arXiv subjects

Yan Gu

Publications and source records attributed to Yan Gu.

At least 91 records · Page 5Linked to original sources

arXiv2021

Efficient Stepping Algorithms and Implementations for Parallel Shortest Paths

In this paper, we study the single-source shortest-path (SSSP) problem with positive edge weights, which is a notoriously hard problem in the parallel context. In practice, the $Δ$-stepping algorithm proposed by Meyer and Sanders has been widely adopted. However, $Δ$-stepping has no known worst-case bounds for general graphs. The performance of $Δ$-stepping also highly relies on the parameter $Δ$. There have also been lots of algorithms with theoretical bounds, such as Radius-stepping, but they either have no implementations available or are much slower than $Δ$-stepping in practice. We propose a stepping algorithm framework that generalizes existing algorithms such as $Δ$-stepping and Radius-stepping. The framework allows for similar analysis and implementations of all stepping algorithms. We also propose a new ADT, lazy-batched priority queue (LaB-PQ), that abstracts the semantics of the priority queue needed by the stepping algorithms. We provide two data structures for LaB-PQ, focusing on theoretical and practical efficiency, respectively. Based on the new framework and LaB-PQ, we show two new stepping algorithms, $ρ$-stepping and $Δ^*$-stepping, that are simple, with non-trivial worst-case bounds, and fast in practice. The stepping algorithm framework also provides almost identical implementations for three algorithms: Bellman-Ford, $Δ^*$-stepping, and $ρ$-stepping. We compare our code with four state-of-the-art implementations. On five social and web graphs, $ρ$-stepping is 1.3--2.5x faster than all the existing implementations. On two road graphs, our $Δ^*$-stepping is at least 14\% faster than existing implementations, while $ρ$-stepping is also competitive. The almost identical implementations for stepping algorithms also allow for in-depth analyses and comparisons among the stepping algorithms in practice.

cs.DS↗

arXiv2021

Analysis of Work-Stealing and Parallel Cache Complexity

Parallelism has become extremely popular over the past decade, and there have been a lot of new parallel algorithms and software. The randomized work-stealing (RWS) scheduler plays a crucial role in this ecosystem. In this paper, we study two important topics related to the randomized work-stealing scheduler. Our first contribution is a simplified, classroom-ready version of analysis for the RWS scheduler. The theoretical efficiency of the RWS scheduler has been analyzed for a variety of settings, but most of them are quite complicated. In this paper, we show a new analysis, which we believe is easy to understand, and can be especially useful in education. We avoid using the potential function in the analysis, and we assume a highly asynchronous setting, which is more realistic for today's parallel machines. Our second and main contribution is some new parallel cache complexity for algorithms using the RWS scheduler. Although the sequential I/O model has been well-studied over the past decades, so far very few results have extended it to the parallel setting. The parallel cache bounds of many existing algorithms are affected by a polynomial of the span, which causes a significant overhead for high-span algorithms. Our new analysis decouples the span from the analysis of the parallel cache complexity. This allows us to show new parallel cache bounds for a list of classic algorithms. Our results are only a polylogarithmic factor off the lower bounds, and significantly improve previous results.

cs.DS↗

arXiv2021

Extended Capture Point and Optimization-based Control for Quadrupedal Robot Walking on Dynamic Rigid Surfaces

Stabilizing legged robot locomotion on a dynamic rigid surface (DRS) (i.e., rigid surface that moves in the inertial frame) is a complex planning and control problem. The complexity arises due to the hybrid nonlinear walking dynamics subject to explicitly time-varying holonomic constraints caused by the surface movement. The first main contribution of this study is the extension of the capture point from walking on a static surface to locomotion on a DRS as well as the use of the resulting capture point for online motion planning. The second main contribution is a quadratic-programming (QP) based feedback controller design that explicitly considers the DRS movement. The stability and robustness of the proposed control approach are validated through simulations of a quadrupedal robot walking on a DRS with a rocking motion. The simulation results also demonstrate the improved walking performance compared with our previous approach based on offline planning and input-output linearizing control that does not explicitly guarantee the feasibility of ground contact constraints.

cs.RO↗

arXiv2021

Invariant Filtering for Bipedal Walking on Dynamic Rigid Surfaces with Orientation-based Measurement Model

