arXiv subjects

Jianzhong Li

Publications and source records attributed to Jianzhong Li.

At least 19 recordsLinked to original sources

arXiv2026

On Delay-robustness of Extremum Seeking of Nonlinear Static Maps with Small Disturbance

Extremum seeking (ES) is a real-time optimization strategy, thus transmission delays in the feedback loop of ES have big impact on its stability. How big delay that ES control systems are able to withstand? This paper provides a potential answer to this problem. We focus on gradient-based ES for nonlinear static maps subject to known constant delays plus a small time-varying delay uncertainty. We also consider the measurement to be subject to a small disturbance. Different from a majority of existing literature addressing quadratic maps with delays by predictor feedback, this paper deals with a wider class of non-quadratic maps without any predictor or observer for delay compensation. Dither signals in modulation and demodulation are carefully designed to handle constant delays and time-varying delay uncertainties. When the nonlinear map is unknown, we offer a rigorously analytical framework of ES convergence and delay-robustness. When some a prior knowledge of nonlinear maps is available, we are able to provide a quantitative estimation on upper bounds of time delay and dither periods to keep ES systems to remain stable. A suitable choice of ES parameters guarantees practical stability for any large known constant delay.

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arXiv2024

Convex-area-wise Linear Regression and Algorithms for Data Analysis

This paper introduces a new type of regression methodology named as Convex-Area-Wise Linear Regression(CALR), which separates given datasets by disjoint convex areas and fits different linear regression models for different areas. This regression model is highly interpretable, and it is able to interpolate any given datasets, even when the underlying relationship between explanatory and response variables are non-linear and discontinuous. In order to solve CALR problem, 3 accurate algorithms are proposed under different assumptions. The analysis of correctness and time complexity of the algorithms are given, indicating that the problem can be solved in $o(n^2)$ time accurately when the input datasets have some special features. Besides, this paper introduces an equivalent mixed integer programming problem of CALR which can be approximately solved using existing optimization solvers.

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arXiv2024

Balanced Partitioning for Optimizing Big Graph Computation: Complexities and Approximation Algorithms

Graph partitioning is a key fundamental problem in the area of big graph computation. Previous works do not consider the practical requirements when optimizing the big data analysis in real applications. In this paper, motivated by optimizing the big data computing applications, two typical problems of graph partitioning are studied. The first problem is to optimize the performance of specific workloads by graph partitioning, which lacks of algorithms with performance guarantees. The second problem is to optimize the computation of motifs by graph partitioning, which has not been focused by previous works. First, the formal definitions of the above two problems are introduced, and the semidefinite programming representations are also designed based on the analysis of the properties of the two problems. For the motif based partitioning problem, it is proved to be NP-complete even for the special case of $k=2$ and the motif is a triangle, and its inapproximability is also shown by proving that there are no efficient algorithms with finite approximation ratio. Finally, using the semidefinite programming and sophisticated rounding techniques, the bi-criteria $O(\sqrt{\log n\log k})$-approximation algorithms with polynomial time cost are designed and analyzed for them.

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arXiv2024

Computing All Restricted Skyline Probabilities on Uncertain Datasets

Restricted skyline (rskyline) query is widely used in multi-criteria decision making. It generalizes the skyline query by additionally considering a set of personalized scoring functions F. Since uncertainty is inherent in datasets for multi-criteria decision making, we study rskyline queries on uncertain datasets from both complexity and algorithm perspective. We formalize the problem of computing rskyline probabilities of all data items and show that no algorithm can solve this problem in truly subquadratic-time, unless the orthogonal vectors conjecture fails. Considering that linear scoring functions are widely used in practical applications, we propose two efficient algorithms for the case where $\calF$ is a set of linear scoring functions whose weights are described by linear constraints, one with near-optimal time complexity and the other with better expected time complexity. For special linear constraints involving a series of weight ratios, we further devise an algorithm with sublinear query time and polynomial preprocessing time. Extensive experiments demonstrate the effectiveness, efficiency, scalability, and usefulness of our proposed algorithms.

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arXiv2024

The PCP-like Theorem for Sub-linear Time Inapproximability

In this paper we propose the PCP-like theorem for sub-linear time inapproximability. Abboud et al. have devised the distributed PCP framework for proving sub-quadratic time inapproximability. Here we try to go further in this direction. Staring from SETH, we first find a problem denoted as Ext-$k$-SAT, which can not be computed in linear time, then devise an efficient MA-like protocol for this problem. To use this protocol to prove the sub-linear time inapproximability of other problems, we devise a new kind of reduction denoted as Ext-reduction, and it is different from existing reduction techniques. We also define two new hardness class, the problems in which can be computed in linear-time, but can not be efficiently approximated in sub-linear time. Some problems are shown to be in the newly defined hardness class.

