arXiv subjects

Siu-Wing Cheng

Publications and source records attributed to Siu-Wing Cheng.

28 records · Page 2Linked to original sources

arXiv2019

Restricted Max-Min Allocation: Approximation and Integrality Gap

Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most $4$. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a $(6 + δ)$-approximation algorithm where $δ$ can be any positive constant, and there is still a gap of roughly $2$. In this paper, we narrow the gap significantly by proposing a $(4+δ)$-approximation algorithm where $δ$ can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is $\mathit{poly}(m,n)\cdot n^{\mathit{poly}(\frac{1}δ)}$ where $n$ is the number of players and $m$ is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to $3 + \frac{21}{26} \approx 3.808$.

cs.DS↗

arXiv2019

Implicit Manifold Reconstruction

Let ${\cal M} \subset \mathbb{R}^d$ be a compact, smooth and boundaryless manifold with dimension $m$ and unit reach. We show how to construct a function $φ: \mathbb{R}^d \rightarrow \mathbb{R}^{d-m}$ from a uniform $(\varepsilon,κ)$-sample $P$ of $\cal M$ that offers several guarantees. Let $Z_φ$ denote the zero set of $φ$. Let $\widehat{\cal M}$ denote the set of points at distance $\varepsilon$ or less from $\cal M$. There exists $\varepsilon_0 \in (0,1)$ that decreases as $d$ increases such that if $\varepsilon \leq \varepsilon_0$, the following guarantees hold. First, $Z_φ\cap \widehat{\cal M}$ is a faithful approximation of $\cal M$ in the sense that $Z_φ\cap \widehat{\cal M}$ is homeomorphic to $\cal M$, the Hausdorff distance between $Z_φ\cap \widehat{\cal M}$ and $\cal M$ is $O(m^{5/2}\varepsilon^{2})$, and the normal spaces at nearby points in $Z_φ\cap \widehat{\cal M}$ and $\cal M$ make an angle $O(m^2\sqrt{κ\varepsilon})$. Second, $φ$ has local support; in particular, the value of $φ$ at a point is affected only by sample points in $P$ that lie within a distance of $O(m\varepsilon)$. Third, we give a projection operator that only uses sample points in $P$ at distance $O(m\varepsilon)$ from the initial point. The projection operator maps any initial point near $P$ onto $Z_φ\cap \widehat{\cal M}$ in the limit by repeated applications.

cs.CG↗

arXiv2019

A note on self-improving sorting with hidden partitions

We study self-improving sorting with hidden partitions. Our result is an optimal algorithm which runs in expected time O(H(π(I)) + n), where I is the given input which contains n elements to be sorted, π(I) is the output which are the ranks of all element in I, and H(π(I)) denotes the entropy of the output.

cs.CG↗

arXiv2018

Adaptive Planar Point Location

We present self-adjusting data structures for answering point location queries in convex and connected subdivisions. Let $n$ be the number of vertices in a convex or connected subdivision. Our structures use $O(n)$ space. For any convex subdivision $S$, our method processes any online query sequence $σ$ in $O(\mathrm{OPT} + n)$ time, where $\mathrm{OPT}$ is the minimum time required by any linear decision tree for answering point location queries in $S$ to process $σ$. For connected subdivisions, the processing time is $O(\mathrm{OPT} + n + |σ|\log(\log^* n))$. In both cases, the time bound includes the $O(n)$ preprocessing time.

cs.CG↗

arXiv2018

Restricted Max-Min Fair Allocation

The restricted max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. It is NP-hard to approximate the problem to a ratio less than 2. Comparing the current best algorithm for estimating the optimal value with the current best for constructing an allocation, there is quite a gap between the ratios that can be achieved in polynomial time: roughly 4 for estimation and roughly $6 + 2\sqrt{10}$ for construction. We propose an algorithm that constructs an allocation with value within a factor of $6 + δ$ from the optimum for any constant $δ> 0$. The running time is polynomial in the input size for any constant $δ$ chosen.

