A. Intro

Download the code for Lab 19 and create a new Eclipse project out of it.

B. Representing a tree with an array

Representing a Tree with an Array

You've seen two approaches to implementing a sequence data structure: either using an array, or using linked nodes. We extended our idea of linked nodes to implement a tree data structure. It turns out we can also use an array to represent a tree.

Here's how we implement a binary tree:

• The root of the tree will be in position 1 of the array (nothing is at position 0). We can define the position of every other node in the tree recursively:
• The left child of a node at position n is at position 2n.
• The right child of a node at position n is at position 2n + 1.
• The parent of a node at position n is at position n/2.

Self-test: ArrayBST Representation

Here is an ArrayBST.

Which of the following trees could be represented by the ArrayBST shown above? (The myValues have been omitted and replaced with ?s)

Option 1

Option 2

Option 3

Option 4

Choose one answer.

Now we have a new ArrayBST:

Which of the trees could represent it? Choose one answer.

Now suppose we have this ArrayBST:

Exercise: Implement ArrayBST

Download the framework code at ArrayBST.java, or listed below. Fill in the methods getLeft, getRight, getParent, setLeft, and setRight being sure not to violate the layers of abstraction (use the setNode method we implemented!) Check that the main method creates the tree shown below.

Self-test: a Naive BST contains

Code for the contains method appears below. Complete the following statement:

public boolean contains (int value) {
    for (Node treeNode : contents) {
        if (treeNode != null && treeNode.myValue == value) {
            return true;
        }
    }
    return false;

The contains method runs in time proportional to in the worst case.

Exercise: Rewrite contains

Rewrite the contains method in ArrayBST.java to run in time proportional to the height of the tree. Make sure to keep with the abstraction (use the getLeft and getRight you wrote!). Augment the main method with simple checks to make sure your contains works.

C. Implementations of queues and priority queues

Queues and Priority Queues

Earlier in the class we looked at queues, a data structure that keeps items in order. Remember they process items in a first-in-first-out system.

Sometimes, however, processing items in the order they arrive is not what we want to do. We may instead want to process items in order of importance or a priority score. Enter the priority queue...

Queues and priority queues have the same operations:

• construct an empty queue
• enqueue: add an element somewhere in the queue
• dequeue: remove an element from the queue (for a normal queue, this is the oldest element in the queue. For a priority queue, this is the one with most priority)
• isEmpty: check if the queue is empty
• peek: get the item at the front of the queue without removing it

Traditionally, Black Friday lines are like a queue...

People are served based on the order they arrive, regardless of what they want to buy.

An emergency room is like a priority queue...

The people waiting with the most pain are served first, regardless of when they arrived.

Why this is important: Priority queues are an integral component of many algorithms for graph processing (which we'll cover in a few labs). For example, the first few weeks of CS 170 typically involve graph algorithms that use priority queues. Look out for priority queues in other CS classes as well — for example, you'll find them invaluable the operating systems class CS 162, where they're used to schedule which processes in a computer to run at what times.

Self-test: Runtimes of Different Priority Queue Implementations

For the following examples, we will follow Java's default priority for a priority queue, which prioritizes items with the smallest value first (assuming the items are Comparable). Note that you can create a priority queue with your own custom priority by using the following priority queue constructor: PriorityQueue(int initialCapacity, Comparator<? super E> comparator).

Consider the implementation of a priority queue with an unordered linked list of N elements, and complete the following sentence:

• In the worst case, the run time to insert something into the priority queue is proportional to .

Consider the implementation of a priority queue with an unordered linked list of N elements, and complete the following sentence:

• In the worst case, the run time to remove the element with highest priority is proportional to .

Consider the implementation of a priority queue with an ordered linked list of N elements, and complete the following sentence:

• In the worst case, the run time to insert something into the priority queue is proportional to .

Consider the implementation of a priority queue with an ordered linked list of N elements, and complete the following sentence:

• In the worst case, the run time to remove the element with highest priority is proportional to .

Consider the implementation of a priority queue with a balanced binary search tree of N elements, and complete the following sentence:

• In the worst case, the run time to insert something into the priority queue is proportional to .

Consider the implementation of a priority queue with a balanced binary search tree of N elements, and complete the following sentence:

• In the worst case, the run time to remove the element with highest priority is proportional to .

Queue and PriorityQueue in java.util

The java.util library contains a Queue interface and a PriorityQueue class that implements Queue. Queue methods include the following:

• offer(element), which adds the element to the queue (this is enqueue)
• peek(), which returns but does not remove the head of the queue
• poll(), which retrieves and removes the head of the queue (returning null if the queue is empty) (this is dequeue)

Java's priority queue is implemented with a data structure called a binary min heap. In the rest of this lab you will study and implement this data structure yourself.

Binary Heaps Intro

Having a fast implementation of a priority queue would be very helpful, but as we saw in the last self-test, the most obvious implementations each have flaws. Enter the binary heap. The binary heap is a data structure that will help us implement a priority queue with fast operations. This means we want to be able to quickly insert objects into the heap with different priority values, and we want to be able to quickly remove the most important item quickly.

There are two flavors of binary heaps: min heaps and max heaps. They're very similar except the former organizes items with the smallest given the most importance, and the latter organizes items with the largest given the most importance. Choose either one depending on what your definition of a priority score is.

In these labs we may work with either a max or min heap. It should be easy to figure out how the other one works by example. But pay attention so you don't get confused which one we're using!

D. Work with binary heaps

Binary Heaps Defined

Binary min heaps are basically just binary trees (but not binary search trees) — they have all of the same invariants of binary trees — with two extra invariants:

• Invariant 1: the tree must be maximally balanced (more on this later)
• Invariant 2: every node is smaller than its descendents (there is another variation called a binary max heap where the reverse is true)

Invariant 2 guarantees that the min element will always be at the root of the tree. This helps us access that item quickly, which is what we need for a priority queue.