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arXiv2023

Testing Higher-order Clusterability on graphs

Analysis of higher-order organizations, usually small connected subgraphs called motifs, is a fundamental task on complex networks. This paper studies a new problem of testing higher-order clusterability: given query access to an undirected graph, can we judge whether this graph can be partitioned into a few clusters of highly-connected motifs? This problem is an extension of the former work proposed by Czumaj et al. (STOC' 15), who recognized cluster structure on graphs using the framework of property testing. In this paper, a good graph cluster on high dimensions is first defined for higher-order clustering. Then, query lower bound is given for testing whether this kind of good cluster exists. Finally, an optimal sublinear-time algorithm is developed for testing clusterability based on triangles.

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arXiv2023

An Efficient Data Analysis Method for Big Data using Multiple-Model Linear Regression

This paper introduces a new data analysis method for big data using a newly defined regression model named multiple model linear regression(MMLR), which separates input datasets into subsets and construct local linear regression models of them. The proposed data analysis method is shown to be more efficient and flexible than other regression based methods. This paper also proposes an approximate algorithm to construct MMLR models based on $(ε,δ)$-estimator, and gives mathematical proofs of the correctness and efficiency of MMLR algorithm, of which the time complexity is linear with respect to the size of input datasets. This paper also empirically implements the method on both synthetic and real-world datasets, the algorithm shows to have comparable performance to existing regression methods in many cases, while it takes almost the shortest time to provide a high prediction accuracy.

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arXiv2023

The Hardness of Optimization Problems on the Weighted Massively Parallel Computation Model

The topology-aware Massively Parallel Computation (MPC) model is proposed and studied recently, which enhances the classical MPC model by the awareness of network topology. The work of Hu et al. on topology-aware MPC model considers only the tree topology. In this paper a more general case is considered, where the underlying network is a weighted complete graph. We then call this model as Weighted Massively Parallel Computation (WMPC) model, and study the problem of minimizing communication cost under it. Two communication cost minimization problems are defined based on different pattern of communication, which are the Data Redistribution Problem and Data Allocation Problem. We also define four kinds of objective functions for communication cost, which consider the total cost, bottleneck cost, maximum of send and receive cost, and summation of send and receive cost, respectively. Combining the two problems in different communication pattern with the four kinds of objective cost functions, 8 problems are obtained. The hardness results of the 8 problems make up the content of this paper. With rigorous proof, we prove that some of the 8 problems are in P, some FPT, some NP-complete, and some W[1]-complete.

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arXiv2023

Approximate sorting and its application in I/O model

The approximate sorting for big data is considered in this paper. The goal of approximate sorting for big data is to generate an approximate sorted result, but using less CPU and I/O cost. For big data, we consider the approximate sorting in I/O model. The existing metrics on permutation space are not available for external approximate sorting algorithms. Thus, we propose a new kind of metric named External metric, which ignores the errors and dislocation that happened in each I/O block.The External Spearmans footrule metric is an example of external metric for Spearmans footrule metric. Furthermore, to facilitate a better evaluation of the approximate sorted result, we propose a new metric, named as errors, which directly states the number of dislocation of the elements. Its external metric external errors is also considered in this paper. Then, according to the rate-distortion relationship endowed by these two metrics, the lower bound of these two metrics on external approximate sorting problem with t I/O operations is proved. We propose a k-pass external approximate sorting algorithm, named as EASORT, and prove that EASORT is asymptotically optimal. Finally, we consider the applications on approximate sorting results. An index for the result of our approximate sorting is proposed and analyze the single and range query on approximate sorted result using this index. Further, the sort-merge join on two relations, where one of the relations is approximate sorted or both relations are approximate sorted, are all discussed in this paper.

