16,019 research outputs found

    Weighted ancestors in suffix trees

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    The classical, ubiquitous, predecessor problem is to construct a data structure for a set of integers that supports fast predecessor queries. Its generalization to weighted trees, a.k.a. the weighted ancestor problem, has been extensively explored and successfully reduced to the predecessor problem. It is known that any solution for both problems with an input set from a polynomially bounded universe that preprocesses a weighted tree in O(n polylog(n)) space requires \Omega(loglogn) query time. Perhaps the most important and frequent application of the weighted ancestors problem is for suffix trees. It has been a long-standing open question whether the weighted ancestors problem has better bounds for suffix trees. We answer this question positively: we show that a suffix tree built for a text w[1..n] can be preprocessed using O(n) extra space, so that queries can be answered in O(1) time. Thus we improve the running times of several applications. Our improvement is based on a number of data structure tools and a periodicity-based insight into the combinatorial structure of a suffix tree.Comment: 27 pages, LNCS format. A condensed version will appear in ESA 201

    Simple and Efficient Fully-Functional Succinct Trees

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    The fully-functional succinct tree representation of Navarro and Sadakane (ACM Transactions on Algorithms, 2014) supports a large number of operations in constant time using 2n+o(n)2n+o(n) bits. However, the full idea is hard to implement. Only a simplified version with O(logn)O(\log n) operation time has been implemented and shown to be practical and competitive. We describe a new variant of the original idea that is much simpler to implement and has worst-case time O(loglogn)O(\log\log n) for the operations. An implementation based on this version is experimentally shown to be superior to existing implementations

    Managing Unbounded-Length Keys in Comparison-Driven Data Structures with Applications to On-Line Indexing

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    This paper presents a general technique for optimally transforming any dynamic data structure that operates on atomic and indivisible keys by constant-time comparisons, into a data structure that handles unbounded-length keys whose comparison cost is not a constant. Examples of these keys are strings, multi-dimensional points, multiple-precision numbers, multi-key data (e.g.~records), XML paths, URL addresses, etc. The technique is more general than what has been done in previous work as no particular exploitation of the underlying structure of is required. The only requirement is that the insertion of a key must identify its predecessor or its successor. Using the proposed technique, online suffix tree can be constructed in worst case time O(logn)O(\log n) per input symbol (as opposed to amortized O(logn)O(\log n) time per symbol, achieved by previously known algorithms). To our knowledge, our algorithm is the first that achieves O(logn)O(\log n) worst case time per input symbol. Searching for a pattern of length mm in the resulting suffix tree takes O(min(mlogΣ,m+logn)+tocc)O(\min(m\log |\Sigma|, m + \log n) + tocc) time, where tocctocc is the number of occurrences of the pattern. The paper also describes more applications and show how to obtain alternative methods for dealing with suffix sorting, dynamic lowest common ancestors and order maintenance

    Homeomorphic Embedding for Online Termination of Symbolic Methods

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    Well-quasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of techniques for program analysis, specialisation, transformation, and verification. In this paper we survey and discuss this use of homeomorphic embedding and clarify the advantages of such an approach over one using well-founded orders. We also discuss various extensions of the homeomorphic embedding relation. We conclude with a study of homeomorphic embedding in the context of metaprogramming, presenting some new (positive and negative) results and open problems
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