Finding approximate repetitions under Hamming distance

The problem of computing tandem repetitions with $K$ possible mismatches is studied. Two main definitions are considered, and for both of them an $O(nK\log K+S)$ algorithm is proposed ($S$ 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

The $k$'th frequency moment of a sequence of integers is defined as $F_k = \sum_j n_j^k$, where $n_j$ is the number of times that $j$ 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 $F_k$ up to relative error $\epsilon$. 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 $F_k$. We describe quantum algorithms for $F_0$, $F_2$ and $F_\infty$ 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

We present a data structure for *spherical range reporting* on a point set $S$, i.e., reporting all points in $S$ that lie within radius $r$ of a given query point $q$. 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 $q$ to the points of $S$, 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 $S$, or where almost all points in $S$ 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 $O(t (n/t)^\rho)$, where $t$ is the number of points to report and $\rho\in (0,1)$ 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 $O(n^\rho+t)$, which is the best we can hope to achieve using LSH. The previously best running time in high dimensions was $\Omega(t n^\rho)$. For many data distributions where the intrinsic dimensionality of the point set close to $q$ 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