Element Distinctness, Frequency Moments, and Sliding Windows

We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than previous lower bounds for comparison-based algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and space efficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows, where the task is to take an input of length 2n-1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a time-space tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to compute the number of distinct elements over sliding windows. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k, k neq 1. This shows that those frequency moments and the decision problem F_0 mod 2 are strictly harder than element distinctness. We complement this lower bound with a T in O(n^2/S) comparison-based deterministic RAM algorithm for exactly computing F_k over sliding windows, nearly matching both our lower bound for the sliding-window version and the comparison-based lower bounds for the single-window version. We further exhibit a quantum algorithm for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437

Approximation of empowerment in the continuous domain

The empowerment formalism offers a goal-independent utility function fully derived from an agent's embodiment. It produces intrinsic motivations which can be used to generate self-organizing behaviours in agents. One obstacle to the application of empowerment in more demanding (esp. continuous) domains is that previous ways of calculating empowerment have been very time consuming and only provided a proof-of-concept. In this paper we present a new approach to efficiently approximate empowerment as a parallel, linear, Gaussian channel capacity problem. We use pendulum balancing to demonstrate this new method, and compare it to earlier approximation methods.Peer reviewe

Murasaki: A Fast, Parallelizable Algorithm to Find Anchors from Multiple Genomes

BACKGROUND: With the number of available genome sequences increasing rapidly, the magnitude of sequence data required for multiple-genome analyses is a challenging problem. When large-scale rearrangements break the collinearity of gene orders among genomes, genome comparison algorithms must first identify sets of short well-conserved sequences present in each genome, termed anchors. Previously, anchor identification among multiple genomes has been achieved using pairwise alignment tools like BLASTZ through progressive alignment tools like TBA, but the computational requirements for sequence comparisons of multiple genomes quickly becomes a limiting factor as the number and scale of genomes grows. METHODOLOGY/PRINCIPAL FINDINGS: Our algorithm, named Murasaki, makes it possible to identify anchors within multiple large sequences on the scale of several hundred megabases in few minutes using a single CPU. Two advanced features of Murasaki are (1) adaptive hash function generation, which enables efficient use of arbitrary mismatch patterns (spaced seeds) and therefore the comparison of multiple mammalian genomes in a practical amount of computation time, and (2) parallelizable execution that decreases the required wall-clock and CPU times. Murasaki can perform a sensitive anchoring of eight mammalian genomes (human, chimp, rhesus, orangutan, mouse, rat, dog, and cow) in 21 hours CPU time (42 minutes wall time). This is the first single-pass in-core anchoring of multiple mammalian genomes. We evaluated Murasaki by comparing it with the genome alignment programs BLASTZ and TBA. We show that Murasaki can anchor multiple genomes in near linear time, compared to the quadratic time requirements of BLASTZ and TBA, while improving overall accuracy. CONCLUSIONS/SIGNIFICANCE: Murasaki provides an open source platform to take advantage of long patterns, cluster computing, and novel hash algorithms to produce accurate anchors across multiple genomes with computational efficiency significantly greater than existing methods. Murasaki is available under GPL at http://murasaki.sourceforge.net

Particle Computation: Complexity, Algorithms, and Logic

We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal (such as gravity or a magnetic field). We show that a maze of obstacles to the environment can be used to create complex systems. We provide a wide range of results for a wide range of questions. These can be subdivided into external algorithmic problems, in which particle configurations serve as input for computations that are performed elsewhere, and internal logic problems, in which the particle configurations themselves are used for carrying out computations. For external algorithms, we give both negative and positive results. If we are given a set of stationary obstacles, we prove that it is NP-hard to decide whether a given initial configuration of unit-sized particles can be transformed into a desired target configuration. Moreover, we show that finding a control sequence of minimum length is PSPACE-complete. We also work on the inverse problem, providing constructive algorithms to design workspaces that efficiently implement arbitrary permutations between different configurations. For internal logic, we investigate how arbitrary computations can be implemented. We demonstrate how to encode dual-rail logic to build a universal logic gate that concurrently evaluates and, nand, nor, and or operations. Using many of these gates and appropriate interconnects, we can evaluate any logical expression. However, we establish that simulating the full range of complex interactions present in arbitrary digital circuits encounters a fundamental difficulty: a fan-out gate cannot be generated. We resolve this missing component with the help of 2x1 particles, which can create fan-out gates that produce multiple copies of the inputs. Using these gates we provide rules for replicating arbitrary digital circuits.Comment: 27 pages, 19 figures, full version that combines three previous conference article