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Self‐Adjusting Data Structures

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Self‐Adjusting Data Structures

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Search(2), Search(2), Search(2) , Search(5), Search(5), Search(5)

ve‐to‐front

Lists [D.D. Sleator, R.E. Tarjan, Amortized Efficiency of List Update Rules, Proc. 16th Annual ACM Symposium on

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[ , j , ff y f p , y p Theory of Computing, 488‐492, 1984] Dictionaries [D.D. Sleator, R.E. Tarjan, Self‐Adjusting Binary Search Trees, Journal of the ACM, 32(3): 652‐686, 1985] → splay trees → splay trees Priority Queues [C.A. Crane, Linear lists and priority queues as balanced binary trees, PhD thesis, Stanford University, 1972] [D.E. Knuth. Searching and Sorting, volume 3 of The Art of Computer Programming, Addison‐Wesley, 1973] l f h → leftist heaps [D.D. Sleator, R.E. Tarjan, Self‐Adjusting Heaps, SIAM Journal of Computing, 15(1): 52‐69, 1986] → skew heaps [C Okasaki Alternatives to Two Classic Data Structures Symposium on Computer Science Education 162‐

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[C. Okasaki, Alternatives to Two Classic Data Structures, Symposium on Computer Science Education, 162 165, 2005] → maxiphobix heaps Okasaki: maxiphobix heaps are an alternative to leftist heaps ... but without the “magic”

MakeHeap FindMin Insert Meld DeleteMin

Heaps (via Binary Heap‐Ordered Trees)

5 2 3

2 2 3

[C.A. Crane, Linear lists and priority queues as balanced binary

MakeHeap, FindMin, Insert, Meld, DeleteMin

Meld Cut root + Meld

= =

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1 1 1 1 2 1

Leftist Heaps

trees, PhD thesis, Stanford University, 1972] [D.E. Knuth. Searching and Sorting, volume 3 of The Art of Computer Programming, Addison‐Wesley, 1973]

8 10 11 12

1 1 2 1

2

2 3 3

Meld ( , ) =

Each node distance to empty leaf Inv Distance right child ≤ left child

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2 2 4 2 3

4 13

2 1 2

7 2

2 2 3

M i h bi H

• Inv. Distance right child ≤ left child

⇒ rightmost path ≤ ⎡log n+1⎤ nodes

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1 1 1 1

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4 1 1

Time O(log n)

= Meld ( , ) Meld ( , )

Maxiphobic Heaps

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y x x

[C. Okasaki, Alternatives to Two Classic Data Structures, Symposium on Computer Science Education, 162‐165, 2005]

( , )

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T3 T1 T2 Ti Tj Tk

largest size two smallest

Max size n → ⅔n Time O(log3/2 n)

x < y

[D.D. Sleator, R.E. Tarjan, Self‐Adjusting Heaps, SIAM Journal of Computing, 15(1): 52‐69, 1986]

Skew Heaps

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Meld Cut root + Meld

= =

Heap ordered binary tree with no balance information MakeHeap, FindMin, Insert, Meld, DeleteMin

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Meld Cut root Meld 2

Meld = merge rightmost paths + swap all siblings on merge path

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Meld ( , ) =

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v heavy if |Tv| > |Tp(v)|/2, otherwise light ⇒ any path ≤ log n light nodes

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Potential Φ = # heavy right children in tree

O(log n) amortized Meld Heavy right child on merge path before meld → replaced by light child l l d f h d

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⇒ 1 potential released for heavy node ⇒ amortized cost 2∙ # light children on rightmost paths before meld

[D.D. Sleator, R.E. Tarjan, Self‐Adjusting Heaps, SIAM Journal of Computing, 15(1): 52‐69, 1986]

Skew Heaps – O(1) time Meld

• Meld = Bottom‐up merg of rightmost paths + swap all siblings on merge path

1 1

Meld ( , ) =

4 2 4 13 2 1 5 3 4 2 6 3

=

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Φ = # heavy right children in tree + 2 ∙ # light children on minor & major path

O(1) amortized Meld Heavy right child on merge path before meld → replaced by light child ⇒ 1 potential released Light nodes disappear from major paths (but might → heavy) ⇒ ≥ 1 potential released and become a heavy or light right children on major path ⇒ potential increase by ≤ 4

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and become a heavy or light right children on major path ⇒ potential increase by ≤ 4 O(log n) amortized DeleteMin Cutting root ⇒ 2 new minor paths, i.e. ≤ 2∙log n new light children on minor & major paths

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[D.D. Sleator, R.E. Tarjan, Self‐Adjusting Binary Search Trees, Journal of the ACM, 32(3): 652‐686, 1985]

Splay Trees

Binary search tree with no balance information splay(x) = rotate x to root (zig/zag, zig‐zig/zag‐zag, zig‐zag/zag‐zig)

i i zig zag

y x C x y A

zig

y x y z

zig‐zig

z D x A y x x z

zig‐zag

z D y

root

Search (splay), Insert (splay predecessor+new root), Delete (splay+cut root+join),

B A C B B A C D C B C B A D C B A

Join (splay max, link), Split (splay+unlink)

v u x

zag‐zag i i insert

y F z A E z C

zig‐zig

u v D y x u v D y

insert

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D C x B A B A C F E D B A z C F E

[D.D. Sleator, R.E. Tarjan, Self‐Adjusting Binary Search Trees, Journal of the ACM, 32(3): 652‐686, 1985]

Splay Trees

The access bounds of splay trees are amortized The access bounds of splay trees are amortized (1) O(log n) ( ) l (2) Static optimal (3) Static finger optimal (4) Working set optimal (proof requires dynamic change of weight) ( ) g p (p q y g g ) Static optimality: Φ = v log |Tv|

Σ

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