Balanced Allocations: A Simple Proof for the Heavily Loaded Case

We provide a relatively simple proof that the expected gap between the maximum load and the average load in the two choice process is bounded by $(1+o(1))\log \log n$, irrespective of the number of balls thrown. The theorem was first proven by Berenbrink et al. Their proof uses heavy machinery from Markov-Chain theory and some of the calculations are done using computers. In this manuscript we provide a significantly simpler proof that is not aided by computers and is self contained. The simplification comes at a cost of weaker bounds on the low order terms and a weaker tail bound for the probability of deviating from the expectation

Selfish traffic allocation for server farms

We study the price of selfish routing in noncooperative networks like the Internet. In particular, we investigate the price of selfish routing using the price of anarchy (a.k.a. the coordination ratio) and other (e.g., bicriteria) measures in the recently introduced game theoretic parallel links network model of Koutsoupias and Papadimitriou. We generalize this model toward general, monotone families of cost functions and cost functions from queueing theory. A summary of our main results for general, monotone cost functions is as follows: 1. We give an exact characterization of all cost functions having a bounded/unbounded price of anarchy. For example, the price of anarchy for cost functions describing the expected delay in queueing systems is unbounded. 2. We show that an unbounded price of anarchy implies an extremely high performance degradation under bicriteria measures. In fact, the price of selfish routing can be as high as a bandwidth degradation by a factor that is linear in the network size. 3. We separate the game theoretic (integral) allocation model from the (fractional) flow model by demonstrating that even a very small or negligible amount of integrality can lead to a dramatic performance degradation. 4. We unify recent results on selfish routing under different objectives by showing that an unbounded price of anarchy under the min-max objective implies an unbounded price of anarchy under the average cost objective and vice versa. Our special focus lies on cost functions describing the behavior of Web servers that can open only a limited number of Transmission Control Protocol (TCP) connections. In particular, we compare the performance of queueing systems that serve all incoming requests with servers that reject requests in case of overload. Our analysis indicates that all queueing systems without rejection cannot give any reasonable guarantee on the expected delay of requests under selfish routing even when the injected load is far away from the capacity of the system. In contrast, Web server farms that are allowed to reject requests can guarantee a high quality of service for every individual request stream even under relatively high injection rates

Einmal Siedlce und wieder zurück – der 18. internationale Kongress der Arachnologen in Polen

In diesem Jahr fand vom 11. bis 17. Juli der 18. Internationale Kongress der Arachnologen in Siedlce, welches ca. 90 km östlich von Warschau liegt, statt. Der Kongress wurde von Marek Zabka, Barbara Patoleta und unzähligen weiteren fleißigen Helferlein organisiert. Schon die Liste derjenigen die einen Hauptvortrag halten würden war vielversprechend. Dabei waren Friedrich Barth (Österreich), William Eberhard (Costa Rica), Mark Elgar (Australien), Gonzalo Giribet (USA), Rudy Jocqué (Belgien), Wayne Maddison (USA), Robert Raven (Australien), Paul Selden (USA), Gabriele Uhl (Deutschland) und Samuel Zschokke (Schweiz)

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on real world Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every $p\in\mathbb{N}$, a family of $L_p$ instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which $n$ points are placed uniformly at random in the unit square $[0,1]^2$. We consider a more advanced model in which the points can be placed independently according to general distributions on $[0,1]^d$, for an arbitrary $d\ge 2$. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number $n$ of points and the maximal density $\phi$ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to $\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by $\tilde{O}(n^{4-1/d}\cdot\phi)$. In addition, we prove an upper bound of $O(\sqrt[d]{\phi})$ on the expected approximation factor with respect to all $L_p$ metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with $\phi=1$ and a smoothed analysis with Gaussian perturbations of standard deviation $\sigma$ with $\phi\sim1/\sigma^d$.Comment: An extended abstract of this work has appeared in the Proc. of the 18th ACM-SIAM Symposium on Discrete Algorithms. The results of this extended abstract have been split into two articles (Algorithmica 2014) and (ACM Transactions on Algorithms 2016). This report is an updated version of the first journal article, in which two minor errors in the proofs of Lemma 8 and Lemma 9 have been correcte

