A Lower Bound for Sampling Disjoint Sets

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}

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

Lower Bounds for Oblivious Near-Neighbor Search

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search ($\mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for $\mathsf{ANN}$. This is the first super-logarithmic $\mathit{unconditional}$ lower bound for $\mathsf{ANN}$ against general (non black-box) data structures. We also show that any oblivious $\mathit{static}$ data structure for decomposable search problems (like $\mathsf{ANN}$) can be obliviously dynamized with $O(\log n)$ overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page

Some Communication Complexity Results and their Applications

Communication Complexity represents one of the premier techniques for proving lower bounds in theoretical computer science. Lower bounds on communication problems can be leveraged to prove lower bounds in several different areas. In this work, we study three different communication complexity problems. The lower bounds for these problems have applications in circuit complexity, wireless sensor networks, and streaming algorithms. First, we study the multiparty pointer jumping problem. We present the first nontrivial upper bound for this problem. We also provide a suite of strong lower bounds under several restricted classes of protocols. Next, we initiate the study of several non-monotone functions in the distributed functional monitoring setting and provide several lower bounds. In particular, we give a generic adversarial technique and show that when deletions are allowed, no nontrivial protocol is possible. Finally, we study the Gap-Hamming-Distance problem and give tight lower bounds for protocols that use a constant number of messages. As a result, we take a well-known lower bound for one-pass streaming algorithms for a host of problems and extend it so it applies to streaming algorithms that use a constant number of passes

Freely Scalable Quantum Technologies using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links

Exquisite quantum control has now been achieved in small ion traps, in nitrogen-vacancy centres and in superconducting qubit clusters. We can regard such a system as a universal cell with diverse technological uses from communication to large-scale computing, provided that the cell is able to network with others and overcome any noise in the interlinks. Here we show that loss-tolerant entanglement purification makes quantum computing feasible with the noisy and lossy links that are realistic today: With a modestly complex cell design, and using a surface code protocol with a network noise threshold of 13.3%, we find that interlinks which attempt entanglement at a rate of 2MHz but suffer 98% photon loss can result in kilohertz computer clock speeds (i.e. rate of high fidelity stabilizer measurements). Improved links would dramatically increase the clock speed. Our simulations employed local gates of a fidelity already achieved in ion trap devices.Comment: corrected typos, additional references, additional figur