An \~{O}(n2)(n^2) Time Matrix Multiplication Algorithm

We show, for the input vectors $(a_0, a_1, ..., a_{n-1})$ and $(b_0, b_1, ..., b_{n-1})$, where $a_i$'s and $b_j$'s are real numbers, after \~{O}$(n)$ time preprocessing for each of them, the vector multiplication $(a_0, a_1, ..., a_{n-1})(b_0, b_1, ..., b_{n-1})^T$ can be computed in \~{O}$(1)$ time. This enables the matrix multiplication of two $n\times n$ matrices to be computed in \~{O}$(n^2)$ time.Comment: Version 11 and Version 12 section 2 laid the foundation of this algorithm but has a problem unresolved. This version corrects the problem in Version 11 and Section 2 of Version 1

Parallel Merging and Sorting on Linked List

We study linked list sorting and merging on the PRAM model. In this paper we show that n real numbers can be sorted into a linked list in constant time with n2+e processors or in ) time with n2 processors. We also show that two sorted linked lists of n integers in {0, 1, …, m}  can be merged into one sorted linked list in O(log(c)n(loglogm)1/2) time using n/(log(c)n(loglogm)1/2)  processors, where c is an arbitrarily large constant

DISTRIBUTION OF PATH DURATION IN WIRELESS AD-HOC NETWORKS AND PATH SELECTION

The performance of routing protocols in wireless ad-hoc networks is determined by a number of factors, among which the path durations are of much importance. Path durations affect the reliability of the network service provided to the applications and the routing overhead incurred. In this dissertation, we study the distribution of path duration in wireless ad-hoc networks and its impact on routing. We focus on identifying a scheme that selects a path with the largest expected duration for data transmission when multiple paths are available between a source-destination pair. To this end, we first study the distribution of path duration. Our analytical result also reveals the relation between link level and path level statistics, which can be used to estimate expected path durations. Our main results show that in a large scale wireless ad-hoc network, as long as the local dependency among link excess lives is not too strong, the path durations can be well approximated by an exponential random variable for paths with sufficiently large hop count. Furthermore, the inverse of the expected duration of a path can be estimated using the sum of the inverses of the expected durations of the links along the path. Based on these analytical results, we propose a new path selection scheme referred to as ``Maximum Expected Duration" (MED) path selection. This scheme can be easily incorporated into existing routing protocols. Information needed for stimating the expected path duration can be collected during path discovery phase. Our simulation results demonstrate that under routing protocols with the MED path selection scheme added, the median value of the path durations can be increased up to $60$ percent compared to those without using the scheme. Moreover, we reduce the delay and overhead during local path recovery by using cached paths that are likely to be available when the primary path breaks down

Sorting Real Numbers in Constant Time Using n^2/log^cn Processors

We study the sorting of real numbers into a linked list on the PRAM (Parallel Random-Access Machine) model. We show that n real numbers can be sorted into a linked list in constant time using n2 processors. Previously n numbers can be sorted into a linked list using n2 processors in O(loglogn) time. We also study the time processor trade-off for sorting real numbers into a linked list on the PRAM (Parallel Random Access Machine) model. We show that n real numbers can be sorted into a linked list with n2/t processors in O(logt) time. Previously n real numbers can be sorted into a linked list using n3 processors in constant time and n2 processors in O(loglogn). And then we show that input array of n real numbers  can be sorted into linked list in constant time using n2/logcn  processors for any positive constant c. We believe that further reduction on the number of processors for sorting real numbers in constant time will be very difficult if not impossible

Investigation of wave propagation in piezoelectric helical waveguides with the spectral finite element method

The dispersion behaviors of wave propagation in waveguides of piezoelectric helical structures are investigated. By using the tensor analysis in the helical curve coordinate, the general strain − displacement relationship of piezoelectric helix is firstly considered. This paper's formulation is based on the spectral finite element which just requires the discretization of the cross-section with high-order spectral elements. The eigenvalue matrix of the dispersion relationship between wavenumbers and frequencies is obtained. Numerical examples on PZT5A and Ba2NaNb5O15 helical waveguides of a wide range of lay angles are presented. The effects of the piezoelectric on the dispersive properties and the variation tendency of dispersion curves on helix angles are shown. The mechanism of mode separation in piezoelectric helical waveguides is further analyzed through studying waves structures of the flexural modes

Resequencing delays under multipath routing -- Asymptotics in a simple queueing model

We study the resequencing delay caused by multipath routing. We use a queueing model which consists of parallel queues to model the network routing behavior. We define a new metric denoted by $gamma$, to study the impact of resequencing on the customer end-to-end delay. Our results characterize some properties of $gamma$ with respect to different service time distributions. In particular, the resequencing delay can be negligible when the delay along each path is light-tailed, but can be of major concern when it is heavy-tailed

EffLiFe: Efficient Light Field Generation via Hierarchical Sparse Gradient Descent

With the rise of Extended Reality (XR) technology, there is a growing need for real-time light field generation from sparse view inputs. Existing methods can be classified into offline techniques, which can generate high-quality novel views but at the cost of long inference/training time, and online methods, which either lack generalizability or produce unsatisfactory results. However, we have observed that the intrinsic sparse manifold of Multi-plane Images (MPI) enables a significant acceleration of light field generation while maintaining rendering quality. Based on this insight, we introduce EffLiFe, a novel light field optimization method, which leverages the proposed Hierarchical Sparse Gradient Descent (HSGD) to produce high-quality light fields from sparse view images in real time. Technically, the coarse MPI of a scene is first generated using a 3D CNN, and it is further sparsely optimized by focusing only on important MPI gradients in a few iterations. Nevertheless, relying solely on optimization can lead to artifacts at occlusion boundaries. Therefore, we propose an occlusion-aware iterative refinement module that removes visual artifacts in occluded regions by iteratively filtering the input. Extensive experiments demonstrate that our method achieves comparable visual quality while being 100x faster on average than state-of-the-art offline methods and delivering better performance (about 2 dB higher in PSNR) compared to other online approaches.Comment: Submitted to IEEE TPAM