A Handy Data Structure

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A Handy Data StructureA Handy Data Structure

Space-Efficient Finger Search on

Degree-Balanced Search Trees

Guy Blelloch, Bruce Maggs, Maverick Woo

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Guibas et al., 1977Guibas et al., 1977Guibas et al., 1977Guibas et al., 1977

Finger points to a key---the “current” key

Live demonstration coming…(Trained professional; closed course; do not

attempt!)

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Totally-ordered list of unique keysTotally-ordered

list of unique keys

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Finger SearchFinger SearchFinger SearchFinger Search

FingerSearch(f, x):Starting from f, search for x, then move f to final destination.

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O(log d) time if finger rank changes by d

What Destination?What Destination?

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O(log d) time if finger rank changes by d

Straightforward for sorted arrays…

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Balanced Search TreesBalanced Search TreesBalanced Search TreesBalanced Search Trees

BSTs can be seen as versatile sorted arraysCan we do finger search on BSTs?

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Brown and Tarjan, 1979Brown and Tarjan, 1979Brown and Tarjan, 1979Brown and Tarjan, 1979

Level-linked 2-3 TreesWorst case bound5 pointers / key

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Tarjan and Van Wyk, 1988Tarjan and Van Wyk, 1988Tarjan and Van Wyk, 1988Tarjan and Van Wyk, 1988

Heterogeneous Finger Search TreesAmortized bound2 pointers / key

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4 12Inverted SpineInverted Spine

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This PaperThis PaperThis PaperThis Paper

How to achieve the best of both worlds? (sort of)Worst case boundAs few pointers as possible

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Programming PerspectiveProgramming PerspectiveProgramming PerspectiveProgramming Perspective

Suppose we are writing a search engine.For each document, assign a unique number.For each term, maintain a document list using a BST.A query corresponds to taking intersections.

To fit everything in main memory,

node size matters.

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Why Finger Search?Why Finger Search?Why Finger Search?Why Finger Search?

Finger search is theoretically appealingMerging of two sorted list of m and n keys takes

time, where m · nOptimal in comparison-based modelsEasy to extend to set intersection and union

Locality is RealityLocality is Reality

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In this paper…In this paper…In this paper…In this paper…

We introduce another way to support finger searchAssume 2 pointers per key (left, right)Worst-case O(log d) boundDuring runtime, maintain an O(log n)-size auxiliary data structure for an n-key degree-balanced BSTInsertions and deletions can be interleaved with finger searches in any order

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Space - Time Price ChartSpace - Time Price ChartSpace - Time Price ChartSpace - Time Price Chart

VersionTree Size

Time Bound Space

Level-link 2,3 [BT79]

n+5n = 6nWorst case O(log

d)O(1)

Hetero. RB [VT88]

n+2n = 3nAmortized O(log

d)O(1)

Skip List [P90] n+2n = 3n Expected O(log d) O(1)

Splay Tree [ST85,C95]

n+2n = 3nAmortized O(log

d)O(1)

Treaps [AS96] n+3n = 4n Expected O(log d) O(1)

Functional List [KT96]

n+~3n = ~4n

Worst case O(log d)

O(1)

Hetero. Treaps [BB]

n+2n = 3n Expected O(log d) O(1)

“This Paper” [BMW03]

n+2n = 3nWorst case O(log

d)O(log n)

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In the rest of this talk…In the rest of this talk…In the rest of this talk…In the rest of this talk…

I will sketch how we designed our “hand” data structure.

Two simplifying assumptions (for this talk only):A full binary search treeFinger will only go forward

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Special Case: In-order WalkSpecial Case: In-order WalkSpecial Case: In-order WalkSpecial Case: In-order Walk

If finger search is possible, then in-order walk must be worst-case O(1) time per step.

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In-order WalkIn-order WalkIn-order WalkIn-order Walk

Need to walk up, e.g., at 5

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Need to walk up, e.g., at 5Classic: use a “Right Parent Stack”

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5

6

8

Rps

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Need to walk down, e.g., at 8

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8

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Need to walk down, e.g., at 8Can’t start chasing pointers after

we’ve arrived at 8.Be Eager…Be Eager…

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Chase them beforehand, one at a time!

When we arrive at 8, 9 has already been “discovered”.8

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leaving 4

leaving 6

leaving 7

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The HandThe HandThe HandThe Hand

Each node on the Rps maintainsa prefix of its right-left spine.(How long?)

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Rps

node

x1

x3

x2

s1

s3

s2

spine

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1. Pop top cell and keep its spine s2. Extend spine of new top cell3. Prepend s to Rps

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Rps

node

x1

x3

x2

s1

s3

s2

spine

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55665566

1. Pop top cell and keep its spine s

2. Extend spine of new top cell3. Prepend s to Rps

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8

Rps

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55665566


6

8

Rps

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712

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55665566


6

8

Rps

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8

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At 6At 6At 6At 6


6

8

Rps

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8

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66776677

1. Pop top cell and keep its spine s

2. Extend spine of new top cell3. Prepend s to Rps

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Rps

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8

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66776677


Rps

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7

8

10

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66776677


7

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Rps

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At 7At 7At 7At 7


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Rps

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During an in-order walk, we know the future.

But that’s not the case in finger search! 8

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Surprise!Surprise!Surprise!Surprise!

Use the Hand!

curr

RP

peer

Rps

s’

scurr

RP

…

s

s’[atlas, peer)

Length of PrefixLength of Prefix

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Finger DestinationsFinger DestinationsFinger DestinationsFinger Destinations

curr

RP

peer

s

s’[atlas, peer)

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Extensions and Future WorkExtensions and Future WorkExtensions and Future WorkExtensions and Future Work

Refer to paper“Real” degree-balanced search treesGoing backwardInterleaving with insertions and deletions

Future WorkExperiments (especially on database prefetching)

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Q & AQ & AQ & AQ & A

Thank you for your attention.

A Handy Data Structure

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