1,912 research outputs found

    Finding approximate repetitions under Hamming distance

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    The problem of computing tandem repetitions with KK possible mismatches is studied. Two main definitions are considered, and for both of them an O(nKlogK+S)O(nK\log K+S) algorithm is proposed (SS the size of the output). This improves, in particular, the bound obtained in \citeLS93. Finally, other possible definions are briefly analyzed.

    The quantum complexity of approximating the frequency moments

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    The kk'th frequency moment of a sequence of integers is defined as Fk=jnjkF_k = \sum_j n_j^k, where njn_j is the number of times that jj occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate FkF_k up to relative error ϵ\epsilon. We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements of the input one at a time, and seek to minimise the amount of storage space, or passes over the data, used to approximate FkF_k. We describe quantum algorithms for F0F_0, F2F_2 and FF_\infty in this model which substantially outperform the best possible classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio

    Parameter-free Locality Sensitive Hashing for Spherical Range Reporting

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    We present a data structure for *spherical range reporting* on a point set SS, i.e., reporting all points in SS that lie within radius rr of a given query point qq. Our solution builds upon the Locality-Sensitive Hashing (LSH) framework of Indyk and Motwani, which represents the asymptotically best solutions to near neighbor problems in high dimensions. While traditional LSH data structures have several parameters whose optimal values depend on the distance distribution from qq to the points of SS, our data structure is parameter-free, except for the space usage, which is configurable by the user. Nevertheless, its expected query time basically matches that of an LSH data structure whose parameters have been *optimally chosen for the data and query* in question under the given space constraints. In particular, our data structure provides a smooth trade-off between hard queries (typically addressed by standard LSH) and easy queries such as those where the number of points to report is a constant fraction of SS, or where almost all points in SS are far away from the query point. In contrast, known data structures fix LSH parameters based on certain parameters of the input alone. The algorithm has expected query time bounded by O(t(n/t)ρ)O(t (n/t)^\rho), where tt is the number of points to report and ρ(0,1)\rho\in (0,1) depends on the data distribution and the strength of the LSH family used. We further present a parameter-free way of using multi-probing, for LSH families that support it, and show that for many such families this approach allows us to get expected query time close to O(nρ+t)O(n^\rho+t), which is the best we can hope to achieve using LSH. The previously best running time in high dimensions was Ω(tnρ)\Omega(t n^\rho). For many data distributions where the intrinsic dimensionality of the point set close to qq is low, we can give improved upper bounds on the expected query time.Comment: 21 pages, 5 figures, due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract appearing here is slightly shorter than that in the PDF fil
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