In standard number-in-hand multi-party communication complexity, performance is measured as the total number of bits transmitted globally in the network. In this paper, we study a variation called local communication complexity in which performance instead measures the maximum number of bits sent or received at any one player. We focus on a simple model where $n$ players, each with one input bit, execute a protocol by exchanging messages to compute a function on the $n$ input bits. We ask what can and cannot be solved with a small local communication complexity in this setting. We begin by establishing a non-trivial lower bound on the local complexity for a specific function by proving that counting the number of $1$'s among the first $17$ input bits distributed among the participants requires a local complexity strictly greater than $1$. We further investigate whether harder counting problems of this type can yield stronger lower bounds, providing a largely negative answer by showing that constant local complexity is sufficient to count the number $1$ bits over the entire input, and therefore compute any symmetric function. In addition to counting, we show that both sorting and searching can be computed in constant local complexity. We then use the counting solution as a subroutine to demonstrate that constant local complexity is also sufficient to compute many standard modular arithmetic operations on two operands, including: comparisons, addition, subtraction, multiplication, division, and exponentiation. Finally we establish that function $GCD(x,y)$ where $x$ and $y$ are in the range $[1,n]$ has local complexity of $O(1)$. Our work highlights both new techniques for proving lower bounds on this metric and the power of even a small amount of local communication.
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We prove a bound of $O( k (n+m)\log^{d-1})$ on the number of incidences between $n$ points and $m$ axis parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points. That is, the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work by a factor of $\log^d n$, for $d >2$. We also study other variants of the problem. For halfspaces, using shallow cuttings, we get a near linear bound in two and three dimensions. Finally, we present near linear bound for the case of shapes in the plane with low union complexity (e.g. fat triangles).
Performance assessment and optimization for networks jointly performing caching, computing, and communication (3C) has recently drawn significant attention because many emerging applications require 3C functionality. However, studies in the literature mostly focus on the particular algorithms and setups of such networks, while their theoretical understanding and characterization has been less explored. To fill this gap, this paper conducts the asymptotic (scaling-law) analysis for the delay-outage tradeoff of noise-limited wireless edge networks with joint 3C. In particular, assuming the user requests for different tasks following a Zipf distribution, we derive the analytical expression for the optimal caching policy. Based on this, we next derive the closed-form expression for the optimum outage probability as a function of delay and other network parameters for the case that the Zipf parameter is smaller than 1. Then, for the case that the Zipf parameter is larger than 1, we derive the closed-form expressions for upper and lower bounds of the optimum outage probability. We provide insights and interpretations based on the derived expressions. Computer simulations validate our analytical results and insights.
This paper studies the problem of online user grouping, scheduling and power allocation in beyond 5G cellular-based Internet of things networks. Due to the massive number of devices trying to be granted to the network, non-orthogonal multiple access method is adopted in order to accommodate multiple devices in the same radio resource block. Different from most previous works, the objective is to maximize the number of served devices while allocating their transmission powers such that their real-time requirements as well as their limited operating energy are respected. First, we formulate the general problem as a mixed integer non-linear program (MINLP) that can be transformed easily to MILP for some special cases. Second, we study its computational complexity by characterizing the NP-hardness of different special cases. Then, by dividing the problem into multiple NOMA grouping and scheduling subproblems, efficient online competitive algorithms are proposed. Further, we show how to use these online algorithms and combine their solutions in a reinforcement learning setting to obtain the power allocation and hence the global solution to the problem. Our analysis are supplemented by simulation results to illustrate the performance of the proposed algorithms with comparison to optimal and state-of-the-art methods.
