Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

In the semialgebraic range searching problem, we are to preprocess $n$ points in $\mathbb{R}^d$ s.t. for any query range from a family of constant complexity semialgebraic sets, all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem can be solved using $S(n)$ space and with $Q(n)$ query time with $S(n)Q^d(n) = \tilde{O}(n^d)$ and this trade-off is almost tight. Consequently, there exists low space structures that use $\tilde{O}(n)$ space with $O(n^{1-1/d})$ query time and fast query structures that use $O(n^d)$ space with $O(\log^{d} n)$ query time. However, for the general semialgebraic ranges, only low space solutions are known, but the best solutions match the same trade-off curve as the simplex queries. It has been conjectured that the same could be done for the fast query case but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semilagebraic range searching and related problems. We show that any data structure for reporting the points between two concentric circles with $Q(n)$ query time must use $S(n)=\Omega(n^{3-o(1)}/Q(n)^5)$ space, meaning, for $Q(n)=O(\log^{O(1)}n)$, $\Omega(n^{3-o(1)})$ space must be used. We also study the problem of reporting the points between two polynomials of form $Y=\sum_{i=0}^\Delta a_i X^i$ where $a_0, \cdots, a_\Delta$ are given at the query time. We show $S(n)=\Omega(n^{\Delta+1-o(1)}/Q(n)^{\Delta^2+\Delta})$. So for $Q(n)=O(\log^{O(1)}n)$, we must use $\Omega(n^{\Delta+1-o(1)})$ space. For the dual semialgebraic stabbing problems, we show that in linear space, any data structure that solves 2D ring stabbing must use $\Omega(n^{2/3})$ query time. This almost matches the linearization upper bound. For general semialgebraic slab stabbing problems, again, we show an almost tight lower bounds.Comment: Submitted to SoCG'21; this version: readjust the table and other minor change

An Optimal Algorithm for Higher-Order Voronoi Diagrams in the Plane: The Usefulness of Nondeterminism

We present the first optimal randomized algorithm for constructing the order-$k$ Voronoi diagram of $n$ points in two dimensions. The expected running time is $O(n\log n + nk)$, which improves the previous, two-decades-old result of Ramos (SoCG'99) by a $2^{O(\log^*k)}$ factor. To obtain our result, we (i) use a recent decision-tree technique of Chan and Zheng (SODA'22) in combination with Ramos's cutting construction, to reduce the problem to verifying an order-$k$ Voronoi diagram, and (ii) solve the verification problem by a new divide-and-conquer algorithm using planar-graph separators. We also describe a deterministic algorithm for constructing the $k$-level of $n$ lines in two dimensions in $O(n\log n + nk^{1/3})$ time, and constructing the $k$-level of $n$ planes in three dimensions in $O(n\log n + nk^{3/2})$ time. These time bounds (ignoring the $n\log n$ term) match the current best upper bounds on the combinatorial complexity of the $k$-level. Previously, the same time bound in two dimensions was obtained by Chan (1999) but with randomization.Comment: To appear in SODA 2024. 16 pages, 1 figur

On Range Summary Queries

We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter ? and a set P of n points in ?^d where each point is assigned a color from a set C, and the goal is to build a structure such that given any geometric range ?, we can efficiently find a list of approximate heavy hitters in ??P, i.e., colors that appear at least ? |??P| times in ??P, as well as their frequencies with an additive error of ? |??P|. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence S of 1+1/? weights such that the i-th weight in S has approximate rank i?|??P|, meaning, rank i?|??P| up to an additive error of ?|??P|. Previously, optimal results were only known in 1D [Wei and Yi, 2011] but a few sub-optimal methods were available in higher dimensions [Peyman Afshani and Zhewei Wei, 2017; Pankaj K. Agarwal et al., 2012]. We study the problems for two important classes of geometric ranges: 3D halfspace and 3D dominance queries. It is known that many other important queries can be reduced to these two, e.g., 1D interval stabbing or interval containment, 2D three-sided queries, 2D circular as well as 2D k-nearest neighbors queries. We consider the real RAM model of computation where integer registers of size w bits, w = ?(log n), are also available. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time O(log n + 1/?). Note that as the output size is 1/?, after investing the initial O(log n) searching time, our structure takes on average O(1) time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra log_w(1/?) (resp. log log_w(1/?)) factor in 3D (resp. 2D). By spending extra log^O(1)(1/?) factors in both time and space, we can also support quantile queries. We remark that it is hopeless to achieve a similar query bound for dimensions 4 or higher unless significant advances are made in the data structure side of theory of geometric approximations