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arXiv2023

Random-Order Enumeration for Self-Reducible NP-Problems

In plenty of data analysis tasks, a basic and time-consuming process is to produce a large number of solutions and feed them into downstream processing. Various enumeration algorithms have been developed for this purpose. An enumeration algorithm produces all solutions of a problem instance without repetition. To be a statistically meaningful representation of the solution space, solutions are required to be enumerated in uniformly random order. This paper studies a set of self-reducible NP-problems in three hierarchies, where the problems are polynomially countable ($Sr_{NP}^{FP}$), admit FPTAS ($Sr_{NP}^{FPTAS}$), and admit FPRAS ($Sr_{NP}^{FPRAS}$), respectively. The trivial algorithm based on a (almost) uniform generator is in fact inefficient. We provide a new insight that the (almost) uniform generator is not the end of the story. More efficient algorithmic frameworks are proposed to enumerate solutions in uniformly random order for problems in these three hierarchies. (1) For problems in $Sr_{NP}^{FP}$, we show a random-order enumeration algorithm with polynomial delay (PDREnum); (2) For problems in $Sr_{NP}^{FPTAS}$, we show a Las Vegas random-order enumeration algorithm with expected polynomial delay (PDLVREnum); (3) For problems in $Sr_{NP}^{FPRAS}$, we devise a fully polynomial delay Atlantic City random-order enumeration algorithm with expected delay polynomial in the input size and the given error probability $δ$ (FPACREnum), which has a probability of at least $1-δ$ becoming a Las Vegas random-order enumeration algorithm. Finally, to further improve the efficiency of the random-order enumeration algorithms, based on the master/slave paradigm, we present a parallelization with 1.5-optimal enumeration delay and running time, along with the theoretical analysis.

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arXiv2022

Turing Machines with Two-level Memory: A Deep Look into the Input/Output Complexity

The input/output complexity, which is the complexity of data exchange between the main memory and the external memory, has been elaborately studied by a lot of former researchers. However, the existing works failed to consider the input/output complexity in a computation model point of view. In this paper we remedy this by proposing three variants of Turing machine that include external memory and the mechanism of exchanging data between main memory and external memory. Based on these new models, the input/output complexity is deeply studied. We discussed the relationship between input/output complexity and the other complexity measures such as time complexity and parameterized complexity, which is not considered by former researchers. We also define the external access trace complexity, which reflects the physical behavior of magnetic disks and gives a theoretical evidence of IO-efficient algorithms.

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arXiv2022

Dynamic Approximate Maximum Independent Set on Massive Graphs

Computing a maximum independent set (MaxIS) is a fundamental NP-hard problem in graph theory, which has important applications in a wide spectrum of fields. Since graphs in many applications are changing frequently over time, the problem of maintaining a MaxIS over dynamic graphs has attracted increasing attention over the past few years. Due to the intractability of maintaining an exact MaxIS, this paper aims to develop efficient algorithms that can maintain an approximate MaxIS with an accuracy guarantee theoretically. In particular, we propose a framework that maintains a $(\fracΔ{2} + 1)$-approximate MaxIS over dynamic graphs and prove that it achieves a constant approximation ratio in many real-world networks. To the best of our knowledge, this is the first non-trivial approximability result for the dynamic MaxIS problem. Following the framework, we implement an efficient linear-time dynamic algorithm and a more effective dynamic algorithm with near-linear expected time complexity. Our thorough experiments over real and synthetic graphs demonstrate the effectiveness and efficiency of the proposed algorithms, especially when the graph is highly dynamic.

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arXiv2022

PCP Theorems, SETH and More: Towards Proving Sub-linear Time Inapproximability

In this paper we propose the PCP-like theorem for sub-linear time inapproximability. Abboud et al. have devised the distributed PCP framework for sub-quadratic time inapproximability. We show that the distributed PCP theorem can be generalized for proving arbitrary polynomial time inapproximability, but fails in the linear case. We prove the sub-linear PCP theorem by adapting from an MA-protocol for the Set Containment problem, and show how to use the theorem to prove both existing and new inapproximability results, exhibiting the power of the sub-linear PCP theorem. Considering the emerging research works on sub-linear time algorithms, the sub-linear PCP theorem is important in guiding the research in sub-linear time approximation algorithms.

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arXiv2022

A New Model for Massively Parallel Computation Considering both Communication and IO Cost

In the research area of parallel computation, the communication cost has been extensively studied, while the IO cost has been neglected. For big data computation, the assumption that the data fits in main memory no longer holds, and external memory must be used. Therefore, it is necessary to bring the IO cost into the parallel computation model. In this paper, we propose the first parallel computation model which takes IO cost as well as non-uniform communication cost into consideration. Based on the new model, we raise several new problems which aim to minimize the IO and communication cost on the new model. We prove the hardness of these new problems, then design and analyze the approximate algorithms for solving them.