Real-world applications of bipedal robot walking require accurate, real-time state estimation. State estimation for locomotion over dynamic rigid surfaces (DRS), such as elevators, ships, public transport vehicles, and aircraft, remains under-explored, although state estimator designs for stationary rigid surfaces have been extensively studied. Addressing DRS locomotion in state estimation is a challenging problem mainly due to the nonlinear, hybrid nature of walking dynamics, the nonstationary surface-foot contact points, and hardware imperfections (e.g., limited availability, noise, and drift of onboard sensors). Towards solving this problem, we introduce an Invariant Extended Kalman Filter (InEKF) whose process and measurement models explicitly consider the DRS movement and hybrid walking behaviors while respectively satisfying the group-affine condition and invariant form. Due to these attractive properties, the estimation error convergence of the filter is provably guaranteed for hybrid DRS locomotion. The measurement model of the filter also exploits the holonomic constraint associated with the support-foot and surface orientations, under which the robot's yaw angle in the world becomes observable in the presence of general DRS movement. Experimental results of bipedal walking on a rocking treadmill demonstrate the proposed filter ensures the rapid error convergence and observable base yaw angle.

cs.RO↗

arXiv2021

Global-Position Tracking Control of 3-D Bipedal Walking via Virtual Constraint Design and Multiple Lyapunov Analysis

A safety-critical measure of legged locomotion performance is a robot's ability to track its desired time-varying position trajectory in an environment, which is herein termed as "global-position tracking". This paper introduces a nonlinear control approach that achieves asymptotic global-position tracking for three-dimensional (3-D) bipedal robot walking. Designing a global-position tracking controller presents a challenging problem due to the complex hybrid robot model and the time-varying desired global-position trajectory. Towards tackling this problem, the first main contribution is the construction of impact invariance to ensure all desired trajectories respect the foot-landing impact dynamics, which is a necessary condition for realizing asymptotic tracking of hybrid walking systems. Thanks to their independence of the desired global position, these conditions can be exploited to decouple the higher-level planning of the global position and the lower-level planning of the remaining trajectories, thereby greatly alleviating the computational burden of motion planning. The second main contribution is the Lyapunov-based stability analysis of the hybrid closed-loop system, which produces sufficient conditions to guide the controller design for achieving asymptotic global-position tracking during fully actuated walking. Simulations and experiments on a 3-D bipedal robot with twenty revolute joints confirm the validity of the proposed control approach in guaranteeing accurate tracking.

cs.RO↗

arXiv2021

Symbolic Powers of Cover Ideals of Graphs and Koszul Property

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

math.AC↗

arXiv2021

Fast Parallel Algorithms for Euclidean Minimum Spanning Tree and Hierarchical Spatial Clustering

This paper presents new parallel algorithms for generating Euclidean minimum spanning trees and spatial clustering hierarchies (known as HDBSCAN$^*$). Our approach is based on generating a well-separated pair decomposition followed by using Kruskal's minimum spanning tree algorithm and bichromatic closest pair computations. We introduce a new notion of well-separation to reduce the work and space of our algorithm for HDBSCAN$^*$. We also present a parallel approximate algorithm for OPTICS based on a recent sequential algorithm by Gan and Tao. Finally, we give a new parallel divide-and-conquer algorithm for computing the dendrogram and reachability plots, which are used in visualizing clusters of different scale that arise for both EMST and HDBSCAN$^*$. We show that our algorithms are theoretically efficient: they have work (number of operations) matching their sequential counterparts, and polylogarithmic depth (parallel time). We implement our algorithms and propose a memory optimization that requires only a subset of well-separated pairs to be computed and materialized, leading to savings in both space (up to 10x) and time (up to 8x). Our experiments on large real-world and synthetic data sets using a 48-core machine show that our fastest algorithms outperform the best serial algorithms for the problems by 11.13--55.89x, and existing parallel algorithms by at least an order of magnitude.

