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An optimal dynamic spanner for points residing in doubling metric spaces. Lee-Ad Gottlieb NYU Weizmann. Liam Roditty Weizmann. Spanners. A spanner for graph G is a subgraph H H contains all vertices in G H contains only some edges of G. G. H. 1. 2. 2. 1. 1. 1. 1. Spanners.

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an optimal dynamic spanner for points residing in doubling metric spaces

An optimal dynamic spanner for points residing in doubling metric spaces

Lee-Ad Gottlieb

NYU Weizmann

Liam Roditty

Weizmann

spanners
Spanners
  • A spanner for graph G is a subgraph H
    • H contains all vertices in G
    • H contains only some edges of G

G

H

1

2

2

1

1

1

1

spanners1
Spanners
  • Some qualities of a spanner
    • degree
    • stretch
    • hop
    • weight
  • Applications: networks, routing…

G

H

1

2

2

1

1

1

1

spanners2
Spanners
  • Our goal:
    • Build (1+e)-stretch spanner for the full graph on S
    • Low degree
    • Maintain dynamically
  • Lower bounds on degree and dynamic maintenance follow…
    • First need to define doubling dimension
doubling dimension
Doubling Dimension
  • Point set X has doubling dimension if
    • the points of X covered by ball B can be covered by 2 balls of half the radius.
    • Where a ball centered at point c is the space within distance r of c.

4

5

3

6

8

2

7

1

lower bound on degree
Lower bound on degree

1

1

1

1

1

  • Low stretch spanner necessitates high degree.
    • Example: A

(2-e)-spanner

is the full graph

  • Lower bounds on degree
    • (1/e)O()
lower bounds on insertions
Lower bounds on insertions

H

  • Lower bound on insertions
    • An insertion of a new point in a (1+e)-spanner subsumes a (1+e)-NNS
search lower bounds
Search lower bounds

e

1

q

e

e

e

e

  • Lower bounds on (1+e)-ANN search (arbitrary metric space)
    • 2O() log n
    • (1/e)O()
nns in low doubling dimension
NNS in Low Doubling Dimension
  • Krauthgamer and Lee (SODA ‘04)
    • considered (1+e)-ANN queries on S having low doubling dimension
    • Created a point hierarchy to solve this problem
  • Hierarchy is composed of levels of d-nets.
    • Packing: Points of each net spaced out
    • Covering: Points of each net cover all points of the previous level
hierarchies
Hierarchies
  • Spanners can be created using point hierarchies (GGN-04)
  • Example…
    • Consider the hierarchy of KL-04, used in nearest neighbor search
    • Hierarchy is composed of levels of d-nets.
      • Packing: Points of each net spaced out
      • Covering: Points of each net cover all points of the previous level
hierarchy
Hierarchy

1-net

2-net

4-net

8-net

hierarchy1
Hierarchy

1-net

2-net

4-net

8-net

Packing

Radius = 1

Covering: all points

are covered

hierarchy2
Hierarchy

1-net

2-net

4-net

8-net

Radius = 2

hierarchy3
Hierarchy

1-net

2-net

4-net

8-net

hierarchy4
Hierarchy

1-net

2-net

4-net

8-net

hierarchy5
Hierarchy

1-net

2-net

4-net

8-net

hierarchy6
Hierarchy

1-net

2-net

4-net

8-net

hierarchy7
Hierarchy

1-net

2-net

4-net

8-net

hierarchy8
Hierarchy

1-net

2-net

4-net

8-net

another perspective
Another Perspective

Spanning Tree defines

Parent-child relationship

Let logD be the aspect

Ratio of the point set.

