I/O-efficient batched union-find and its applications to terrain analysis
Despite extensive study over the last four decades and numerous applications, no I/O-efficient algorithm is known for the union-find problem. In this paper 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 modem 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 paper, we study two terrain analysis problems that benefit from a unionfind 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. Finally, we report some preliminary experimental results, showing that our algorithms give order-of-magnitude improvement over previous methods on large data sets that do not fit in memory. Copyright 2006 ACM.
Agarwal, PK; Arge, L; Yi, K
Proceedings of the Annual Symposium on Computational Geometry
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