Round Algorithm Research Articles

A Spectral Approach to Network Design

We present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional non-negative linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, this provides an alternative spectral algorithm to achieve constant factor approximation for the classical survivable network design problem, and partially answers a question of Bansal about survivable network design with concentration property. We also show many other applications of the spectral rounding results, including weighted experimental design and additive spectral sparsification.

Incentivizing Federated Learning Under Long-Term Energy Constraint via Online Randomized Auctions

Mobile users are often reluctant to participate in federated learning to train models, due to the excessive consumption of the limited resources such as the mobile devices’ energy. We propose an auction-based online incentive mechanism, FLORA, which allows users to submit bids dynamically and repetitively and compensates such bids subject to each user’s long-term battery capacity. We formulate a nonlinear mixed-integer program to capture the social cost minimization in the federated learning system. Then we design multiple polynomial-time online algorithms, including a fractional online algorithm and a randomized rounding algorithm to select winning bids and control training accuracy, as well as a payment allocation algorithm to calculate the remuneration based on the bid-winning probabilities. Maintaining the satisfiable quality of the global model that is trained, our approach works on the fly without relying on the unknown future inputs, and achieves provably a sublinear regret and a sublinear fit over time while attaining the economic properties of truthfulness and individual rationality in expectation. Extensive trace-driven evaluations have confirmed the practical superiority of FLORA over existing alternatives.

Distributed Graph Realizations

We study graph realization problems for the first time from a distributed perspective. Graph realization problems are encountered in distributed construction of overlay networks that must satisfy certain degree or connectivity properties. We study them in the node capacitated clique (NCC) model of distributed computing, recently introduced for representing peer-to-peer overlay networks. We focus on two central variants, <i>degree-sequence realization</i> and <i>minimum threshold-connectivity</i> realization. In the degree sequence problem, each node <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> is associated with a degree <inline-formula><tex-math notation="LaTeX">$d(v)$</tex-math></inline-formula> , and the resulting degree sequence is realizable if it is possible to construct an overlay network in which the degree of each node <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> is <inline-formula><tex-math notation="LaTeX">$d(v)$</tex-math></inline-formula> . The minimum threshold-connectivity problem requires us to construct an overlay network that satisfies connectivity constraints specified between every pair of nodes. Overlay network realizations can be either explicit or implicit. Explicit realizations require both endpoints of any edge in the realized graph to be aware of the edge. In implicit realizations, on the other hand, at least one endpoint of each edge of the realized graph needs to be aware of the edge. The main realization algorithms we present are the following. (Note that all our algorithms are randomized Las Vegas algorithms unless specified otherwise. The stated running times hold with high probability.) 1) An <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\min \lbrace \sqrt{m},\Delta \rbrace)$</tex-math></inline-formula> time algorithm for implicit realization of a degree sequence. Here, <inline-formula><tex-math notation="LaTeX">$\Delta = \max _v d(v)$</tex-math></inline-formula> is the maximum degree and <inline-formula><tex-math notation="LaTeX">$m = (1/2) \sum _v d(v)$</tex-math></inline-formula> is the number of edges in the final realization. 2) <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\Delta)$</tex-math></inline-formula> time algorithm for an explicit realization of a degree sequence. We first compute an implicit realization and then transform it into an explicit one in <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\Delta)$</tex-math></inline-formula> additional rounds. 3) An <inline-formula><tex-math notation="LaTeX">$\tilde{O}(\Delta)$</tex-math></inline-formula> time algorithm for the threshold connectivity problem that obtains an explicit solution and an improved <inline-formula><tex-math notation="LaTeX">$\tilde{O}(1)$</tex-math></inline-formula> algorithm for implicit realization when all nodes know each other’s IDs. These algorithms yield 2-approximations w.r.t. the number of edges. We complement our upper bounds with lower bounds to show that the above algorithms are tight up to factors of <inline-formula><tex-math notation="LaTeX">$\log n$</tex-math></inline-formula> . Additionally, we provide algorithms for realizing trees (including a procedure for obtaining a tree with a minimal diameter), an <inline-formula><tex-math notation="LaTeX">$\tilde{O}(1)$</tex-math></inline-formula> round algorithm for approximate degree sequence realization and finally an <inline-formula><tex-math notation="LaTeX">$O(\log ^2 n)$</tex-math></inline-formula> algorithm for degree sequence realization in the non-preassigned case namely, where the input degree sequence may be permuted among the nodes.