cs.DS↗

arXiv2021

A Parallel Batch-Dynamic Data Structure for the Closest Pair Problem

We propose a theoretically-efficient and practical parallel batch-dynamic data structure for the closest pair problem. Our solution is based on a serial dynamic closest pair data structure by Golin et al., and supports batches of insertions and deletions in parallel. For a data set of size $n$, our data structure supports a batch of insertions or deletions of size $m$ in $O(m(1+\log ((n+m)/m)))$ expected work and $O(\log (n+m)\log^*(n+m))$ depth with high probability, and takes linear space. The key techniques for achieving these bounds are a new work-efficient parallel batch-dynamic binary heap, and careful management of the computation across sets of points to minimize work and depth. We provide an optimized multicore implementation of our data structure using dynamic hash tables, parallel heaps, and dynamic $k$-d trees. Our experiments on a variety of synthetic and real-world data sets show that it achieves a parallel speedup of up to 38.57x (15.10x on average) on 48 cores with hyper-threading. In addition, we also implement and compare four parallel algorithms for static closest pair problem, for which we are not aware of any existing practical implementations. On 48 cores with hyper-threading, the static algorithms achieve up to 51.45x (29.42x on average) speedup, and Rabin's algorithm performs the best on average. Comparing our dynamic algorithm to the fastest static algorithm, we find that it is advantageous to use the dynamic algorithm for batch sizes of up to 20\% of the data set. As far as we know, our work is the first to experimentally evaluate parallel closest pair algorithms, in both the static and the dynamic settings.

cs.DS↗

arXiv2021

Parallel In-Place Algorithms: Theory and Practice

Many parallel algorithms use at least linear auxiliary space in the size of the input to enable computations to be done independently without conflicts. Unfortunately, this extra space can be prohibitive for memory-limited machines, preventing large inputs from being processed. Therefore, it is desirable to design parallel in-place algorithms that use sublinear (or even polylogarithmic) auxiliary space. In this paper, we bridge the gap between theory and practice for parallel in-place (PIP) algorithms. We first define two computational models based on fork-join parallelism, which reflect modern parallel programming environments. We then introduce a variety of new parallel in-place algorithms that are simple and efficient, both in theory and in practice. Our algorithmic highlight is the Decomposable Property introduced in this paper, which enables existing non-in-place but highly-optimized parallel algorithms to be converted into parallel in-place algorithms. Using this property, we obtain algorithms for random permutation, list contraction, tree contraction, and merging that take linear work, $O(n^{1-ε})$ auxiliary space, and $O(n^ε\cdot\text{polylog}(n))$ span for $0<ε<1$. We also present new parallel in-place algorithms for scan, filter, merge, connectivity, biconnectivity, and minimum spanning forest using other techniques. In addition to theoretical results, we present experimental results for implementations of many of our parallel in-place algorithms. We show that on a 72-core machine with two-way hyper-threading, the parallel in-place algorithms usually outperform existing parallel algorithms for the same problems that use linear auxiliary space, indicating that the theory developed in this paper indeed leads to practical benefits in terms of both space usage and running time.

cs.DC↗

arXiv2021

Theoretically-Efficient and Practical Parallel DBSCAN

The DBSCAN method for spatial clustering has received significant attention due to its applicability in a variety of data analysis tasks. There are fast sequential algorithms for DBSCAN in Euclidean space that take $O(n\log n)$ work for two dimensions, sub-quadratic work for three or more dimensions, and can be computed approximately in linear work for any constant number of dimensions. However, existing parallel DBSCAN algorithms require quadratic work in the worst case, making them inefficient for large datasets. This paper bridges the gap between theory and practice of parallel DBSCAN by presenting new parallel algorithms for Euclidean exact DBSCAN and approximate DBSCAN that match the work bounds of their sequential counterparts, and are highly parallel (polylogarithmic depth). We present implementations of our algorithms along with optimizations that improve their practical performance. We perform a comprehensive experimental evaluation of our algorithms on a variety of datasets and parameter settings. Our experiments on a 36-core machine with hyper-threading show that we outperform existing parallel DBSCAN implementations by up to several orders of magnitude, and achieve speedups by up to 33x over the best sequential algorithms.

