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Algorithm & data structure lec2


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  • 1. Algorithm & Data Structure CSC-112 Fall 2011 Syed Muhammad Raza
  • 2. Algorithm Requirements Requirements for an algorithm:  Input  Output  Unambiguous  Generality  Correctness  Finite  Efficiency
  • 3. Algorithm Representation Pseudo-code Flow chart
  • 4. Pseudo-Code Pseudo-code is a semi-formal, English-like language with a limited vocabulary that can be used to design and describe algorithms. The main purpose of a pseudo-code is to define the procedural logic of an algorithm in a simple, easy-to-understand manner for its readers, who may or may not be proficient in computer programming.
  • 5. Pseudo-Code Used in designing algorithms. Used in communicating to users. Used in implementing algorithms as programs. Used in debugging logic errors in programs. Used in documenting programs for future maintenance and expansion purposes.
  • 6. Pseudo-Code Must have a limited vocabulary. Must be easy to learn. Must produce simple, English-like narrative notation. Must be capable of describing all algorithms, regardless of their complexity.
  • 7. Control Structures Sequence Selection Repetition
  • 8. Sequence Series of steps or statements that are executed in the order they are written. Example: Read taxable income Read filing status Compute income tax Print income tax
  • 9. Selection Defines one or two courses of action depending on the evaluation of a condition. A condition is an expression that is either true or false. Example if condition (is true) then-part else else-part end_if
  • 10. Nested Selection if status is equal to 1 print “Single” else if status is equal to 2 print “Married filing jointly” else if status is equal to 3 print “Married filing separately end_if end_if end_if
  • 11. Repetition Specifies a block of one or more statements that are repeatedly executed until a condition is satisfied. Example: while condition loop-body end_while
  • 12. Conventions Each pseudo-code statement consists of keywords that describe operations and some appropriate, English-like description of operands. Each statement should be written on a separate line. Continuation lines should be indented
  • 13. Conventions II Sequence statements should begin with unambiguous words (compute, set, initialize). Selection statements then-part and else-part should be indented. Selection statements end with the keyword end_if.
  • 14. Convention III Repetition statements end with end_while. Loop-bodies are indented. All words in a pseudo-code statement must be chosen to be unambiguous, and as easy as possible to understand by non- programmers. Enclose comments between /* and */
  • 15. Example 1If students grade is greater than or equal to 60 Print "passed"else Print "failed“End_if
  • 16. Example 2Set total to zeroSet grade counter to oneWhile grade counter is less than or equal to ten Input the next grade Add the grade into the totalEnd_whileSet the class average to the total divided by tenPrint the class average.
  • 17. Example 3initialize passes to zeroinitialize failures to zeroinitialize student to onewhile student counter is less than or equal to ten input the next exam result if the student passed add one to passes else add one to failures End_if add one to student counterEnd_whileprint the number of passesprint the number of failuresif eight or more students passed print “good result“End_if
  • 18.  Set asterik counter to one Set outer while counter to one While out counter is less than or equal to 4 set spaces counter to 1 while spaces counter is less than 4 print “ ” increment space by one end_while set asterik counter to one while asterik counter is less than or equal to odd multiple of outer counter print “*” increment asterik counter by one end_while print new lineEnd_while
  • 19. Basic Symbols Rounded box - use it to represent an event which occurs automatically. Such an event will trigger a subsequent action, for example `receive telephone call’, or describe a new state of affairs. Rectangle or box - use it to represent an event which is controlled within the process. Typically this will be a step or action which is taken. In most flowcharts this will be the most frequently used symbol. Diamond - use it to represent a decision point in the process. Typically, the statement in the symbol will require a `yes or `no response and branch to different parts of the flowchart accordingly. Circle - use it to represent a point at which the flowchart connects with another process. The name or reference for the other process should appear within the symbol.
  • 20. Flowchart A flowchart is a diagrammatic representation that illustrates the sequence of operations to be performed to get the solution of a problem
  • 21. Guide Lines Flowcharts are usually drawn using some standard symbols; however, some special symbols can also be developed when required Start or end of the program. Computational steps or processing function of a program Input or output operation, Decision making and branching Connector or joining of two parts of program Magnetic Tape Off-page connector Flow line Annotation Display
  • 22. Example Draw a flowchart to find the largest of three numbers A,B, and C.
  • 23. Example Draw a flowchart to find the sum of first 50 natural numbers.
