Data Structures and Algorithms
Lecture 1: Asymptotic Notations
Basic Terminologies
• Algorithm
– Outline
– Essence of a computational procedure
– Step by step instructions

• Program
– ...
Algorithmic Problem
Specification of
Input

?

Specification of
Output as a
function of Input

• Infinite number of input ...
Algorithmic Solution
Input instance
adhering to
Specification

Algorithm

Output related
to the Input as
required

• Algor...
Characteristics of a Good Algorithm
• Efficient
– Small Running Time
– Less Memory Usage

• Efficiency as a function of in...
Measuring the running time
• Experimental Study
– Write a program that implements the algorithm
– Run the program with dat...
Measuring the running time
• Limitations of Experimental Study
– It is necessary to implement and test the
algorithm in or...
Beyond Experimental Study
• We will develop a general methodology for
analyzing the running time of algorithms. This
appro...
Pseudo - Code
• A mixture of natural language and high level
programming concepts that describes the main
ideas behind a g...
Pseudo - Code
• It is more structured that usual prose but less
formal than a programming language
• Expressions:
– Use st...
Pseudo - Code
• Programming Constructs:
–
–
–
–
–

Decision structures: if…then….[else….]
While loops: while…..do…..
Repea...
Analysis of Algorithms
• Primitive Operation:
– Low level operation independent of programming
language.
– Can be identifi...
Example: Sorting
OUPUT
Permutation of the sequence of
numbers in non-decreasing order

INPUT
Sequence of
numbers

a1,a2,a3...
Insertion Sort

• Used while playing cards
3
4
6
8
9
A
1
i j
Strategy
• Start “empty
handed”
• Insert a card in the
right ...
Analysis of Insertion Sort
• For j
2 to n do
•
key A[j]
•
Insert A[j] into sorted
sequence A*1…j-1]
•
i
j-1
•
while i >0 a...
Best / Average/ Worst Case:
Difference made by tj
• Total Time = n(c1 + c2 + c3 + c7) +
+c3 +c5+c6+c7)
• Best Case

(c4 + ...
Best / Average/ Worst Case
• For a Specific input size say n

Running Time (sec)

5

Worst Case

4

Average Case

3

Best ...
Best / Average/ Worst Case
• Varying input size
Worst case

Running Time (sec)

50

Average Case
40
Best Case

30
20

10
1...
Best / Average/ Worst Case
• Worst Case:
– Mostly used
– It is an Upper bound of how bad a system can be.
– In cases like ...
Asymptotic Analysis
• Goal: Simplify analysis of running time by getting
rid of details which may be affected by specific
...
Asymptotic Notation
• The “big Oh” (O)
Notation
– Asymptotic Upper Bound
– f(n) = O(g(n)), if there
exists constants c and...
Examples
• E.g. 1:
–
–
–
–
–

f(n) = 2n + 6
g(n) = n
c =4
n0 = 3
Thus, f(n) = O(n)

• E.g. 2:
– f(n) = n2
– g(n) = n
– f(n...
Asymptotic Notation
• Simple Rules:
– Drop lower order terms and constant factors
• 50 nlogn is O(nlogn)
• 7n – 3 is O(n)
...
Comparing Asymptotic Analysis of
Running Time
• Hierarchy of f(n)
– Log n < n < n2 < n3 < 2n

• Caution!!!!!
– An algorith...
Examples of Asymptotic Analysis
• Algorithm: prefixAvg1(X)
• Input: An n element array X of numbers
• Output: An n element...
Examples of Asymptotic Analysis
• Algorithm: prefixAvg2(X)
• Input: An n element array X of numbers
• Output: An n element...
Asymptotic Notation (Terminologies)
• Special cases of Algorithms
– Logarithmic : O(log n)
– Linear: O(n)
– Quadratic: O(n...
Asymptotic Notation
• The big Omega (Ω) Notation
– Asymptotic Lower bound
– f(n) = Ω(g(n), if there exists
constants c and...
Asymptotic Notation
• The Big Theta (Θ) notation
– Asymptotically tight bound
– f(n) = Θ(g(n)), if there exists
constants ...
Asymptotic Notation
• There are 2 more notations
– Little oh notation (o), f(n) = o(g(n))
• For every c there should exist...
Comparison of Running times
Assignment
• Write a C program to implement i) Insertion
sort and ii) Bubble Sort.
• Perform an Experimental Study and
gra...
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asymptotic analysis and insertion sort analysis

