asymptotic analysis and insertion sort analysis
Upcoming SlideShare
Loading in...5
×
 

Like this? Share it with your network

Share

asymptotic analysis and insertion sort analysis

on

  • 798 views

 

Statistics

Views

Total Views
798
Views on SlideShare
798
Embed Views
0

Actions

Likes
1
Downloads
26
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

asymptotic analysis and insertion sort analysis Presentation Transcript

  • 1. Data Structures and Algorithms Lecture 1: Asymptotic Notations
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Comparison of Running times
  • 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.