Unit 11: Data Structures & Complexity

1. Unit 11: Data Structures & Complexity • We discuss in this unit • Graphs and trees • Binary search trees • Hashing functions • Recursive sorting: quicksort, mergesort syllabus basic programming concepts object oriented programming topics in computer science

2. Graphs and Trees Graph: a data representation which includes nodes and edges, where each edge connects two nodes Example: the internet Tree: a connected graph with no loops, and a root Example: inheritance tree

3. Graphs and Trees b) a) ROOT c) internal vertices LEAF NODES

4. Binary Trees • A rooted tree is called a binary tree if every internal vertex has no more than 2 children The tree is called a full binary tree if every internal vertex has exactly 2 children • An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered; we call the children of a vertex the left child and the right child, if they exist • A rooted binary tree of height H is called balanced if all its leaves are at levels H or H-1

5. Binary tree: example Lou Hal Max Ken Sue Ed Joe Ted

6. Tree Properties Theorem: A tree with N vertices has N-1 edges Theorem: There are at most 2H leaves in a binary tree of height H Corallary: If a binary tree with L leaves is full and balanced, then its height is H = log2 L  Theorem: There are at most (2H+1–1) nodes in a binary tree of height H

7. Binary Search Tree (BST) A special kind of binary tree in which: 1. Each vertex contains a distinct key value 2. The key values in the tree can be compared using “greater than” and “less than” 3. The key value of each vertex in the tree is less than every key value in its left subtree, and greater than every key value in its right subtree

8. Example: Binary Search Tree Lou Hal Max Ken Sue Ed Joe Ted

9. ‘J’ Shape of a BST Depends on its key values and their order of insertion: • Insert the elements ‘J’ ‘E’ ‘F’ ‘T’ ‘A’ in that order. • The first value to be inserted is put into the root.

10. ‘J’ ‘E’ Inserting ‘E’ into the BST Thereafter, each value to be inserted begins by comparing itself to the value in the root, moving left it is less, or moving right if it is greater. This continues at each level until it can be inserted as a new leaf.

11. ‘J’ ‘E’ Inserting ‘F’ into the BST Begin by comparing ‘F’ to the value in the root, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf. ‘F’

12. ‘J’ ‘T’ ‘E’ Inserting ‘T’ into the BST Begin by comparing ‘T’ to the value in the root, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf. ‘F’

13. ‘J’ ‘T’ ‘E’ ‘A’ ‘F’ Inserting ‘A’ into the BST Begin by comparing ‘A’ to the value in the root, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.

14. ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ Order of insertion what BST is obtained by inserting the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order?

15. ‘J’ Another binary search tree ‘T’ ‘E’ ‘A’ ‘H’ ‘M’ ‘P’ ‘K’ Add nodes containing these values in this order: ‘D’ ‘B’ ‘L’ ‘Q’ ‘S’ ‘V’ ‘Z’

16. ‘D’ ‘Z’ ‘K’ ‘B’ ‘L’ ‘Q’ ‘S’ Task: is ‘F’ in the tree? ‘J’ ‘T’ ‘E’ ‘A’ ‘V’ ‘M’ ‘H’ ‘P’

17. Search(x) • start at the root of the tree which contains y: • the tree is empty  x is not present • x = y (the item at the root)  the root is returned • x < y  recursively search the left subtree • x > y  recursively search the right subtree

18. Operations • Search(x) • Insert(x) • Delete(x) tree algs/demo

19. Complexity • Search(x) – O(H) • Insert(x) – O(H) • Delete(x) – O(H) • worst case O(n) when tree is a list best case O(log n) when tree is full and balanced

20. Traversal Algorithms • A traversal algorithm is a procedure for systematically visiting every vertex of an ordered binary tree • Tree traversals are defined recursively • Three traversals are named: preorder inorder postorder

21. PREORDER Traversal Algorithm Let T be an ordered binary tree with root r: • If T has only r, then r is the preorder traversal • Otherwise, suppose T1, T2 are the left and right subtrees at r; the preorder traversal is • visit r • traverse T1 in preorder • traverse T2 in preorder

22. ‘J’ Preorder Traversal Visit first ROOT ‘T’ ‘E’ ‘A’ ‘H’ ‘M’ ‘Y’ Visit left subtree second Visit right subtree last result: J E A H T M Y