  1. 1. Data Structures and Algorithms Lecture 1: Asymptotic Notations
  2. 2. Basic Terminologies • Algorithm – Outline – Essence of a computational procedure – Step by step instructions • Program – Implementation of an Algorithm in some programming language • Data Structure – Organization of Data needed to solve the problem effectively – Data Structure you already know: Array
  3. 3. Algorithmic Problem Specification of Input ? Specification of Output as a function of Input • Infinite number of input instances satisfying the specification • E.g.: Specification of Input: • A sorted non-decreasing sequence of natural numbers of non-zero, finite length: • 1, 20,908,909,1000,10000,1000000 • 3
  4. 4. Algorithmic Solution Input instance adhering to Specification Algorithm Output related to the Input as required • Algorithm describes the actions on the input instances • Infinitely many correct algorithms may exist for the same algorithmic problem
  5. 5. Characteristics of a Good Algorithm • Efficient – Small Running Time – Less Memory Usage • Efficiency as a function of input size – The number of bits in a data number – The number of data elements
  6. 6. Measuring the running time • Experimental Study – Write a program that implements the algorithm – Run the program with data sets of varying size and composition – Use system defined functions like clock() to get an measurement of the actual running time
  7. 7. Measuring the running time • Limitations of Experimental Study – It is necessary to implement and test the algorithm in order to determine its running time. – Experiments can be done on a limited set of inputs, and may not be indicative of the running time on other inputs not included in the experiment – In order to compare 2 algorithms same hardware and software environments must be used.
  8. 8. Beyond Experimental Study • We will develop a general methodology for analyzing the running time of algorithms. This approach – Uses a high level description of the algorithm instead of testing one of its implementations – Takes into account all possible inputs – Allows one to evaluate the efficiency of any algorithm in a way that is independent of the hardware and software environment.
  9. 9. Pseudo - Code • A mixture of natural language and high level programming concepts that describes the main ideas behind a generic implementation of a data structure or algorithm. • E.g. Algorithm arrayMax(A,n) – Input: An array A storing n integers – Output: The maximum element in A. currMax A[0] for i 1 to n-1 do if currMax < A[i] then currMax return currMax A[i]
  10. 10. Pseudo - Code • It is more structured that usual prose but less formal than a programming language • Expressions: – Use standard mathematical symbols to describe numeric and Boolean expressions – Use ‘ ’ for assignment instead of “=” – Use ‘=’ for equality relationship instead of “==” • Function Declarations: – Algorithm name (param1 , param2)
  11. 11. Pseudo - Code • Programming Constructs: – – – – – Decision structures: if…then….[else….] While loops: while…..do….. Repeat loops: Repeat………until……. For loop: for……do……….. Array indexing: A[i], A[I,j] • Functions: – Calls: return_type function_name(param1 ,param2) – Returns: return value
  12. 12. Analysis of Algorithms • Primitive Operation: – Low level operation independent of programming language. – Can be identified in a pseudo code. • For e.g. – Data Movement (assign) – Control (branch, subroutine call, return) – Arithmetic and Logical operations (e.g. addition comparison) – By inspecting the pseudo code, we can count the number of primitive operations executed by an algorithm.
  13. 13. Example: Sorting OUPUT Permutation of the sequence of numbers in non-decreasing order INPUT Sequence of numbers a1,a2,a3,a4,….an 2, 5 , 4, 10, 7 Sort Correctness (Requirements for the output) For any given input the algorithm halts with the output: • b1 < b2 < b3 < b4 < ….. < bn • b1, b2, b3, b4,…. bn is a permutation of a1, a2, a3, a4, …. an b1,b2,b3,b4,….bn 2, 4 , 5, 7, 10 Running Time depends on • No. of elements (n) • How sorted (partial) the numbers are? • Algorithm
  14. 14. Insertion Sort • Used while playing cards 3 4 6 8 9 A 1 i j Strategy • Start “empty handed” • Insert a card in the right position of the already sorted hand • Continue until all cards are inserted or sorted 7 2 5 1 n INPUT: An array of Integers A[1..n] OUTPUT: A permutation of A such that A[1] <= A[2] <= A[3]<= ..<=A[n] For j 2 to n do key A[j] Insert A*j+ into sorted sequence A*1…j-1] i j-1 while i >0 and A[i] >key do A[i+1] A[i] i- A[i+1] key
  15. 15. Analysis of Insertion Sort • For j 2 to n do • key A[j] • Insert A[j] into sorted sequence A*1…j-1] • i j-1 • while i >0 and A[i] >key • do A[i+1] A[i] • i- • A[i+1] key Total Time = n(c1 + c2 + c3 + c7) + Cost c1 c2 Times n n-1 c3 c4 c5 c6 c7 n-1 n-1 (c4 + c5 + c6)–( c2 +c3 +c5+c6+c7) tj counts the number of times the values have to be shifted in one iteration