Financial and Economic Sequence Forecasting Based on Time Slot Allocation Algorithm

Financial and economic series prediction was very important for the study of economic time series. Because the traditional time series analysis method was mainly suitable for stable data and cannot be applied to unstable economic variables, this paper proposed a knowledge model of financial and economic series prediction time slot algorithm, which provided the basis for solving unstable financial data. Firstly, based on the analysis of relevant financial and economic series prediction methods at home and abroad, this paper applied the time slot allocation algorithm to economic series analysis and prediction and designed a path level time slot allocation algorithm, which introduced a fixed time slot to shorten the time required to complete the path level time slot allocation. Secondly, this paper abstracted the calculation of slot allocation as a convex optimization problem and proposed a round slot allocation algorithm based on constraints and optimal solution. The algorithm can improve the end-to-end throughput and the effective utilization of slots. Finally, the model proposed in this paper was used to predict the indicators of relevant industries, and the simulation results were compared with the standard data. The experimental results showed that the financial and economic series prediction system based on time slot allocation algorithm proposed in this paper can effectively improve the accuracy of financial and economic series prediction, which verified the effectiveness and feasibility of this model.

A Subquadratic-Time Distributed Algorithm for Exact Maximum Matching

For a graph G=(V,E), finding a set of disjoint edges that do not share any vertices is called a matching problem, and finding the maximum matching is a fundamental problem in the theory of distributed graph algorithms. Although local algorithms for the approximate maximum matching problem have been widely studied, exact algorithms has not been much studied. In fact, no exact maximum matching algorithm that is faster than the trivial upper bound of O(n^2) rounds is known for the general instance. In this paper, we propose a randomized O(s_{max}^{3/2}+log n)-round algorithm in the CONGEST model, where s_{\max} is the size of maximum matching. This is the first exact maximum matching algorithm in o(n^2) rounds for general instances in the CONGEST model. The key technical ingredient of our result is a distributed algorithms of finding an augmenting path in O(s_{\max}) rounds, which is based on a novel technique of constructing a sparse certificate of augmenting paths, which is a subgraph of the input graph preserving at least one augmenting path. To establish a highly parallel construction of sparse certificates, we also propose a new characterization of sparse certificates, which might also be of independent interest.

Low-congestion shortcut and graph parameters

Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of {tilde{Omega }}(sqrt{n} + D) rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class {mathcal {C}}, an f-round algorithm for constructing shortcuts of quality q for any instance in {mathcal {C}} results in {tilde{O}}(q + f)-round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for {mathcal {C}}. The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking {tilde{O}}(sqrt{n}+D)-round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.

Improved distributed Delta -coloring

We present a randomized distributed algorithm that computes a Delta -coloring in any non-complete graph with maximum degree Delta ge 4 in O(log Delta ) + 2^{O(sqrt{log log n})} rounds, as well as a randomized algorithm that computes a Delta -coloring in O((log log n)^2) rounds when Delta in [3, O(1)]. Both these algorithms improve on an O(log ^3 n / log Delta )-round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an Omega (log log n) round lower bound of Brandt et al. (STOC’16).

