I/O-Efficient Batched Union-Find and Its Applications to Terrain Analysis

I/O-Efficient Batched Union-Find and Its I/O-Efficient Batched Union-Find and Its

Applications to Terrain AnalysisApplications to Terrain Analysis

Pankaj K. Agarwal, Lars Arge, and Ke YiPankaj K. Agarwal, Lars Arge, and Ke Yi

Duke UniversityDuke UniversityUniversity of AarhusUniversity of Aarhus

The Union-Find ProblemThe Union-Find Problem

• A universe of N elements: x1, x2, …, xN

• Initially N singleton sets: {x1}, {x2 }, …, {xN}

• Each set has a representative

• Maintain the partition under– Union(xi, xj) : Joins the sets containing xi and xj

– Find(xi) : Returns the representative of the set containing xi

The SolutionThe Solution

d

b j a

e g

h

f l

n

m

i

s r cz k

p

representatives

d

b j a

e g

h

f l

n

m

Union(d, h) :

link-by-rank

d

b j a

e g

h

f l n

Find(n) :

path compression

m

ComplexityComplexity

• O(N α(N)) for a sequence of N union and find operations [Tarjan 75]

– α(•) : Inverse Ackermann function (very slow!)– Optimal in the worst case [Tarjan79, Fredman

and Saks 89]

• Batched (Off-line) version– Entire sequence known in advance– Can be improved to linear on RAM [Gabow and

Tarjan 85]– Not possible on a pointer machine [Tarjan79]

Simple and Good, as long as …Simple and Good, as long as …

The entire data structure fits in memory

The I/O ModelThe I/O Model

Main memory of size M

Disk of infinite size

One I/O transfers B items between memory and disk

Our ResultsOur Results

• An I/O-efficient algorithm for the batched union-find problem using O(sort(N)) = O(N/B logM/B(N/B)) I/Os expected– Same as sorting– optimal in the worst case

• A practical algorithm using O(sort(N) log(N/M)) I/Os• Applications to terrain analysis

– Topological persistence : O(sort(N)) I/Os– Contour trees : O(sort(N)) I/Os

I/O-Efficient Batched Union-FindI/O-Efficient Batched Union-Find

• Assumption: No redundant unions– Each union must join two different sets– Will remove later

• Two-stage algorithm– Convert to interval union-find

• Compute an order on the elements s.t. each union joins two adjacent sets

– Solve batched interval union-find

Union GraphUnion Graph

r

a b

c d e f

g h i

1: Union(d, g)2: Union(a, c)3: Union(r, b)4: Union(a, e)5: Union(e, i)6: Union(r, a)7: Union(a, d) g8: Union(d, h) r9: Union(b, f)

3

1

2 4

5

7

8

9

6r

a b

c d e

f

g

h

i

3

1

2 4

5

7

8

96

Equivalent union trees

(Tree if no redundant unions)

Transforming the Union TreeTransforming the Union Treer

a b

c d e f

g h i

3

1

2 4

5

7

8

9

6r

a b

c d e f

g

h

i

3

1

2 4

5

7

8

9

6r

a b

c

d

e fg

h

i

3

1 2 4

5

78

9

6

r

a b

c

d

e

f

g

h

i

3

1 2 4 5

78

96

Weights along root-to-leafpath decrease

Formulating as a Batched ProblemFormulating as a Batched Problem

r

a b

c d e f

g h i

3

1

2 4

5

7

8

9

6

r

a b

c

d

e

f

g

h

i

3

1 2 4 5

78

96

For each edge, find the lowest ancestor edgewith a higher weight

Cast in a Geometry SettingCast in a Geometry Settingr

a b

c d e f

g h i

3

1

2 4

5

7

8

9

6

Euler Tour

In O(sort(N)) I/Os [Chiang et al. 95]

12

3

45

6

78

9

x: positions in the toury: weight

Cast in a Geometry SettingCast in a Geometry Settingr

a b

c d e f

g h i

3

1

2 4

5

7

8

9

6

12

3

45

6

78

9

For each edge, find the lowest ancestor edgewith a higher weight

For each segment, find the shortest segment above and containing it

Distribution SweepingDistribution SweepingM/B vertical slabs

checked here

checkedrecursively

Total cost:O(sort(N))

In-Order TraversalIn-Order Traversalr

ab

c

d

e

f

g

h

i

3

12 4 5

796Weights along root-to-leaf

path decrease

At u, with child u1,…, uk (in increasing order of weight)

1. Recursively visit subtree at u1

2. Return u3. For i=2 ,…, k

Recursively visit subtree at ui

b r

8

ac e i g d h f

Claim: this traversalproduces the right order

Solving Interval Union-FindSolving Interval Union-Find

Union:x: two operands y: time stamp

Find:x: operand y: time stamp

representative




Four instances of batched ray shooting: O(sort(N))




