325 research outputs found
Error-Correction in Flash Memories via Codes in the Ulam Metric
We consider rank modulation codes for flash memories that allow for handling
arbitrary charge-drop errors. Unlike classical rank modulation codes used for
correcting errors that manifest themselves as swaps of two adjacently ranked
elements, the proposed \emph{translocation rank codes} account for more general
forms of errors that arise in storage systems. Translocations represent a
natural extension of the notion of adjacent transpositions and as such may be
analyzed using related concepts in combinatorics and rank modulation coding.
Our results include derivation of the asymptotic capacity of translocation rank
codes, construction techniques for asymptotically good codes, as well as simple
decoding methods for one class of constructed codes. As part of our exposition,
we also highlight the close connections between the new code family and
permutations with short common subsequences, deletion and insertion
error-correcting codes for permutations, and permutation codes in the Hamming
distance
Combinatorial Methods in Coding Theory
This thesis is devoted to a range of questions in applied mathematics and signal processing motivated by applications in error correction, compressed sensing, and writing on non-volatile memories. The underlying thread of our results is the use of diverse combinatorial methods originating in coding theory and computer science.
The thesis addresses three groups of problems. The first of them is
aimed at the construction and analysis of codes for error correction. Here we examine properties of codes that are constructed using random and structured graphs and hypergraphs, with the main purpose of devising new decoding algorithms as well as estimating the distribution of Hamming weights in the resulting codes. Some of the results obtained give the best known estimates of the number of correctable errors for codes whose decoding relies on local operations on the graph.
In the second part we address the question of constructing sampling
operators for the compressed sensing problem. This topic has been
the subject of a large body of works in the literature. We propose
general constructions of sampling matrices based on ideas from coding theory that act as near-isometric maps on almost all sparse signal. This matrices can be used for dimensionality reduction and compressed sensing.
In the third part we study the problem of reliable storage of information in non-volatile memories such as flash drives. This problem gives rise to a writing scheme that relies on relative magnitudes of neighboring cells, known as rank modulation. We establish the exact asymptotic behavior of the size of codes for rank modulation and suggest a number of new general constructions of such codes based on properties of finite fields as well as combinatorial considerations
LNCS
We construct a perfectly binding string commitment scheme whose security is based on the learning parity with noise (LPN) assumption, or equivalently, the hardness of decoding random linear codes. Our scheme not only allows for a simple and efficient zero-knowledge proof of knowledge for committed values (essentially a Σ-protocol), but also for such proofs showing any kind of relation amongst committed values, i.e. proving that messages m_0,...,m_u, are such that m_0=C(m_1,...,m_u) for any circuit C.
To get soundness which is exponentially small in a security parameter t, and when the zero-knowledge property relies on the LPN problem with secrets of length l, our 3 round protocol has communication complexity O(t|C|l log(l)) and computational complexity of O(t|C|l) bit operations. The hidden constants are small, and the computation consists mostly of computing inner products of bit-vectors
Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation
We propose GM-QAOA, a variation of the Quantum Alternating Operator Ansatz
(QAOA) that uses Grover-like selective phase shift mixing operators. GM-QAOA
works on any NP optimization problem for which it is possible to efficiently
prepare an equal superposition of all feasible solutions; it is designed to
perform particularly well for constraint optimization problems, where not all
possible variable assignments are feasible solutions. GM-QAOA has the following
features: (i) It is not susceptible to Hamiltonian Simulation error (such as
Trotterization errors) as its operators can be implemented exactly using
standard gate sets and (ii) Solutions with the same objective value are always
sampled with the same amplitude.
We illustrate the potential of GM-QAOA on several optimization problem
classes: for permutation-based optimization problems such as the Traveling
Salesperson Problem, we present an efficient algorithm to prepare a
superposition of all possible permutations of numbers, defined on
qubits; for the hard constraint -Vertex-Cover problem, and for an
application to Discrete Portfolio Rebalancing, we show that GM-QAOA outperforms
existing QAOA approaches
Deep Hashing for Image Similarity Search
Hashing for similarity search is one of the most widely used methods to solve the approximate nearest neighbor search problem. In this method, one first maps data items from a real valued high-dimensional space to a suitable low dimensional binary code space and then performs the approximate nearest neighbor search in this code space instead. This is beneficial because the search in the code space can be solved more efficiently in terms of runtime complexity and storage consumption. Obviously, for this method to succeed, it is necessary that similar data items be mapped to binary code words that have small Hamming distance. For real-world data such as images, one usually proceeds as follows. For each data item, a pre-processing algorithm removes noise and insignificant information and extracts important discriminating information to generate a feature vector that captures the important semantic content. Next, a vector hash function maps this real valued feature vector to a binary code word. It is also possible to use the raw feature vectors afterwards to further process the search result candidates produced by binary hash codes. In this dissertation we focus on the following. First, developing a learning based counterpart for the MinHash hashing algorithm. Second, presenting a new unsupervised hashing method UmapHash to map the neighborhood relations of data items from the feature vector space to the binary hash code space. Finally, an application of the aforementioned hashing methods for rapid face image recognition
On the Quantum Complexity of Closest Pair and Related Problems
The closest pair problem is a fundamental problem of computational geometry:
given a set of points in a -dimensional space, find a pair with the
smallest distance. A classical algorithm taught in introductory courses solves
this problem in time in constant dimensions (i.e., when ).
This paper asks and answers the question of the problem's quantum time
complexity. Specifically, we give an algorithm in constant
dimensions, which is optimal up to a polylogarithmic factor by the lower bound
on the quantum query complexity of element distinctness. The key to our
algorithm is an efficient history-independent data structure that supports
quantum interference.
In dimensions, no known quantum algorithms perform
better than brute force search, with a quadratic speedup provided by Grover's
algorithm. To give evidence that the quadratic speedup is nearly optimal, we
initiate the study of quantum fine-grained complexity and introduce the Quantum
Strong Exponential Time Hypothesis (QSETH), which is based on the assumption
that Grover's algorithm is optimal for CNF-SAT when the clause width is large.
We show that the na\"{i}ve Grover approach to closest pair in higher dimensions
is optimal up to an factor unless QSETH is false. We also study the
bichromatic closest pair problem and the orthogonal vectors problem, with
broadly similar results.Comment: 46 pages, 3 figures, presentation improve
Reconstruction Codes for DNA Sequences with Uniform Tandem-Duplication Errors
DNA as a data storage medium has several advantages, including far greater
data density compared to electronic media. We propose that schemes for data
storage in the DNA of living organisms may benefit from studying the
reconstruction problem, which is applicable whenever multiple reads of noisy
data are available. This strategy is uniquely suited to the medium, which
inherently replicates stored data in multiple distinct ways, caused by
mutations. We consider noise introduced solely by uniform tandem-duplication,
and utilize the relation to constant-weight integer codes in the Manhattan
metric. By bounding the intersection of the cross-polytope with hyperplanes, we
prove the existence of reconstruction codes with greater capacity than known
error-correcting codes, which we can determine analytically for any set of
parameters.Comment: 11 pages, 2 figures, Latex; version accepted for publicatio
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