LECTURE 18:SCREW THAT, I WANT IT ALL

CSC 213 – Large Scale Programming

Today’s Goal

Review Map & Dictionary implementations What do they do well? When would they be

used? Why do they suck so much another is

needed? Discuss how we can get it all NOW with BSTs Why greedy approach could be useful &

helpful Seeing how idea of splay tree not balanced

or Zen What & when to splay node in a BST

Map & Dictionary ADT

Implementation

Searching Adding Removing

Ordered List O(log n) O(n) O(n)Unordered List

O(n) O(n)/O(1) O(n)

Hash O(n) O(n) O(n) if lucky/good

O(1) O(1) O(1)

BST O(n) O(n) O(n)AVL /

balancedO(log n) O(log n) O(log n)

Map & Dictionary ADT

Implementation

Searching Adding Removing

Ordered List O(log n) O(n) O(n)Unordered List

O(n) O(n)/O(1) O(n)

Hash O(n) O(n) O(n) if lucky/good

O(1) O(1) O(1)

BST O(n) O(n) O(n)AVL /

balancedO(log n) O(log n) O(log n)

Ideal: Hash Without Withdrawal Want O(1) times that hash provides

At the same time scared of slowdowns from bad hash

Or lack hash function that can work with your data

BSTs provide safety that array-based hash lacks Worst-case only O(log n) as long as tree is

balanced Requires way to keep tree balanced (e.g.,

AVL) and Need O(1) access to tree’s nodes, but how?

Implementing Map with Tree

Accessing root much faster than going to leaves In real-world, really want important data

near root Which key best at root of Tree of NHL

teams? BST: Key for Entry added first

Implementing Map with Tree

Accessing root much faster than going to leaves In real-world, really want important data

near root Which key best at root of Tree of NHL

teams? BST: Key for Entry added first

AVLTree: Random Entry near midpoint

Implementing Map with Tree

Accessing root much faster than going to leaves In real-world, really want important data

near root Which key best at root of Tree of NHL

teams? BST: Key for Entry added first

AVLTree: Random Entry near midpoint

SplayTree: Most recently used key

Building a SplayTree

Another approach which builds upon BST Not an AVLTree, however, but a new BST

subclass

Concept Behind Splay Trees

Splay trees do NOT maintain balance Recently used nodes clustered near top of BST

Most recently accessed nodes take O(1) time

Other nodes may need O(n) time to find, however

Concept Behind Splay Trees

Splay trees do NOT maintain balance Recently used nodes clustered near top of BST

Most recently accessed nodes take O(1) time

Other nodes may need O(n) time to find, however

SplayTree Complexity

Without balancing, keeps BST's O(n) complexity Worst-case performance is like all

unbalanced trees But splaying gives expected O(log n)

complexity Averages complexity of O(1) & O(n)

operations If work concentrated on small subset, time

is faster Worst-case hard to create without knowing

tree

Be Kind: Splay Your Tree

Assumes nodes reused soon after initial use At end of each method, moves node up to

root Using node now O(1) and will only slowly

drop in tree Splay tree with each find, insert & remove AVL-like restructuring to reorganize nodes

in tree But

Be Kind: Splay Your Tree

Assumes nodes reused soon after initial use At end of each method, moves node up to

root Using node now O(1) and will only slowly drop

in tree Splay tree with each find, insert & remove AVL-like restructuring to reorganize nodes in

tree But

continues rotationsuntil node becomes

root

How To Splay

How To Splay

Uses trinode restructuring but not like AVL does Not balancing: selects node, parent, &

grandparent Always move node to root of subtree when

splaying When splaying, new types restructures

also exist Node & parent always used in these

rotations Rotations will also use grandparent, if it

exists Moving node to tree root is goal of

splaying May get a balanced tree, but WANT IT

ALL, ASAP

When To Use Splay Tree

What applications are good for a splay tree?

Where would splay trees be a BAD IDEA?

Splay Node Rotations

Uses different nodes than AVL tree rotations AVLTree moves node down to balance

tree's tao Hedonistic ideal moves node to SplayTree's root

Splay Node Rotations

Uses different nodes than AVL tree rotations AVLTree moves node down to balance

tree's tao Hedonistic ideal moves node to SplayTree's root

AVLTree

Splay Node Rotations

Uses different nodes than AVL tree rotations AVLTree moves node down to balance

tree's tao Hedonistic ideal moves node to SplayTree's root

AVLTree SplayTree

Zig-Zag When Splaying a Tree When node median of parent &

grandparent Just like in AVL tree, perform trinode

restructuring Use 7(+1) variables to set node's parent &

childrenparent

node

T2 T3

T4

grandparent

T1

parent

node

T2 T3 T4

grandparent

T1

Zig-Zig When Splaying a Tree When node, parent, & grandparent in a

line Rotation differs from AVL tree's trinode

restructure

parent

node

grandparent

T1

T4

T2

T3

Zig-Zig When Splaying a Tree When node, parent, & grandparent in a

line Rotation differs from AVL tree's trinode

restructure Greed is very good: splay node to subtree

rootparent

node

T2

T3

T4grandparent

T1

parent

node

grandparent

T1

T4

T2

T3

Splaying Right Child Of Root Simplest process is when parent is the

root Single rotation completes splaying process

Splaying does not stop until you reach the top Rotation not always used, may only need

restructuresroot

node

T3

T1

T2

Splaying Right Child Of Root Simplest process is when parent is the

root Single rotation completes splaying process

Splaying does not stop until you reach the top Rotation not always used, may only need

restructuresroot

node

T3

T1

T2

Splaying Right Child Of Root Simplest process is when parent is the

root Single rotation completes splaying process

Splaying does not stop until you reach the top Rotation not always used, may only need

restructures root

node

T2

T3

T1

root

node

T3

T1

T2

Splaying Left Child Of Root

Simplest process is when parent is the root Single rotation completes splaying process

Splaying does not stop until you reach the top Rotation not always used, may only need

restructures

T2

T3

T1

root

node

T3

T1

T2

node

root

Which Node Gets Splayed?

Method

Node to splay

findIf found, splay node containing matchIf not found, splay last node in treeSearch

add Splay node for new Entry

remove

If found, splay parent of external node removedIf not found, splay last node in treeSearch

Which Node Gets Splayed?

Method

Node to splay

findIf found, splay node containing matchIf not found, splay last node in treeSearch

add Splay node for new Entry

remove

If found, splay parent of external node removedIf not found, splay last node in treeSearch

Only internal nodes can be splayed When operation end with leaf, splay its

parent

For Next Lecture

Weekly assignment #7 now available on Angel

Project #1 due Friday at midnight

Remember class on Friday is BST Problem Day

Midterm #1 in class week from Friday Open-book, open-note exam (as usual) Use any lecture’s slides IF you have notes

on them No computers, calculators, or friends,

however