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1. Maintaining the Union of Unit Discs under Insertions with Near-Optimal Overhead

   https://doi.org/10.1145/3527614  

   Agarwal, Pankaj K. ; Cohen, Ravid ; Halperin, Dan ; Mulzer, Wolfgang ( July 2022 , ACM Transactions on Algorithms)

   We present efficient dynamic data structures for maintaining the union of unit discs and the lower envelope of pseudo-lines in the plane. More precisely, we present three main results in this paper: (i) We present a linear-size data structure to maintain the union of a set of unit discs under insertions. It can insert a disc and update the union in O (( k +1)log 2 n ) time, where n is the current number of unit discs and k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. It can also compute, within the same time bound, the area of the union after the insertion of each disc. (ii) We propose a linear-size data structure for maintaining the lower envelope of a set of x -monotone pseudo-lines. It can handle insertion/deletion of a pseudo-line in O (log 2 n ) time; for a query point x 0 ∈ ℝ, it can report, in O (log n ) time, the point on the lower envelope with x -coordinate x 0 ; and for a query point q ∈ ℝ 2 , it can return all k pseudo-lines lying below q in time O (log n + k log 2 n ). (iii) We present a linear-size data structure for storing a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), so that for a query unit disc D , all input arcs intersecting D can be reported in O ( n 1/2+ɛ + k ) time, where k is the output size and ɛ > 0 is an arbitrarily small constant. A unit-circle arc can be inserted or deleted in O (log 2 n ) time. 

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2. Fair Representation Clustering with Several Protected Classes

   https://doi.org/10.1145/3531146.3533146  

   Dai, Zhen ; Makarychev, Yury ; Vakilian, Ali ( June 2022 , 2022 ACM Conference on Fairness, Accountability, and Transparency)

   We study the problem of fair k-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation k-median problem, we are given a set of points X in a metric space. Each point x ∈ X belongs to one of ℓ groups. Further, we are given fair representation parameters αj and β_j for each group j ∈ [ℓ]. We say that a k-clustering C_1, ⋅⋅⋅, C_k fairly represents all groups if the number of points from group j in cluster C_i is between α_j |C_i| and β_j |C_i| for every j ∈ [ℓ] and i ∈ [k]. The goal is to find a set of k centers and an assignment such that the clustering defined by fairly represents all groups and minimizes the ℓ_1-objective ∑_{x ∈ X} d(x, ϕ(x)). We present an O(log k)-approximation algorithm that runs in time n^{O(ℓ)}. Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both k and ℓ. We also consider an important special case of the problem where and for all j ∈ [ℓ]. For this special case, we present an O(log k)-approximation algorithm that runs in time. 

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3. Issues in the Reproducibility of Deep Learning Results

   https://doi.org/10.1109/SPMB47826.2019.9037840  

   Jean-Paul, S. ; Elseify, T. ; Obeid, I. ; Picone, J. ( December 2019 , IEEE Signal Processing in Medicine and Biology Symposium (SPMB))

   Obeid, I. ; Selesnik, I. ; Picone, J. (Ed.)

