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

Stacks and Queues

Stack-based Recursion Computers use stacks to manage function calls/function returns. When a function is called, data is pushed onto the stack When a function finishes, data is popped off the stack Rather than using the computer’s stack, we can use our own to implement recursion! Binary tree traversal

Stacks and Recursion Any recursive algorithm can be implemented non-recursively. Do you have to use a function that calls itself? No. You can use a stack to implement recursion Do you have to use a recursive algorithm? Yes. How do you traverse a binary tree without a recursive algorithm?

Queues A queue is a “First-In, First-Out” = “FIFO” buffer. e.g. line-ups people enter from the back of the line people are served (exit) from the front of the line When elements are added, they are enqueued to the back. When elements are removed, they are dequeued from the front. enqueue() and dequeue() are the two defining functions of a queue. Example Link-based Queue

Priority Queues Previous queues were based on entry order (e.g. LIFO, FIFO). Priority queues are based on item value. Stacks and queues aren’t designed for searches. BST could work, but there is more overhead than we need. don’t need completely sorted queue – only need first element

Heaps and Heapsort

Key Points Heaps Make a heap Heapsort

Heaps Complete binary tree Partially ordered max heap – the value of every node is greater than its children min heap – the value of every node is smaller than its children No relationship between siblings, only ancestors (up-down) BST has left-right relationships.

Removing the Minimum Value Minimum value is at root, so we should remove this element. How do we maintain a complete binary tree? Put last value at the root and sift down Switch the value with the smallest of its children – continue Heap property is maintained.

Inserting a new value How do we maintain a complete binary tree? Put value at the bottom and sift up Heap property is maintained.

Complexity Analysis Remove minimum Insert Best  value stays at depth 1  O(1) Worst  value goes back to bottom  O(logn) Average  value goes half way O(logn) Insert Best  value stays at bottom  O(1) Worst  value goes to top  O(logn)

Making a Heap Array representation Make a heap Analysis of building a heap Heapsort