Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers

Competitive Boolean Function Evaluation: Beyond Monotonicity, and the Symmetric Case

We study the extremal competitive ratio of Boolean function evaluation. We provide the first non-trivial lower and upper bounds for classes of Boolean functions which are not included in the class of monotone Boolean functions. For the particular case of symmetric functions our bounds are matching and we exactly characterize the best possible competitiveness achievable by a deterministic algorithm. Our upper bound is obtained by a simple polynomial time algorithm.Comment: 15 pages, 1 figure, to appear in Discrete Applied Mathematic

Lower Bounds on Quantum Query Complexity

Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.Comment: survey, 23 page

On the Computational Power of Radio Channels

Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature

Lower bounds for uniform read-once threshold formulae in the randomized decision tree model

We investigate the randomized decision tree complexity of a specific class of read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, every internal node of which is labeled by a threshold function $T_k^n$ (with output 1 only when at least $k$ out of $n$ input bits are 1) and each leaf by a distinct variable. Such a tree defines a Boolean function in a natural way. We focus on the randomized decision tree complexity of such functions, when the underlying tree is a uniform tree with all its internal nodes labeled by the same threshold function. We prove lower bounds of the form $c(k,n)^d$, where $d$ is the depth of the tree. We also treat trees with alternating levels of AND and OR gates separately and show asymptotically optimal bounds, extending the known bounds for the binary case