6 research outputs found
Approximate Clustering via Metric Partitioning
In this paper we consider two metric covering/clustering problems -
\textit{Minimum Cost Covering Problem} (MCC) and -clustering. In the MCC
problem, we are given two point sets (clients) and (servers), and a
metric on . We would like to cover the clients by balls centered at
the servers. The objective function to minimize is the sum of the -th
power of the radii of the balls. Here is a parameter of the
problem (but not of a problem instance). MCC is closely related to the
-clustering problem. The main difference between -clustering and MCC is
that in -clustering one needs to select balls to cover the clients.
For any \eps > 0, we describe quasi-polynomial time (1 + \eps)
approximation algorithms for both of the problems. However, in case of
-clustering the algorithm uses (1 + \eps)k balls. Prior to our work, a
and a approximation were achieved by
polynomial-time algorithms for MCC and -clustering, respectively, where is an absolute constant. These two problems are thus interesting examples of
metric covering/clustering problems that admit (1 + \eps)-approximation
(using (1+\eps)k balls in case of -clustering), if one is willing to
settle for quasi-polynomial time. In contrast, for the variant of MCC where
is part of the input, we show under standard assumptions that no
polynomial time algorithm can achieve an approximation factor better than
for .Comment: 19 page
The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex
We consider the problem of finding a subcomplex K\u27 of a simplicial complex K such that K\u27 is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K\u27 with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^(o(sqrt(k)))) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K\u27 can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k)
Compressibility-Aware Quantum Algorithms on Strings
Sublinear time quantum algorithms have been established for many fundamental
problems on strings. This work demonstrates that new, faster quantum algorithms
can be designed when the string is highly compressible. We focus on two popular
and theoretically significant compression algorithms -- the Lempel-Ziv77
algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT),
and obtain the results below.
We first provide a quantum algorithm running in time
for finding the LZ77 factorization of an input string with
factors. Combined with multiple existing results, this yields an
time quantum algorithm for finding the RL-BWT encoding
with BWT runs. Note that . We complement these
results with lower bounds proving that our algorithms are optimal (up to
polylog factors).
Next, we study the problem of compressed indexing, where we provide a
time quantum algorithm for constructing a recently
designed space structure with equivalent capabilities as the
suffix tree. This data structure is then applied to numerous problems to obtain
sublinear time quantum algorithms when the input is highly compressible. For
example, we show that the longest common substring of two strings of total
length can be computed in time, where is the
number of factors in the LZ77 factorization of their concatenation. This beats
the best known time quantum algorithm when is
sufficiently small
Persistent Laplacians: properties, algorithms and implications
We present a thorough study of the theoretical properties and devise
efficient algorithms for the \emph{persistent Laplacian}, an extension of the
standard combinatorial Laplacian to the setting of pairs (or, in more
generality, sequences) of simplicial complexes , which was
recently introduced by Wang, Nguyen, and Wei. In particular, in analogy with
the non-persistent case, we first prove that the nullity of the -th
persistent Laplacian equals the -th persistent Betti number
of the inclusion . We then present an initial algorithm
for finding a matrix representation of , which itself helps
interpret the persistent Laplacian. We exhibit a novel relationship between the
persistent Laplacian and the notion of Schur complement of a matrix which has
several important implications. In the graph case, it both uncovers a link with
the notion of effective resistance and leads to a persistent version of the
Cheeger inequality. This relationship also yields an additional, very simple
algorithm for finding (a matrix representation of) the -th persistent
Laplacian which in turn leads to a novel and fundamentally different algorithm
for computing the -th persistent Betti number for a pair which can
be significantly more efficient than standard algorithms. Finally, we study
persistent Laplacians for simplicial filtrations and present novel stability
results for their eigenvalues. Our work brings methods from spectral graph
theory, circuit theory, and persistent homology together with a topological
view of the combinatorial Laplacian on simplicial complexes
Space Efficient Data Structures and Algorithms in the Word-RAM Model
In modern computation the volume of data-sets has increased dramatically.
Since the majority of these data-sets are stored in internal memory, reducing their storage requirement is an important research topic.
One way of reducing storage is using succinct and compact data structures which maintain the data in compressed form with extra data structures over it in a way that allows efficient access and query of the data.
In this thesis we study space-efficient data structures for various combinatorial objects.
We focus on succinct and compact data structures.
Succinct data structures are data structures whose size is within the information theoretic lower bound plus a lower order term,
whereas compact data structures are data structures whose size is a constant factor from the information theoretic lower bound