cs.DS↗

arXiv2020

Optimal (Randomized) Parallel Algorithms in the Binary-Forking Model

In this paper we develop optimal algorithms in the binary-forking model for a variety of fundamental problems, including sorting, semisorting, list ranking, tree contraction, range minima, and ordered set union, intersection and difference. In the binary-forking model, tasks can only fork into two child tasks, but can do so recursively and asynchronously. The tasks share memory, supporting reads, writes and test-and-sets. Costs are measured in terms of work (total number of instructions), and span (longest dependence chain). The binary-forking model is meant to capture both algorithm performance and algorithm-design considerations on many existing multithreaded languages, which are also asynchronous and rely on binary forks either explicitly or under the covers. In contrast to the widely studied PRAM model, it does not assume arbitrary-way forks nor synchronous operations, both of which are hard to implement in modern hardware. While optimal PRAM algorithms are known for the problems studied herein, it turns out that arbitrary-way forking and strict synchronization are powerful, if unrealistic, capabilities. Natural simulations of these PRAM algorithms in the binary-forking model (i.e., implementations in existing parallel languages) incur an $Ω(\log n)$ overhead in span. This paper explores techniques for designing optimal algorithms when limited to binary forking and assuming asynchrony. All algorithms described in this paper are the first algorithms with optimal work and span in the binary-forking model. Most of the algorithms are simple. Many are randomized.

cs.DS↗

arXiv2020

Sage: Parallel Semi-Asymmetric Graph Algorithms for NVRAMs

Non-volatile main memory (NVRAM) technologies provide an attractive set of features for large-scale graph analytics, including byte-addressability, low idle power, and improved memory-density. NVRAM systems today have an order of magnitude more NVRAM than traditional memory (DRAM). NVRAM systems could therefore potentially allow very large graph problems to be solved on a single machine, at a modest cost. However, a significant challenge in achieving high performance is in accounting for the fact that NVRAM writes can be much more expensive than NVRAM reads. In this paper, we propose an approach to parallel graph analytics using the Parallel Semi-Asymmetric Model (PSAM), in which the graph is stored as a read-only data structure (in NVRAM), and the amount of mutable memory is kept proportional to the number of vertices. Similar to the popular semi-external and semi-streaming models for graph analytics, the PSAM approach assumes that the vertices of the graph fit in a fast read-write memory (DRAM), but the edges do not. In NVRAM systems, our approach eliminates writes to the NVRAM, among other benefits. To experimentally study this new setting, we develop Sage, a parallel semi-asymmetric graph engine with which we implement provably-efficient (and often work-optimal) PSAM algorithms for over a dozen fundamental graph problems. We experimentally study Sage using a 48-core machine on the largest publicly-available real-world graph (the Hyperlink Web graph with over 3.5 billion vertices and 128 billion edges) equipped with Optane DC Persistent Memory, and show that Sage outperforms the fastest prior systems designed for NVRAM. Importantly, we also show that Sage nearly matches the fastest prior systems running solely in DRAM, by effectively hiding the costs of repeatedly accessing NVRAM versus DRAM.

cs.DC↗

arXiv2019

Impact-Aware Online Motion Planning for Fully-Actuated Bipedal Robot Walking

The ability to track a general walking path with specific timing is crucial to the operational safety and reliability of bipedal robots for avoiding dynamic obstacles, such as pedestrians, in complex environments. This paper introduces an online, full-body motion planner that generates the desired impact-aware motion for fully-actuated bipedal robotic walking. The main novelty of the proposed planner lies in its capability of producing desired motions in real-time that respect the discrete impact dynamics and the desired impact timing. To derive the proposed planner, a full-order hybrid dynamic model of fully-actuated bipedal robotic walking is presented, including both continuous dynamics and discrete lading impacts. Next, the proposed impact-aware online motion planner is introduced. Finally, simulation results of a 3-D bipedal robot are provided to confirm the effectiveness of the proposed online impact-aware planner. The online planner is capable of generating full-body motion of one walking step within 0.6 second, which is shorter than a typical bipedal walking step.

cs.RO↗

arXiv2019

Improved Parallel Cache-Oblivious Algorithms for Dynamic Programming and Linear Algebra