Long-term efficacy of hydrotherapy on balance function in patients with Parkinson’s disease: a systematic review and meta-analysis

BackgroundHydrotherapy can improve the motor and non-motor symptoms of Parkinson’s disease (PD), but the long-term effects of hydrotherapy on PD are still unclear.ObjectiveThe purpose of this systematic evaluation and meta-analysis was to explore the long-term effects of hydrotherapy on balance function in PD patients.MethodsA systematic search of five databases was conducted to identify appropriate randomized controlled trials (RCTs) according to the established inclusion and exclusion criteria. The general characteristics and outcome data (balance, exercise, mobility, quality of life, etc.) of the included studies were extracted, and the quality of the included studies was evaluated using the Cochrane risk of bias assessment tool. Finally, the outcome data were integrated for meta-analysis.ResultsA total of 149 articles were screened, and 5 high-quality RCTs involving 135 PD patients were included. The results of the meta-analysis showed positive long-term effects of hydrotherapy on balance function compared to the control group (SMD = 0.69; 95% CI = 0.21, 1.17; p = 0.005; I2 = 44%), However, there were no significant long-term effects of hydrotherapy on motor function (SMD = 0.06; 95% CI = −0.33, 0.44; p = 0.77; I2 = 0%), mobility and quality of life (SMD = −0.21; 95% CI = −0.98, 0.57; p = 0.6; I2 = 71%). Interestingly, the results of the sensitivity analysis performed on mobility showed a clear continuation effect of hydrotherapy on mobility compared to the control group (SMD = −0.80; 95% CI = −1.23, −0.37; p < 0.001; I2 = 0%).ConclusionThe long-term effects of hydrotherapy on PD patients mainly focus on balance function, and the continuous effects on motor function, mobility, and quality of life are not obvious

On Semialgebraic Range Reporting

In the problem of semialgebraic range searching, we are to preprocess a set of points in $\mathbb{R}^D$ such that the subset of points inside a semialgebraic region described by $O(1)$ polynomial inequalities of degree $\Delta$ can be found efficiently. Relatively recently, several major advances were made on this problem. Using algebraic techniques, "near-linear space" structures [AMS13,MP15] with almost optimal query time of $Q(n)=O(n^{1-1/D+o(1)})$ were obtained. For "fast query" structures (i.e., when $Q(n)=n^{o(1)}$), it was conjectured that a structure with space $S(n) = O(n^{D+o(1)})$ is possible. The conjecture was refuted recently by Afshani and Cheng [AC21]. In the plane, they proved that $S(n) = \Omega(n^{\Delta+1 - o(1)}/Q(n)^{(\Delta+3)\Delta/2})$ which shows $\Omega(n^{\Delta+1-o(1)})$ space is needed for $Q(n) = n^{o(1)}$. While this refutes the conjecture, it still leaves a number of unresolved issues: the lower bound only works in 2D and for fast queries, and neither the exponent of $n$ or $Q(n)$ seem to be tight even for $D=2$, as the current upper bound is $S(n) = O(n^{\boldsymbol{m}+o(1)}/Q(n)^{(\boldsymbol{m}-1)D/(D-1)})$ where $\boldsymbol{m}=\binom{D+\Delta}{D}-1 = \Omega(\Delta^D)$ is the maximum number of parameters to define a monic degree-$\Delta$ $D$-variate polynomial, for any $D,\Delta=O(1)$. In this paper, we resolve two of the issues: we prove a lower bound in $D$-dimensions and show that when $Q(n)=n^{o(1)}+O(k)$, $S(n)=\Omega(n^{\boldsymbol{m}-o(1)})$, which is almost tight as far as the exponent of $n$ is considered in the pointer machine model. When considering the exponent of $Q(n)$, we show that the analysis in [AC21] is tight for $D=2$, by presenting matching upper bounds for uniform random point sets. This shows either the existing upper bounds can be improved or a new fundamentally different input set is needed to get a better lower bound.Comment: Full version of the SoCG'22 pape