   The Neuronix high-performance computing cluster allows us to conduct extensive machine learning experiments on big data [1]. This heterogeneous cluster uses innovative scheduling technology, Slurm [2], that manages a network of CPUs and graphics processing units (GPUs). The GPU farm consists of a variety of processors ranging from low-end consumer grade devices such as the Nvidia GTX 970 to higher-end devices such as the GeForce RTX 2080. These GPUs are essential to our research since they allow extremely compute-intensive deep learning tasks to be executed on massive data resources such as the TUH EEG Corpus [2]. We use TensorFlow [3] as the core machine learning library for our deep learning systems, and routinely employ multiple GPUs to accelerate the training process. Reproducible results are essential to machine learning research. Reproducibility in this context means the ability to replicate an existing experiment – performance metrics such as error rates should be identical and floating-point calculations should match closely. Three examples of ways we typically expect an experiment to be replicable are: (1) The same job run on the same processor should produce the same results each time it is run. (2) A job run on a CPU and GPU should produce identical results. (3) A job should produce comparable results if the data is presented in a different order. System optimization requires an ability to directly compare error rates for algorithms evaluated under comparable operating conditions. However, it is a difficult task to exactly reproduce the results for large, complex deep learning systems that often require more than a trillion calculations per experiment [5]. This is a fairly well-known issue and one we will explore in this poster. Researchers must be able to replicate results on a specific data set to establish the integrity of an implementation. They can then use that implementation as a baseline for comparison purposes. A lack of reproducibility makes it very difficult to debug algorithms and validate changes to the system. Equally important, since many results in deep learning research are dependent on the order in which the system is exposed to the data, the specific processors used, and even the order in which those processors are accessed, it becomes a challenging problem to compare two algorithms since each system must be individually optimized for a specific data set or processor. This is extremely time-consuming for algorithm research in which a single run often taxes a computing environment to its limits. Well-known techniques such as cross-validation [5,6] can be used to mitigate these effects, but this is also computationally expensive. These issues are further compounded by the fact that most deep learning algorithms are susceptible to the way computational noise propagates through the system. GPUs are particularly notorious for this because, in a clustered environment, it becomes more difficult to control which processors are used at various points in time. Another equally frustrating issue is that upgrades to the deep learning package, such as the transition from TensorFlow v1.9 to v1.13, can also result in large fluctuations in error rates when re-running the same experiment. Since TensorFlow is constantly updating functions to support GPU use, maintaining an historical archive of experimental results that can be used to calibrate algorithm research is quite a challenge. This makes it very difficult to optimize the system or select the best configurations. The overall impact of all of these issues described above is significant as error rates can fluctuate by as much as 25% due to these types of computational issues. Cross-validation is one technique used to mitigate this, but that is expensive since you need to do multiple runs over the data, which further taxes a computing infrastructure already running at max capacity. GPUs are preferred when training a large network since these systems train at least two orders of magnitude faster than CPUs [7]. Large-scale experiments are simply not feasible without using GPUs. However, there is a tradeoff to gain this performance. Since all our GPUs use the NVIDIA CUDA® Deep Neural Network library (cuDNN) [8], a GPU-accelerated library of primitives for deep neural networks, it adds an element of randomness into the experiment. When a GPU is used to train a network in TensorFlow, it automatically searches for a cuDNN implementation. NVIDIA’s cuDNN implementation provides algorithms that increase the performance and help the model train quicker, but they are non-deterministic algorithms [9,10]. Since our networks have many complex layers, there is no easy way to avoid this randomness. Instead of comparing each epoch, we compare the average performance of the experiment because it gives us a hint of how our model is performing per experiment, and if the changes we make are efficient. In this poster, we will discuss a variety of issues related to reproducibility and introduce ways we mitigate these effects. For example, TensorFlow uses a random number generator (RNG) which is not seeded by default. TensorFlow determines the initialization point and how certain functions execute using the RNG. The solution for this is seeding all the necessary components before training the model. This forces TensorFlow to use the same initialization point and sets how certain layers work (e.g., dropout layers). However, seeding all the RNGs will not guarantee a controlled experiment. Other variables can affect the outcome of the experiment such as training using GPUs, allowing multi-threading on CPUs, using certain layers, etc. To mitigate our problems with reproducibility, we first make sure that the data is processed in the same order during training. Therefore, we save the data from the last experiment and to make sure the newer experiment follows the same order. If we allow the data to be shuffled, it can affect the performance due to how the model was exposed to the data. We also specify the float data type to be 32-bit since Python defaults to 64-bit. We try to avoid using 64-bit precision because the numbers produced by a GPU can vary significantly depending on the GPU architecture [11-13]. Controlling precision somewhat reduces differences due to computational noise even though technically it increases the amount of computational noise. We are currently developing more advanced techniques for preserving the efficiency of our training process while also maintaining the ability to reproduce models. In our poster presentation we will demonstrate these issues using some novel visualization tools, present several examples of the extent to which these issues influence research results on electroencephalography (EEG) and digital pathology experiments and introduce new ways to manage such computational issues. 

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4. Geometric Hitting Set for Line-Constrained Disks

   Liu, Gang ; Wang, Haitao ( August 2023 , Proceedings of the 18th Algorithms and Data Structures Symposium (WADS 2023))

   Given a set P of n weighted points and a set S of m disks in the plane, the hitting set problem is to compute a subset 𝑃′ of points of P such that each disk contains at least one point of 𝑃′ and the total weight of all points of 𝑃′ is minimized. The problem is known to be NP-hard. In this paper, we consider a line-constrained version of the problem in which all disks are centered on a line ℓ. We present an 𝑂((𝑚+𝑛)log(𝑚+𝑛)+𝜅log𝑚) time algorithm for the problem, where 𝜅 is the number of pairs of disks that intersect. For the unit-disk case where all disks have the same radius, the running time can be reduced to 𝑂((𝑛+𝑚)log(𝑚+𝑛)). In addition, we solve the problem in 𝑂((𝑚+𝑛)log(𝑚+𝑛)) time in the 𝐿∞ and 𝐿1 metrics, in which a disk is a square and a diamond, respectively. 

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5. One Hop Greedy Permutations

   Sheehy, Donald R. ( January 2020 , Proceedings of the 32nd Canadian Conference on Computational Geometry)

   Keil, Mark ; Mondal, Debajyoti (Ed.)

   We adapt and generalize a heuristic for k-center clustering to the permutation case, where every prex of the ordering is a guaranteed approximate solution. The one-hop greedy permutations work by choosing at each step the farthest unchosen point and then looking in its local neighborhood for a point that covers the most points at a certain scale. This balances the competing demands of reducing the coverage radius and also covering as many points as possible. This idea first appeared in the work of Garcia-Diaz et al. and their algorithm required O(n2 log n) time for a fixed k (i.e. not the whole permutation). We show how to use geometric data structures to approximate the entire permutation in O(n log D
   ) time for metrics sets with spread D. Notably, this running time is asymptotically the same as the running time for computing the ordinary greedy permutation. 

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