  • 24. Imagination is more important than knowledge Knowledge is Limited. Imagination encircles the world Einstein
  • 25. Efficiency of an Algorithm The Amount of Computer Time and Space that an Algorithm Requires We are more interested in time complexity. Why? Efficiency of algorithms helps comparing different methods of solution for the same problem Algorithm Analysis Should be Independent of  Specific Implementation languages,  Computers and  Data Efficiency is a concern for large problems (large data size)
  • 26. Algorithm Growth Rate Algorithm growth rate is a measure of it’s efficiency Growth Rate: Measure an Algorithm’s Time Requirement as a Function of the Problem Size (N) How quickly the Algorithm’s Time Requirement Grows as a Function of the Problem Size The Way to Measure a Problem’s Size Depends on the Application  Number of Nodes in a Linked List  The Size of an Array  Number of Items in the Stack  Etc.
  • 27. Example Suppose we have two algorithms A and B for solving the same problem. Which one do we use? Suppose  Algorithm A Requires Time Proportional to N2 (expressed as O(N2 ). This is Big O Notation.)  Algorithm B Requires Time Proportional to N , i.e. B is O(N) Obviously B is a better choice
  • 28. Big O Notation If Algorithm A Requires Time Proportional to f(N), Algorithm A is Said to be Order f(N), Which is Denoted as O(f(N)); f(N) is Called Algorithm’s Growth-Rate Function
  • 29. Growth Rates Comparison
  • 30. Growth Rates Comparison (Tabular)
  • 31. Common Growth Rates Constant : O(1)  Time Requirement is Constant  Independent of Problem Size  Example: Accessing an array element Logarithmic : O(log N)  Time Requirement Increases Slowly for a Logarithmic Function  Typical algorithm: Solves a Problem by Solving a Smaller Constant Fraction of the Problem  Example: Binary search Linear : O(N)  Time Requirement Increases Directly with the size of the Problem  Example: Retrieving an element from a linked list
  • 32. Common Growth Rates O(N log N)  Time Requirement Increases More Rapidly Than a Linear Algorithm  Typical algorithms : Divide a Problem into Smaller Problems That are Each Solved Separately Quadratic : O(N2)  Increases Rapidly with the Size of the Problem  Typical Algorithms : Algorithms that Use Two Nested Loops  Practical for Only Small Problems Cubic : O(N3)  Increases Rapidly with the Size of the Problem  Typical Algorithms : Algorithms that Use Three Nested Loops
  • 33. Common Growth Rates Exponential time: O( 2N)  Very, very, very bad, even for small N  Usually Increases too Rapidly to be Practical
  • 34. Sorting Algorithms Sorting is the process of rearranging your data elements/Item in ascending or descending order  Unsorted Data 4 3 2 7 1 6 5 8 9  Sorted Data (Ascending) 1 2 3 4 5 6 7 8 9  Sorted Data (Descending) 9 8 7 6 5 4 3 2 1
  • 35. Sorting Algorithms They are many  Bubble Sort  Selection Sort  Insertion Sort  Shell sort  Comb Sort  Merge Sort  Heap Sort  Quick Sort  Counting Sort  Bucket Sort  Radix Sort  Distribution Sort  Time Sort Source: Wikipedia
  • 36. Bubble Sort Compares Adjacent Items and Exchanges Them if They are Out of Order When You Order Successive Pairs of Elements, the Largest Element Bubbles to the Top(end) of the Array Bubble Sort (Usually) Requires Several Passes Through the Array
  • 37. Bubble Sort Pass 1 Pass 229 10 14 37 13 10 14 29 13 3710 29 14 37 13 10 14 29 13 3710 14 29 37 13 10 14 29 13 3710 14 29 37 13 10 14 13 29 3710 14 29 13 37
  • 38. Selection Sort To Sort an Array into Ascending Order, First Search for the Largest Element Because You Want the Largest Element to be in the Last Position of the Array, You Swap the Last Item With the Largest Item to be in the Last Position of the Array, You Swap the Last Item with the Largest Item, Even if These Items Appear to be Identical Now, Ignoring the Last (Largest) Item of the Array, search Rest of the Array For Its Largest Item and Swap it With Its Last Item, Which is the Next-to-Last Item in the original Array You Continue Until You Have Selected and Swapped N-1 of the N Items in the Array The Remaining Item, Which is Now in the First Position of the Array, is in its Proper Order
  • 39. Selection Sort Initial Array 29 10 14 37 13 After1st Swap 29 10 14 13 37 After2nd Swap 13 10 14 29 37 After3rd Swap 13 10 14 29 37 After4th Swap 10 13 14 29 37
  • 40. Insertion Sort Divide Array Into Sorted Section At Front (Initially Empty), Unsorted Section At End Step Iteratively Through Array, Moving Each Element To Proper Place In Sorted Section Sorted Unsorted ….. ….. 0 i N-1 After i Iterations
  • 41. Insertion SortInitial Array 29 10 14 37 13 Copy 10 29 29 14 37 13 Shift 29 10 29 14 37 13 Insert 10, Copy 14 10 29 29 37 13 Shift 29 Insert 14; Copy 37 10 14 29 37 13 Insert 37 on Itself 10 14 29 37 13 Copy 13 10 14 14 29 37 Shift 14, 29, 37Sorted Array 10 13 14 29 37 Insert 13
  • 42. Shell Sort Improved and efficient version of insertion sort It iterates the data elements/items like insertion sort, but instead of shifting it makes the swaps.