Emerging non-volatile main memory (NVRAM) technologies provide byte-addressability, low idle power, and improved memory-density, and are likely to be a key component in the future memory hierarchy. However, a critical challenge in achieving high performance is in accounting for the asymmetry that NVRAM writes can be significantly more expensive than NVRAM reads. In this paper, we consider a large class of cache-oblivious algorithms for dynamic programming (DP) and linear algebra, and try to reduce the writes in the asymmetric setting while maintaining high parallelism. To achieve that, our key approach is to show the correspondence between these problems and an abstraction for their computation, which is referred to as the $k$-d grids. Then by showing lower bound and new algorithms for computing $k$-d grids, we show a list of improved cache-oblivious algorithms of many DP recurrences and in linear algebra in the asymmetric setting, both sequentially and in parallel. Surprisingly, even without considering the read-write asymmetry (i.e., setting the write cost to be the same as the read cost in the algorithms), the new algorithms improve the existing cache complexity of many problems. We believe the reason is that the extra level of abstraction of $k$-d grids helps us to better understand the complexity and difficulties of these problems. We believe that the novelty of our framework is of interests and leads to many new questions for future work.

cs.DS↗

arXiv2019

Measuring outcome correlation for spin-s Bell cat-state and geometric phase induced spin parity effect

In terms of quantum probability statistics the Bell inequality (BI) and its violation are extended to spin-$s$ entangled Schrödinger cat-state (called the Bell cat-state) with both parallel and antiparallel spin-polarizations. The BI is never ever violated for the measuring outcome probabilities evaluated over entire two-spin Hilbert space except the spin-$1/2$ entangled states. A universal Bell-type inequality (UBI) denoted by $p_{s}^{lc}\leq0$ is formulated with the local realistic model under the condition that the measuring outcomes are restricted in the subspace of spin coherent states. A spin parity effect is observed that the UBI can be violated only by the Bell cat-states of half-integer but not the integer spins. The violation of UBI is seen to be a direct result of non-trivial Berry phase between the spin coherent states of south- and north-pole gauges for half-integer spin, while the geometric phase is trivial for the integer spins. A maximum violation bound of UBI is found as $p_{s}^{\max}$=1, which is valid for arbitrary half-integer spin-$s$ states.

quant-ph↗

arXiv2019

Alternating Direction Method of Multipliers with Variable Metric Indefinite Proximal Terms for Convex Optimization

This paper studies a proximal alternating direction method of multipliers (ADMM) with variable metric indefinite proximal terms for linearly constrained convex optimization problems. The proximal ADMM plays an important role in many application areas, since the subproblems of the method are easy to solve. Recently, it is reported that the proximal ADMM with a certain fixed indefinite proximal term is faster than that with a positive semidefinite term, and still has the global convergence property. On the other hand, Gu and Yamashita studied a variable metric semidefinite proximal ADMM whose proximal term is generated by the BFGS update. They reported that a slightly indefinite matrix also makes the algorithm work well in their numerical experiments. Motivated by this fact, we consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, we propose a new indefinite proximal term based on the BFGS update which can satisfy the conditions for the global convergence.

math.OC↗

arXiv2019

An Alternating Direction Method of Multipliers with the BFGS Update for Structured Convex Quadratic Optimization

The alternating direction method of multipliers (ADMM) is an effective method for solving wide fields of convex problems. At each iteration, the classical ADMM solves two subproblems exactly. However, in many applications, it is expensive or impossible to obtain the exact solutions of the subproblems. To overcome the difficulty, some proximal terms are added to the subproblems. This class of methods normally solves the original subproblem approximately, and thus takes more iterations. This fact urges us to consider that a special proximal term can lead to a better result as the classical ADMM. In this paper, we propose a proximal ADMM whose regularized matrix in the proximal term is generated by the BFGS update (or limited memory BFGS) at every iteration. These types of matrices use second-order information of the objective function. The convergence of the proposed method is proved under certain assumptions. Numerical results are presented to show the effectiveness of the proposed proximal ADMM.

math.OC↗

arXiv2019

Symbolic powers of edge ideals of graphs

Let $G$ be a graph and let $I = I(G)$ be its edge ideal. When $G$ is unicyclic, we give a decomposition of symbolic powers of $I$ in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of $I$. When $G$ is an odd cycle, we explicitly compute the regularity of $I^{(s)}$ for all $s \in \mathbb{N}$. In doing so, we also give a natural lower bound for the regularity function $\text{reg } I^{(s)}$, for $s \in \mathbb{N}$, for an arbitrary graph $G$.

math.AC↗