We need to make sure binary heap methods maintain the above two invariants. Here's how we do it:

Add an item

1. Put the item you're adding in the next available spot in the bottom row
2. Swap the item you just added with its parent until it is larger than its parent, or until it is the new root. This is called bubbling up.

Remove the min item

1. Replace the item at the root with the item in the last bottom spot
2. Bubble down the new root until it is smaller than both its children. If you reach a point where you can either bubble down through the left or right child, you must choose the smaller of the two.

Maximally Balanced Trees

There are a couple different notions of what it means for a tree to be well balanced. A binary heap must always be what is called maximally balanced (also sometimes called complete).

A maximally balanced tree:

• Has all available positions for nodes filled, except for possibly the last row, which must be filled left-to-right

Below are some examples of trees that are maximally balanced:

And some examples of trees that are NOT balanced:

Play with Heaps

We just covered a lot of new concepts for heaps. To test yourself and see if you understand them, try drawing a heap. Draw what happens when you insert numbers into the heap and when you decide to remove the smallest (or biggest) one.

Also, check out this interactive animation of a min heap. You can type in numbers to insert, or remove the min element (ignore the BuildHeap button for now; we'll talk about that shortly). You can use this animation to check your own drawings too — there is no ambiguity when constructing heaps, so if you insert numbers in the same order, yours should look exactly the same as this animation.

Exercise: Writing Heap Methods

First, switch which partner is coding for this lab, if you haven't recently.

The class ArrayHeap implements a binary max heap using an ArrayList (just like ArrayBST implemented a BST). Fill in the missing methods in ArrayHeap.java.

Respect the abstraction! — insert, removeMin, and changePriority may use the methods bubbleUp and bubbleDown. bubbleUp and bubbleDown may use getLeft, getRight, and getParent (which should be the same as what you wrote for ArrayBST).

E. Binary heap brainteasers

Self-test: Heaps and BSTs

Consider binary trees that are both max heaps and binary search trees.

How many nodes can such a tree have? Check all that apply.

Self-test: Determining if a Tree is Maximally Balanced

It's not obvious how to verify that a linked binary tree is maximally balanced (aka complete). A CS 61B student suggests the following recursive algorithm:

1. A one-node tree is complete.
2. A tree with two or more nodes is complete if its left subtree is complete and has depth k for some k, and its right subtree is complete and has depth k or k-1

Here are some example trees. Think about whether or not the student's proposed algorithm works correctly on them.

Check all that apply to test your understanding of the proposed algorithm.

Discussion: Complete Trees and BFS

Another CS 61B student claims that a binary tree is complete if all the leaves in the tree come at the end of a breadth-first traversal of the tree. Give a counterexample to this claim.

Self-test: The Third Biggest Element in a Max Heap

Here's an example heap.

Which nodes could contain the third largest elements in the heap assuming that the heap does not contain any duplicates?

Which nodes could contain the third largest elements in the heap assuming that the heap can contain any duplicates? (NOTE: Heaps CAN contain duplicates)

F. Heapsort

Sort Some Numbers Using a Heap

The main method of ArrayHeap.java adds some numbers to a heap. Add code to the main method to print these numbers out from smallest to largest. It is fine if this modifies the heap in the process.

Notice we have just sorted the numbers!

Heapsort

Suppose you have an array of N numbers that you want to sort smallest-to-largest. As we saw in the last slide, one algorithm for doing this as as follows:

1. Put all of the numbers in a min heap
2. Repeatedly remove the min element from the heap, and store them in an array in that order

This is called heapsort.

Since each insertion takes proprotional to log N comparisons once the heap gets large enough, and each removal may also take proportional to log N comparisons, the whole process takes proportional to N log N comparisons.

It turns out we can actually make step 1 of heapsort run faster — propotional to N comparisons — using a process called heapify, which we'll cover on the next slide. (Unfortunately, we can't make step 2. run any faster than N log N, so the overall heapsort must take N log N time.)

Heapify

The algorithm for taking an arbitrary array and making it into a min (or max) heap in time proprotional to N is called heapify. Pseudocode for this algorithm is below:

def heapify(array):
    index = N/2
    while index < 0:
        bubble down item at index
        index--

Conceptually, you can think of this as building a heap from the bottom up. To get a visualization of this algorithm working, click on the BuildHeap button on the animation we linked earlier. This loads a pre-set array and then runs heapify on it.

It is probably not immediately clear to you why this heapify runs in O(N). Don't worry about it — the math required for explaining it is beyond what you are expected to use in this course. For those who are curious, you can check out an explanation on Wikipedia here.

Finding the K Largest Elements

In the following problem you will reason about an algorithm that collects the K largest elements in a list of length N. Before you look at the questions, try to design an algorithm that could accomplish this in the asymptotically lowest amount of time.

What data structure would you use?

Self-test: Finding the K Largest Elements

You have N elements and you want to collect the K largest. You can use a heap to solve this problem. How large of a heap would you need? Choose one answer.

What type of heap would be best to use? Choose one answer.

What would the run-time be proportional to for this algorithm? Choose one answer.

This is a tricky algorithm! If you still can't picture how it works, draw out the heap, add all the list elements to it, and see what happens.

G. Conclusion

Submission

IMPORTANT!: Before this lab is due, please fill out an evaluation on your group members' performances for project 2. You can find it here. This is to be done individually. You will get 0 points on the project if you do not fill it out, so please don't forget. Thanks!

Submit the following files as lab19.

• ArrayBST.java
• ArrayHeap.java

Reading

For next lab, read DSA chapter 13.1 - 13.4 (Graphs).