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

Abstract

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 log M/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 R 2 , 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.