A Handy Data Structure

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A 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., 1977
Totally-ordered list of unique keys

• Finger points to a key---the current key
• Live demonstration coming
• (Trained professional closed course do not
  attempt!)

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Finger Search

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

What Destination?

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

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Finger Search

• Straightforward for sorted arrays

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

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Balanced Search Trees

• BSTs can be seen as versatile sorted arrays
• Can we do finger search on BSTs?

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Brown and Tarjan, 1979

• Level-linked 2-3 Trees
• Worst case bound
• 5 pointers / key

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Tarjan and Van Wyk, 1988

• Heterogeneous Finger Search Trees
• Amortized bound
• 2 pointers / key

Inverted Spine
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This Paper

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

?
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Programming 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?
Locality is Reality

• Finger search is theoretically appealing
• Merging of two sorted list of m and n keys
  takestime, where m n
• Optimal in comparison-based models
• Easy to extend to set intersection and union

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In this paper

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

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Space - Time Price Chart
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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 tree
• Finger will only go forward

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Special 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 Walk

• Need to walk up, e.g., at 5

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In-order Walk

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

Rps
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In-order Walk

• Need to walk down, e.g., at 8

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In-order Walk

• Need to walk down, e.g., at 8
• Cant start chasing pointers after weve arrived
  at 8.

Be Eager
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In-order Walk

• Chase them beforehand, one at a time!
• When we arrive at 8, 9 has already been
  discovered.

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leaving 4
4
12
leaving 6
2
14
10
6
leaving 7
1
15
3
13
11
9
7
5
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The Hand
node
spine
x3
s3
x2
s2

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

x1
s1
Rps
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In-order Walk
node
spine
x3
s3
x2
s2

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

x1
s1
Rps
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5?6

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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5?6

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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5?6

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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At 6

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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6?7

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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6?7

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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6?7

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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At 7

• 1. Pop top cell and keep its spine s
• 2. Extend spine of new top cell
• 3. Prepend s to Rps

Rps
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Finger Search

• During an in-order walk, we know the future.
• But thats not the case in finger search!

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Surprise!

• Use the Hand!

RP
Rps
s atlas, peer)
curr
peer
Length of Prefix
s
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Finger Destinations
?
?
?
RP
s atlas, peer)
curr
peer
s
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Extensions and Future Work

• Refer to paper
• Real degree-balanced search trees
• Going backward
• Interleaving with insertions and deletions
• Future Work
• Experiments (especially on database prefetching)

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Q A
Thank you for your attention.
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Database Application

• Queries often involve columns that are not
  indexed.
• Need to scan the whole column
• Prefetching
• Imagine doing an in-order walk, each node
  corresponding to a page of rows.
• Our in-order walk algorithm is a prefetching
  scheme!