Lower bounds for on-line graph colorings

We propose two strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use $2\log_2 n - 10$ colors, where $n$ is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs graphs that contain neither $C_3$ nor $C_5$ as a subgraph and forces $\Omega(\frac{n}{\log n}^\frac{1}{3})$ colors. The best known on-line coloring algorithm for these graphs uses $O(n^{\frac{1}{2}})$ colors

Characteristics of Glial Reaction in the Perinatal Rat Cortex: Effect of Lesion Size in the ‘Critical Period’

In this study we investigate the capability of lesions, performed between embryonic day E18 and postnatal day P6, to provoke glial reaction. Two different lesion types were applied: ‘severe’ lesion (tissue defect) and ’light’ lesion (stab wound). The glial reaction was detected with immunostain[ng against glial fibrillary acidic protein. When performed as early as P0, severe lesions could result in reactive gliosis, which persisted even after a month. The glial reaction was detected at P6/P7 and became strong by P8, regardless of the age when the animals were lesioned between P0 and P5. Namely, a strict limit could be estimated for the age when reactive glia were already found rather than for the age when glial reaction-provoking lesions could occur. After prenatal lesions, no glial reaction developed, but the usual glia limitans covered the deformed brain, surface. Light lesions provoked glial reactions when performed at P6. In conclusion, three scenarios were found, depending on the age of the animal at injury: (i) healing without glial reaction, regardless of the remaining deformation; (ii) depending on the size of the lesion, either healing without residuum or with remaining tissue defect plus reactive gliosis; and (iii) healing always with reactive gliosis. The age limits between them were at P0 and P5. The glial reactivity seemingly appears after the end of the neuronal migration and just precedes the massive transformation of the radial glia into astrocytes. Estimating the position of the appearance of glial reactivity among the events of cortical maturation can help to adapt the experimental results to humans

On graphs with a large chromatic number containing no small odd cycles

In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles

The Set of Continuous Functions with the Everywhere Convergent Fourier Series

This paper deals with the descriptive set theoretic properties of the class EC of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in C(T). A natural coanalytic rank function on EC is studied that assigns to each ƒ Є EC a countable ordinal number, which measures the "complexity" of the convergence of the Fourier series of ƒ. It is shown that there exist functions in EC (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on EC with ω_1 distinct levels

Finding the Median (Obliviously) with Bounded Space

We prove that any oblivious algorithm using space $S$ to find the median of a list of $n$ integers from $\{1,...,2n\}$ requires time $\Omega(n \log\log_S n)$. This bound also applies to the problem of determining whether the median is odd or even. It is nearly optimal since Chan, following Munro and Raman, has shown that there is a (randomized) selection algorithm using only $s$ registers, each of which can store an input value or $O(\log n)$-bit counter, that makes only $O(\log\log_s n)$ passes over the input. The bound also implies a size lower bound for read-once branching programs computing the low order bit of the median and implies the analog of $P \ne NP \cap coNP$ for length $o(n \log\log n)$ oblivious branching programs

Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search

By applying Grover's quantum search algorithm to the lattice algorithms of Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and Stehl\'{e}, we obtain improved asymptotic quantum results for solving the shortest vector problem. With quantum computers we can provably find a shortest vector in time $2^{1.799n + o(n)}$, improving upon the classical time complexity of $2^{2.465n + o(n)}$ of Pujol and Stehl\'{e} and the $2^{2n + o(n)}$ of Micciancio and Voulgaris, while heuristically we expect to find a shortest vector in time $2^{0.312n + o(n)}$, improving upon the classical time complexity of $2^{0.384n + o(n)}$ of Wang et al. These quantum complexities will be an important guide for the selection of parameters for post-quantum cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page

Rigidity and Non-recurrence along Sequences

Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences. A variety of classes of sequences with various properties are considered showing the complicated and rich structure of rigid and non-recurrent sequences

Efficient Distributed Quantum Computing

We provide algorithms for efficiently addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with low overhead by the more realistic model of a distributed quantum computer. As a result, the circuit model can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit. In addition, we apply our results to existing memory intensive quantum algorithms. We present a parallel quantum search algorithm and improve the time-space trade-off for the Element Distinctness and Collision problems.Comment: Some material rearranged and references adde

Perfectly Secure Oblivious RAM without Random Oracles

We present an algorithm for implementing a secure oblivious RAM where the access pattern is perfectly hidden in the information theoretic sense, without assuming that the CPU has access to a random oracle. In addition we prove a lover bound on the amount of randomness needed for information theoretically secure oblivious RAM.

Computational Indistinguishability between Quantum States and Its Cryptographic Application

We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is "secure" against any polynomial-time quantum adversary. Our problem, QSCDff, is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonly-used distinction problem between two probability distributions in computational cryptography. As our major contribution, we show that QSCDff has three properties of cryptographic interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff coincides with its worst-case hardness; and (iii) QSCDff is computationally at least as hard as the graph automorphism problem in the worst case. These cryptographic properties enable us to construct a quantum public-key cryptosystem, which is likely to withstand any chosen plaintext attack of a polynomial-time quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail proofs and follow-up of recent wor