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Algorithms -- What we’ll do

Algorithms -- What we’ll do

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Algorithms -- What we’ll do

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  1. Algorithms -- What we’ll do • Accomplishing tasks with algorithms • Working with existing algorithms: • Selection Sort • Insertion Sort • MergeSort • Binary Search • Euclid’s algorithm

  2. Knowing all of the rules of English, grammar and spelling, will not help you give directions from place a to place b if you do not know how to get there. • In systems, an analyst can describe a method in more abstract terms to a programmer without knowing the exact syntax of the programming language

  3. Programs are typically based on one or more algorithms.

  4. An algorithm is an abstract and formalstep by step recipe that tells how to perform a certain task or solve a certain problem on a computer • An algorithm can describe a method for accomplishing a task without relying on any particular programming or any particular computer model

  5. Flowcharts assist in the decomposition of a problem and are frequently used although not in the most formal form as illustrated in handout (flowchart) • Pseudocode is a solution in a loosely formatted style of the actual software, JAVA, code but without the syntax. This is the shorthand that developers use to flesh out a solution.

  6. Bubble Sort • Sort an array in ascending or descending order by evaluating the nth element against the nth+1 element. If they are not in the prescribed order, swap them. When we reach the end of the array, all of the items will be sorted. • This sort traverses the array swapping all the pairs of adjacent elements that are out of order. • The total number of comparisons is (n-1)n/2 • This is an O(n^2) sort

  7. BUBBLE SORT 8 5 10 1 6

  8. PASS 1 8 5 10 1 6

  9. PASS 1 5 8 10 1 6

  10. PASS 1 5 8 10 1 6

  11. PASS 1 5 8 10 1 6

  12. PASS 1 5 8 1 10 6

  13. PASS 1 5 8 1 10 6

  14. PASS 1 5 8 1 6 10

  15. 5 8 1 6 10

  16. PASS 2 5 8 1 6 10

  17. PASS 2 5 8 1 6 10

  18. PASS 2 5 1 8 6 10

  19. PASS 2 5 1 8 6 10

  20. PASS 2 5 1 6 8 10

  21. 5 1 6 8 10

  22. PASS 3 5 1 6 8 10

  23. PASS 3 1 5 6 8 10

  24. PASS 3 1 5 6 8 10

  25. 1 5 6 8 10

  26. PASS 4 1 5 6 8 10

  27. 1 5 6 8 10

  28. 1 5 6 8 10

  29. Selection Sort • For example, an algorithm for arranging the elements of an array in ascending order • Find the largest element on the array and swap it with the last element, then continue the process for the first n-1 elements

  30. 1st iteration takes the LARGEST ARRAY element and swaps it with the LAST array element • The largest element is now in its correct place and will not be moved again

  31. We logically reduce the size of the array and ignore the “last” element(s)

  32. Steps in selection sort: • Initialize a variable, n , to the size of the array • Find the largest among the first n elements • Make it swap places with the nth element • Decrement n by 1 • Repeat steps 2 to 4 while n >= 2

  33. SEE ALGORITHM.JAVA

  34. Go through a sample algorithm (output in class notes)

  35. SELECTION SORT OUTPUT: initial array: 2 97 17 37 12 46 10 55 80 42 39 selection sort in progress... 2 39 17 37 12 46 10 55 80 42 97 2 39 17 37 12 46 10 55 42 80 97 2 39 17 37 12 46 10 42 55 80 97 2 39 17 37 12 42 10 46 55 80 97

  36. SELECTION SORT OUTPUT: initial array: 2 97 17 37 12 46 10 55 80 42 39 selection sort in progress... 2 10 12 17 37 39 42 46 55 80 97

  37. The same procedure can be used to sort the array in descending order by finding the SMALLEST element in the array

  38. Slowest and most predictable sort • There are at most n swap operations performed. • This is still an O(n^2) sort because counting the number of comparisons yields a progression similar to the bubble sort. • Quadratic Time • The number of operations is proportional to the size of the task SQUARED 20 tasks = 400

  39. TPS: How would you code for the following: • How would you Insert an element in an Unsorted Array • How would you Insert an element in a Sorted Array

  40. Insertion Sort O(n^2) • Places one element at a time into position. • The idea with this sort is to keep the beginning part of the array sorted and insert each next element into the correct place in it • We must always start with an empty or an ordered list to traverse • We are now placing the elements as we find them

  41. The approach is analogous to the fashion in which many people pick up playing cards and order them in their hands, one by one • You pick up the cards in order and compare the n th card with the cards already held until the appropriate place is found searching from the end of your current hand and working backwards

  42. for (count = 1; count < numberofelements –1; count++) { assign value of A[count] to a variable temp check each of the array elements to the left of A[count] if the array element is GREATER than temp, shift it right 1 position when an array element is found that is NOT GREATER than temp, temp goes into the open position }

  43. Example: ordered list: 1 3 5 7 (hand of cards) list to order: 2 8 9 4 6 0 (n th card to pick up) • Take 2 and try to swap it with the next lower position until it finds the proper position or until the array is fully traversed

  44. Example: • In evaluating the list to order’s 1st element, 2 1 3 5 2 7 1 3 2 5 7 1 2 3 5 7

  45. This sort works better with Linked Lists • Best case is an O(n) sort --- occurs when list is already sorted • However, in an average case, the math proves this is still an O(n^2) sort.

  46. Example: For (count = 1; count < numberofelements –1; count++) { assign value of A[count] to a variable temp check each of the array elements to the left of A[count] if the array element is GREATER than temp, shift it right 1 position when an array element is found that is NOT GREATER than temp, temp goes into the open position }

  47. Code Example in file Algorithm.java

  48. Mergesort O(n log n) • Mergesort O(n log n) time • A “Divide and Conquer” sort • If we had 2 sorted arrays we could combine (merge) them into 1 sorted array --- this serves as the basis for the mergesort

  49. Recursively, the algorithm works as follows: • Split the array into 2 equal halves • Sort the first half • Sort the second half • Merge the 2 halfs • The base case occurs when the array has only 1 element

  50. This is an O(n log n) sort (performance is similar for best and worst cases) 20 tasks = 86.4 • Requires a temporary workspace of n elements • The merge uses a temp (array) to store the halves of the array