1. © 2006 Pearson Addison-Wesley. All rights reserved8 A-1 CMPT 225 Final Review

2. © 2006 Pearson Addison-Wesley. All rights reserved8 A-2 Time and Place Exam is at 7:00-10:00pm on April 10 Room: HCC 1315 Exception... for the 3 students with a conflict in Burnaby, the room will be AQ4120

3. © 2006 Pearson Addison-Wesley. All rights reserved8 A-3 Topics Covered (before midterm) Java review Software Engineering Recursion –(and a little algorithm efficiency) Abstract Data Types Linked Lists Stacks Queues

4. © 2006 Pearson Addison-Wesley. All rights reserved8 A-4 Topics Covered (after midterm) Java Interfaces and Generics Trees Tables and Priority Queues Hashing Graphs Sorting

5. © 2006 Pearson Addison-Wesley. All rights reserved8 A-5 Topics Covered Roughly: chapters 1-14 Minus chapter 6... though we did some of it in class We didn’t read the second half of ch. 7, but we covered it in class Didn’t read first half of ch. 10, but we covered the material in class

6. © 2006 Pearson Addison-Wesley. All rights reserved8 A-6 Java Review You should be able to read Java code –You don’t need to memorize a bunch of java.util classes You should be able to write Java code –Defining your own classes, interfaces, methods, etc.

7. © 2006 Pearson Addison-Wesley. All rights reserved8 A-7 Software Engineering What is a good solution to a problem? Object oriented design –Read a UML diagram –Write a UML diagram Modularity, modifiability, readability, ease of use, fail-safe programming…

8. © 2006 Pearson Addison-Wesley. All rights reserved8 A-8 Recursion Write recurrence relations –e.g. rabbits(n) = rabbits(n-1) + rabbits(n-2) Write recursive (Java) methods Solve problems with simple recursive solutions Backtracking Relation with induction

9. © 2006 Pearson Addison-Wesley. All rights reserved8 A-9 Algorithm Efficiency Big-O notation: –What does O(n) mean? –What about O(n 2 )? Count the steps of an algorithm, translate to big-O

10. © 2006 Pearson Addison-Wesley. All rights reserved8 A-10 Abstract Data Types Difference: –ADT vs. data structure How do you specify ADTs? Designing the right ADT for a problem ADT List and SortedList

11. © 2006 Pearson Addison-Wesley. All rights reserved8 A-11 Linked Lists Object references Using nodes and references Programming with linked lists –inserting, deleting, traversing Variations –tail references, circular lists, doubly linked

12. © 2006 Pearson Addison-Wesley. All rights reserved8 A-12 Stacks ADT Stack –LIFO Implementations Problem solving with Stacks

13. © 2006 Pearson Addison-Wesley. All rights reserved8 A-13 Queues ADT Queue –FIFO Implementations Problem solving with Queues

14. © 2006 Pearson Addison-Wesley. All rights reserved8 A-14 Java Generics What is a generic class? Why do you use them? How do you define one?

15. © 2006 Pearson Addison-Wesley. All rights reserved8 A-15 Trees General trees Binary trees Binary search trees Traversals (in-order, pre-order, post-order) Implementations Efficiency of search

16. © 2006 Pearson Addison-Wesley. All rights reserved8 A-16 Tables Basic implementations –array, reference Faster implementations –binary search tree –2-3 trees –red-black trees –hash functions

17. © 2006 Pearson Addison-Wesley. All rights reserved8 A-17 Hashing What is a good has function? What is a collision? Collision resolution

18. © 2006 Pearson Addison-Wesley. All rights reserved8 A-18 Graphs Definitions and basic properties Implementations –adjacency list –adjacency matrix Traversals and spanning trees –depth-first –breadth-first

19. © 2006 Pearson Addison-Wesley. All rights reserved8 A-19 Sorting Quadratic sorts: –Selection, insertion, bubble Log linear sorts: –merge, quick Special case: radix Should be able to explain/sketch/understand different sorts

20. © 2006 Pearson Addison-Wesley. All rights reserved8 A-20 Example 1 What can go wrong with this method? How can you protect yourself? public static double computation(double x) { return java.lang.Math.sqrt(x) / java.lang.Math.cos(x); }

21. © 2006 Pearson Addison-Wesley. All rights reserved8 A-21 Example 1 - Solution Two hazards: 1)the argument x cannot be a negative number because computation() calls the sqrt() method with x as the argument. 2)since zero is in the range of the method cos(x), the result of this computation must be tested prior to using it as a divisor in order to avoid a "Divide by Zero error".

