# I/O-efficient batched union-find and its applications to terrain analysis

Journal Article

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(N/B logM/B N/B) 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(N/M ))-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. © 2010 ACM.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Arge, L; Yi, K

### Published Date

- November 1, 2010

### Published In

### Volume / Issue

- 7 / 1

### Electronic International Standard Serial Number (EISSN)

- 1549-6333

### International Standard Serial Number (ISSN)

- 1549-6325

### Digital Object Identifier (DOI)

- 10.1145/1868237.1868249

### Citation Source

- Scopus