Data Structures for Sets:Conclusions

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Conclusions

We have presented a number of algorithms and data structures for supporting (largely) basic set-theoretic operations. Aside from the union-find problem, which has been extensively studied, relatively little research has been done into these problems. For instance, even the most basic problem, that of finding a general-purpose data structure that supports basic set operations, is not yet satisfactorily solved. The problem becomes more acute if one is concerned about the space usage of the data structures—for example, it is not known whether one can solve set equality testing efficiently in linear space.

Due to the proliferation of unstructured data, set operations are increasingly important. For instance, many search engines return a set of documents that match a given boolean keyword query by means of set operations on the sets of documents that contain each of the keywords in the query. The characteristics of this kind of application also suggest directions for research. For example, given the large data-sets that could be involved, it is a little surprising that work on external-memory algorithms for these problems is somewhat limited. Another issue is that these sets usually have patterns (e.g. the number of sets that contain a given keyword may satisfy a power law; certain sets of keywords may be more likely to be queried together etc.), which should be exploited by efficient algorithms.

With the latter motivation in mind, Demaine et al. [10] have considered the adaptive complexity of these problems. They assume that they are given a collection of sorted lists that comprise the sets, and need to compute unions, intersections and differences of these sets. If one is only interested in worst-case complexity (across all instances) then this problem is uninteresting (it essentially boils down to merging). However, some instances can be harder than others: for instance, computing the intersection of two sets when all elements in one set are smaller than the other is much easier than for sets that interleave substantially. Building on this idea, they develop a notion of the complexity of a given instance of a problem and develop algorithms that, for each particular instance, are efficient with respect to the difficulty of that instance.

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