1. Compsci 201 Midterm 2 Review Jimmy Wei Make sure you sign in on the paper being passed around!

2. List of Topics Linked Lists – Comparison with ArrayList – Linked List practice Trees – Traversals – BSTs – Tree practice Recursion – Three steps of recursion – Recursion practice Recursive Backtracking – The extra steps of backtracking – Recursive backtracking practice Big O Overview – Significance – Big O for common operations

3. Linked Lists How do we build a LinkedList? public class Node { public int value; public Node next; public Node ( int v, Node n ) { value = v; next = n; } Traverse by storing pointer to HEAD and referencing its NEXT pointer

4. Linked Lists Comparison with ArrayList: – ArrayList works by storing an array and resizing it as necessary, hence why it is so fast—we can access different indices and make changes just as we would with an array – Removal in ArrayList is a bit slower since we need to recopy the entire array to eliminate an element

5. Linked Lists Big OArrayListLinkedList Access Insertion Removal What are the runtime differences in these two implementations?

6. Linked Lists Big OArrayListLinkedList AccessO(1)O(n) Insertion Removal What are the runtime differences in these two implementations? – Access is just grabbing an element at an index.

7. Linked Lists Big OArrayListLinkedList AccessO(1)O(n) InsertionO(1) Removal What are the runtime differences in these two implementations? – Access is just grabbing an element at an index. – Insertion is adding at the end of the list

8. Linked Lists Big OArrayListLinkedList AccessO(1)O(n) InsertionO(1) RemovalO(n) What are the runtime differences in these two implementations? – Access is just grabbing an element at an index. – Insertion is adding at the end of the list – Removal is removing from anywhere in the list How can ArrayList insert be O(1)? – Insert is O(1) amortized When would you want to use Linked Lists?

9. Linked Lists Linked List practice! Write the following method: public Node removeDuplicates(Node head) { // your code here } removeDuplicates takes in the head of a Linked List and removes all duplicate values from that list, returns the head

10. Trees Linked Lists on steroids Now instead of one node having one next pointer, it has multiple children; we will focus on binary trees (two children) What do we change from the LinkedList node? public class Node { public int value; public Node left; public Node right; public Node ( int v, Node l, Node r ) { value = v; left = l; right = r; } }

11. Trees How do we navigate across a tree? Three types of traversals: 1)Preorder (my favorite) : Root Left Right 2)Inorder : Left Root Right 3)Postorder : Left Right Root

12. Trees How do we express a tree in terms of its traversal? Think of every node as the root of a subtree, and things become much easier. Lets find the preorder traversal of this tree on the right. 1057121361

13. Preorder traversal: [ Root ][ Left ] [ Right ] Trees Preorder traversal is [Root] [Left] [Right] 1057121361

14. Preorder traversal: 10 [ Left ] [ Right ] Trees Preorder traversal is [Root] [Left] [Right] Root is easy; it’s 10 1057121361

15. Preorder traversal: 10 [[Root][Left][Right]] [ Right ] Trees Preorder traversal is [Root] [Left] [Right] Root is easy; it’s 10 Replace [Left] with the preorder traversal of the left subtree. 1057121361

16. Preorder traversal: 10 5 7 12 [ Right ] Trees Preorder traversal is [Root] [Left] [Right] Root is easy; it’s 10 Replace [Left] with the preorder traversal of the left subtree. Now we can view this subtree as a subset of the main traversal and find its preorder 1057121361

17. Preorder traversal: 10 5 7 12 13 6 1 Trees Preorder traversal is [Root] [Left] [Right] Root is easy; it’s 10 Replace [Left] with the preorder traversal of the left subtree. Now we can view this subtree as a subset of the main traversal and find its preorder 1057121361

18. Trees What makes a binary search tree special? – The tree is sorted: All left children are less than root and all right children are greater than root, repeated values are not allowed – Allows us to search the tree easily – Inserting also not any harder than before, we just move left or right by comparing our new value to the old value at each node

19. Trees What are the runtime complexities? – Depends on whether or not the tree is balanced. OperationBalancedUnbalanced Insert Search Remove

20. Trees What are the runtime complexities? – Depends on whether or not the tree is balanced. OperationBalancedUnbalanced InsertO(log n)O(n) Search Remove

21. Trees What are the runtime complexities? – Depends on whether or not the tree is balanced. OperationBalancedUnbalanced InsertO(log n)O(n) SearchO(log n)O(n) Remove

22. Trees What are the runtime complexities? – Depends on whether or not the tree is balanced. OperationBalancedUnbalanced InsertO(log n)O(n) SearchO(log n)O(n) RemoveO(log n)O(n)

23. Trees Tree practice! Write the following method: public boolean validate(Node root) { // your code here } Validate takes in the root of a binary search tree and returns true if that tree is a valid BST, and false otherwise

24. Trees Tree practice! Need a harder question? Write the following method: public Node remove(Node root, int value) { // your code here } Remove takes in a Node root and an int value and removes the node with that value from the tree with root, if it exists. This is tricky! Just finding the node isn’t enough, we must also resort the tree to still be a valid BST afterwards! – Hint: if you replace a removed node with a node from its left subtree, you must use the rightmost node in that subtree! On the other hand, if you replace from the right subtree, you must use the leftmost node in that subtree!

