Fractional Cascading: A Data Structuring Technique with Geometric Applications

We examine the problem of searching for a given item in several sets. Let U be a linearly ordered universe and C be a finite collection of subsets of U; given an arbitrary query (x, H) with x e U and H\(\subseteq\)C, search for x in each set of H. This operation, termed iterative search, is the bottleneck of a large number of retrieval problems. To perform it efficiently, we introduce a new technique, called fractional cascading. We demonstrate its versatility by applying it to a number of different geometric problems. Among the major applications of fractional cascading, we find improved methods for answering range queries, searching in the past, computing functions on d-ranges, intersection searching, solving general extensions of classical retrieval problems, answering visibility questions in the context of ray-tracing, etc.