Asymptotics of Rounding Algorithms for Integer Control Problems in the Presence of Convexified Pointwise Mixed Constraints

AbstractThe analysis of computationally cheap algorithms to approximate the infimum of optimal control problems with discrete‐valued controls has been improved in recent literature. The overall method is often referred to as combinatorial integral approximation decomposition and the algorithms are often referred to as rounding algorithms.The studied algorithms have in common that they seek to minimize or at least bound a quantity called integrated control deviation. The algorithms operate on a grid (often called rounding grid) and the integrated control deviation vanishes when the grid constant of the grid on tends to zero. Assuming a suitable spatial coherence property of the grid refinement implies weak∗ convergence of the computed controls in L∞.When the integer optimal control problem features additional pointwise mixed state‐control constraints, the aforementioned rounding algorithms cannot be applied directly. There exist examples for which the observed infeasibility of the constraint after applying the algorithms does not vanish in an L∞‐sense when the grid constant is driven to zero. While modifications of the algorithms exist that resolve this problem, the asymptotics of the infeasibility without those modifications have not been analyzed so far. This work closes this gap in the existing literature and we show that the infeasibility vanishes in a pointwise almost everywhere sense.

A Performance Study of Some Approximation Algorithms for Computing a Small Dominating Set in a Graph

We implement and test the performances of several approximation algorithms for computing the minimum dominating set of a graph. These algorithms are the standard greedy algorithm, the recent Linear programming (LP) rounding algorithms and a hybrid algorithm that we design by combining the greedy and LP rounding algorithms. Over the range of test data, all algorithms perform better than anticipated in theory, and have small performance ratios, measured as the size of output divided by the LP objective lower bound. However, each have advantages over the others. For instance, LP rounding algorithm normally outperforms the other algorithms on sparse real-world graphs. On a graph with 400,000+ vertices, LP rounding took less than 15 s of CPU time to generate a solution with performance ratio 1.011, while the greedy and hybrid algorithms generated solutions of performance ratio 1.12 in similar time. For synthetic graphs, the hybrid algorithm normally outperforms the others, whereas for hypercubes and k-Queens graphs, greedy outperforms the rest. Another advantage of the hybrid algorithm is to solve very large problems that are suitable for application of LP rounding (sparse graphs) but LP formulations become formidable in practice and LP solvers crash, as we observed on a real-world graph with 7.7 million+ vertices and a planar graph on 1,000,000 vertices.

Low-Congestion shortcuts without embedding

Distributed optimization algorithms are frequently faced with solving sub-problems on disjoint connected parts of a network. Unfortunately, the diameter of these parts can be significantly larger than the diameter of the underlying network, leading to slow running times. Recent work by [Ghaffari and Hauepler; SODA'16] showed that this phenomenon can be seen as the broad underlying reason for the pervasive $\Omega(\sqrt{n} + D)$ lower bounds that apply to most optimization problems in the CONGEST model. On the positive side, this work also introduced low-congestion shortcuts as an elegant solution to circumvent this problem in certain topologies of interest. Particularly, they showed that there exist good shortcuts for any planar network and more generally any bounded genus network. This directly leads to fast $O(D \log^{O(1)} n)$ distributed algorithms for MST and Min-Cut approximation, given that one can efficiently construct these shortcuts in a distributed manner. Unfortunately, the shortcut construction of [Ghaffari and Hauepler; SODA'16] relies heavily on having access to a genus embedding of the network. Computing such an embedding distributedly, however, is a hard problem - even for planar networks. No distributed embedding algorithm for bounded genus graphs is in sight. In this work, we side-step this problem by defining a restricted and more structured form of shortcuts and giving a novel construction algorithm which efficiently finds a shortcut which is, up to a logarithmic factor, as good as the best shortcut that exists for a given network. This new construction algorithm directly leads to an $O(D \log^{O(1)} n)$-round algorithm for solving optimization problems like MST for any topology for which good restricted shortcuts exist - without the need to compute any embedding. This includes the first efficient algorithm for bounded genus graphs.