Four instances of batched ray shooting: O(sort(N))

Handling Redundant UnionsHandling Redundant Unions

• Union tree becomes a general graph

• Compute the minimum spanning tree– O(sort(N)) I/Os (randomized) [Chiang et al. 95]

O(sort(N) loglog B) I/Os (deterministic) [Arge et al. 04]

– Deterministic O(sort(N)) I/Os if graph is planar– Only MST edges are non-redundant

ApplicationsApplications

1.1. Topological PersistenceTopological Persistence

2.2. Contour TreesContour Trees

Application: Application: Topological PersistenceTopological Persistence

• Introduced by Edelsbrunner et al. 2000• Measure importance on a surface

– Feature extraction– Topological de-noising

• Many applications– Surface modeling– Shape analysis– Terrain analysis– Computational Biology

Topological Persistence IllustratedTopological Persistence Illustrated

Formulated as Batched Union-FindFormulated as Batched Union-Find• Represented as a triangulated mesh

• Consider minimum-saddle pairs• When reach

– A minimum or maximum: do nothing– A regular point u: Issue union(u,v) for a lower neighbor v– A saddle u: let v and w be nodes from u’s two connected

pieces in its lower link Issue: find(v), find(w), union(u,v), union(u,w)

lower link

Experiment 1:Experiment 1:Random Union-FindRandom Union-Find

128MBmemory

Experiment 2: Topological Experiment 2: Topological Persistence on Terrain DataPersistence on Terrain Data

Neuse River Basin of North Carolina: ~ 0.5 billion points


Entire data set (0.5b): IM fails and EM takes 10 hours

128MBmemory

Contour TreesContour Trees

SummarySummary

• An I/O-efficient algorithm for the batched union-find problem using O(sort(N)) = O(N/B logM/B(N/B)) I/Os– optimal in the worst case

• A practical algorithm using O(sort(N) log(N/M)) I/Os• Applications to terrain analysis

– Topological persistence : O(sort(N)) I/Os– Contour trees : O(sort(N)) I/Os

• Open Question: – On-line case: Can we get below O(N α(N)) I/Os?

Thank you!Thank you!

Previous ResultsPrevious Results

• Directly maintain contours– O(N log N) time [van Kreveld et al. 97]

– Needs union-split-find for circular lists– Do not extend to higher dimensions

• Two sweeps by maintaining components, then merge– O(N log N) time [Carr et al. 03]

– Extend to arbitrary dimensions

Join Tree and Split TreeJoin Tree and Split Tree

9

8

76

5

4

32

1

Join tree

9

8

76

5

4

32

1

Split tree

Qualified nodes

9

8

76

5

4

3

1

Join tree

9

8

76

5

4

3

1

Split tree

Final Contour TreeFinal Contour Tree

9

8

76

5

4

32

1

Join tree

9

8

76

5

4

32

1

Split tree

9

8

76

5

4

32

1

Contour tree

Hard to BATCH!

Another CharacterizationAnother Characterization

9

8

76

5

4

32

1

Join tree

9

8

76

5

4

32

1

Split tree

9

8

76

5

4

32

1

Contour tree

u

vw

u

vw

u

uw

Let w be the highest node that is a descendant of v in join treeand ancestor of u in split tree, (u, w) is a contour tree edge

Now can BATCH!

Map to RectanglesMap to Rectangles

9

8

76

5

4

32

1

Join tree

9

8

76

5

4

32

1

Split tree

u

vw

u

vw

u

v

w

Can be solved in O(sort(N)) I/Os(practical, too)

Topological PersistenceTopological Persistence

Label Nodes with IntervalsLabel Nodes with Intervals

Using Euler tour (O(sort(N) I/Os)

9

8

76

5

4

32

1

Map to RectanglesMap to Rectangles

9

8

76

5

4

32

1

Join tree

9

8

76

5

4

32

1

Split tree

u

vw

u

vw

u

v

w

Can be solved in O(sort(N)) I/Os(practical, too)

Formulated as Batched Union-FindFormulated as Batched Union-Find• Represented as a triangulated mesh

• Consider minimum-saddle pairs• When reach

– A minimum or maximum: do nothing– A regular poin u: Issue union(u,v) for a lower neighbor v– A saddle u: let v and w be nodes from u’s two

connected pieces in its lower link Issue: find(v), find(w), union(u,v), union(u,w)

lower link

Experiment 1:Experiment 1:Random Union-FindRandom Union-Find



Documents

I/O-Efficient Batched Union-Find and Its Applications to Terrain Analysis