Lecture 39: Greek Tragedy & Balanced Trees

LECTURE 39: GREEK TRAGEDY & BALANCED TREES

CSC 213 – Large Scale Programming

Today’s Goals

Review a new search tree algorithm is needed What real-world problems occur with old

tree? Why does garbage collection make problem

worse? What was ideal approach? How could we

force this? Consider how to create other search

tree types Not limit nodes to 1 element & what could

happen? How to perform insertions on multi-nodes? What about withdrawal? How can we

remove data? Can this sound dirtier? And do I hear

banjos playing?

Dictionary ADT

Dictionary and Map maps keys to values O(1) time with hash, but only if hash is good Can guarantee better -- O(log n) with

balanced BST Assumes data fits in memory since locality

will suck But, honestly, how big can a tree be?

Dictionary ADT




Library of Congress – 20 TB in text database

Amazon.com – 42 TB of combined data ChoicePoint – 250 TB of data on everyday

Americans World Data Center for Climate – 4 PB of

climate data

Dictionary ADT




Library of Congress – 20 TB in text database

Amazon.com – 42 TB of combined data ChoicePoint – 250 TB of data on everyday

Americans World Data Center for Climate – 4 PB of

climate data(Numbers gathered from Feb. 2007 article)

Optimal Tree Partition

Optimal Tree Partition

But no GC algorithm produces this!

Real-World Big Search Trees

Excellent way to test roommates system





(a,b) Trees to the Rescue!

General solution to frequent hikes to Germany Linux & MacOS to track files & directories MySQL & other databases use this to hold

all the data Found in many other places where paging

occurs Simple rules define working of any (a,b)

tree Grows upward so that all leaves found at

same level At least a children for each internal node Every internal node has at most b children

What is “the BTree?”

Common multi-way tree implementation Describe B-Tree using order (“BTree of order

m”) m/2 to m children per internal node Root node can have m or fewer elements

Many variants exist to improve some failing Each variant is specialized for some niche

use Minor differences only between each

variant Will just describe most basic B-Tree during

lecture

BTree Order

Select order minimizing paging when created Elements & references to kids in full node fills

page Nodes have at least m/2 elements, even at

their smallest In memory guarantees each page is at least

50% full How many pages touched by each

operation?

Multi-Way Search Tree

Nodes contain multiple elements Tree grows up with leaves always at same

level Each internal node:

At least 2 children 1 fewer Entrys than children Entrys sorted from smallest to largest

11 24

2 6 8 15 27 30

Multi-Way Search Tree

Children v1 v2 v3 … vd & keys k1 k2 … kd-1

Keys in subtree v1 smaller than k1

Keys in subtree vi between ki-1 and k2

Keys in subtree vd greater than kd-1

11 24

27 302 6 8 15

Inorder Traversal

Visit each child, vi , before visiting Entry ei

As with BST, visits keys in increasing order

11 24

2 6 8 15 27 301 2 3 7

4 6

5 8

Multi-Way Searching

Similar to BST treeSearch finding a keyfor i = 0 to numChildren – 1 doif k < e[i].getKey() then return search(child[i])if k == e[i].getKey() then return e[i]

endfor

if k > e[e.length-1].getKey() then return search(child[child.length-1])11 24

2 6 8 15 27 30

Multi-Way Searching

for i = 0 to numChildren – 1 doif k < e[i].getKey() then return search(child[i])if k == e[i].getKey() then return e[i]

endfor

if k > e[e.length-1].getKey() then return search(child[child.length-1])

Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

2 6 8 15 27 30

Multi-Way Searching


endfor


Example: find(8)

11 24

15 27 302 6 8

(2,4) Trees

Multi-way search tree with 2 properties: Node-Size Property

Internal nodes have at most 4 children

Depth PropertyAll external nodes at same depth

Nodes are either 2-node, 3-node or 4-node Node’s number of children used as basis

for name

10 15 24

2 8 12 27 3218

Insertion

Start by searching for key k Entry added to last internal node

searched Depth property preserved by enforcing this

Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 3218

Insertion



Example: insert(30)

10 15 24

2 8 12 27 30 3218

Insertion

Insertion may cause overflow! 5-node created by the insertion This would make it violate Node-Size

property

27 32 35

15 24

12 18

Insertion

Insertion may cause overflow! 5-node created by the insertion This would make it violate Node-Size

property

27 30 32 35

15 24

12 18

In Case Of Overflow Split Node Split 5-node into 2 new nodes

Entrys e1 e2 & children v1 v2 v3 become a 3-node 2-node created with Entry e4 & children v4 v5

