On Parity Decision Trees for Fourier-Sparse Boolean Functions

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Our contributions are as follows. Let f : ??? ? {-1, 1} be a Boolean function with Fourier support ? and Fourier sparsity k. - We prove via the probabilistic method that there exists a parity decision tree of depth O(?k) that computes f. This matches the best known upper bound on the parity decision tree complexity of Boolean functions (Tsang, Wong, Xie, and Zhang, FOCS 2013). Moreover, while previous constructions (Tsang et al., FOCS 2013, Shpilka, Tal, and Volk, Comput. Complex. 2017) build the trees by carefully choosing the parities to be queried in each step, our proof shows that a naive sampling of the parities suffices. - We generalize the above result by showing that if the Fourier spectra of Boolean functions satisfy a natural "folding property", then the above proof can be adapted to establish existence of a tree of complexity polynomially smaller than O(? k). More concretely, the folding property we consider is that for most distinct ?, ? in ?, there are at least a polynomial (in k) number of pairs (?, ?) of parities in ? such that ?+? = ?+?. We make a conjecture in this regard which, if true, implies that the communication complexity of an XOR function is bounded above by the fourth root of the rank of its communication matrix, improving upon the previously known upper bound of square root of rank (Tsang et al., FOCS 2013, Lovett, J. ACM. 2016). - Motivated by the above, we present some structural results about the Fourier spectra of Boolean functions. It can be shown by elementary techniques that for any Boolean function f and all (?, ?) in binom(?,2), there exists another pair (?, ?) in binom(?,2) such that ? + ? = ? + ?. One can view this as a "trivial" folding property that all Boolean functions satisfy. Prior to our work, it was conceivable that for all (?, ?) ? binom(?,2), there exists exactly one other pair (?, ?) ? binom(?,2) with ? + ? = ? + ?. We show, among other results, that there must exist several ? ? ??? such that there are at least three pairs of parities (??, ??) ? binom(?,2) with ??+?? = ?. This, in particular, rules out the possibility stated earlier

Distributed Hierarchical SVD in the Hierarchical Tucker Format

We consider tensors in the Hierarchical Tucker format and suppose the tensor data to be distributed among several compute nodes. We assume the compute nodes to be in a one-to-one correspondence with the nodes of the Hierarchical Tucker format such that connected nodes can communicate with each other. An appropriate tree structure in the Hierarchical Tucker format then allows for the parallelization of basic arithmetic operations between tensors with a parallel runtime which grows like $\log(d)$, where $d$ is the tensor dimension. We introduce parallel algorithms for several tensor operations, some of which can be applied to solve linear equations $\mathcal{A}X=B$ directly in the Hierarchical Tucker format using iterative methods like conjugate gradients or multigrid. We present weak scaling studies, which provide evidence that the runtime of our algorithms indeed grows like $\log(d)$. Furthermore, we present numerical experiments in which we apply our algorithms to solve a parameter-dependent diffusion equation in the Hierarchical Tucker format by means of a multigrid algorithm

Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page

Medians and Beyond: New Aggregation Techniques for Sensor Networks

Wireless sensor networks offer the potential to span and monitor large geographical areas inexpensively. Sensors, however, have significant power constraint (battery life), making communication very expensive. Another important issue in the context of sensor-based information systems is that individual sensor readings are inherently unreliable. In order to address these two aspects, sensor database systems like TinyDB and Cougar enable in-network data aggregation to reduce the communication cost and improve reliability. The existing data aggregation techniques, however, are limited to relatively simple types of queries such as SUM, COUNT, AVG, and MIN/MAX. In this paper we propose a data aggregation scheme that significantly extends the class of queries that can be answered using sensor networks. These queries include (approximate) quantiles, such as the median, the most frequent data values, such as the consensus value, a histogram of the data distribution, as well as range queries. In our scheme, each sensor aggregates the data it has received from other sensors into a fixed (user specified) size message. We provide strict theoretical guarantees on the approximation quality of the queries in terms of the message size. We evaluate the performance of our aggregation scheme by simulation and demonstrate its accuracy, scalability and low resource utilization for highly variable input data sets