SPRING: A Sparsity-Aware Reduced-Precision Monolithic 3D CNN Accelerator Architecture for Training and Inference

Convolutional neural networks (CNNs) outperform traditional machine learning algorithms across a wide range of applications, such as object recognition, image segmentation, and autonomous driving. However, their ever-growing computational complexity makes it necessary to design efficient hardware accelerators. Most CNN accelerators focus on exploring various dataflow styles and designs that exploit computational parallelism. However, potential performance improvement (speedup) from sparsity has not been adequately addressed. The computation and memory footprint of CNNs can be significantly reduced if sparsity is exploited in network evaluations. To further improve performance and energy efficiency, some accelerators evaluate CNNs with limited precision. However, this is limited to the inference phase since reduced precision sacrifices network accuracy if used in training. In addition, CNN evaluation is usually memory-intensive, especially during training. The performance bottleneck arises from the fact that the memory cannot feed the computational units enough data, resulting in idling of these computational units and thus low utilization ratios. In this article, we propose SPRING, a SParsity-aware Reduced-precision Monolithic 3D CNN accelerator for trainING and inference. SPRING supports both CNN training and inference. It uses a binary mask scheme to encode sparsities in activations and weights. It uses the stochastic rounding algorithm to train CNNs with reduced precision without accuracy loss. To alleviate the memory bottleneck in CNN evaluation, especially during training, SPRING uses an efficient monolithic 3D nonvolatile memory interface to increase memory bandwidth. Compared to Nvidia GeForce GTX 1080 Ti, SPRING achieves 15.6×, 4.2×, and 66.0× improvements in performance, power reduction, and energy efficiency, respectively, for CNN training, and 15.5×, 4.5×, and 69.1× improvements in performance, power reduction, and energy efficiency, respectively, for inference.

Rounding Meshes in 3D

Let $${\mathcal {P}}$$ be a set of n polygons in $${\mathbb {R}}^3$$, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps $${\mathcal {P}}$$ to a simplicial complex $${\mathcal {Q}}$$ whose vertices have integer coordinates. Every face of $${\mathcal {P}}$$ is mapped to a set of faces (or edges or vertices) of $${\mathcal {Q}}$$ and the mapping from $${\mathcal {P}}$$ to $${\mathcal {Q}}$$ can be done through a continuous motion of the faces such that: (i) the $$L_\infty $$ Hausdorff distance between a face and its image during the motion is at most 3/2, and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case the size of $${\mathcal {Q}}$$ is $$O(n^{13})$$ and the time complexity of the algorithm is $$O(n^{15})$$ but, under reasonable assumptions, these complexities decrease to $$O(n^{4}\sqrt{n})$$ and $$O(n^{5})$$. Furthermore, these complexities are likely not tight and we expect, in practice on non-pathological data, $$O(n\sqrt{n})$$ space and time complexities.

Concurrent Multipath Routing Optimization in Named Data Networks

To achieve maximum throughput in named data networking (NDN) with consideration of fairness among each source-sink pair, we model the routing problem in NDN as a mixed-integer maximum concurrent flow problem. Taking advantage of the smart forwarding framework in NDN, we obtain an approximate optimal routing and forwarding solution with our improved Fleischer and random rounding (FRR) algorithm. We analyze the complexity of the Fleischer algorithm more comprehensively and give the best scaling method to achieve the least iterations. In addition, we further reduce the complexity of the Fleischer algorithm so that it iterates only once, thus getting a heuristic algorithm which is called multicommodity multipath (MCMP). Using ndnSIM-based simulation, we compare our proposed routing algorithms with the existing routing algorithms, including flooding, k-shortest path, and best single path routing, and the simulation results prove that FRR and MCMP can outperform other algorithms both in throughput and latency, and it can also be concluded that the best number of paths for k-shortest path routing should be close to and less than the average node degree of the network. For studying routing optimization with caching, we conduct simulations with different in-network cache capacity and different caching policies. The simulation results show that although the LRU can achieve the best performance under the same cache capacity, the cache capacity plays a decisive role. All routing algorithms produce better performance with caching, but the FRR still achieves the best performance, and follows by MCMP.

