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Lecture 6: Binary Search Trees (BST) and Red-Black Trees (RBTs) - PowerPoint PPT Presentation

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Lecture 6: Binary Search Trees (BST) and Red-Black Trees (RBTs) Instructor: Saravanan Thirumuruganathan CSE 5311 Saravanan Thirumuruganathan Outline 1 Data Structures for representing Dynamic Sets Binary Search Trees (BSTs) Balanced Search

  1. Lecture 6: Binary Search Trees (BST) and Red-Black Trees (RBTs) Instructor: Saravanan Thirumuruganathan CSE 5311 Saravanan Thirumuruganathan

  2. Outline 1 Data Structures for representing Dynamic Sets Binary Search Trees (BSTs) Balanced Search Trees Balanced Binary Trees - Red Black Trees (RBTs) CSE 5311 Saravanan Thirumuruganathan

  3. In-Class Quizzes URL: http://m.socrative.com/ Room Name: 4f2bb99e CSE 5311 Saravanan Thirumuruganathan

  4. Data Structures Key Things to Know for Data Structures Motivation Distinctive Property Major operations Key Helper Routines Representation Algorithms for major operations Applications CSE 5311 Saravanan Thirumuruganathan

  5. Trees Non-Linear Data Structures: Very common and useful category of data structures Most popular one is hierarchical CSE 5311 Saravanan Thirumuruganathan

  6. Trees - Applications 1 Family Tree: 1 http://interactivepython.org/runestone/static/pythonds/Trees/ trees.html CSE 5311 Saravanan Thirumuruganathan

  7. Trees - Applications 2 Taxonomy Tree: 2 http://interactivepython.org/runestone/static/pythonds/Trees/ trees.html CSE 5311 Saravanan Thirumuruganathan

  8. Trees - Applications 3 Directory Tree: 3 http://interactivepython.org/runestone/static/pythonds/Trees/ trees.html CSE 5311 Saravanan Thirumuruganathan

  9. Trees - Applications 4 HTML DOM (Parse) Tree: 4 http://interactivepython.org/runestone/static/pythonds/Trees/ trees.html CSE 5311 Saravanan Thirumuruganathan

  10. Tree - Terminology Node Edge Root Children Parent Sibling Subtree Leaf/External node Internal node Level (node) Height (tree) Arity CSE 5311 Saravanan Thirumuruganathan

  11. Tree - Abstract Representation 5 5 http://interactivepython.org/runestone/static/pythonds/Trees/ trees.html CSE 5311 Saravanan Thirumuruganathan

  12. Motivation Store dynamic set efficiently Use good ideas from ordered list (OL) and ordered doubly linked list (ODLL) Use hierarchical storage to avoid pitfalls of OL and ODLL First attempt at hierarchical data structure that tries to implement all 7 operations efficiently CSE 5311 Saravanan Thirumuruganathan

  13. Binary Trees Each node has at most 2 children Commonly referred to as left and right child The descendants of left child constitute left subtree The descendants of right child constitute right subtree CSE 5311 Saravanan Thirumuruganathan

  14. BST Property For every node x , the keys of left subtree ≤ key ( x ) For every node x , the keys of right subtree ≥ key ( x ) 6 6 http: //www.cs.princeton.edu/~rs/AlgsDS07/08BinarySearchTrees.pdf CSE 5311 Saravanan Thirumuruganathan

  15. BST Examples 7 7 http://en.wikipedia.org/wiki/Binary_search_tree CSE 5311 Saravanan Thirumuruganathan

  16. BST Height 8 There exists multiple possible BSTs to store same set of elements Minimum and Maximum Height: 8 https://engineering.purdue.edu/~ee608/handouts/lec10.pdf CSE 5311 Saravanan Thirumuruganathan

  17. BST Height 8 There exists multiple possible BSTs to store same set of elements Minimum and Maximum Height: lg n and n Best and worst case analysis (or specify analysis wrt height) 8 https://engineering.purdue.edu/~ee608/handouts/lec10.pdf CSE 5311 Saravanan Thirumuruganathan

  18. Representation - I key: Stores key information that is used to compare two nodes value: Stores satellite/auxillary information parent: Pointer to parent node. parent(root) = NULL left: Pointer to left child if it exists. Else NULL right: Pointer to right child if it exists. Else NULL CSE 5311 Saravanan Thirumuruganathan

  19. Representation - I 9 9 CLRSFig10.9 CSE 5311 Saravanan Thirumuruganathan

  20. Representation - II CSE 5311 Saravanan Thirumuruganathan

  21. Major Operations Search Insert Minimum/Maximum Successor/Predecessor Deletion Traversals CSE 5311 Saravanan Thirumuruganathan

  22. Key Helper Routines Successor/Predecessor Traversals Case by Case Analysis: Analysis by number of children : 0 , 1 , 2 Analysis by type of children: left , right CSE 5311 Saravanan Thirumuruganathan

  23. BST: Search Search: 4 CSE 5311 Saravanan Thirumuruganathan

  24. BST: Search CSE 5311 Saravanan Thirumuruganathan

  25. BST: Search Tree-Search(x, k): if x == NULL or k == x.key return x if k < x.key return Tree-Search(x.left, k) else return Tree-Search(x.right, k) Analysis: CSE 5311 Saravanan Thirumuruganathan

  26. BST: Search Tree-Search(x, k): if x == NULL or k == x.key return x if k < x.key return Tree-Search(x.left, k) else return Tree-Search(x.right, k) Analysis: O ( h ) Best Case: lg n and Worst Case: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  27. BST: Insert Insert: 13 CSE 5311 Saravanan Thirumuruganathan

