ITEC 2620M Introduction to Data Structures

1. ITEC 2620MIntroduction to Data Structures Instructor: Prof. Z. Yang Course Website: http://people.math.yorku.ca/~zyang/itec2620m.htm Office: DB 3049

2. Sorting

3. Key Points • Recursive sorting algorithms • Achieving leverage • Quicksort • Mergesort

4. Review • Previous sorting algorithms were O(n2) on average. • What if we could get O(nlogn) • Where have we seen O(logn) before? • Binary search • How did binary search work? • first query explores 1 element • second query explores 2 elements • third query explores 4 elements

5. Recursive Sorting • Split the elements into smaller sub-groups • Partially sort each sub-group • Trust recursion to put everything back together

6. Quicksort Algorithm • Pick an element • partially sort them • move all larger elements on one side, and smaller elements on the other • have to look at all elements to get one element in position • Pick one element in each sub-division (2) • partially sort them • move all elements as before (twice) • have to look at half of the elements to get each new element into position • Pick one element in each sub-division (4) • partially sort them • move all elements as before (four times)

7. Quicksort Algorithm (Cont’d) • Each sub-division is being sorted by the same algorithm as the overall set • recursion • base case is 0 or 1 elements – already sorted • Work to sort each element is cut in half each level down • Get twice as much done for our effort • How many times can we cut something in half? • O(logn) • Pseudocode

8. Mergesort Algorithm • Divide what you have to do into two halves • sort each half • merge the two halves into a fully sorted set • Each half will be sorted by the same algorithm as the overall set • recursion • base case is 0 or 1 elements – already sorted • Each upward merge sorts twice as much • How many times can we cut something in half? • O(logn) • Pseudocode

9. Binary Search Trees

10. Key Points • Inserting and Deleting into Linked Structures • Linked lists • Basic BST operations • Deleting from BSTs

11. Codes • Inserting into a Linked List • Deleting from a Linked List • Inserting into a BST • insertion always happens at the bottom of the tree • new node will be a leaf node • Deleting from a BST – target node has less than two children • Deleting from a BST – target node has two children • Deleting from a BST – a better way