  • 43. Shell Sort 3 1 1 1 5 5 2 2 1 3 3 3 2 2 5 4 4 4 4 5
  • 44. Merge Sort Divide and Conquer Algorithm Recursively split the input in half Then recursively merge pairs of pieces Recursive Steps:  Divide the Array in Half  Sort the Halves  Merge The Halves inside a Temporary Array  Copy Temporary Array to the appropriate locations in original array
  • 45. Merge Sort The Recursive calls Continue to divide the Array into Pieces Until Each Piece Contains Only One Item An Array of One Item is Obviously Sorted The Algorithm Then Merges These small Pieces Until One Sorted Array Results
  • 46. Merge Sort 38 16 27 39 12 27 38 16 27 39 12 27 Recursive Calls38 16 27 39 12 27 to Mergesort38 16 39 1216 38 12 39 Merge 16 27 38 12 27 39 Steps 12 16 27 27 38 39
  • 47. Quick Sort Divide and Conquer algorithm Quicksort Works by Partitioning the Array into Two Pieces Separated by a Single Element That is Greater Than all the Elements in the Left part and Smaller Than all the Elements in the right part This Guarantees That, the Single Element , Called the Pivot Element, is in its Correct position Then the Algorithm Proceeds, Applying the Same Method to the Two parts Separately
  • 48. Quick Sort Partition (Divide)  Choose a pivot  Find the position for the pivot so that  all elements to the left are less  all elements to the right are greater < pivot pivot >= pivot
  • 49. Quick Sort Conquer  Apply the same algorithm to each half < pivot >= pivot < p’ p’ > p’ pivot < p” p” >= p”
  • 50. Partitioning Method Must Arrange Items Into Two regions S1, the Set of Items Less Than the Pivot, and S2, the Set of Items Greater Than or Equal to Pivot Different algorithms for Choice of a Pivot Retain Pivot in A[F] position The Items That await Placement are in Another Region , Called the Unknown Region S1 S2 Unknown P <P >=P ? F LastS1 FirstUnknown L At Each Step of the partition Algorithm you Examine One Item from Unknown Region and Place it in S1 or S2
  • 51.  To Move A[FirstUnknown] into S1 Swap S1 Unknown S2 P <P >=P >=P <P ? F LastS1 LastS1+1 FirstUnknown L  Swap A[FirstUnknown] With the First Item of S2, Which is A[LastS1+1], and Then Increment S1 by 1  Thus the Item That Was in A[FirstUnknown] will be at the Rightmost Position of S1  Item of S2 That was Moved to A[FirstUnknown]: If you Increment FirstUnknown by 1, That Item Becomes the Rightmost Member of S2  Thus, Statements for the Following Steps are Required  Swap A [FirstUnknown] with A[lastS1+1]  Increment LastS1  Increment FirstUnknown
  • 52.  To Move A[FirstUnknown] into S2 S1 S2 Unknown P <P >=P ? F LastS1 FirstUnknown L  Region S2 and Unknown are Adjacent  Simply Increment FirstUnknown by 1, S2 Expands to the Right To Move Pivot Between S1 and S2  Interchange A[LastS1], the Rightmost Item in S1 with Pivot  Thus, Pivot Would be in its Actual Location
  • 53. Quick Sort Example (only one step)Pivot Choose Pivot, keep it in A[F] 27 38 12 39 27 16Pivot Unknown FirstUnknown = 1(Points to 38 27 38 12 39 27 16 38 Belongs in S2Pivot S2 Unknown S1 is Empty 27 38 12 39 27 16 12 Belongs in S1, swap38 & 12Pivot S1 S2 Unknown 27 12 38 39 27 16 39 Belongs in S2Pivot S1 S2 Unknown 27 12 38 39 27 16 27 Belongs in S2Pivot S1 S2 Unknown 27 12 38 39 27 16 16 Belongs in S1, Swap 38 & 16Pivot S1 S2 27 12 16 39 27 38 No more Unknown S1 Pivot S2 16 12 27 39 27 38 Place Pivot between S1 and S2
  • 54. Radix Sort 329 720 720 329 457 355 329 355 657 436 436 436 839 457 839 457 436 657 355 657 720 329 457 720 355 839 657 839
  • 55. Comparison of Sorting Algorithms Worst CaseSelection Sort N2Bubble Sort N2Insertion Sort N2Mergesort N * log NQuicksort N2Radix Sort N
  • 56. Searching Algorithms Searching is the process of determining whether or not a given value exists in a data structure or a storage media. We will study two searching algorithms  Linear Search  Binary Search
