arXiv subjects

Jesper Sindahl Nielsen

Publications and source records attributed to Jesper Sindahl Nielsen.

4 recordsLinked to original sources

arXiv2018

Fast Exact k-Means, k-Medians and Bregman Divergence Clustering in 1D

The $k$-Means clustering problem on $n$ points is NP-Hard for any dimension $d\ge 2$, however, for the 1D case there exists exact polynomial time algorithms. Previous literature reported an $O(kn^2)$ time dynamic programming algorithm that uses $O(kn)$ space. It turns out that the problem has been considered under a different name more than twenty years ago. We present all the existing work that had been overlooked and compare the various solutions theoretically. Moreover, we show how to reduce the space usage for some of them, as well as generalize them to data structures that can quickly report an optimal $k$-Means clustering for any $k$. Finally we also generalize all the algorithms to work for the absolute distance and to work for any Bregman Divergence. We complement our theoretical contributions by experiments that compare the practical performance of the various algorithms.

cs.DS↗

arXiv2016

Data Structure Lower Bounds for Document Indexing Problems

We study data structure problems related to document indexing and pattern matching queries and our main contribution is to show that the pointer machine model of computation can be extremely useful in proving high and unconditional lower bounds that cannot be obtained in any other known model of computation with the current techniques. Often our lower bounds match the known space-query time trade-off curve and in fact for all the problems considered, there is a very good and reasonable match between the our lower bounds and the known upper bounds, at least for some choice of input parameters. The problems that we consider are set intersection queries (both the reporting variant and the semi-group counting variant), indexing a set of documents for two-pattern queries, or forbidden- pattern queries, or queries with wild-cards, and indexing an input set of gapped-patterns (or two-patterns) to find those matching a document given at the query time.

cs.DS↗

arXiv2016

Applications of incidence bounds in point covering problems

In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed. First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).

cs.CG↗

arXiv2015

Strictly Implicit Priority Queues: On the Number of Moves and Worst-Case Time

The binary heap of Williams (1964) is a simple priority queue characterized by only storing an array containing the elements and the number of elements $n$ - here denoted a strictly implicit priority queue. We introduce two new strictly implicit priority queues. The first structure supports amortized $O(1)$ time Insert and $O(\log n)$ time ExtractMin operations, where both operations require amortized $O(1)$ element moves. No previous implicit heap with $O(1)$ time Insert supports both operations with $O(1)$ moves. The second structure supports worst-case $O(1)$ time Insert and $O(\log n)$ time (and moves) ExtractMin operations. Previous results were either amortized or needed $O(\log n)$ bits of additional state information between operations.

cs.DS↗