On the Complexity of Computing Two Nonlinearity Measures

We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time $2^{O(n)}$ given the truth table of length $2^n$, in fact under the same assumption it is impossible to approximate the multiplicative complexity within a factor of $(2-\epsilon)^{n/2}$. When given a circuit, the problem of determining the multiplicative complexity is in the second level of the polynomial hierarchy. For nonlinearity, we show that it is #P hard to compute given a function represented by a circuit

The Frequent Items Problem in Online Streaming under Various Performance Measures

In this paper, we strengthen the competitive analysis results obtained for a fundamental online streaming problem, the Frequent Items Problem. Additionally, we contribute with a more detailed analysis of this problem, using alternative performance measures, supplementing the insight gained from competitive analysis. The results also contribute to the general study of performance measures for online algorithms. It has long been known that competitive analysis suffers from drawbacks in certain situations, and many alternative measures have been proposed. However, more systematic comparative studies of performance measures have been initiated recently, and we continue this work, using competitive analysis, relative interval analysis, and relative worst order analysis on the Frequent Items Problem.Comment: IMADA-preprint-c

Reflection, Interrupted: Material Mirror Work in the Confessio Amantis

The Confessio Amantis concludes with a revelatory scene in which Venus holds up a mirror to Amans, allowing him to recognize John Gower the poet— a moment that is often read as a mimetic and healing counterpoint to the Confessio’s sickness and self-questioning. My intention in this paper is to very slightly modify certain aspects of this narrative, to consider how the materiality of the mirror can inform its metaphoric deployments in the Confessio. I organize my discussion around two seemingly contrasting moments in the poem in which the self is seen and in different ways recognized through a reflective surface: the “Tale of Narcissus,” and the concluding moment in which Amans looks into the mirror to see, eventually, John Gower. Drawing in particular on the production and dissemination of mirrors in the Middle Ages, as well as basic properties of reflection, I point to certain challenges facing the medieval mirror: the hazy reflective properties of the lead mirror, and the impurities of the precariously made, limitedly accessible glass mirror. I ultimately suggest that, more than a revelation through reflective recognition, the Confessio’s ending would have proven most resonant for its portrayal of seeing through a complicated medium

The Advice Complexity of a Class of Hard Online Problems

The advice complexity of an online problem is a measure of how much knowledge of the future an online algorithm needs in order to achieve a certain competitive ratio. Using advice complexity, we define the first online complexity class, AOC. The class includes independent set, vertex cover, dominating set, and several others as complete problems. AOC-complete problems are hard, since a single wrong answer by the online algorithm can have devastating consequences. For each of these problems, we show that $\log\left(1+(c-1)^{c-1}/c^{c}\right)n=\Theta (n/c)$ bits of advice are necessary and sufficient (up to an additive term of $O(\log n)$) to achieve a competitive ratio of $c$. The results are obtained by introducing a new string guessing problem related to those of Emek et al. (TCS 2011) and B\"ockenhauer et al. (TCS 2014). It turns out that this gives a powerful but easy-to-use method for providing both upper and lower bounds on the advice complexity of an entire class of online problems, the AOC-complete problems. Previous results of Halld\'orsson et al. (TCS 2002) on online independent set, in a related model, imply that the advice complexity of the problem is $\Theta (n/c)$. Our results improve on this by providing an exact formula for the higher-order term. For online disjoint path allocation, B\"ockenhauer et al. (ISAAC 2009) gave a lower bound of $\Omega (n/c)$ and an upper bound of $O((n\log c)/c)$ on the advice complexity. We improve on the upper bound by a factor of $\log c$. For the remaining problems, no bounds on their advice complexity were previously known.Comment: Full paper to appear in Theory of Computing Systems. A preliminary version appeared in STACS 201