15 24

12 18 27 30 32 35

In Case Of Overflow Split Node Split 5-node into 2 new nodes

Entrys e1 e2 & children v1 v2 v3 become a 3-node 2-node created with Entry e4 & children v4 v5

Promote e3 to parent node If overflow occurs in root node, create new root Overflow can cascade when parent already was

4-node15 24

12 18 27 30 32 35 12 27 3018 35

15 24 32

Parent Overflow

In case of cascade, repeat overflow process Works identically to when children are

external Example: insert(29)

27 32 3512 18 25

15 24 26

Parent Overflow



27 32 3512 18 25

15 24 26

Parent Overflow



27 32 3512 18 25

15 24 26

Parent Overflow



27 32 3512 18 25

15 24 26

Parent Overflow



27 29 32 3512 18 25

15 24 26

Parent Overflow



27 29 32 3512 18 25

15 24 26

Parent Overflow



27 29 32 3512 18 25

15 24 26

Parent Overflow



12 18 25

15 24 26 32

27 29 35

Parent Overflow



12 18 25

15 24 26 32

27 29 35

Parent Overflow



12 18 25

15 24

27 29 35

32

26

Parent Overflow



12 18 25

15 24

27 29 35

32

26

Adding to MultiWay Tree

Leaves all at same level, so trees grow upwards Add to last internal node from initial tree

search Addition not done – check if node too

large Push 1 item up into parent and split into 2

nodes May pass problem along, so check if parent

too large Traverse until overflow stops or made new

root node

Deletion

Must first find Entry to be deleted Remove Entry & an external child if it is on

leaf Example: delete(27)

10 15 24

2 8 12 18 27 32 35

Deletion



10 15 24

2 8 12 18 27 32 35

Deletion



10 15 24

2 8 12 18 27 32 35

Deletion



10 15 24

2 8 12 18 27 32 35

Deletion



10 15 24

2 8 12 18 27 32 35

Deletion



10 15 24

2 8 12 18 27 32 35

10 15 24

2 8 12 18 32 35

Deletion

If Entry's child internal, replace with successor Go 1 to right and then go left just like with

a BST

Deletion

If Entry's child internal, replace with successor Go 1 to right and then as far left as

possible; like BST Example: delete(24)

10 15 24

2 8 12 18 27 32 35

Deletion

If Entry's child internal, replace with successor Go 1 to right and then go left just like with

a BST Example: delete(24)

10 15 24

2 8 12 18 27 32 35

10 15 27

2 8 12 18 32 35

15

9 14

Underflow and Fusion

Entry deletion may cause underflow Node becomes 1-node Choice of solution depends on situation

Example: remove(15)

102 5 7

15

9 14



Example: remove(15)

102 5 7

15

9 14



Example: remove(15)

102 5 7


Example: remove(15)

9 14


102 5 7


Example: remove(15)

9 14


102 5 7

Case 1: Transfer

Has adjacent 3- or 4-node sibling Steal parent’s Entry closest to the 1-node Prevent loneliness & promote sibling’s Entry

No further processing needed in this case Example: remove(10)

4 9

6 82 10

Case 1: Transfer



4 9

6 82 10

Case 1: Transfer



4 9

6 82 10

Case 1: Transfer



4 9

6 82 10

Case 1: Transfer



4 9

6 82

Case 1: Transfer



4 9

6 82

Case 1: Transfer



4

6 82 9

Case 1: Transfer



4

6 82 9

Case 1: Transfer



4 8

62 9

Case 1: Transfer



4 8

62 9

Emptied node has adjacent 2-node sibling Merge node & sibling into one Look to parent and steal Entry between

siblings May propagate underflow to parent!

Example: remove(15)

Case 2: Fusion Mom

9 14

102 5 7



Example: remove(15)9 14

Case 2: Fusion

102 5 7

Mom



Example: remove(15)9

Case 2: Fusion

10 142 5 7

Mom




Case 2: Fusion

10 142 5 7

Mom




Case 2: Fusion

10 142 5 7

Mom

Deletion from MultiWay Tree

Removal like BST: swap element to legal node If removal causes underflow, check its nearest

siblings If 3-node or 4-node as sibling, then solution is

easy…… move sibling up and bring parent down into node

Merge with sibling & parent data if no big neighbor…… but must then check if parent has an underflow

For Next Lecture

Wednesday will be quiz on real-world stuff Garbage collection, cache behavior & trees

(oh, my) End of day Wednesday lab project is

due Will have regular hour during lab, too

At end of day on Friday will have Project #3 due

Documents

Lecture 39: Greek Tragedy & Balanced Trees