22. © 2006 Pearson Addison-Wesley. All rights reserved8 A-22 Example 1 – Solution double computation( double x ) { double denominator = Math.cos(x); double result = 0; if ( x < 0 ) {System.out.println("Cannot take square root of negative number"); result = -1; } if ( denominator == 0 ) {System.out.println("Cannot divide by zero"); result = -1; } if ( result != -1 ) result = Math.sqrt(x)/denominator; return result; }

23. © 2006 Pearson Addison-Wesley. All rights reserved8 A-23 Example 2 Write the loops invariant for this: void printN(int n) { int i = 20; while(i>1) i--; }

24. © 2006 Pearson Addison-Wesley. All rights reserved8 A-24 Example 2 - Solution 20 >= i >= 1

25. © 2006 Pearson Addison-Wesley. All rights reserved8 A-25 Example 3 Write a recursive method that will compute the sum of the first n integers in an array of at least n integers.

26. © 2006 Pearson Addison-Wesley. All rights reserved8 A-26 Example 3 - Solution int computeSum(final int a[], int n) { // base case if(n <= 0) return 0; else // reduce the problem size return a[n - 1] + computeSum(a, n - 1); } // end computeSum

27. © 2006 Pearson Addison-Wesley. All rights reserved8 A-27 Example 4 Write a recursive Java method that writes the digits of a positive decimal integer in reverse order.

28. © 2006 Pearson Addison-Wesley. All rights reserved8 A-28 Example 4 - Solution void reverseDigits(int number){ if(number >= 0) // check for input bounds { // base case if(number < 10) System.out.print(number); else { // print out rightmost digit System.out.print(number % 10); // pass remainder of digits to next call reverseDigits(number / 10); } // end if } // end reverseDigits

29. © 2006 Pearson Addison-Wesley. All rights reserved8 A-29 Example 5 Implement a method “swap” for lists... use the ADT list, and handle the exception if the positions being swapped do not exist.

30. © 2006 Pearson Addison-Wesley. All rights reserved8 A-30 Example 5 - Solution void swap(List aList, int i, int j) { Object first, second; try{ first = aList.get(i); second = aList.get(j); } catch(Exception e){ System.out.println(e); } aList.remove(i); aList.add(i, second); aList.remove(j); aList.add(j, first); } // end Swap

31. © 2006 Pearson Addison-Wesley. All rights reserved8 A-31 Example 6 Trace the selection sort as it sorts the following array: 7, 12, 24, 4, 19, 32

32. © 2006 Pearson Addison-Wesley. All rights reserved8 A-32 Example 6 - Solution 7 12 24 4 19 32 4 12 24 7 19 32 4 7 19 12 24 32 4 7 12 19 24 32

33. © 2006 Pearson Addison-Wesley. All rights reserved8 A-33 Example 7 Draw the complete binary tree obtained by inserting the following values in the given order: 4, 13, 5, 3, 7, 30

34. © 2006 Pearson Addison-Wesley. All rights reserved8 A-34 Example 7 - Solution 4 / \ 13 5 / \ / 3 7 30

35. © 2006 Pearson Addison-Wesley. All rights reserved8 A-35 Example 8 Draw the binary search tree obtained by inserting the following values in the given order: 4, 13, 5, 3, 7, 30

36. © 2006 Pearson Addison-Wesley. All rights reserved8 A-36 Example 7 - Solution 4 / \ 3 13 / \ 5 30 \ 7

37. © 2006 Pearson Addison-Wesley. All rights reserved8 A-37 Example 9 A general tree can be implemented by having an array of child references. 1) What are the advantages and disadvantages of this implementation? 2) Write a recursive preorder traversal method for a general tree.

