Search

Given a real number α ∈ (0 , 1) , we define the Webster sequence of density α to be W α = (⌈ (n − 1 / 2) / α ⌉) n ∈ N , where ⌈ x ⌉ is the ceiling function. It is known that if α and β are irrational with α + β = 1 , then W α and W β partition N. On the other hand, an analogous result for three-part partitions does not hold: There does not exist a partition of N into sequences W α , W β , W γ with α , β , γ irrational. In this paper, we consider the question of how close one can come to a three-part partition of N into Webster sequences with prescribed densities α , β , γ. We show that if α , β , γ are irrational with α + β + γ = 1 , there exists a partition of N into sequences of densities α , β , γ , in which one of the sequences is a Webster sequence and the other two are "almost" Webster sequences that are obtained from Webster sequences by perturbing some elements by at most 1. We also provide two efficient algorithms to construct such partitions. The first algorithm outputs the n th element of each sequence in O (1) time and the second algorithm gives the assignment of the m th positive integer to a sequence in O (1) time. We show that the results are best-possible in several respects. Moreover, we describe applications of these results to apportionment and optimal scheduling problems. [ABSTRACT FROM AUTHOR]

Semantic segmentation has been a major topic in research and industry in recent years. However, due to the computation complexity of pixel-wise prediction and backpropagation algorithm, semantic segmentation has been demanding in computation resources, resulting in slow training and inference speed and large storage space to store models. Existing schemes that speed up segmentation network change the network structure and come with noticeable accuracy degradation. However, neural network quantization can be used to reduce computation load while maintaining comparable accuracy and original network structure. Semantic segmentation networks are different from traditional deep convolutional neural networks (DCNNs) in many ways, and this topic has not been thoroughly explored in existing works. In this paper, we propose a new quantization framework for training and inference of segmentation networks, where parameters and operations are constrained to 8-bit integer-based values for the first time. Full quantization of the data flow and the removal of square and root operations in batch normalization give our framework the ability to perform inference on fixed-point devices. Our proposed framework is evaluated on mainstream semantic segmentation networks like FCN-VGG16 and DeepLabv3-ResNet50, achieving comparable accuracy against floating-point framework on ADE20K dataset and PASCAL VOC 2012 dataset. [ABSTRACT FROM AUTHOR]