  28. BST: Insert Insert: 13 CSE 5311 Saravanan Thirumuruganathan

  29. BST: Analysis: CSE 5311 Saravanan Thirumuruganathan

  30. BST: Analysis: O ( h ) Best Case: lg n and Worst Case: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  31. BST: Minimum CSE 5311 Saravanan Thirumuruganathan

  32. BST: Minimum CSE 5311 Saravanan Thirumuruganathan

  33. BST: Minimum Tree-Minimum(x) while x.left is not NULL x = x.left return x Analysis: O ( h ) Best Case: lg n and Worst Case: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  34. BST: Maximum CSE 5311 Saravanan Thirumuruganathan

  35. BST: Maximum CSE 5311 Saravanan Thirumuruganathan

  36. BST: Maximum Tree-Maximum(x) while x.right is not NULL x = x.right return x Analysis: O ( h ) Best Case: lg n and Worst Case: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  37. BST: Successor Successor: 15 CSE 5311 Saravanan Thirumuruganathan

  38. BST: Successor Successor: 15 CSE 5311 Saravanan Thirumuruganathan

  39. BST: Successor Successor: 13 CSE 5311 Saravanan Thirumuruganathan

  40. BST: Successor Successor: 13 CSE 5311 Saravanan Thirumuruganathan

  41. BST: Successor CSE 5311 Saravanan Thirumuruganathan

  42. BST: Tree-Successor(x): if x.right is not NULL return Tree-Minimum(x.right) y = x.parent while y is not NULL and x == y.right x = y y = y.parent return y BST Property allowed us to find successor without comparing keys Analysis: O ( h ) Best Case: lg n and Worst Case: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  43. BST: Predecessor Predecessor: 6 CSE 5311 Saravanan Thirumuruganathan

  44. BST: Predecessor Predecessor: 6 CSE 5311 Saravanan Thirumuruganathan

  45. BST: Predecessor Predecessor: 17 CSE 5311 Saravanan Thirumuruganathan

  46. BST: Predecessor Predecessor: 17 CSE 5311 Saravanan Thirumuruganathan

  47. BST: Tree-Predecessor(x): if x.left is not NULL return Tree-Maximum(x.left) y = x.parent while y is not NULL and x == y.left x = y y = y.parent return y BST Property allowed us to find predecessor without comparing keys Analysis: O ( h ) Best Case: lg n and Worst Case: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  48. BST: Deletion Trickiest Operation! Suppose we want to delete node z 1 z has no children: Replace z with NULL 2 z has one children c : Promote c to z ’s place 3 z has two children: (a) Let z ’s successor be y (b) y is either a leaf or has only right child (c) Promote y to z ’s place (d) Fix y ’s loss via Cases 1 or 2 CSE 5311 Saravanan Thirumuruganathan

  49. BST: Deletion Case I 10 10 https: //www.cs.rochester.edu/u/gildea/csc282/slides/C12-bst.pdf CSE 5311 Saravanan Thirumuruganathan

  50. BST: Deletion Case II 11 11 https: //www.cs.rochester.edu/u/gildea/csc282/slides/C12-bst.pdf CSE 5311 Saravanan Thirumuruganathan

  51. BST: Deletion Case III 12 12 https: //www.cs.rochester.edu/u/gildea/csc282/slides/C12-bst.pdf CSE 5311 Saravanan Thirumuruganathan

  52. Digression Perfectly fine if you cannot do deletion by memory Things will become hairier in RBT As long as you remember the key ideas and operations, you will be fine CSE 5311 Saravanan Thirumuruganathan

  53. BST: Traversal Traversal: Visit all nodes in a tree Many possible traversal strategies Three are most popular: Pre-Order, In-Order, Post-Order CSE 5311 Saravanan Thirumuruganathan

  54. BST: Traversals In-Order-Walk(x): if x == NULL return In-Order-Walk(x.left) Print x.key In-Order-Walk(x.right) Analysis: CSE 5311 Saravanan Thirumuruganathan

  55. BST: Traversals In-Order-Walk(x): if x == NULL return In-Order-Walk(x.left) Print x.key In-Order-Walk(x.right) Analysis: O ( n ) Holds true for all three traversals CSE 5311 Saravanan Thirumuruganathan

  56. BST: Traversal In-Order: CSE 5311 Saravanan Thirumuruganathan

  57. BST: Traversal In-Order: 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 11 , 12 , 15 , 19 , 20. Pre-Order: CSE 5311 Saravanan Thirumuruganathan

  58. BST: Traversal In-Order: 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 11 , 12 , 15 , 19 , 20. Pre-Order: 7 , 4 , 2 , 3 , 6 , 5 , 12 , 9 , 8 , 11 , 19 , 15 , 20. Pre-Order: CSE 5311 Saravanan Thirumuruganathan

  59. BST: Traversal In-Order: 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 11 , 12 , 15 , 19 , 20. Pre-Order: 7 , 4 , 2 , 3 , 6 , 5 , 12 , 9 , 8 , 11 , 19 , 15 , 20. Pre-Order: 3 , 2 , 5 , 6 , 4 , 8 , 11 , 9 , 15 , 20 , 19 , 12 , 7. CSE 5311 Saravanan Thirumuruganathan

  60. BST: Traversals Notice anything special about In-Order traversal? CSE 5311 Saravanan Thirumuruganathan

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