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, Ke YiPankaj K. Agarwal, Lars Arge, Ke Yi Duke UniversityDuke University

University 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

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representatives

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Union(d, h) :

link-by-rank

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Find(n) :

path compression

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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

Sources of “Non-Locality”Sources of “Non-Locality”

• Two operands in a union

• Nodes on a leaf-to-root path

• Operands in consecutive operations– Cannot remove for the on-line case

Need to eliminate all of them in order to get less than one I/O per operation!

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– Same as sorting– optimal in the worst case

• A practical algorithm using O(sort(N) log(N/M)) I/Os– Implemented

• Applications to terrain analysis– Topological persistence : O(sort(N)) I/Os

• Implemented

– 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 TreeUnion Tree

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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)

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Equivalent union trees

Transforming the Union TreeTransforming the Union Treer

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Weights along root-to-leafpath decrease

Formulating as a Batched ProblemFormulating as a Batched Problem

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

Cast in a Geometry SettingCast in a Geometry Settingr

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Euler Tour

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

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x: weighty: positions in the tour

Cast in a Geometry SettingCast in a Geometry Settingr

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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

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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

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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

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

Handling Redundant UnionsHandling Redundant Unions

• Union tree becomes a 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

A Practical AlgorithmA Practical Algorithm

• Previous algorithm too complicated– 2 Euler tours– 4 instances of batched ray shooting– MST

• A simple and practical algorithm– Divide-and-conquer– O(sort(N) log(N/M)) I/Os– Implemented

ApplicationsApplications

1.1. Topological PersistenceTopological Persistence

2.2. Contour TreesContour Trees

Topological PersistenceTopological Persistence

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

Contour TreesContour Trees

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

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Join tree

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Split tree

Qualified nodes

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Join tree

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Final Contour TreeFinal Contour Tree

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Join tree

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Contour tree

Hard to BATCH!

Another CharacterizationAnother Characterization

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Split tree

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Contour tree

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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!

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

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

Neuse River Basin of NC

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

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!