Fast k-means based on KNN Graph

In the era of big data, k-means clustering has been widely adopted as a basic processing tool in various contexts. However, its computational cost could be prohibitively high as the data size and the cluster number are large. It is well known that the processing bottleneck of k-means lies in the operation of seeking closest centroid in each iteration. In this paper, a novel solution towards the scalability issue of k-means is presented. In the proposal, k-means is supported by an approximate k-nearest neighbors graph. In the k-means iteration, each data sample is only compared to clusters that its nearest neighbors reside. Since the number of nearest neighbors we consider is much less than k, the processing cost in this step becomes minor and irrelevant to k. The processing bottleneck is therefore overcome. The most interesting thing is that k-nearest neighbor graph is constructed by iteratively calling the fast $k$-means itself. Comparing with existing fast k-means variants, the proposed algorithm achieves hundreds to thousands times speed-up while maintaining high clustering quality. As it is tested on 10 million 512-dimensional data, it takes only 5.2 hours to produce 1 million clusters. In contrast, to fulfill the same scale of clustering, it would take 3 years for traditional k-means

Development of Rotary Variable Damping and Stiffness Magnetorheological Dampers and their Applications on Robotic Arms and Seat Suspensions

This thesis successfully expanded the idea of variable damping and stiffness (VSVD) from linear magnetorheological dampers (MR) to rotary magnetorheological dampers; and explored the applications of rotary MR dampers on the robotic arms and seat suspension. The idea of variable damping and stiffness has been proved to be able to reduce vibration to a large degree. Variable damping can reduce the vibration amplitude and variable stiffness can shift the natural frequency of the system from excitation and prevent resonance. Linear MR dampers with the capacity of variable damping and stiffness have been studied by researchers. However, Linear MR dampers usually require larger installation space than rotary MR dampers, and need more expensive MR fluids to fill in their chambers. Furthermore, rotary MR dampers are inherently more suitable than linear MR dampers in rotary motions like braking devices or robot joints. Hence, rotary MR dampers capable of simultaneously varying the damping and stiffness are very attractive to solve angular vibration problems. Out of this motivation, a rotary VSVD MR damper was designed, prototyped, with its feature of variable damping and stiffness verified by experimental property tests in this thesis. Its mathematical model was also built with the parameters identified. The experimental tests indicated that it has a 141.6% damping variation and 618.1% stiffness variation. This damper’s successful development paved the way for the applications of rotary MR dampers with the similar capability of variable damping and stiffness

On the Min-Max-Delay Problem: NP-completeness, Algorithm, and Integrality Gap

We study a delay-sensitive information flow problem where a source streams information to a sink over a directed graph G(V,E) at a fixed rate R possibly using multiple paths to minimize the maximum end-to-end delay, denoted as the Min-Max-Delay problem. Transmission over an edge incurs a constant delay within the capacity. We prove that Min-Max-Delay is weakly NP-complete, and demonstrate that it becomes strongly NP-complete if we require integer flow solution. We propose an optimal pseudo-polynomial time algorithm for Min-Max-Delay, with time complexity O(\log (Nd_{\max}) (N^5d_{\max}^{2.5})(\log R+N^2d_{\max}\log(N^2d_{\max}))), where N = \max\{|V|,|E|\} and d_{\max} is the maximum edge delay. Besides, we show that the integrality gap, which is defined as the ratio of the maximum delay of an optimal integer flow to the maximum delay of an optimal fractional flow, could be arbitrarily large