Advanced Implementations of Tables: Balanced Search Trees and - - PowerPoint PPT Presentation

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Advanced Implementations of Tables: Balanced Search Trees and - - PowerPoint PPT Presentation

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Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees Binary search tree operations such as insert, delete, retrieve, etc. depend on the length of the path to the desired node The path length can

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Advanced Implementations of Tables: Balanced Search Trees and Hashing

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Binary searching & introduction to trees

2 CMPS 12B, UC Santa Cruz

Balanced Search Trees

Binary search tree operations such as insert, delete,

retrieve, etc. depend on the length of the path to the desired node

The path length can vary from log2(n+1) to O(n)

depending on how balanced or unbalanced the tree is

The shape of the tree is determined by the values of

the items and the order in which they were inserted

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Binary searching & introduction to trees

3 CMPS 12B, UC Santa Cruz

Examples

40 20 60 50 30 10 70 10 40 20 70 50 60 30 Can you get the same tree with different insertion orders?

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Binary searching & introduction to trees

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2-3 Trees

Each internal node has two or three children All leaves are at the same level 2 children = 2-node, 3 children = 3-node

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2-3 Trees (continued)

2-3 trees are not binary trees (duh) A 2-3 tree of height h always has at least 2h-1 nodes

i.e. greater than or equal to a binary tree of height h

A 2-3 tree with n nodes has height less than or equal

to log2(n+1)

i.e less than or equal to the height of a full binary tree with

n nodes

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Binary searching & introduction to trees

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Definition of 2-3 Trees

T is a 2-3 tree of height h if T is empty (height 0), OR T is of the form Where r is a node that contains one data item and TL

and TR are 2-3 trees, each of height h-1, and the search key of r is greater than any in TL and less than any in TR, OR

r TL TR

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Binary searching & introduction to trees

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Definition of 2-3 Trees (continued)

T is of the form Where r is a node that contains two data items and

TL, TM, and TR are 2-3 trees, each of height h-1, and the smaller search key of r is greater than any in TL and less than any in TM and the larger search key in r is greater than any in TM and smaller than any in TR.

r TL TR TM

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Binary searching & introduction to trees

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Placing Data in a 2-3 tree

  • 1. A 2-node must contain
  • a single data item whose value is
  • greater than any in its left subtree, and
  • smaller than any in its right subtree
  • 2. A 3-mode must contain two data items,
  • the smaller of which is
  • greater than any in its left subtree, and
  • smaller than any in its middle subtree, and
  • the greater of which is
  • greater than any in its middle subtree, and
  • smaller than any in its right subtree
  • 3. A leaf may contain either one or two data items
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Binary searching & introduction to trees

9 CMPS 12B, UC Santa Cruz

2-3 Node Code

import searchkeys.*; public class Tree23Node { private KeyedItem smallitem; private KeyedItem largeItem; private Tree23Node leftChild; private Tree23Node middleChild; private Tree23Node rightChild; }

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Binary searching & introduction to trees

10 CMPS 12B, UC Santa Cruz

Traversing a 2-3 Tree

Just like a binary tree: preorder, inorder, postorder

50 90 20 120 150 100 110 30 40 10 130 140 70 160 60 80

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Binary searching & introduction to trees

11 CMPS 12B, UC Santa Cruz

Searching a 2-3 tree

Same efficiency as a balanced binary search tree:

O(logn)

BUT, easy to keep balanced, unlike binary search

trees

This means that as you insert elements, the balance is

easily maintained, and worst case performance remains O(logn)

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Binary searching & introduction to trees

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Inserting 39 Into A 2-3 Tree

50 30 70 90 60 10 20 80 160 40

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Binary searching & introduction to trees

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Inserting 38

50 30 70 90 60 10 20 80 160 39 40

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Binary searching & introduction to trees

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Inserting 38 (continued)

50 30 70 90 60 10 20 80 160 38 39 40

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Binary searching & introduction to trees