38. © 2006 Pearson Addison-Wesley. All rights reserved8 A-38 Example 9 - Solution Advantages: Access to the children is easy; direct access to each child is available. Disadvantages: Limits the number of children to a fixed maximum. If the maximum number of children possible is higher than the average number of children, then memory is wasted. If the children are kept in order, insertions and deletions may require shifting pointers in the array.

39. © 2006 Pearson Addison-Wesley. All rights reserved8 A-39 Example 9 - Solution public void traverse(TreeNode tNode); {if (tNode != null) { queue.enqueue(tNode.getItem()); for (int i=0; i<tNode.numChildren; ++i) traverse(tNode.child[i]); } // end if } // end traverse

40. © 2006 Pearson Addison-Wesley. All rights reserved8 A-40 Example 10 What are the advantages of implementing ADT Table with a 2-3 tree instead of a binary search tree?

41. © 2006 Pearson Addison-Wesley. All rights reserved8 A-41 Example 10- Solution The height of a binary search tree depends on the order in which the items are inserted. Advantage: In a tree with N items, the height can range between N (equivalent to a linked list) and log2 ( N+1) (a fully balanced tree). A 2-3 tree is always balanced and thus has height proportional to log N. So, a 2-3 tree is a more efficient ADT table implementation than a binary search tree. Advantage: Maintaining a balanced binary search tree may become very expensive in the face of frequent inserts and deletions as the entire tree must be rebuilt in the worst case.

42. © 2006 Pearson Addison-Wesley. All rights reserved8 A-42 Example 11 Write pseudocode for the tableDelete operation when the implementation uses a hash-table with seperate chaining.

43. © 2006 Pearson Addison-Wesley. All rights reserved8 A-43 Example 11 - Solution tableDelete(Object item) hashval = h(item.searchKey) let LL = the linked list at table[hashval] LL.delete(item)

44. © 2006 Pearson Addison-Wesley. All rights reserved8 A-44 Example 12 Briefly explain/define these terms: –2-3 tree –hash function –adjacency list

45. © 2006 Pearson Addison-Wesley. All rights reserved8 A-45 Example 12 Briefly explain/define these terms: –2-3 tree tree where each node contains either 1 or 2 data items. nodes containing 1 data item have 2 children, nodes containing 2 data items have 3. –hash function a function that maps a search key to a position in an array, to implement an ADT table with efficient search/retrieval –adjacency list a way to implement the ADT graph in which the nodes adjacent to each node are stored in a linked list

46. © 2006 Pearson Addison-Wesley. All rights reserved8 A-46 Example 13 Given the following running times for input n, what is the big-O measure of efficiency –n + 6000 –n (log n) + n(n 2 + 6000) –log n + 6000n

47. © 2006 Pearson Addison-Wesley. All rights reserved8 A-47 Example 13 - Solution Given the following running times for input n, what is the big-O measure of efficiency –n + 6000 ----------------------- O(n) –n (log n) + n(n 2 + 6000) ------------ O(n 3 ) –log n + 6000n -------------------- O(n)

48. © 2006 Pearson Addison-Wesley. All rights reserved8 A-48 Example 14 Which ADT is appropriate for each type of data: –A pile of shoeboxes in a warehouse –The World Wide Web –A collection of employees at a company

49. © 2006 Pearson Addison-Wesley. All rights reserved8 A-49 Example 14 - Solution Which ADT is appropriate for each type of data: –STACK –DIRECTED GRAPH –TABLE

50. © 2006 Pearson Addison-Wesley. All rights reserved8 A-50 Some more notes The final will look a lot like the midterm.... same kinds of questions Some people on the midterm seemed to have trouble with recursion... it will be back There will be no cheat sheet of ADT operations this time

51. © 2006 Pearson Addison-Wesley. All rights reserved8 A-51 More types of examples Displaying contents of a Queue or Stack Writing data structure for an ADT... say the Bag from earlier... need to decide data members Inserting nodes in doubly linked/circular linked lists