Sparse matrix multiplication and triangle listing in the Congested Clique model

We show how to multiply two n×n matrices S and T over semirings in the Congested Clique model, where n nodes communicate in a fully connected synchronous network using O(log⁡n)-bit messages, within O(nz(S)1/3nz(T)1/3/n+1) rounds of communication, where nz(S) and nz(T) denote the number of non-zero elements in S and T, respectively. By leveraging the sparsity of the input matrices, our algorithm greatly reduces communication costs compared with general multiplication algorithms [Censor-Hillel et al. (2015) [9]], and thus improves upon the state-of-the-art for matrices with o(n2) non-zero elements. Moreover, our algorithm exhibits the additional strength of surpassing previous solutions also in the case where only one of the two matrices is such. Particularly, this allows to efficiently raise a sparse matrix to a power greater than 2. As applications, we show how to speed up the computation on non-dense graphs of 4-cycle counting and all-pairs-shortest-paths.Our algorithmic contribution is a new deterministic method of restructuring the input matrices in a sparsity-aware manner, which assigns each node with element-wise multiplication tasks that are not necessarily consecutive but guarantee a balanced element distribution, providing for communication-efficient multiplication.Moreover, this new deterministic method for restructuring matrices may be used to restructure the adjacency matrix of input graphs, enabling faster deterministic solutions for graph related problems. As an example, we present a new sparsity aware, deterministic algorithm which solves the triangle listing problem in O(m/n5/3+1) rounds, a complexity that was previously obtained by a randomized algorithm [Pandurangan et al. (2018) [26]], and that matches the known lower bound of Ω˜(n1/3) when m=n2 of [Izumi and Le Gall (2017) [19], Pandurangan et al. (2018) [26]]. Naturally, our triangle listing algorithm also implies triangle counting within the same complexity of O(m/n5/3+1) rounds, which is (possibly more than) a cubic improvement over the previously known deterministicO(m2/n3)-round algorithm [Dolev et al. (2012) [12]].

Multi‐dimensional Sum‐Up Rounding using Hilbert curve iterates

AbstractMixed‐integer optimal control problems can be reformulated by means of partial outer convexification, which introduces binary‐valued switching functions for the different realizations of a discrete‐valued control variable. They can be relaxed naturally by allowing them to take values in [0, 1]. Sum‐Up Rounding (SUR) algorithms approximate feasible switching functions of the relaxation with binary ones. If the controls are distributed in one dimension, the approximants are known to converge in the weak∗ topology of L∞. We show that this still holds true for controls that are distributed in more than one dimension if an appropriate grid refinement strategy that is coupled with a deliberate ordering of the grid cells is chosen. This condition is satisfied by the iterates of space‐filling curves, e.g. the Hilbert curve.

Centrality Analysis in $d$ -Regular Directed Acyclic Random Networks and Its Applications in Top-$k$ Recommendations

Centrality analysis has been a very important research topic in online social networks. Traditionally, centrality analysis is carried out for a specific network. In this article, we conduct centrality analysis in random networks. For this, we consider the class of $d$ -regular directed acyclic random networks generated by the configuration model. We consider two centralities: the PageRank and the in-component centrality (defined as the size of the in-component of a node). For the PageRank, we derive a closed-form solution in the complete directed acyclic network and show that it can be used as an approximation in a $d$ -regular directed acyclic random network. For the in-component centrality, we derive upper bounds for its first moment and its second moment in a $d$ -regular directed acyclic random network. These results have interesting applications in top- $k$ recommendations. In particular, we propose a qualifying round algorithm that outputs a set of nodes to include the top- $k$ nodes with high probability. Various experiments are conducted to show the effectiveness of the algorithm.

Evaluation of lossless and lossy algorithms for the compression of scientific datasets in netCDF-4 or HDF5 files

Abstract. The increasing volume of scientific datasets requires the use of compression to reduce data storage and transmission costs, especially for the oceanographic or meteorological datasets generated by Earth observation mission ground segments. These data are mostly produced in netCDF files. Indeed, the netCDF-4/HDF5 file formats are widely used throughout the global scientific community because of the useful features they offer. HDF5 in particular offers a dynamically loaded filter plugin so that users can write compression/decompression filters, for example, and process the data before reading or writing them to disk. This study evaluates lossy and lossless compression/decompression methods through netCDF-4 and HDF5 tools on analytical and real scientific floating-point datasets. We also introduce the Digit Rounding algorithm, a new relative error-bounded data reduction method inspired by the Bit Grooming algorithm. The Digit Rounding algorithm offers a high compression ratio while keeping a given number of significant digits in the dataset. It achieves a higher compression ratio than the Bit Grooming algorithm with slightly lower compression speed.