23. INORDER Traversal Algorithm Let T be an ordered binary tree with root r: • If T has only r, then r is the inorder traversal • Otherwise, suppose T1, T2 are the left and right subtrees at r; then • traverse T1 in inorder • visit r • traverse T2 in inorder

24. ‘J’ Inorder Traversal Visit second ROOT ‘T’ ‘E’ ‘A’ ‘H’ ‘M’ ‘Y’ Visit left subtree first Visit right subtree last result: A E H J M T Y

25. POSTORDER Traversal Algorithm Let T be an ordered binary tree with root r: • If T has only r, then r is the postorder traversal • Otherwise, suppose T1, T2 are the left and right subtrees at r; then • traverse T1 in postorder • traverse T2 in postorder • visit r

26. ‘J’ Postorder Traversal Visit last ROOT ‘T’ ‘E’ ‘A’ ‘H’ ‘M’ ‘Y’ Visit left subtree first Visit right subtree second result: A H E M Y T J

27. A Binary Expression Tree ROOT ‘-’ ‘8’ ‘5’ INORDER TRAVERSAL: 8 - 5 has value 3 PREORDER TRAVERSAL: - 8 5 POSTORDER TRAVERSAL:8 5 -

28. Binary Expression Tree A special kind of binary tree in which: 1. Each leaf node contains a single operand 2. Each nonleaf node contains asingle binary operator 3. The left and right subtrees of an operator node represent subexpressions that must be evaluated before applying the operator at the root of the subtree

29. ‘*’ ‘3’ ‘+’ ‘2’ ‘4’ A Binary Expression Tree What value does it have? ( 4 + 2 ) * 3 = 18

30. ‘*’ ‘3’ ‘+’ ‘2’ ‘4’ A Binary Expression Tree What infix, prefix, postfix expressions does it represent?

31. ‘*’ ‘3’ ‘+’ ‘2’ ‘4’ A Binary Expression Tree Infix: ( ( 4 + 2 ) * 3 ) Prefix: * + 4 2 3 evaluate from right Postfix: 4 2 + 3 * evaluate from left

32. Levels Indicate Precedence When a binary expression tree is used to represent an expression, the levels of the nodes in the tree indicate their relative precedence of evaluation Operations at higher levels of the tree are evaluated later than those below them;the operation at the root is always the last operation performed

33. ‘/’ ‘3’ ‘+’ ‘2’ ‘4’ Example ‘*’ ‘-’ ‘5’ ‘8’ What infix, prefix, postfix expressions does it represent?

34. ‘*’ ‘/’ ‘-’ ‘8’ ‘5’ ‘3’ ‘+’ ‘2’ ‘4’ A binary expression tree Infix: ( ( 8 - 5 ) * ( ( 4 + 2 ) / 3 ) ) Prefix: * - 8 5 / + 4 2 3 Postfix: 8 5 - 4 2 + 3 / * has operators in order used

35. Hash tables • Goal: accesses data with complexity O(1), even when the data needs to be dynamically administered (mostly by inserting new or deleting existing data) • Solution: array • Problem: access is only O(1) if the index of the element is known – otherwise O(log n) • Solution: each data item to be stored is associated with a hash value which gives array index: • generate array of size m, where m is prime and sufficiently large • assign unique numerical value N(k) to key of element k • h(k) = N(k) mod m

36. Hash tables - cont • Problem: hash value is not unique anymore! • Solution: a collision procedure must determine the position for the new object hasshing

37. Operations • Search(x) • Insert(x) • Delete(x) Complexity: • worst case O(n) when hash value is always the same (and table is really a list) • best case O(1) when hash value is always distinct

38. Example: keys are names • list of unique identifiers • washington  103288042987600 • lincoln  5201793578 • bush  151444 • the range currently is all positive integers, which is not possible to implement • for m=10007 • washington  3249 • lincoln  4873 • bush  1339

39. why modulus prime number? • underlying reason: the new code numbers should appear random • technical reason: the modulus operation should be a “field” • example: • original codes are all even numbers • m is even • only even buckets in the table will get filled, half the table will be empty

40. Back to sorting... Two recursive algorithms: • MergeSort • QuickSort

41. MergeSort MergeSort • divide array to two equal parts • recursively sort left part • recursively sort right part • merge the two sorted lists

42. quicksort QuickSort • choose pivot arbitrarily • divide to left part (< = than pivot) and right part (> than pivot) • sort each part recursively

43. Complexity • MergeSort worst case O(n * log n) • QuickSort worst case O(n2) average case O(n * log n)