  Bubble Sort  
Exponential  Traveling Salesman (Dynamic Programming)  
Factorial  Traveling Salesman (Brute Force) 
Asymptotic Complexity Shapes
What does Asymptotic Complexity Really Say?
 Larger asymptotic complexity values equate to longer run times given that the input size is sufficiently large.
 Consider two theoretical algorithms
 \( \sigma  T(n) = n^2 \)  asymptotically quadratic
 \( \alpha  T(n) = 50n \)  asymptotically linear
Question: Does the linear algorithm always execute fewer machine operations?
What does Asymptotic Complexity Really Say?
Answer: No, for values of \( n \) lower than 50, \( \sigma \) has fewer operations that \( \alpha \)
Asymptotic Complexity
One Last Time:
The goal is to concisely express how an algorithm will scale (growth rate)
Asymptotic Notation
 Known as Landau Symbols outside of CS
 Express asymptotic concepts concisely
 \( \Theta \)  Big Theta
 \( O \)  Big O
 \( \Omega \)  Big Omega
\( \Theta \)  Big Theta

Asymptotic Tight Bound
 Asymptotic complexity is the same regardless of input
 It’s not possible to specify \( \Theta \) for all algorithms
 Recall findDuplicates
 Lower Bound \( T(n) = 3n^2+n+2 \)
 Upper Bound \( T(n) = 2n^2+n+2 \)
 Asymptotic complexity for upper and lower is \( T(n) = n^2 \)
 findDuplicates = \( \Theta(n^2) \)
\( O \)  Big O
 Asymptotic Upper Bound
 Worst Case Scenario
 O stands for Order of or more precisely the German word Ordnung
 \( \Theta \) is a stronger statement than \( O \)
 if findDuplicates = \( \Theta(n^2) \) then findDuplicates = \( O(n^2) \)
 Big O makes no indication of the best case scenario
\( \Omega \)  Big Omega
 Asymptotic Lower Bound
 Best Case Scenario
 \( \Theta \) is a stronger statement than \( \Omega \)
 if findDuplicates = \( \Theta(n^2) \) then findDuplicates = \( \Omega(n^2) \)
 Big \( \Omega \) makes no indication of the worst case scenario
Example Analysis
function bubbleSort(A)
sorted ← false
index ← A − 2
while sorted is false do
sorted←true
for i ← to index do
if A[i] > A[i + 1] then
swap A[i] and A[i + 1]
sorted ← false
end if
end for
index ← index  1
end while
end function
Lower \( 2n + 5 \) \( \Omega(n) \)
Upper \( 4n^2 + 3n + 3 \) \( O(n^2) \)
* Continues to loop until the for loop on line 6 makes a complete revolution without triggering the if statement on line 7. The extra operation is the exit condition (sorted is true)
** The number of iterations is reduced by 1 for each iteration of the outer loop by virtue of line 12. So it’s technically \( \sum_{i=1}^{n} i \) ; however, this is a bit of minutia that’s best left rounded up to \( n^2 \).
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Asymptotic Complexity
By Dale Alleshouse
Asymptotic Complexity
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