The tree has logD levels.

spanner construction
Spanner Construction

Tree

Parent-child

edge

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner construction1
Spanner Construction

Tree

Lateral

edge

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner paths
Spanner Paths

Tree

Path

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner paths1
Spanner Paths

Tree

Path

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner paths2
Spanner Paths

Tree

Path

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner paths3
Spanner Paths

Tree

Path

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner paths4
Spanner Paths

Tree

Path

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner paths5
Spanner Paths

Tree

Path

  • Edges
    • Parent-child
    • Lateral
  • Path:
    • Up, across, down
spanner construction2
Spanner Construction

1-net

2-net

4-net

8-net

Identify 1-net

points

spanner construction3
Spanner Construction

1-net

2-net

4-net

8-net

Connect 1-net

Points within

radius 3

Call these lateral

connections

spanner construction4
Spanner Construction

1-net

2-net

4-net

8-net

Identify 2-net

points

Note that we

got all parent-

child connections

for free

spanner construction5
Spanner Construction

1-net

2-net

4-net

8-net

Connect 2-net

Points within

radius 6

spanner construction6
Spanner Construction

1-net

2-net

4-net

8-net

Identify 4-net

points

spanner construction7
Spanner Construction

1-net

2-net

4-net

8-net

Connect 4-net

Points within

radius 12

analysis
Analysis

1-net

2-net

4-net

8-net

What’s the stretch

between these two

Points?

analysis1
Analysis

1-net

2-net

4-net

8-net

What’s the stretch

between these two

Points?

Key to proof: blue

points are connected,

but white points aren’t

analysis2
Analysis

1-net

2-net

4-net

8-net

Stretch: dspanner/d =

(2+4+(d’+4+4)+4+2)/(d’-2-2) =

(d’+20)/(d’-4) < 13

d’>6

analysis3
Analysis
  • We connected d-net points within distance 3d.
  • More generally
    • Connect d-net points that are within distance cd
      • Degree is cO()
    • Let j be the last level at which parents of the points are not connected. Stretch:
      • dspanner/d < ((d’+2j+1+2j+1)+2j+2+2j+2)/(d’-2j-1-2j-1) <

(c2j+2j+2+2j+2+2j+2)/(c2j-2j) =

(c+12)/(c-1) = 1+13/(c-1) = (1+e)

degree
Degree

What’s the degree of the spanner in the previous example?

degree1
Degree

1-net

2-net

4-net

8-net

Problem: degree is logD

degree2
Degree
  • Problem: This
  • node appears
  • at every level.
  • Solution: Why
  • require each level
  • to be subset of
  • the next one?
dynamic hierarchy
Dynamic hierarchy
  • Another problem: Need fast dynamic updates
    • CG-06 showed how to support a hierarchy under dynamic update in 2O() log n time.
    • But doesn’t support deletions!
  • Goals:
    • Replace deleted points
    • Low Degree:
      • each point should appear only O(1) times in the hierarchy
dynamic hierarchy1
Dynamic hierarchy

A look

at the new

hierarchy

dynamic hierarchy2
Dynamic hierarchy

Step 1: Remove leaf nodes of deleted points

dynamic hierarchy3
Dynamic hierarchy

Step 1: Remove leaf nodes of deleted points

dynamic hierarchy4
Dynamic hierarchy

Step 2: Compress single child paths

dynamic hierarchy5
Dynamic hierarchy

Step 2: Compress single child paths

dynamic hierarchy6
Dynamic hierarchy

Step 2: Compress single child paths

dynamic hierarchy7
Dynamic hierarchy

Step 2: Compress single child paths

dynamic hierarchy8
Dynamic hierarchy

Step 2: Compress single child paths

dynamic hierarchy9
Dynamic hierarchy

Tree with

degree at

least 2

Step 2: Compress single child paths

dynamic hierarchy10
Dynamic hierarchy
  • Replacement scheme:
  • Eliminates deleted points
  • Each point appears
    • O(1) times
  • Adds a small cost to the stretch
extracting a spanner
Extracting a spanner
  • Spanner edges:
  • Have parent-child edges
  • Missing some lateral edges

parent-child

edges

Missing

lateral

edge

spanner
Spanner

Possible solution?

extracting a spanner1
Extracting a spanner
  • Replacing lateral edges

Problem: Too many

replacement edges

incident on a single

node

Solution: Assign x

as a “step child” of

the lowest covering

point

x

conclusion
Conclusion
  • Final spanner
    • (1+e) stretch
    • (1/e)O() degree (optimal)
    • (1/e)O() log n update time (optimal?)
  • Thank you!
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