25. Recursion What is recursion? – See: https://piazza.com/class#spring2013/cs201/736 – Strategy for solving harder problems by breaking them down into smaller subproblems – Three easy steps: 1)Recursion step : how do we break a problem into easier subproblems? 2)Resolution step : how do we combine the answers to our subproblems into an answer for our harder problem? 3)Base step : when do we stop recursing?

26. Recursion Let’s start with an easy example: finding the nth fibonacci number. We will write the following method: public int fibonacci(int n) { // returns the nth fibonacci number }

27. Recursion public int fibonacci(int n) { } What is the “hard” problem here? – Finding the nth fibonacci number. How can we break this down to easier subproblems? – Finding the n-1th and n-2th fibonacci number.

28. Recursion public int fibonacci(int n) { int a = fibonacci(n-1); int b = fibonacci(n-2); } Let’s start by adding our recursion step. How do we break the hard problem down in code?

29. Recursion public int fibonacci(int n) { int a = fibonacci(n-1); int b = fibonacci(n-2); int result = a + b; } Next comes the resolution step. How do we combine these subproblem answers into our hard problem answer?

30. Recursion public int fibonacci(int n) { if (n == 1 || n == 2) { return 1; } int a = fibonacci(n-1); int b = fibonacci(n-2); int result = a + b; } Finally, the base step. When do we stop recursing?

31. Recursion Recursion practice! Write the following method: public String countDown(int n) { // your code here } countDown takes in an int and prints out a String counting down from that int to 0 and then back up to that int. – For example, countDown(3) should return “3…2…1…0…1…2…3”

32. Recursion Recursion practice! Need a harder problem? Write the following method: public ArrayList factorize(int n) { // your code here } Factorize takes in an int and returns an ArrayList containing all of that number’s prime factors. Assume that the number is not prime and its prime factors all belong to the set [2, 3, 5, 7, 9, 11].

33. Recursive Backtracking Time to add in backtracking (shit just got real). Let’s face it, recursion can be super inefficient. Backtracking refers to the idea that while we are recursing, we might not always want to recurse to the end, so that we can save time. Requires us to add some steps to our three easy steps of recursion.

34. Recursive Backtracking Let’s amend our steps to recursion as such: 1)Recursion step : how do we break a problem into easier subproblems? 2)Memorization step : how do we save our current state in memory? 3)Resolution step : how do we combine the answers to our subproblems into an answer for our harder problem? 4)Backtrack step : how do we go back to our old state 5)Base step : when do we stop recursing?

35. Recursive Backtracking Let’s solve a backtracking problem! Simplified sudoku: one nxn board We will write the following method: public boolean solve(int[][] board) { // your code here } Solve takes in a 2D int array representing an incomplete sudoku board, and returns true if it can be solved – Empty spaces are represented by -1 – This method is DESTRUCTIVE!!

36. Recursive Backtracking public boolean solve(int[][] board) { } What is the “hard” problem here? – Solving the sudoku board. How can we break this down to easier subproblems? – Solving the sudoku board with one more space filled in. What makes this problem harder than regular recursion? – We must be able to revert our board to a previous state if we find ourselves filling in the wrong answers!

37. Recursive Backtracking public int[] findNextOpenSpace(int[][] board) { // returns int array representing coordinates // of next open space } public boolean checkValid(int[][] board) { // returns true if board is valid, false otherwise } To help you solve this problem, here’s a present—two black- box helper methods!

38. Recursive Backtracking public boolean solve(int[][] board) { int[] next = findNextOpenSpace(board); int i = next[0]; int j = next[1]; for (int v=1; v<=board.length; v++) { board[i][j] = v; boolean newBoardValid = solve(board); } } Let’s start by adding our recursion step. How do we break the hard problem down in code?

39. Recursive Backtracking public boolean solve(int[][] board) { int[] next = findNextOpenSpace(board); int i = next[0]; int j = next[1]; for (int v=1; v<=board.length; v++) { board[i][j] = v; boolean newBoardValid = solve(board); } Let’s next add our memorization step. How do we save our current state so that we can revert to it later if needed?

40. Recursive Backtracking public boolean solve(int[][] board) { int[] next = findNextOpenSpace(board); int i = next[0]; int j = next[1]; for (int v=1; v<=board.length; v++) { board[i][j] = v; boolean newBoardValid = solve(board); if (newBoardValid) { return true; } } return false; } Now the resolution step. How do we take our subproblem answer and turn it into a hard problem answer?

41. Recursive Backtracking public boolean solve(int[][] board) { int[] next = findNextOpenSpace(board); int i = next[0]; int j = next[1]; for (int v=1; v<=board.length; v++) { board[i][j] = v; boolean newBoardValid = solve(board); if (newBoardValid) { return true; } board[i][j] = -1; } return false; } Almost done! Next comes the backtrack step. How do we revert to our old state if we find our answer is not correct so far?

