arXiv subjects

Majid Mirzanezhad

Publications and source records attributed to Majid Mirzanezhad.

6 recordsLinked to original sources

arXiv2023

Realizability of Free Spaces of Curves

The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often, the question arises whether a certain pattern in the free space diagram is "realizable", i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore, we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram? Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in $\mathbb{R}^{> 2}$, showing $\exists\mathbb{R}$-hardness. We use this to show that for curves in $\mathbb{R}^{\ge 2}$, the realizability problem is $\exists\mathbb{R}$-complete, both for continuous and for discrete Fréchet distance. We prove that the continuous case in $\mathbb{R}^1$ is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in $\mathbb{R}^1$, we show that the problem becomes solvable in polynomial time.

cs.CG↗

arXiv2022

Estimating Demand for Online Delivery using Limited Historical Observations

Driven in part by the COVID-19 pandemic, the pace of online purchases for at-home delivery has accelerated significantly. However, responding to this development has been challenging given the lack of public data. The existing data may be infrequent, and a significant portion of data may be missing because of survey participant non-responses. This data paucity renders conventional predictive models unreliable. We address this shortcoming by developing algorithms for data imputation and synthetic demand estimation for future years without the actual ground truth data. We use 2017 Puget Sound Regional Council (PSRC) and National Household Travel Survey (NHTS) data and impute from the NHTS for the Seattle-Tacoma-Bellevue MSA where delivery data is relatively more frequent. Our imputation has the mean-squared error $\mathsf{MSE} \approx 0.65$ to NHTS with mean $\approx 1$ and standard deviation $\approx 3.5$ and provides a similarity matching between the two data sources' samples. Given the unavailability of NHTS data for 2021, we use the temporal fidelity of PSRC data sources (2017 and 2021) to project the resolution onto the NHTS providing a synthetic estimate of NHTS deliveries. Beyond the improved reliability of the estimates, we report explanatory variables that were relevant in determining the volume of deliveries. This work furthers existing methods in demand estimation for goods deliveries by maximizing available sparse data to generate reasonable estimates that could facilitate policy decisions.

stat.ME↗

arXiv2021

Minimum-Complexity Graph Simplification under Fréchet-Like Distances

Simplifying graphs is a very applicable problem in numerous domains, especially in computational geometry. Given a geometric graph and a threshold, the minimum-complexity graph simplification asks for computing an alternative graph of minimum complexity so that the distance between the two graphs remains at most the threshold. In this paper, we propose several NP-hardness and algorithmic results depending on the type of input and simplified graphs, the vertex placement of the simplified graph, and the distance measures between them (graph and traversal distances [1,2]). In general, we show that for arbitrary input and output graphs, the problem is NP-hard under some specific vertex-placement of the simplified graph. When the input and output are trees, and the graph distance is applied from the simplified tree to the input tree, we give an $O(kn^5)$ time algorithm, where $k$ is the number of the leaves of the two trees that are identical and $n$ is the number of vertices of the input.

cs.CG↗

arXiv2021

On the Approximate Nearest Neighbor Queries among Curves under the Fréchet Distance

Approximate near-neighbors search (\textsc{ANNS}) is a long-studied problem in computational geometry. %that has received considerable attention by researchers in the community. In this paper, we revisit the problem and propose the first data structure for curves under the (continuous) Fréchet distance in $\Reals^d$. Given a set $¶$ of $n$ curves of size at most $m$ each in $\Reals^d$, and a real fixed $δ>0$, we aim to preprocess $¶$ into a data structure so that for any given query curve $Q$ of size $k$, we can efficiently report all curves in $¶$ whose Fréchet distances to $Q$ are at most $δ$. In the case that $k$ is given in the preprocessing stage, for any $\eps>0$ we propose a deterministic data structure whose space is $n \cdot O\big(\max\big\{\big(\frac{\sqrt{d}}{\eps}\big)^{kd}, \big(\frac{\D\sqrt{d}}{\eps^2}\big)^{kd}\big\}\big)$ that can answer \textsc{$(1+\eps)δ$-ANNS} queries in $O(kd)$ query time, where $\D$ is the diameter of $¶$. Considering $k$ as part of the query slightly changes the space to $n \cdot O\big(\frac{1}{\eps}\big)^{md} $ with $O(kd)$ query time within an approximation factor of $5+\eps$. We show that our generic data structure for ANNS can give an alternative treatment of the approximate subtrajectory range searching problem studied by de Berg et al. [8]. We also revisit the time-window data structure for spatial density maps in [6]. Given $θ>0$, and $n$ time-stamped points spread over $m$ regions in a map, for any query window $W$, we propose a data structure of size $O(n/\eps^2)$ and construction time $O((n+m)/\eps^2)$ that can approximately return the regions containing at least $θ$ points whose times are within $W$ in $O(1)$ query time.

cs.CG↗

arXiv2020

Global Curve Simplification

Due to its many applications, \emph{curve simplification} is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve $P$ with $n$ vertices, the goal is to find another polygonal curve $P'$ with a smaller number of vertices such that $P'$ is sufficiently similar to $P$. Quality guarantees of a simplification are usually given in a \emph{local} sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work, we aim to provide a systematic overview of curve simplification problems under \emph{global} distance measures that bound the distance between $P$ and $P'$. We consider six different curve distance measures: three variants of the \emph{Hausdorff} distance and three variants of the \emph{Fréchet} distance. And we study different restrictions on the choice of vertices for $P'$. We provide polynomial-time algorithms for some variants of the global curve simplification problem and show NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area.

cs.CG↗

arXiv2019

Fast Frechet Distance Between Curves With Long Edges

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve.

cs.CG↗