CS221Week 8 - Friday

Last time

What did we talk about last time? Practiced with 2-3 trees Red-black trees

Questions?

Project 3

Assignment 4

Red-black Trees

Building the tree

We can do an insertion with a red-black tree using a series of rotations and recolors

We do a regular BST insert Then, we work back up the tree as the recursion unwinds

If the right child is red and the left is black, we rotate the current node left

If the left child is red and the left child of the left child is red, we rotate the current node right

If both children are red, we recolor them black and the current node red

You have to do all these checks, in order! Multiple rotations can happen

It doesn't make sense to have a red root, so we always color the root black after the insert

Left rotation

We perform a left rotation when the right

child is red

Y

X

B

A

C

Current

Y

X

BA

C

Current

Right rotation

We perform a right rotation when the left child is red and

its left child is red

Z

Y

BA

D

Current

X C

Z

Y

BA D

Current

X

C

Recolor

We recolor both children and the current node when both children

are red

Y

X

BA D

Current

Z

C

Y

X

BA D

Current

Z

C

Red-black tree practice

Add the following keys to a red-black tree: 62 11 32 7 45 24 88 25 28 90

Analysis of red-black trees

The height of a red-black tree is no more than 2 log n

Find is Θ(height), so find is Θ(log n) Since we only have to go down that

path and back up to insert, insert is Θ(log n)

Delete in red-black trees is messy, but it is also actually Θ(log n)

AVL Trees

AVL trees

Another approach to balancing is the AVL tree Named for its inventors G.M. Adelson-Velskii and

E.M. Landis Invented in 1962

An AVL tree is better balanced than a red-black tree, but it takes more work to keep such a good balance It's faster for finds (because of the better balance) It's slower for inserts and deletes Like a red-black tree, all operations are Θ(log n)

AVL definition

An AVL tree is a binary search tree where

The left and right subtrees of the root have heights that differ by at most one

The left and right subtrees are also AVL trees

AVL tree?

10

6 14

1 9 17

72 15

Balance factor

The balance factor of a tree (or subtree) is the height of its right subtree minus the height of its left

In an AVL tree, the balance factor of every subtree is -1, 0, or +1

What’s the balance factor of this tree?

6

1 9

2

How do we build an AVL tree?

Carefully. Every time we do an add, we have to

make sure that the tree is still balanced

There are 4 cases1. Left Left2. Right Right3. Left Right4. Right Left

Left Left

3

2

1

BA

C

D

3

2

1

BA C D

Right Rotation

Right Right

2

1

B

A

C D

3

3

2

1

BA C D

Left Rotation

Left Right

3

2

1

B

A

C D

3

2

1

B AC D

Left Rotation + Right Rotation

Right Left

2

1

B

A

CD

3 3

2

1

BA CD

Right Rotation + Left

Rotation

Let’s make an AVL tree!

Let’s just add these numbers in order:1, 2, 3, 4, 5, 6, 7, 8, 9, 10

What about these?7, 24, 92, 32, 2, 57, 67, 84, 66, 75

Balancing a Tree by Construction

How to make a balanced tree

What if we knew ahead of time which numbers we were going to put into a tree?

How could we make sure the tree is balanced?

Answer: Sort the numbers Recursively add the numbers such that

the tree stays balanced

Balanced insertion

Write a recursive method that adds a sorted array of data such that the tree stays balanced

Assume that an add(int key) method exists Use the usual convention that start is the

beginning of a range and end is the location after the last legal element in the range

public void balance(int[] data, int start, int end )

DSW algorithm

Takes a potentially unbalanced tree and turns it into a balanced tree

Step 1 Turn the tree into a degenerate tree (backbone or

vine) by right rotating any nodes with left children Step 2

Turn the degenerate tree into a balanced tree by doing sequences of left rotations starting at the root

The analysis is not obvious, but it takes O(n) total rotations

Which method is best?

How much time does it take to insert n items with: Red-black or AVL tree Balanced insertion method Unbalanced insertion + DSW rebalance

How much space does it take to insert n items with: Red-black or AVL tree Balanced insertion method Unbalanced insertion + DSW rebalance

Self-Adjusting Trees

Self-Adjusting Trees

Balanced binary trees ensure that accessing any element takes O(log n) time

But, maybe only a few elements are accessed repeatedly

In this case, it may make more sense to restructure the tree so that these elements are closer to the root

Restructuring strategies

1. Single rotation Rotate a child about its parent if an

element in a child is accessed, unless it's the root

2. Moving to the root Repeat the child-parent rotation until the

element being accessed is the root

Splaying

A concept called splaying takes the rotate to the root idea further, doing special rotations depending on the relationship of the child R with its parent Q and grandparent P

Cases:1. Node R's parent is the root

Do a single rotation to make R the root2. Node R is the left child of Q and Q is the left child of P

(respectively right and right) Rotate Q around P Rotate R around Q

3. Node R is the left child of Q and Q is the right child of P (respectively right and left)

Rotate R around Q Rotate R around P

Danger, Will Robinson!

We are not going deeply into self-organizing trees

It's an interesting concept, and rigorous mathematical analysis shows that splay trees should perform well, in terms of Big O bounds

In practice, however, they usually do not An AVL tree or a red-black tree is generally the

better choice If you come upon some special problem where

you repeatedly access a particular element most of the time, only then should you consider splay trees

Hash TablesStudent Lecture

Hash Tables

Recall: Symbol table ADT We can define a symbol table ADT with a few

essential operations: put(Key key, Value value)▪ Put the key-value pair into the table

get(Key key):▪ Retrieve the value associated with key

delete(Key key)▪ Remove the value associated with key

contains(Key key)▪ See if the table contains a key

isEmpty() size()

It's also useful to be able to iterate over all keys

Unordered symbol table

We have been talking a lot about trees and other ways to keep ordered symbol tables

Ordered symbol tables are great, but we may not always need that ordering

Keeping an unordered symbol table might allow us to improve our running time

Hash tables: motivation

Balanced binary search trees give us: O(log n) time to find a key O(log n) time to do insertions and

deletions

Can we do better? What about:

O(1) time to find a key O(1) to do an insertion or a deletion

Hash tables: theory

We make a huge array, so big that we’ll have more spaces in the array than we expect data values

We use a hashing function that maps keys to indexes in the array

Using the hashing function, we know where to put each key but also where to look for a particular key

Hash table: example

Let’s make a hash table to store integer keys

Our hash table will be 13 elements long

Our hashing function will be simply modding each value by 13

Hash table: example

Insert these keys: 3, 19, 7, 104, 89

0 1 2 3 4 5 6 7 8 9 10 11 12

104 3 19 7 89

0 1 2 3 4 5 6 7 8 9 10 11 12

Hash table: example

Find these keys:

19 YES!

88 NO!

16 NO!

104 3 19 7 89

0 1 2 3 4 5 6 7 8 9 10 11 12

Hash table: issues

We are using a hash table for a space/time tradeoff

Lots of space means we can get down to O(1)

How much space do we need? How do we pick a good hashing

function? What happens if two values collide

(map to the same location)

Upcoming

Next time…

Hashing functions

Reminders

Keep working on Project 3 Sign up for your teams by today!

Finish Assignment 4 Due tonight by midnight