Heaps and Priority Queues Study Guide
Author: Josh Hug

## Overview

Priority Queue. A Max Priority Queue (or PQ for short) is an ADT that supports at least the insert and delete-max operations. A MinPQ supposert insert and delete-min.

Heaps. A max (min) heap is an array representation of a binary tree such that every node is larger (smaller) than all of its children. This definition naturally applies recursively, i.e. a heap of height 5 is composed of two heaps of height 4 plus a parent.

Tree Representations. Know that there are many ways to represent a tree, and that we use Approach 3b (see lecture slides) for representing heaps, since we know they are complete.

Running times of various PQ implementations. Know the running time of the three primary PQ operations for an unordered array, ordered array, and heap implementation.

Note: The reason I’ve given lots of problems here is not because this is a more important topic, but because there are just so many interesting problems.

### C level

1. Is an array that is sorted in descending order also a max-oriented heap?
2. Textbook 2.4.2 (assume we’d also like to support delete operations)
3. Problem 3 from Princeton’s Fall 2009 midterm or Problem 4 from Princeton’s Fall 2008 midterm.
4. Why do we leave the 0th position empty in our array representation for heaps?

### B level

1. (Textbook 2.4.7) The largest item in a heap must appear in position 1, and the second largest must appear in position 2 or 3. Give the list of positions in a heap of size 31 where the kth largest CAN appear, and where the kth largest CANNOT appear for k=2, 3, 4. Assume values are distinct.
2. (Textbook 2.4.10) Suppose we wish to avoid wasting one position in a heap-ordered array pq[], putting the largest value in pq, its children in pq and pq, and so forth, proceeding in level order. Where are the parents and children of pq[k]?
3. (Textbook 2.4.21) Explain how to use a priority queue to implement the stack and queue data types.
4. (Adapted from Textbook 2.4.27). Add a min() method to a maximum-oriented PQ. Your implementation should use constant time and constant extra space.
5. (Textbook 2.4.14) What is the minimum number of items that must be exchanged during a remove-the-max operationin a heap of size N with no duplicate keys? Give a heap of size 15 for which the minimum is achieved. Answer the same qusetion for two and three successive remove-the-maximum operations.
6. Problem 4 from Princeton’s Spring 2008 midterm.
7. Problem 6 from my Fall 2014 midterm.
8. Problem 4 from Princeton’s 2012 midterm.
9. Problem 5a and 5b of my Spring 2013 midterm.
10. Problem 3 from my Fall 2014 midterm.

### A level

1. Design a data type that supports insert in O(log N) time, find-the-median in O(1) time, and delete-the-median in O(log N) time.
2. Design a data type that supports insert in O(log N) time, delete-the-max in O(log N) time, and delete-the-minimum in O(log N) time.
3. Problem 7 from Princeton’s Fall 2010 midterm.
4. Problem 8 from Princeton’s Spring 2012 midterm.