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arXiv2022

Rank-Regret Minimization

Multi-criteria decision-making often requires finding a small representative set from the database. A recently proposed method is the regret minimization set (RMS) query. RMS returns a size $r$ subset $S$ of dataset $D$ that minimizes the regret-ratio (the difference between the score of top-1 in $S$ and the score of top-1 in $D$, for any possible utility function). RMS is not shift invariant, causing inconsistency in results. Further, existing work showed that the regret-ratio is often a made-up number and users may mistake its absolute value. Instead, users do understand the notion of rank. Thus it considered the problem of finding the minimal set $S$ with a rank-regret (the rank of top-1 tuple of $S$ in the sorted list of $D$) at most $k$, called the rank-regret representative (RRR) problem. Corresponding to RMS, we focus on the min-error version of RRR, called the rank-regret minimization (RRM) problem, which finds a size $r$ set to minimize the maximum rank-regret for all utility functions. Further, we generalize RRM and propose the restricted RRM (i.e., RRRM) problem to optimize the rank-regret for functions restricted in a given space. Previous studies on both RMS and RRR did not consider the restricted function space. The solution for RRRM usually has a lower regret level and can better serve the specific preferences of some users. Note that RRM and RRRM are shift invariant. In 2D space, we design a dynamic programming algorithm 2DRRM to return the optimal solution for RRM. In HD space, we propose an algorithm HDRRM that introduces a double approximation guarantee on rank-regret. Both 2DRRM and HDRRM are applicable for RRRM. Extensive experiments on the synthetic and real datasets verify the efficiency and effectiveness of our algorithms. In particular, HDRRM always has the best output quality in experiments.

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arXiv2021

Sublinear-time Reductions for Big Data Computing

With the rapid popularization of big data, the dichotomy between tractable and intractable problems in big data computing has been shifted. Sublinear time, rather than polynomial time, has recently been regarded as the new standard of tractability in big data computing. This change brings the demand for new methodologies in computational complexity theory in the context of big data. Based on the prior work for sublinear-time complexity classes \cite{DBLP:journals/tcs/GaoLML20}, this paper focuses on sublinear-time reductions specialized for problems in big data computing. First, the pseudo-sublinear-time reduction is proposed and the complexity classes \Pproblem and \PsT are proved to be closed under it. To establish \PsT-intractability for certain problems in \Pproblem, we find the first problem in $\Pproblem \setminus \PsT$. Using the pseudo-sublinear-time reduction, we prove that the nearest edge query is in \PsT but the algebraic equation root problem is not. Then, the pseudo-polylog-time reduction is introduced and the complexity class \PsPL is proved to be closed under it. The \PsT-completeness under it is regarded as an evidence that some problems can not be solved in polylogarithmic time after a polynomial-time preprocessing, unless \PsT = \PsPL. We prove that all \PsT-complete problems are also \Pproblem-complete, which gives a further direction for identifying \PsT-complete problems.

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arXiv2021

Sublinear Time Nearest Neighbor Search over Generalized Weighted Manhattan Distance

Nearest Neighbor Search (NNS) over generalized weighted distances is fundamental to a wide range of applications. The problem of NNS over the generalized weighted square Euclidean distance has been studied in previous work. However, numerous studies have shown that the Manhattan distance could be more effective than the Euclidean distance for high-dimensional NNS, which indicates that the generalized weighted Manhattan distance is possibly more practical than the generalized weighted square Euclidean distance in high dimensions. To the best of our knowledge, no prior work solves the problem of NNS over the generalized weighted Manhattan distance in sublinear time. This paper achieves the goal by proposing two novel hashing schemes ($d_w^{l_1},l_2$)-ALSH and ($d_w^{l_1},θ$)-ALSH.

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arXiv2021

Efficient Trajectory Compression and Range Query Processing

Nowadays, there are ubiquitousness of GPS sensors in various devices collecting, transmitting and storing tremendous trajectory data. However, such an unprecedented scale of GPS data has posed an urgent demand for not only an effective storage mechanism but also an efficient query mechanism. Line simplification in online mode, searving as a mainstream trajectory compression method, plays an important role to attack this issue. But for the existing algorithms, either their time cost is extremely high, or the accuracy loss after the compression is completely unacceptable. To attack this issue, we propose $ε\_$Region based Online trajectory Compression with Error bounded (ROCE for short), which makes the best balance among the accuracy loss, the time cost and the compression rate. The range query serves as a primitive, yet quite essential operation on analyzing trajectories. Each trajectory is usually seen as a sequence of discrete points, and in most previous work, a trajectory is judged to be overlapped with the query region R iff there is at least one point in this trajectory falling in R. But this traditional criteria is not suitable when the queried trajectories are compressed, because there may be hundreds of points discarded between each two adjacent points and the points in each compressed trajectory are quite sparse. And many trajectories could be missing in the result set. To address this, in this paper, a new criteria based on the probability and an efficient Range Query processing algorithm on Compressed trajectories RQC are proposed. In addition, an efficient index \emph{ASP\_tree} and lots of novel techniques are also presented to accelerate the processing of trajectory compression and range queries obviously. Extensive experiments have been done on multiple real datasets, and the results demonstrate superior performance of our methods.

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