  16. 16. Best / Average/ Worst Case: Difference made by tj • Total Time = n(c1 + c2 + c3 + c7) + +c3 +c5+c6+c7) • Best Case (c4 + c5 + c6)–( c2 – Elements already sorted – tj = 1 – Running time = f(n) i.e. Linear Time • Worst Case – Elements are sorted in inverse order – tj = j – Running time = f(n2) i.e. Quadratic Time • Average Case – tj = j/2 – Running Time = f(n2) i.e. Quadratic Time
  17. 17. Best / Average/ Worst Case • For a Specific input size say n Running Time (sec) 5 Worst Case 4 Average Case 3 Best Case 2 1 A B C D E F G H I J K L M N Input instance
  18. 18. Best / Average/ Worst Case • Varying input size Worst case Running Time (sec) 50 Average Case 40 Best Case 30 20 10 10 20 30 40 Input Size 50 60
  19. 19. Best / Average/ Worst Case • Worst Case: – Mostly used – It is an Upper bound of how bad a system can be. – In cases like surgery or air traffic control knowing the worst case complexity is of crucial importance. • Average case is often as bad as worst case. • Finding an average case can be very difficult.
  20. 20. Asymptotic Analysis • Goal: Simplify analysis of running time by getting rid of details which may be affected by specific implementation and hardware – Like rounding 1000001 to 1000000 – 3n2 to n2 • Capturing the essence: How the running time of an algorithm increases with the size of the input in the limit – Asymptotically more efficient algorithms are best for all but small inputs.
  21. 21. Asymptotic Notation • The “big Oh” (O) Notation – Asymptotic Upper Bound – f(n) = O(g(n)), if there exists constants c and n0 such that f(n) ≤ cg(n) for n ≥ n0 – f(n) and g(n) are functions over non negative nondecreasing integers • Used for worst case analysis c. g(n) f(n) n0 n f(n) = O(g(n))
  22. 22. Examples • E.g. 1: – – – – – f(n) = 2n + 6 g(n) = n c =4 n0 = 3 Thus, f(n) = O(n) • E.g. 2: – f(n) = n2 – g(n) = n – f(n) is not O(n) because there exists no constants c and n0 such that f(n) ≤ cg(n) for n ≥ n0 • E.g. 3: – – – – – f(n) = n2 + 5 g(n) = n2 c=2 n0 = 2 Thus, f(n) = O(n2)
  23. 23. Asymptotic Notation • Simple Rules: – Drop lower order terms and constant factors • 50 nlogn is O(nlogn) • 7n – 3 is O(n) • 8n2logn + 5n2 + n is O(n2logn) – Note that 50nlogn is also O(n5) but we will consider it to be O(nlogn) it is expected that the approximation should be of as lower value as possible.
  24. 24. Comparing Asymptotic Analysis of Running Time • Hierarchy of f(n) – Log n < n < n2 < n3 < 2n • Caution!!!!! – An algorithm with running time of 1,000,000 n is still O(n) but might be less efficient than one running in time 2n2, which is O(n2) when n is not very large.
  25. 25. Examples of Asymptotic Analysis • Algorithm: prefixAvg1(X) • Input: An n element array X of numbers • Output: An n element array A of numbers such that A[i] is the average of elements X[0]…X[i]. for i 0 to n-1 do a 0 for j 0 to i do a a+X[j] A[i] a/(i+1) return A n iterations 1 Step i iterations i = 0 , 1, …., i-1 • Analysis: Running Time O(n2) roughly.
  26. 26. Examples of Asymptotic Analysis • Algorithm: prefixAvg2(X) • Input: An n element array X of numbers • Output: An n element array A of numbers such that A[i] is the average of elements X[0]…X[i]. s 0 for i 0 to n do s s + X[i] A[i] s/(i+1) return A • Analysis: Running time is O(n)
  27. 27. Asymptotic Notation (Terminologies) • Special cases of Algorithms – Logarithmic : O(log n) – Linear: O(n) – Quadratic: O(n2) – Polynomial: O(nk), k ≥ 1 – Exponential: O(an), a>1 • “Relatives” of Big Oh – Big Omega (Ω(f(n)) : Asymptotic Lower Bound – Big Theta (Θ(f(n)): Asymptotic Tight Bound
  28. 28. Asymptotic Notation • The big Omega (Ω) Notation – Asymptotic Lower bound – f(n) = Ω(g(n), if there exists constants c and n0 such that c.g(n) ≤ f(n) for n > n0 • Used to describe best case running times or lower bounds of asymptotic problems – E.g: Lower bound of searching in an unsorted array is Ω(n).
  29. 29. Asymptotic Notation • The Big Theta (Θ) notation – Asymptotically tight bound – f(n) = Θ(g(n)), if there exists constants c1, c2 and n0 such that c1.g(n) ≤ f(n) ≤ c2.g(n) for n > n0 – f(n) = Θ(g(n)), iff f(n) = O(g(n)) and f(n) = Ω(g(n), – O(f(n)) is often misused instead of Θ(f(n))
  30. 30. Asymptotic Notation • There are 2 more notations – Little oh notation (o), f(n) = o(g(n)) • For every c there should exist a no. n0 such that f(n) ≤ o(g(n) for n > n0. – Little Omega notation (ω) • Analogy with real numbers – – – – – f(n) = O(g(n)) , f(n) = Ω(g(n)), f(n) = Θ(g(n)), f(n) = o(g(n)), f(n) = ω(g(n)), f≤g f≥g f=g f<g f>g • Abuse of Notation: – f(n) = O(g(n)) actually means f(n) є O(g(n))
  31. 31. Comparison of Running times
  32. 32. Assignment • Write a C program to implement i) Insertion sort and ii) Bubble Sort. • Perform an Experimental Study and graphically show the time required for both the program to run. • Consider varied size of inputs and best case and worst case for each of the input sets.
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