CSE 326: Data StructuresLecture #11

AVL and Splay Trees

Steve Wolfman

Winter Quarter 2000

Today’s Outline

• Extra operations in AVL trees • Splaying and Splay Trees

AVL Tree Dictionary Data Structure

4

121062

115

8

14137 9

• Binary search tree properties– binary tree property

– search tree property

• Balance property– balance of every node is:

-1 b 1– result:

• depth is (log n)

15

Deletion (Hard Case #1)

2092

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121 100

2 2

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00

Delete(12)

Single Rotation on Deletion

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2 2

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3092

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1 00

2 1

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0

Something very bad happened!

Deletion (Hard Case #2-4)

Delete(9)

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Double Rotation on Deletion

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0180

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01800

Deletion with Propagation: Choose Your Own Adventure!

We get to choose whether to single or double rotate!

If you take the gold and single rotate, flip to Slide 10.

If you decide to search further into the scary cavern, continue on to the next slide.

If you decide not to rotate atall, jump to Slide 11.

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Propagated Double Rotation

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Propagated Single Rotation

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You Didn’t Rotate

Casually walking away from the AVL tree, minding your own business, you fall into a sewer and are eaten by wild Red-Black trees (some people say they grow as big as B-Trees in the sewers!).

You die.

AVL Deletion Algorithm• Recursive1. Search downward for

node

2. Delete node

3. Unwind stack,

correcting heights

a. If imbalance #1,

single rotate

b. If imbalance #2,

double rotate

• Iterative1. Search downward for

node, stacking

parent nodes

2. Delete node

3. Unwind stack,

correcting heights

a. If imbalance #1,

single rotate

b. If imbalance #2,

double rotate

AVL buildTree8 10 15 20 30 35 405

17

17

8 10155 20303540

Divide & Conquer– Divide the problem into parts

– Solve each part recursively

– Merge the parts into a general solution

How long does divide & conquer take?

BuildTree Example

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1 0

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8 10 15 20 30 35 405 17

8 10 155

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30 35 4020

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BuildTree Analysis(Approximate)

T(1) = 1

T(n) = 2T(n/2) + 1

T(n) = 2(2T(n/4)+1) + 1

T(n) = 4T(n/4) + 2 + 1

T(n) = 4(2T(n/8)+1) + 2 + 1

T(n) = 8T(n/8) + 4 + 2 + 1

T(n) = 2kT(n/2k) +

let 2k = n, log n = k

T(n) = nT(1) +

T(n) = (n)

k

i

i

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22

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n

i

ilog

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1n

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1n

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1n

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1n

BuildTree Analysis (Exact)

Precise Analysis: T(0) = bT(n) = T( ) + T( ) + c

By induction on n:

T(n) = (b+c)n + b

Base case:

T(0) = b = (b+c)0 + b

Induction step:

T(n) = (b+c) + b +

(b+c) + b + c

= (b+c)n + b

QED: T(n) = (b+c)n + b = (n)

12

1

2

1

nnn

Thinking About AVL

• Observations+ Worst case height of an AVL tree is about 1.44 log n

+ All operations supported in worst case O(log n)

+ Only one (single or double) rotation needed on insertion

– O(log n) rotations needed on deletion

– Height fields must be maintained (or 2-bit balance)

? Coding complexity

Splay Trees

• Problems with AVL Trees– extra storage/complexity for height fields

– ugly delete code

• Solution: splay trees– blind adjusting version of AVL trees

– amortized time for all operations is O(log n)

– worst case time is O(n)

– insert/find always rotates node to the root!

Idea

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You’re forced to make a really deep access:

Since you’re down there anyway,fix up a lot of deep nodes!

Zig-Zig*

*I told you it was a technical term!

n

Z

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p

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g

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Zig-Zag*

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*This is just a double rotation

n

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Zig

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root root

Splaying Example

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Find(6)

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zig-zig

Still Splaying 6

zig-zig2

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Almost There, Stay on Target*

zig

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Splay Again

Find(4)

zig-zag

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Example Splayed Out

zig-zag

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To Do

• Turn in Project II• Read chapter 4 in the book• Do the new worksheet• Prepare for the midterm• Finish HW#4

– Just kidding

Coming Up

• Polish off Splay Trees• Talk about Project III• Second project due (today)• Midterm (Friday)