  • 57. Linear Search: O(n) The linear (or sequential) search algorithm on an array is:  Start from beginning of an array/list and continues until the item is found or the entire array/list has been searched.  Sequentially scan the array, comparing each array item with the searched value.  If a match is found; return the index of the matched element; otherwise return –1. Note: linear search can be applied to both sorted and unsorted arrays.
  • 58. Linear Search bool LinSearch(double x[ ], int n, double item) { for(int i=0;i<n;i++) { if(x[i]==item) { return true; } else { return false; } } return false; }
  • 59. Linear Search Tradeoffs Benefits  Easy to understand  Array can be in any order Disadvantages  Inefficient for array of N elements Examines N/2 elements on average for value in array, N elements for value not in array
  • 60. Binary Search: O(log2 n) Binary search looks for an item in an array/list using divide and conquer strategy
  • 61. Binary Search  Binary search algorithm assumes that the items in the array being searched is sorted  The algorithm begins at the middle of the array in a binary search  If the item for which we are searching is less than the item in the middle, we know that the item won’t be in the second half of the array  Once again we examine the “middle” element  The process continues with each comparison cutting in half the portion of the array where the item might be
  • 62. Binary Search Binary search uses a recursive method to search an array to find a specified value The array must be a sorted array: a[0]≤a[1]≤a[2]≤. . . ≤ a[finalIndex] If the value is found, its index is returned If the value is not found, -1 is returned Note: Each execution of the recursive method reduces the search space by about a half
  • 63. Pseudocode for Binary Search
  • 64. Execution of Binary Search
  • 65. Execution of Binary Search
  • 66. Key Points in Binary Search1. There is no infinite recursion • On each recursive call, the value of first is increased, or the value of last is decreased • If the chain of recursive calls does not end in some other way, then eventually the method will be called with first larger than last2. Each stopping case performs the correct action for that case • If first > last, there are no array elements between a[first] and a[last], so key is not in this segment of the array, and result is correctly set to -1 • If key == a[mid], result is correctly set to mid
  • 67. Key Points in Binary Search3. For each of the cases that involve recursion, if all recursive calls perform their actions correctly, then the entire case performs correctly • If key < a[mid], then key must be one of the elements a[first] through a[mid-1], or it is not in the array • The method should then search only those elements, which it does • The recursive call is correct, therefore the entire action is correct
  • 68. Key Points in Binary Search • If key > a[mid], then key must be one of the elements a[mid+1] through a[last], or it is not in the array • The method should then search only those elements, which it does • The recursive call is correct, therefore the entire action is correctThe method search passes all three tests:Therefore, it is a good recursive method definition
  • 69. Efficiency of Binary Search The binary search algorithm is extremely fast compared to an algorithm that tries all array elements in order  About half the array is eliminated from consideration right at the start  Then a quarter of the array, then an eighth of the array, and so forth
  • 70. Efficiency of Binary Search Given an array with 1,000 elements, the binary search will only need to compare about 10 array elements to the key value, as compared to an average of 500 for a serial search algorithm The binary search algorithm has a worst-case running time that is logarithmic: O(log n)  A serial search algorithm is linear: O(n) If desired, the recursive version of the method search can be converted to an iterative version that will run more efficiently
  • 71. Binary Search