For tabletop rearrangement problems with overhand grasps, storage space outside the tabletop workspace, or buffers, can temporarily hold objects which greatly facilitates the resolution of a given rearrangement task. This brings forth the natural question of how many running buffers are required so that certain classes of tabletop rearrangement problems are feasible. In this work, we examine the problem for both the labeled (where each object has a specific goal pose) and the unlabeled (where goal poses of objects are interchangeable) settings. On the structural side, we observe that finding the minimum number of running buffers (MRB) can be carried out on a dependency graph abstracted from a problem instance, and show that computing MRB on dependency graphs is NP-hard. We then prove that under both labeled and unlabeled settings, even for uniform cylindrical objects, the number of required running buffers may grow unbounded as the number of objects to be rearranged increases; we further show that the bound for the unlabeled case is tight. On the algorithmic side, we develop highly effective algorithms for finding MRB for both labeled and unlabeled tabletop rearrangement problems, scalable to over a hundred objects under very high object density. Employing these algorithms, empirical evaluations show that random labeled and unlabeled instances, which more closely mimics real-world setups, have much smaller MRBs.
Distributed dataflow systems like Spark and Flink enable the use of clusters for scalable data analytics. While runtime prediction models can be used to initially select appropriate cluster resources given target runtimes, the actual runtime performance of dataflow jobs depends on several factors and varies over time. Yet, in many situations, dynamic scaling can be used to meet formulated runtime targets despite significant performance variance. This paper presents Enel, a novel dynamic scaling approach that uses message propagation on an attributed graph to model dataflow jobs and, thus, allows for deriving effective rescaling decisions. For this, Enel incorporates descriptive properties that capture the respective execution context, considers statistics from individual dataflow tasks, and propagates predictions through the job graph to eventually find an optimized new scale-out. Our evaluation of Enel with four iterative Spark jobs shows that our approach is able to identify effective rescaling actions, reacting for instance to node failures, and can be reused across different execution contexts.
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.
For many applications, drones are required to operate entirely or partially autonomously. To fly completely or partially on their own, drones need access to location services to get navigation commands. While using the Global Positioning System (GPS) is an obvious choice, GPS is not always available, can be spoofed or jammed, and is highly error-prone for indoor and underground environments. The ranging method using beacons is one of the popular methods for localization, specially for indoor environments. In general, localization error in this class is due to two factors: the ranging error and the error induced by the relative geometry between the beacons and the target object to localize. This paper proposes OPTILOD (Optimal Beacon Placement for High-Accuracy Indoor Localization of Drones), an optimization algorithm for the optimal placement of beacons deployed in three-dimensional indoor environments. OPTILOD leverages advances in Evolutionary Algorithms to compute the minimum number of beacons and their optimal placement to minimize the localization error. These problems belong to the Mixed Integer Programming (MIP) class and are both considered NP-Hard. Despite that, OPTILOD can provide multiple optimal beacon configurations that minimize the localization error and the number of deployed beacons concurrently and time efficiently.
In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding a symmetric positive-definite term to the mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. For linear problems, we show that our mass-scaling method produces no adverse effects in terms of spatial accuracy and orders of convergence. We illustrate these properties in one, two and three spatial dimension, for quadrilateral elements and triangular elements, and for up to fourth order polynomials basis functions. To extend the method to non-linear problems, we introduce a linear approximation and show that a sizeable increase in critical time-step size can be achieved while only causing minor (even beneficial) influences on the dynamic response.
We consider the problem of coded distributed computing using polar codes. The average execution time of a coded computing system is related to the error probability for transmission over the binary erasure channel in recent work by Soleymani, Jamali and Mahdavifar, where the performance of binary linear codes is investigated. In this paper, we focus on polar codes and unveil a connection between the average execution time and the scaling exponent $\mu$ of the family of codes. The scaling exponent has emerged as a central object in the finite-length characterization of polar codes, and it captures the speed of convergence to capacity. In particular, we show that (i) the gap between the normalized average execution time of polar codes and that of optimal MDS codes is $O(n^{-1/\mu})$, and (ii) this upper bound can be improved to roughly $O(n^{-1/2})$ by considering polar codes with large kernels. We conjecture that these bounds could be improved to $O(n^{-2/\mu})$ and $O(n^{-1})$, respectively, and provide a heuristic argument as well as numerical evidence supporting this view.
In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.
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