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Inserting 37

50 30 39 70 90 60 10 20 80 160 38 40

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Binary searching & introduction to trees

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Inserting 36

50 30 39 70 90 60 10 20 80 160 37 38 40

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Binary searching & introduction to trees

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Inserting 36 (continued)

50 30 39 70 90 60 10 20 80 160 36 37 38 40

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Binary searching & introduction to trees

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Inserting 36 (continued)

50 30 37 39 70 90 60 10 20 80 160 36 40 38

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Binary searching & introduction to trees

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Inserting 75

37 50 30 70 90 60 10 20 80 39 160 36 40 38

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Binary searching & introduction to trees

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Inserting 77

37 50 30 70 90 60 10 20 75 80 39 160 36 40 38

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Binary searching & introduction to trees

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Inserting 77 (continued)

37 50 30 70 90 60 10 20 75 77 80 39 160 36 40 38

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Binary searching & introduction to trees

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Inserting 77 (continued)

37 50 30 70 77 90 60 10 20 75 39 160 36 40 38 80

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Binary searching & introduction to trees

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Inserting 77 (continued)

37 50 77 30 70 60 10 20 75 39 160 36 40 38 80 90

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Binary searching & introduction to trees

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Inserting 77 (continued)

37 30 70 60 10 20 75 39 160 36 40 38 80 90 77 50

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Binary searching & introduction to trees

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Inserting Into A 2-3 Tree

Insert into the leaf node in which the search key

belongs

If the leaf has two values, stop If the leaf has three values, split the node into two

nodes with the smallest and largest values, and

Push the middle value into the parent node Continue with the parent node until either

you push a value into a node that had only one value, or you create a new root node

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Binary searching & introduction to trees

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Deleting From A 2-3 Tree

The inverse of inserting Delete the value (in a leaf), then Merge empty nodes If necessary, delete empty root

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Binary searching & introduction to trees

27 CMPS 12B, UC Santa Cruz

Redistribute I

P S L L S P

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Binary searching & introduction to trees

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Merge I

L S S L

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Binary searching & introduction to trees

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Redistribute II

P S L L S P

a b c d a b c d

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Binary searching & introduction to trees

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Merge II

L S S L

a b c a b c

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Delete

S L

a b c

S L

a b c

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2-3 Trees: Results

Slightly more complicated than binary search trees,

BUT

2-3 Trees are always balanced Every operation takes O(logn)

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2-3-4 Trees

Slightly less complicated than 2-3 Trees Each node can contain 1–3 values and have 1–4

children

Inserting

Split 4-nodes on the way down Insert into leaf

Deleting:

Only delete from 3-node or 4-node

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Red-Black Trees

2-3 Trees are always balanced

O(logn) time for all operations

2-3-4 Trees are always balanced

O(logn) time for all operations, and Insertion and deletion can be done in a single pass from

root to leaf

But, require slightly more storage per node

Red-Black Trees have the advantage of 2-3-4 trees,

without the overhead

Represent the 2-3-4 Tree as a binary tree with colored

references (red or black)

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Red-Black Representation of a 4-Node

S M L

a b c d

M S L

a b c d

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Red-Black Representation of a 3-Node

S L

a b c

L S

a b c

S L

b c a

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Binary searching & introduction to trees

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2-3-4 Tree

37 50 30 35 70 90 60 10 20 80 39 160 32 33 34 40 38 36

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Binary searching & introduction to trees

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Equivalent Red-Black Tree

50 30 90 70 20 100 39 80 33 40 38 36 37 35 10 32 34 60

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Red-Black Tree Node

public class RBTreeNode { public static final int RED = 0; public static final int BLACK = 1; private KeyedItem item; private RBTreeNode leftChild; private RBTreeNode rightChild; private int leftColor; private int rightColor; }

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Searching and Traversing Red-Black Trees