On Range Summary Queries

We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter $\varepsilon$ and a set $P$ of $n$ points in $\mathbb{R}^d$ where each point is assigned a color from a set $C$, and we want to build a structure s.t. given any geometric range $\gamma$, we can efficiently find a list of approximate heavy hitters in $\gamma\cap P$, i.e., colors that appear at least $\varepsilon |\gamma \cap P|$ times in $\gamma \cap P$, as well as their frequencies with an additive error of $\varepsilon |\gamma \cap P|$. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence $S$ of $1+1/\varepsilon$ weights s.t. the $i$-th weight in $S$ has approximate rank $i\varepsilon|\gamma\cap P|$, meaning, rank $i\varepsilon|\gamma\cap P|$ up to an additive error of $\varepsilon|\gamma\cap P|$. Previously, optimal results were only known in 1D [WY11] but a few sub-optimal methods were available in higher dimensions [AW17, ACH+12]. We study the problems for 3D halfspace and dominance queries. We consider the real RAM model with integer registers of size $w=\Theta(\log n)$ bits. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time $O(\log n + 1/\varepsilon)$. Note that as the output size is $\frac{1}{\varepsilon}$, after investing the initial $O(\log n)$ searching time, our structure takes on average $O(1)$ time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra $\log_w\frac{1}{\varepsilon}$ (resp. $\log\log_w\frac{1}{\varepsilon}$) factor in 3D (resp. 2D). By spending extra $\log^{O(1)}\frac{1}{\varepsilon}$ factors in time and space, we can also support quantile queries.Comment: To appear in ICALP'2

Structural Safety Analysis of Cantilever External Shading Components of Buildings under Extreme Wind Environment

The high intensity of solar radiation and long sunshine time in the Turpan area lead to the necessity of sunshade construction. Sunshade components can effectively block direct solar radiation and the secondary heating of buildings. Through the analysis of the importance and sensitivity of sunshade components, it was found that the importance of sunshade components accounts for the largest proportion of multi-parameters, and the sensitivity of sunshade components accounts for about 60% of the total. At the same time, the change in sunshade length has an important influence on the proportion of air conditioning energy consumption and space comfort when the sunshade length reached the 0.6 m–0.8 m range. The energy consumption curve of air conditioning no longer decreased and tended to be horizontal, which showed that a sunshade could effectively reduce the energy consumption of air conditioning, while the PMV comfort curve gradually increased and tended to be horizontal, indicating that a sunshade could effectively improve indoor comfort; therefore, a sunshade could reduce direct solar radiation, reduce the energy consumption of air conditioning and improve indoor thermal comfort. In view of the extremely harsh climate characteristics of Turpan, although Turpan needs to carry out shading design, as a typical wind-sensitive component, the structural safety of the visor under the action of an extreme wind environment is the primary focus of designers. The design requires wind loads as control loads. Based on the ANSYS Workbench platform, this study used the fluid–structure coupling technology to calculate and solve for the wind load stress and strain of a horizontal sunshade and a vertical sunshade in a cantilevered external sunshade of different buildings orientations. In this study, by solving for the maximum principal stress and maximum principal elastic strain under 10 working conditions, the results showed that the maximum principal stress of the sun visor under all working conditions was 0.39 MPa, which is much smaller than the tensile strength of C25 concrete. The calculated maximum principal elastic strain of the sun visor was 0.12 × 10−4, which is much smaller than the maximum strain value of concrete. Therefore, the wind load under this research condition had no great influence on the structural safety of the concrete sunshade, which proves the structural feasibility of the building sunshade member in the Turpan area, and provides a reference for the future practical engineering of cantilever members in the Turpan area

Super-tough artificial nacre based on graphene oxide via synergistic interface interactions of Pi-Pi stacking and hydrogen bonding

Inspired by interfacial interactions of protein matrix and the crystal platelets in nacre, herein, a supertough artificial nacre was produced through constructing the synergistic interface interactions of p-p interaction and hydrogen bonding between graphene oxide(GO) nanosheets and sulfonated styreneethylene/butylene-styrene copolymer synthesized with multifunctional benzene. The resultant GO based artificial nacre showed super-high toughness of 15.3 ± 2.5 MJ/m3, superior to natural nacre andother GO-based nanocomposites. The ultra-tough property of the novel nacre was attributed to synergistic effect of Pi-Pi stacking interactions and hydrogen bonding. This bioinspired synergistic toughening strategy opens a new avenue for constructing high performance GO-based nanocomposites in the near future