Flux sampling is a powerful tool to study metabolism under changing environmental conditions

The development of high-throughput ‘omic techniques has sparked a rising interest in genome-scale metabolic models, with applications ranging from disease diagnostics to crop adaptation. Efficient and accurate methods are required to analyze large metabolic networks. Flux sampling can be used to explore the feasible flux solutions in metabolic networks by generating probability distributions of steady-state reaction fluxes. Unlike other methods, flux sampling can be used without assuming a particular cellular objective. We have undertaken a rigorous comparison of several sampling algorithms and concluded that the coordinate hit-and-run with rounding (CHRR) algorithm is the most efficient based on both run-time and multiple convergence diagnostics. We demonstrate the power of CHRR by using it to study the metabolic changes that underlie photosynthetic acclimation to cold of Arabidopsis thaliana plant leaves. In combination with experimental measurements, we show how the regulated interplay between diurnal starch and organic acid accumulation defines the plant acclimation process. We confirm fumarate accumulation as a requirement for cold acclimation and further predict γ–aminobutyric acid to have a key role in metabolic signaling under cold conditions. These results demonstrate how flux sampling can be used to analyze the feasible flux solutions across changing environmental conditions, whereas eliminating the need to make assumptions which introduce observer bias.

Improved time quantum length estimation for round robin scheduling algorithm using neural network

In most cases, the quantum time length is taken to be fix in all applications that use Round Robin (RR) scheduling algorithm. Many attempts aim to determination of the optimal length of the quantum that results in a small average turnaround time, but the unknown nature of the tasks in the ready queue make the problem more complicated: Considering a large quantum length makes the RR algorithm behave like a First Come First Served (FIFO) scheduling algorithm, and a small quantum length cause high number of contexts switching. In this paper we propose a RR scheduling algorithm based on Neural Network Models for predicting the optimal quantum length which lead to a minimum average turnaround time. The quantum length depends on tasks burst times available in the ready queue. Rather than conventional traditional methods using fixed quantum length, this one giving better results by minimizing the average turnaround time for almost any set of jobs in the ready queue.

Quantizing Euclidean Motions via Double-Coset Decomposition.

Concepts from mathematical crystallography and group theory are used here to quantize the group of rigid-body motions, resulting in a “motion alphabet” with which robot motion primitives are expressed. From these primitives it is possible to develop a dictionary of physical actions. Equipped with an alphabet of the sort developed here, intelligent actions of robots in the world can be approximated with finite sequences of characters, thereby forming the foundation of a language in which robot motion is articulated. In particular, we use the discrete handedness-preserving symmetries of macromolecular crystals (known in mathematical crystallography as Sohncke space groups) to form a coarse discretization of the space SE(3) of rigid-body motions. This discretization is made finer by subdividing using the concept of double-coset decomposition. More specifically, a very efficient, equivolumetric quantization of spatial motion can be defined using the group-theoretic concept of a double-coset decomposition of the form Γ\\SE(3)/Δ, where Γ is a Sohncke space group and Δ is a finite group of rotational symmetries such as those of the icosahedron. The resulting discrete alphabet is based on a very uniform sampling of SE(3) and is a tool for describing the continuous trajectories of robots and humans. An efficient coarse-to-fine search algorithm is presented to round off any motion sampled from the continuous group of motions to the nearest element of our alphabet. It is shown that our alphabet and this efficient rounding algorithm can be used as a geometric data structure to accelerate the performance of other sampling schemes designed for desirable dispersion or discrepancy properties. Moreover, the general “signals to symbols” problem in artificial intelligence is cast in this framework for robots moving continuously in the world.