Binary Search Trees. Ric Glassey

Transcription

1 Binary Search Trees Ric Glassey

2 Outline Binary Search Trees Aim: Demonstrate how a BST can maintain order and fast performance relative to its height Properties Operations Min/Max Search Insertion *Deletion Summary of performance 2

3 PROPERTIES 3

4 Binary Search Tree Ordered binary tree Binary search tree property Keys in left sub-tree of P are < k Keys in right sub-tree of P are > k P k < k > k 4

5 Example Binary Search Tree Any sub- tree of the BST should sa5sfy the binary- search- tree property 5

6 Applications of BST Store a set of ordered keys Fast operations and maintain order Priority Queues Ordered Maps/Dictionaries e.g. Java s TreeMap n.b. if order is not important, regular hash tables still offer better average time complexity O(1) 6

7 MIN/MAX 7

8 Finding Min/Max in a BST What is the fastest path to find the minimum and maximum values in a BST?

9 Finding Min/Max in a BST min(node) while node.left!= null: node = node.left return node max(node) while node.right!= null: node = node.right return node What is the expected 5me complexity of these opera5ons? And the worst case? 9

10 Pathological BSTs 44 height of tree = number of nodes insert(46) 46 insert(54) 54 insert(67) 67 insert(71) 71 insert(88) Order of inserts cannot be predicted in advance Pathological cases can emerge where some opera5ons will take O(n) Solu5ons include Randomised BSTs/Treaps and Balancing Procedures 88 10

11 SEARCH 11

12 Finding x within a BST x = How many opera5ons will it take? N? more/less? 12

13 Finding x within a BST x = 41 x < x > x > x = 41 Once more, the complexity was propor5onal to the height of the tree O(h) 13

14 Recursive approach: BST Search Algorithms search(node, x) if node == null or x == node.key: return node if x < node.key: return search(node.left, x) else: return search(node.right, x) Iterative option (see homework) Why is this the better option? 14

15 BST Search Algorithms Further operations use search(n,x) as a subroutine insert(k, v) remove(k) As a consequence, all primary operations on a BST are considered fast, proportional to the height of the tree An exception is in-order-traversal which should be expected to be O(n) as all nodes must be visited 15

16 INSERTION 16

17 Search for key Two cases: Insertion into a BST Key exists, update value Key does not exist, extend leaf at end of failed search with new node (key, value) 17

18 Insertion into a BST: insert(68, value)

19 Insertion into a BST: Find position > < > < < 82??? 80 19

20 Insertion into a BST: Extend Leaf > < > < <

21 BST Insertion Algorithm We can use search(x) as a subroutine to simplify insertion into a BST n.b. homework demands iterative solution! insert(key, value) leaf = search(root, key) if key == leaf.key: leaf.value = value else if key < leaf.key: leaf.left = node(key, value) else: leaf.right = node(key, value) 21

22 *DELETION 22

23 *Deletion for a BST Insertions always occur at a leaf (trivial) Deletions can occur anywhere within a tree Three cases to consider for remove(z): Case 1: z has no children Case 2: z has one child Case 3: z has two children 23

24 *Case 1: z has no children Simply remove from tree by setting z s parent s pointer to null z 24

25 *Case 2: z has one child R R Z LC LC R R Z RC RC 25

26 *Case 3: z has two children (i) R R Z X LC X LC Y Y X is promoted, linked to parent R of Z, and Z s LC is linked as X s leo child 26

27 *Case 3: z has two children (ii) R R R Z Z Y LC X LC Y LC X Y X W W W We transplant Y with X first, and relink W to X to maintain the BST- property Finally, we link Z s LC to Y and Y to the root R Hint: subs5tute valid integers to show why this works 27

28 Summary of BST Performance Data Structure Array Linked List Hash Table Binary Search Tree average worst expected worst Opera5on Search* O(1) O(n) O(1) O(n) O(h) O(n) Insert O(n) O(1) O(1) O(n) O(h) O(n) Delete O(n) O(1) O(1) O(n) O(h) O(n) * Index or Key based search Assume doubly linked list 28

29 Readings Algorithms and Data Structures *required* Stefan Nilsson s text on the Binary Search Tree Introduction to Algorithms, 3 rd Edition Chapter 12: Binary Search Trees Full text available via KTH Library KTH:KTH_SFX Data Structures and Algorithms in Java, 6 th Edition Goodrich et al. Chapter 11: Search Trees Full text available via KTH Library KTH:KTH_SFX

30 Feedback! This week, I will mostly be asking about: Topics of today s lecture Analysis of algorithms Tipjar Survey will appear at this link Also on the course web page after this lecture n.b. I will switch survey service for P4 :-) 30