42. Recursive Backtracking public boolean solve(int[][] board) { if (!checkValid(board)) { return false; } int[] next = findNextOpenSpace(board); int i = next[0]; int j = next[1]; if (i == -1 && j == -1) { return checkValid(board); } for (int v=1; v<=board.length; v++) { board[i][j] = v; boolean newBoardValid = solve(board); if (newBoardValid) { return true; } board[i][j] = -1; } return false; } Last step! The base step. How do we know when to stop recursing?

43. Recursive Backtracking Recursive backtracking practice! So SimpleSudoku was too easy, huh? Extend the logic such that we only return true if there exists a *unique* solution: The method should remain destructive, i.e. it will change the underlying 2D array when it is run. – Hint: the method should return an int representing the number of solutions – Hint: you should consider a nonrecursive solve() method and a recursive helper

44. Recursive Backtracking Recursive backtracking practice! Need an even harder problem? Let’s try a 2-player backtracking game. Write the following method: public int whoWinsPilesList(ArrayList piles) { // your code here } whoWinsPilesList takes in an ArrayList of Integers that represents the piles list and returns who wins, assuming both play perfectly – During their turn, each player removes either 1 or 2 tokens from 1 pile in the piles list – The first player who cannot move loses

45. Big O What is the point of Big O? – Analyze runtime complexity of algorithms – Tells us how our algorithm’s runtime changes as our input size grows – Smaller Big O is better (but an O(1) algorithm will not always run faster than an O(n)!) Is it even important? – Yes! This is one of the most important things you will learn in this class!

46. Big O Data StructureBest-CaseWorst-CaseExpected Array ArrayList LinkedList HashSet BST Heap Access – looking up an element in the data structure

47. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(1) ArrayList LinkedList HashSet BST Heap Access – looking up an element in the data structure

48. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(1) ArrayListO(1) LinkedList HashSet BST Heap Access – looking up an element in the data structure

49. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(1) ArrayListO(1) LinkedListO(1)O(n) HashSet BST Heap Access – looking up an element in the data structure

50. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(1) ArrayListO(1) LinkedListO(1)O(n) HashSetO(1) BST Heap Access – looking up an element in the data structure

51. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(1) ArrayListO(1) LinkedListO(1)O(n) HashSetO(1) BSTO(1)O(log n) Heap Access – looking up an element in the data structure

52. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(1) ArrayListO(1) LinkedListO(1)O(n) HashSetO(1) BSTO(1)O(log n) HeapO(1)O(n) Access – looking up an element in the data structure

53. Big O Data StructureBest-CaseWorst-CaseExpected Array ArrayList LinkedList HashSet BST Heap Add – adding an element to the data structure

54. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayList LinkedList HashSet BST Heap Add – adding an element to the data structure

55. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(1)O(n)O(1) LinkedList HashSet BST Heap Add – adding an element to the data structure

56. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(1)O(n)O(1) LinkedListO(1)O(n) HashSet BST Heap Add – adding an element to the data structure

57. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(1)O(n)O(1) LinkedListO(1)O(n) HashSetO(1) BST Heap Add – adding an element to the data structure

58. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(1)O(n)O(1) LinkedListO(1)O(n) HashSetO(1) BSTO(log n)O(n)O(log n) Heap Add – adding an element to the data structure

59. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(1)O(n)O(1) LinkedListO(1)O(n) HashSetO(1) BSTO(log n)O(n)O(log n) HeapO(log n) Add – adding an element to the data structure

60. Big O Data StructureBest-CaseWorst-CaseExpected Array ArrayList LinkedList HashSet BST Heap Remove – removing an element from the data structure

61. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayList LinkedList HashSet BST Heap Remove – removing an element from the data structure

62. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(n) LinkedList HashSet BST Heap Remove – removing an element from the data structure

63. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(n) LinkedListO(1)O(n) HashSet BST Heap Remove – removing an element from the data structure

64. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(n) LinkedListO(1)O(n) HashSetO(1) BST Heap Remove – removing an element from the data structure

65. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(n) LinkedListO(1)O(n) HashSetO(1) BSTO(log n)O(n)O(log n) Heap Remove – removing an element from the data structure

66. Big O Data StructureBest-CaseWorst-CaseExpected ArrayO(n) ArrayListO(n) LinkedListO(1)O(n) HashSetO(1) BSTO(log n)O(n)O(log n) HeapO(log n) Remove – removing an element from the data structure

67. Big O Few notes on Big O: – Don’t forget that big O tells us the rate at which our runtime grows as our input grows! – O(1) algorithms can take ages to run if they are written poorly: public int findSum(int a, int b) { int sum = 0; for (int i=0; i<1000000; i++) { sum += a; sum += b; } int result = sum / 1000000; return result; } – The above algorithm is O(1)! But it sucks majorly!

68. Good luck!