Сучасні тенденції в методиці викладання англійської мови

In the interference scheduling problem, one is given a set of n communication requests described by sourcedestination pairs of nodes from a metric space. The nodes correspond to devices in a wireless network. Each pair must be assigned a power level and a color such that the pairs in each color class can communicate simultaneously at the specified power levels. The feasibility of simultaneous communication within a color class is defined in terms of the Signal to Interference plus Noise Ratio (SINR) that compares the strength of a signal at a receiver to the sum of the strengths of other signals. The objective is to minimize the number of colors as this corresponds to the time needed to schedule all requests. We introduce an instance-based measure of interference, denoted by I, that enables us to improve on previous results for the interference scheduling problem. We prove upper and lower bounds in terms of I on the number of steps needed for scheduling a set of requests. For general power assignments, we prove a lower bound of Ω(I/(log ∆ log n)) steps, where ∆ denotes the aspect ratio of the metric. When restricting to the two-dimensional Euclidean space (as previous work) the bound improves to Ω(I / log ∆). Alternatively, when restricting to linear power assignments, the lower bound improves even to Ω(I). The lower bounds are complemented by an efficient algorithm computing a schedule for linear power assignments using onl

Balanced Allocation on Graphs: A Random Walk Approach

In this paper we propose algorithms for allocating $n$ sequential balls into $n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where $d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round $t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and performs a non-backtracking random walk of length $l$ from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that $G$ has a sufficiently large girth and $d=\omega(\log n)$. Then we establish an upper bound for the maximum number of balls at any bin after allocating $n$ balls by the algorithm, called {\it maximum load}, in terms of $l$ with high probability. We also show that the upper bound is at most an $O(\log\log n)$ factor above the lower bound that is proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and $G$ has girth at least $\omega(l)$, then the maximum load attained by the algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

Российская этика бизнеса в контексте национальных особенностей

Анализируются основные элементы бизнес этики, позволяющие утверждать, что этические воззрения не могут носить универсальный характер, поскольку в основе лежат принципиальные культурные различия. Установлены элементы, указывающие на отличия в бизнес этике России, США и европейских государств. Сделаны выводы о культурных предпосылках, которые позволяют увидеть причину, по которой бизнесмены России и США будут по-разному реагировать на аналогичную этическую дилемму

The rhodopsin-retinochrome system for retinal re-isomerization predates the origin of cephalopod eyes

Background: The process of photoreception in most animals depends on the light induced isomerization of the chromophore retinal, bound to rhodopsin. To re-use retinal, the all-trans-retinal form needs to be re-isomerized to 11-cis-retinal, which can be achieved in different ways. In vertebrates, this mostly includes a stepwise enzymatic process called the visual cycle. The best studied re-isomerization system in protostomes is the rhodopsin-retinochrome system of cephalopods, which consists of rhodopsin, the photoisomerase retinochrome and the protein RALBP functioning as shuttle for retinal. In this study we investigate the expression of the rhodopsin-retinochrome system and functional components of the vertebrate visual cycle in a polyplacophoran mollusk, Leptochiton asellus, and examine the phylogenetic distribution of the individual components in other protostome animals. Results: Tree-based orthology assignments revealed that orthologs of the cephalopod retinochrome and RALBP are present in mollusks outside of cephalopods. By mining our dataset for vertebrate visual cycle components, we also found orthologs of the retinoid binding protein RLBP1, in polyplacophoran mollusks, cephalopods and a phoronid. In situ hybridization and antibody staining revealed that L. asellus retinochrome is co-expressed in the larval chiton photoreceptor cells (PRCs) with the visual rhodopsin, RALBP and RLBP1. In addition, multiple retinal dehydrogenases are expressed in the PRCs, which might also contribute to the rhodopsin-retinochrome system. Conclusions: We conclude that the rhodopsin-retinochrome system is a common feature of mollusk PRCs and predates the origin of cephalopod eyes. Our results show that this system has to be extended by adding further components, which surprisingly, are shared with vertebrates.publishedVersio