The Four Bars Problem

A four-bar linkage is a mechanism consisting of four rigid bars which are joined by their endpoints in a polygonal chain and which can rotate freely at the joints (or vertices). We assume that the linkage lies in the 2-dimensional plane so that one of the bars is held horizontally fixed. In this paper we consider the problem of reconfiguring a four-bar linkage using an operation called a \emph{pop}. Given a polygonal cycle, a pop reflects a vertex across the line defined by its two adjacent vertices along the polygonal chain. Our main result shows that for certain conditions on the lengths of the bars of the four-bar linkage, the neighborhood of any configuration that can be reached by smooth motion can also be reached by pops. The proof relies on the fact that pops are described by a map on the circle with an irrational number of rotation.Comment: 18 page

Orderly broadcasting in multidimensional tori

In this thesis, we describe an ordering of the vertices of a multidimensional torus and study the upper bound on the orderly broadcast time. Along with messy broadcasting, orderly broadcasting is another model where the nodes of the network have limited knowledge about their local neighborhood. However, while messy broadcasting explores the worst-case performance of broadcast schemes, orderly broadcasting, like the classical broadcast model, is concerned with finding an ordering of the vertices of a graph that will minimize the overall broadcast time

Efficient Multi-Robot Coverage of a Known Environment

This paper addresses the complete area coverage problem of a known environment by multiple-robots. Complete area coverage is the problem of moving an end-effector over all available space while avoiding existing obstacles. In such tasks, using multiple robots can increase the efficiency of the area coverage in terms of minimizing the operational time and increase the robustness in the face of robot attrition. Unfortunately, the problem of finding an optimal solution for such an area coverage problem with multiple robots is known to be NP-complete. In this paper we present two approximation heuristics for solving the multi-robot coverage problem. The first solution presented is a direct extension of an efficient single robot area coverage algorithm, based on an exact cellular decomposition. The second algorithm is a greedy approach that divides the area into equal regions and applies an efficient single-robot coverage algorithm to each region. We present experimental results for two algorithms. Results indicate that our approaches provide good coverage distribution between robots and minimize the workload per robot, meanwhile ensuring complete coverage of the area.Comment: In proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 201

OC-NMN: Object-centric Compositional Neural Module Network for Generative Visual Analogical Reasoning

A key aspect of human intelligence is the ability to imagine -- composing learned concepts in novel ways -- to make sense of new scenarios. Such capacity is not yet attained for machine learning systems. In this work, in the context of visual reasoning, we show how modularity can be leveraged to derive a compositional data augmentation framework inspired by imagination. Our method, denoted Object-centric Compositional Neural Module Network (OC-NMN), decomposes visual generative reasoning tasks into a series of primitives applied to objects without using a domain-specific language. We show that our modular architectural choices can be used to generate new training tasks that lead to better out-of-distribution generalization. We compare our model to existing and new baselines in proposed visual reasoning benchmark that consists of applying arithmetic operations to MNIST digits

Theta-3 is connected

In this paper, we show that the $\theta$-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.Comment: 11 pages, to appear in CGT

Using Graph Algorithms to Pretrain Graph Completion Transformers

Recent work on Graph Neural Networks has demonstrated that self-supervised pretraining can further enhance performance on downstream graph, link, and node classification tasks. However, the efficacy of pretraining tasks has not been fully investigated for downstream large knowledge graph completion tasks. Using a contextualized knowledge graph embedding approach, we investigate five different pretraining signals, constructed using several graph algorithms and no external data, as well as their combination. We leverage the versatility of our Transformer-based model to explore graph structure generation pretraining tasks (i.e. path and k-hop neighborhood generation), typically inapplicable to most graph embedding methods. We further propose a new path-finding algorithm guided by information gain and find that it is the best-performing pretraining task across three downstream knowledge graph completion datasets. While using our new path-finding algorithm as a pretraining signal provides 2-3% MRR improvements, we show that pretraining on all signals together gives the best knowledge graph completion results. In a multitask setting that combines all pretraining tasks, our method surpasses the latest and strong performing knowledge graph embedding methods on all metrics for FB15K-237, on MRR and Hit@1 for WN18RRand on MRR and hit@10 for JF17K (a knowledge hypergraph dataset)

The Distance Geometry of Music

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG '05), University of Windsor, Canada, 200