Strong Robustness of Randomized Rumor Spreading Protocols

Randomized rumor spreading is a classical protocol to disseminate information across a network. At SODA 2008, a quasirandom version of this protocol was proposed and competitive bounds for its run-time were proven. This prompts the question: to what extent does the quasirandom protocol inherit the second principal advantage of randomized rumor spreading, namely robustness against transmission failures? In this paper, we present a result precise up to $(1 \pm o(1))$ factors. We limit ourselves to the network in which every two vertices are connected by a direct link. Run-times accurate to their leading constants are unknown for all other non-trivial networks. We show that if each transmission reaches its destination with a probability of $p \in (0,1]$, after (1+\e)(\frac{1}{\log_2(1+p)}\log_2n+\frac{1}{p}\ln n) rounds the quasirandom protocol has informed all $n$ nodes in the network with probability at least 1-n^{-p\e/40}. Note that this is faster than the intuitively natural $1/p$ factor increase over the run-time of approximately $\log_2 n + \ln n$ for the non-corrupted case. We also provide a corresponding lower bound for the classical model. This demonstrates that the quasirandom model is at least as robust as the fully random model despite the greatly reduced degree of independent randomness.Comment: Accepted for publication in "Discrete Applied Mathematics". A short version appeared in the proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009). Minor typos fixed in the second version. Proofs of Lemma 11 and Theorem 12 fixed in the third version. Proof of Lemma 8 fixed in the fourth versio

The stable roommates problem with globally-ranked pairs

We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rankmaximal (weakly stable) matching. This is the first generalization of the algorithm due to Irving et al. [17] to a non-bipartite setting. Also, we prove several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs

The Stable Roommates Problem with Globally-Ranked Pairs

We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of the algorithm due to Irving et al. [18] to a non-bipartite setting. Also, we prove several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs

Anarchy Is Free in Network Creation

<p>The Internet has emerged as perhaps the most important network in modern computing, but rather miraculously, it was created through the individual actions of a multitude of agents rather than by a central planning authority. This motivates the game theoretic study of network formation, and our paper considers one of the most-well studied models, originally proposed by Fabrikant et al. In it, each of <em>n</em> agents corresponds to a vertex, which can create edges to other vertices at a cost of <em>α</em> each, for some parameter <em>α</em>. Every edge can be freely used by every vertex, regardless of who paid the creation cost. To reflect the desire to be close to other vertices, each agent’s cost function is further augmented by the sum total of all (graph theoretic) distances to all other vertices.</p> <p>Previous research proved that for many regimes of the (<em>α</em>,<em>n</em>) parameter space, the total social cost (sum of all agents’ costs) of every Nash equilibrium is bounded by at most a constant multiple of the optimal social cost. In algorithmic game theoretic nomenclature, this approximation ratio is called the price of anarchy. In our paper, we significantly sharpen some of those results, proving that for all constant non-integral <em>α</em> > 2, the price of anarchy is in fact 1 + <em>o</em>(1), i.e., not only is it bounded by a constant, but it tends to 1 as <em>n</em> → ∞. For constant integral <em>α</em> ≥ 2, we show that the price of anarchy is bounded away from 1. We provide quantitative estimates on the rates of convergence for both results.</p

Pathogenesis of Acute Viral Disease Induced in Fish by Carp Interstitial Nephritis and Gill Necrosis Virus

A lethal disease of koi and common carp (species Cyprinus carpio) has afflicted many fish farms worldwide since 1998, causing severe financial losses. Morbidity and mortality are restricted to common carp and koi and appear in spring and autumn, when water temperatures are 18 to 28°C. We have isolated the virus causing the disease from sick fish, propagated it in koi fin cell culture, and shown that virus from a single clone causes lethal disease in carp and koi upon infection. Intraperitoneal virus injection or bathing the fish in virus-containing water kills 85 to 100% of the fish within 7 to 21 days. This virus is similar to the previously reported koi herpesvirus; however, it has characteristics inconsistent with the herpesvirus family, and thus we have called it carp interstitial nephritis and gill necrosis virus. We examined the pathobiology of this disease in carp by using immunohistochemistry and PCR. We found large amounts of the virus in the kidneys of sick fish and smaller amounts in liver and brain. A rapid increase in the viral load in the kidneys was detected by using both immunofluorescence and semiquantitative PCR. Histological analyses of fish at various times after infection revealed signs of interstitial nephritis as early as 2 days postinfection, which increased in severity up to 10 days postinfection. There was severe gill disease evidenced by loss of villi with accompanying inflammation in the gill rakers. Minimal focal inflammation was noted in livers and brains. This report describes the etiology and pathology of a recently described viral agent in fish