Worst-case Analysis of Set Union Algorithms
Journal of the ACM (JACM)
The input/output complexity of sorting and related problems
Communications of the ACM
The cell probe complexity of dynamic data structures
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Contour trees and small seed sets for isosurface traversal
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Construction of contour trees in 3D in O(n log n) steps
Proceedings of the fourteenth annual symposium on Computational geometry
External-memory graph algorithms
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
An improved equivalence algorithm
Communications of the ACM
Hierarchical morse complexes for piecewise linear 2-manifolds
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
External memory algorithms and data structures: dealing with massive data
ACM Computing Surveys (CSUR)
Introduction to Algorithms
Computing contour trees in all dimensions
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
A Generic Approach to Bulk Loading Multidimensional Index Structures
VLDB '97 Proceedings of the 23rd International Conference on Very Large Data Bases
The I/O - Complexity of Ordered Binary - Decision Diagram Manipulation
ISAAC '95 Proceedings of the 6th International Symposium on Algorithms and Computation
Implementing I/O-efficient Data Structures Using TPIE
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
An asymptotically optimal multiversion B-tree
The VLDB Journal — The International Journal on Very Large Data Bases
External memory data structures
Handbook of massive data sets
Topological persistence and simplification
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On external-memory MST, SSSP and multi-way planar graph separation
Journal of Algorithms
Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics)
Topology for Computing (Cambridge Monographs on Applied and Computational Mathematics)
Top-Down Analysis of Path Compression
SIAM Journal on Computing
Extreme Elevation on a 2-Manifold
Discrete & Computational Geometry
TerraStream: from elevation data to watershed hierarchies
Proceedings of the 15th annual ACM international symposium on Advances in geographic information systems
External-memory computational geometry
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
I/O-efficient construction of constrained delaunay triangulations
ESA'05 Proceedings of the 13th annual European conference on Algorithms
A topological hierarchy for functions on triangulated surfaces
IEEE Transactions on Visualization and Computer Graphics
Removing local extrema from imprecise terrains
Computational Geometry: Theory and Applications
Simplifying massive contour maps
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Computing highly occluded paths on a terrain
Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
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In this article we present an I/O-efficient algorithm for the batched (off-line) version of the union-find problem. Given any sequence of N union and find operations, where each union operation joins two distinct sets, our algorithm uses O(SORT(N)) = O(&frac;NB logM/B&frac;NB) I/Os, where M is the memory size and B is the disk block size. This bound is asymptotically optimal in the worst case. If there are union operations that join a set with itself, our algorithm uses O(SORT(N) + MST(N)) I/Os, where MST(N) is the number of I/Os needed to compute the minimum spanning tree of a graph with N edges. We also describe a simple and practical O(SORT(N) log(&frac;NM))-I/O algorithm for this problem, which we have implemented. We are interested in the union-find problem because of its applications in terrain analysis. A terrain can be abstracted as a height function defined over R2, and many problems that deal with such functions require a union-find data structure. With the emergence of modern mapping technologies, huge amount of elevation data is being generated that is too large to fit in memory, thus I/O-efficient algorithms are needed to process this data efficiently. In this article, we study two terrain-analysis problems that benefit from a union-find data structure: (i) computing topological persistence and (ii) constructing the contour tree. We give the first O(SORT(N))-I/O algorithms for these two problems, assuming that the input terrain is represented as a triangular mesh with N vertices.