Red-Black trees are binary search trees

Just search them the same way you would any other binary

search tree

Inserting

Split 4-nodes on the way down by changing paired red

child references to black

Insert into a leaf

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Splitting a 4-node root

M S L

a b c d

M S L

a b c d

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Splitting a 4-node

M S L

a b c d

M S L

a b c d

P

e

P

e

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Splitting a 4-node

M S L

b c d e

M S L

b c d e

P

a

P

a

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Splitting a 4-node

M S L

a b c d

P

e

Q

f

M S L

a b c d

P

e

Q

f

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Splitting a 4-node

M S L

a b c d

P

e

Q

f

M S L

a b c d

P

e

Q

f

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Binary searching & introduction to trees

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Splitting a 4-node

M S L

b c d e

Q

f

P

a

P L

d e

M

a

Q

f

S

b c

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AVL Trees

Insert in the appropriate spot Rotate as necessary to restore balance

Rotate your way back up the tree

Every operation is O(logn)

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Hashing

Trees allow for efficient table operations, but What if you want O(1) behavior? What if time is much more critical than space? Basic idea:

Same for insert, lookup, delete

Hash Function Search Key Number in 0..n-1 Array

index

“Hash Table”

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Issues

How to create a hash function

Easy or difficult, depending upon the desired properties

How large the array should be

Factors: how many items will be stored, how much

memory you have, how fast you want the operations to be

Static or dynamic Hash collisions: what if two input values produce the

same hash value?

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Hash Functions

Goals

Fast, easy to compute Distributes hash values evenly in the target range

Possible functions for an n-entry hash table

h(k) = random number in 0..n-1 h(k) = the sum of the digits in k h(k) = the first m digits of k, where m = log10(n) h(k) = k×mod(n)

Are these good or bad? Why?

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Hash Functions on Strings

Need to convert string to a number Possible solutions:

Add the binary representations of the characters Concatenate the binary representations to get a very large

number

Hello = H×324 + e×323 + l×322 + l×32 + o

Horner’s rule: (((H×32 + e) × 32 + l) × 32 + l) × 32 + o This is a very big number, so (A × B) mod n = (A mod n × B mod n) mod n (A + B) mod n = (A mod n + B mod n) mod n Apply the modulo operator early and often to keep the

number small

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Dealing with Hash Collisions

Open addressing: Collision ⇒ try to place the object

in other locations in a predictable sequence

Linear probing: search starting from the current location

Issues: wrapping, slow, empty/deleted entries, clustering

Quadratic probing: search +1,4,9,16,25modn Double hashing: Use second hash function to determine

the size of the steps:

h2(k) ≠ 0 h2 ≠ h1 Example: h1(k) = k mod 11, h2(k) = 7 – (k mod 7)

Note: size of steps and size of table must be relatively

prime so that all entries get visited

Use prime size, step, etc.

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Increase The Size of the Hash Table

Increasing the actual size is infeasible

Everything must be rehashed and moved

Chain-bucket hashing

Each entry in a hash table is a bucket into which multiple

values may be added

Each bucket can be implemented as a chain, or linked list

Now the size of the hash table is variable

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Efficiency of Hashing

Load factor α is a measure of how full the table is

Small α ⇒ little chance of collision and low search time Large α ⇒ high chance of collision and high search time

Unsuccessful searches require more time than

successful searches

table

  • f

Size items

  • f

Number _ _ _ _ = α

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Linear Probing

Successful search Unsuccessful search

      − + α 1 1 1 2 1

( ) 

     − +

2

1 1 1 2 1 α

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Quadratic Probing and Double Hashing

Successful search Unsuccessful search

( )

α α − − 1 loge

α − 1 1

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Chain-Bucket Hashing

Successful search Unsuccessful search

2 1 α +

α

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How well does a hash function work?

How fast is it to compute? How well does it scatter random data? How well does it scatter non-random data?

This can be very important It is always possible to construct a worst case

General principles:

The hash function should involve the whole search key If a hash function involves a modulo operation, the base

should be prime

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Hashing vs. Binary Trees

Hashing supports efficient insert and remove Hashing does not support efficient sorting Hashing does not support efficient range queries