cs.DS↗

arXiv2018

Integrality Gap of the Configuration LP for the Restricted Max-Min Fair Allocation

The max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. Each player $p$ has a non-negative value $v_{pr}$ on resource $r$. In the restricted case, we have $v_{pr}\in \{v_r, 0\}$. That is, a resource $r$ is worth value $v_r$ for the players who desire it and value 0 for the other players. In this paper, we consider the configuration LP, a linear programming relaxation for the restricted problem. The integrality gap of the configuration LP is at least $2$. Asadpour, Feige, and Saberi proved an upper bound of $4$. We improve the upper bound to $23/6$ using the dual of the configuration LP. Since the configuration LP can be solved to any desired accuracy $δ$ in polynomial time, our result leads to a polynomial-time algorithm which estimates the optimal value within a factor of $23/6+δ$.

cs.DS↗

arXiv2017

Denoising a Point Cloud for Surface Reconstruction

Surface reconstruction from an unorganized point cloud is an important problem due to its widespread applications. White noise, possibly clustered outliers, and noisy perturbation may be generated when a point cloud is sampled from a surface. Most existing methods handle limited amount of noise. We develop a method to denoise a point cloud so that the users can run their surface reconstruction codes or perform other analyses afterwards. Our experiments demonstrate that our method is computationally efficient and it has significantly better noise handling ability than several existing surface reconstruction codes.

cs.GR↗

arXiv2015

Minimax Regret 1-Median Problem in Dynamic Path Networks

This paper considers the minimax regret 1-median problem in dynamic path networks. In our model, we are given a dynamic path network consisting of an undirected path with positive edge lengths, uniform positive edge capacity, and nonnegative vertex supplies. Here, each vertex supply is unknown but only an interval of supply is known. A particular assignment of supply to each vertex is called a scenario. Given a scenario s and a sink location x in a dynamic path network, let us consider the evacuation time to x of a unit supply given on a vertex by s. The cost of x under s is defined as the sum of evacuation times to x for all supplies given by s, and the median under s is defined as a sink location which minimizes this cost. The regret for x under s is defined as the cost of x under s minus the cost of the median under s. Then, the problem is to find a sink location such that the maximum regret for all possible scenarios is minimized. We propose an O(n^3) time algorithm for the minimax regret 1-median problem in dynamic path networks with uniform capacity, where n is the number of vertices in the network.

cs.DS↗

arXiv2014

A Faster Algorithm for Computing Straight Skeletons

We present a new algorithm for computing the straight skeleton of a polygon. For a polygon with $n$ vertices, among which $r$ are reflex vertices, we give a deterministic algorithm that reduces the straight skeleton computation to a motorcycle graph computation in $O(n (\log n)\log r)$ time. It improves on the previously best known algorithm for this reduction, which is randomized, and runs in expected $O(n \sqrt{h+1}\log^2 n)$ time for a polygon with $h$ holes. Using known motorcycle graph algorithms, our result yields improved time bounds for computing straight skeletons. In particular, we can compute the straight skeleton of a non-degenerate polygon in $O(n (\log n) \log r + r^{4/3+\varepsilon})$ time for any $\varepsilon>0$. On degenerate input, our time bound increases to $O(n (\log n) \log r + r^{17/11+\varepsilon})$.

cs.CG↗

arXiv2007

Delaunay Edge Flips in Dense Surface Triangulations

Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of triangulations which are not full dimensional is surface triangulations in three dimensions. In this paper we address the question of converting a surface triangulation to a subcomplex of the Delaunay triangulation with edge flips. We show that the surface triangulations which closely approximate a smooth surface with uniform density can be transformed to a Delaunay triangulation with a simple edge flip algorithm. The condition on uniformity becomes less stringent with increasing density of the triangulation. If the condition is dropped completely, the flip algorithm still terminates although the output surface triangulation becomes "almost Delaunay" instead of exactly Delaunay.

cs.CG↗