arXiv subjects

Jie Xue

Publications and source records attributed to Jie Xue.

63 records · Page 4Linked to original sources

arXiv2018

New bounds for range closest-pair problems

Given a dataset $S$ of points in $\mathbb{R}^2$, the range closest-pair (RCP) problem aims to preprocess $S$ into a data structure such that when a query range $X$ is specified, the closest-pair in $S \cap X$ can be reported efficiently. The RCP problem can be viewed as a range-search version of the classical closest-pair problem, and finds applications in many areas. Due to its non-decomposability, the RCP problem is much more challenging than many traditional range-search problems. This paper revisits the RCP problem, and proposes new data structures for various query types including quadrants, strips, rectangles, and halfplanes. Both worst-case and average-case analyses (in the sense that the data points are drawn uniformly and independently from the unit square) are applied to these new data structures, which result in new bounds for the RCP problem. Some of the new bounds significantly improve the previous results, while the others are entirely new.

cs.CG↗

arXiv2018

Graphs with small diameter determined by their $D$-spectra

Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},...,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of $G,$ where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $λ_{1}(D)\geqλ_{2}(D)\geq\cdots\geqλ_{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. In this paper, we give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra.

math.CO↗

arXiv2017

Graphs determined by their $A_α$-spectra

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_α(G)=αD(G)+(1-α)A(G) $$ for any real $α\in [0,1]$. The collection of eigenvalues of $A_α(G)$ together with multiplicities are called the \emph{$A_α$-spectrum} of $G$. A graph $G$ is said to be \emph{determined by its $A_α$-spectrum} if all graphs having the same $A_α$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by its $A_α$-spectrum for $0\leqα<1$, including the complete graph $K_m$, the star $K_{1,n-1}$, the path $P_n$, the union of cycles and the complement of the union of cycles, the union of $K_2$ and $K_1$ and the complement of the union of $K_2$ and $K_1$, and the complement of $P_n$. Setting $α=0$ or $\frac{1}{2}$, those graphs are determined by $A$- or $Q$-spectra. Secondly, when $G$ is regular, we show that $G$ is determined by its $A_α$-spectrum if and only if the join $G\vee K_m$ is determined by its $A_α$-spectrum for $\frac{1}{2}<α<1$. Furthermore, we also show that the join $K_m\vee P_n$ is determined by its $A_α$-spectrum for $\frac{1}{2}<α<1$. In the end, we pose some related open problems for future study.

math.CO↗

arXiv2017

More results on the distance (signless) Laplacian eigenvalues of graphs

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$ and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $\mathcal{L}(G)=Tr(G)-D(G)$. The distance signless Laplacian matrix of $G$ is defined as $\mathcal{Q}(G)=Tr(G)+D(G)$. In this paper, we give a lower bound on the distance Laplacian spectral radius in terms of $D_1$, as a consequence, we show that $\partial_1^L(G)\geq n+\lceil\frac{n}ω\rceil$ where $ω$ is the clique number of $G$. Furthermore, we give some graft transformations, by using them, we characterize the extremal graph attains the maximum distance spectral radius in terms of $n$ and $ω$. Moreover, we also give bounds on the distance signless Laplacian eigenvalues of $G$, and give a confirmation on a conjecture due to Aouchiche and Hansen.

math.CO↗

arXiv2017

On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.

cs.CG↗

arXiv2016

Stochastic closest-pair problem and most-likely nearest-neighbor search in tree spaces

Let $T$ be a tree space (or tree network) represented by a weighted tree with $t$ vertices, and $S$ be a set of $n$ stochastic points in $T$, each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the $\ell$-threshold probability and the expectation of the closest-pair distance of a realization of $S$. We propose the first algorithm to compute the $\ell$-threshold probability in $O(t+n\log n+ \min\{tn,n^2\})$ time for any given threshold $\ell$, which immediately results in an $O(t+\min\{tn^3,n^4\})$-time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a $(1+\varepsilon)$-approximation for the expected closest-pair distance in $O(t+\varepsilon^{-1}\min\{tn^2,n^3\})$ time, by arguing that the expected closest-pair distance can be approximated via $O(\varepsilon^{-1}n)$ threshold probability queries. For the latter, we study the $k$ most-likely nearest-neighbor search ($k$-LNN) via a notion called $k$ most-likely Voronoi Diagram ($k$-LVD). We show that the size of the $k$-LVD $\varPsi_T^S$ of $S$ on $T$ is bounded by $O(kn)$ if the existence probabilities of the points in $S$ are constant-far from 0. Furthermore, we establish an $O(kn)$ average-case upper bound for the size of $\varPsi_T^S$, by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers $k$-LNN queries in $O(\log n+k)$ time using average-case $O(t+k^2n)$ space, and worst-case $O(t+kn^2)$ space if the existence probabilities are constant-far from 0. Finally, we also give an $O(t+ n^2\log n+n^2k)$-time algorithm to construct the LVD data structure.

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arXiv2016

On the Separability of Stochastic Geometric Objects, with Applications

In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let $S=S_R \cup S_B$ be a given set of stochastic bichromatic points, and define $n = \min\{|S_R|, |S_B|\}$ and $N = \max\{|S_R|, |S_B|\}$. We show that the separable-probability (SP) of $S$ can be computed in $O(nN^{d-1})$ time for $d \geq 3$ and $O(\min\{nN \log N, N^2\})$ time for $d=2$, while the expected separation-margin (ESM) of $S$ can be computed in $O(nN^{d})$ time for $d \geq 2$. In addition, we give an $Ω(nN^{d-1})$ witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in $O(nN^{d})$ and $O(nN^{d+1})$ time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.

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arXiv2015

On the second largest distance eigenvalue of a graph

Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that $λ_{1}(D(G))\geqλ_{2}(D(G))\geq\cdots\geqλ_{n}(D(G))$ are the distance spectrum of $G$. A graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph with the same spectrum as $G$ is isomorphic to $G$. In this paper, we consider spectral characterization on the second largest distance eigenvalue $λ_{2}(D(G))$ of graphs, and prove that the graphs with $λ_{2}(D(G))\leq\frac{17-\sqrt{329}}{2}\approx-0.5692$ are determined by their $D$-spectra.

math.CO↗

arXiv2014

Sufficient spectral conditions on Hamiltonian and traceable graphs

In this paper, we give sufficient conditions on the spectral radius for a bipartite graph to Hamiltonian and traceable, which expand the results of Lu, Liu and Tian (2012) [10]. Furthermore, we also present tight sufficient conditions on the signless Laplacian spectral radius for a graph to Hamiltonian and traceable, which improve the results of Yu and Fan (2012) [12].

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