arXiv subjects

Arash Vaezi

Publications and source records attributed to Arash Vaezi.

10 recordsLinked to original sources

arXiv2025

Line Aspect Ratio

We address the problem of covering a target segment $\overline{uv}$ using a finite set of guards $\mathcal{S}$ placed on a source segment $\overline{xy}$ within a simple polygon $\mathcal{P}$, assuming weak visibility between the target and source. Without geometric constraints, $\mathcal{S}$ may be infinite, as shown by prior hardness results. To overcome this, we introduce the {\it line aspect ratio} (AR), defined as the ratio of the \emph{long width} (LW) to the \emph{short width} (SW) of $\mathcal{P}$. These widths are determined by parallel lines tangent to convex vertices outside $\mathcal{P}$ (LW) and reflex vertices inside $\mathcal{P}$ (SW), respectively. Under the assumption that AR is constant or polynomial in $n$ (the polygon's complexity), we prove that a finite guard set $\mathcal{S}$ always exists, with size bounded by $\mathcal{O}(\text{AR})$. This AR-based framework generalizes some previous assumptions, encompassing a broader class of polygons. Our result establishes a framework guaranteeing finite solutions for segment guarding under practical and intuitive geometric constraints.

cs.CG↗

arXiv2024

Quantum Complexity vs Classical Complexity: A Survey

Scientists have demonstrated that quantum computing has presented novel approaches to address computational challenges, each varying in complexity. Adapting problem-solving strategies is crucial to harness the full potential of quantum computing. Nonetheless, there are defined boundaries to the capabilities of quantum computing. This paper concentrates on aggregating prior research efforts dedicated to solving intricate classical computational problems through quantum computing. The objective is to systematically compile an exhaustive inventory of these solutions and categorize a collection of demanding open problems that await further exploration. Through statistical analysis, we help the researchers with their further investigations.

cs.CC↗

arXiv2023

The Loop of the Rings: A Fully Decentralized Cooperative System (The Concept)

We introduce $LoR$, a secure, fully decentralized, and distributed cooperative system, where $LoR$ stands for "the Loop of the Rings". Distinct from conventional transaction-oriented systems, $LoR$ prioritizes {\it cooperation} using its ring-based structure, making it possible to be used both as a cooperative workspace as well as a versatile platform for service provisioning, accommodating various roles such as freelancers, IoT management systems, and even managing 5G-related services. Within this system, users have access to a secure and reliable environment, enabling them to offer a specific set of services to a potentially vast number of users. Our main contribution is to introduce the new structure along with its operating rules and principles, applying a combination of randomized procedures, in such a way that the whole system can be modeled fairly accurately in mathematically rigorous terms. This, in particular, is used to provide rigorous proof for the facts that $LoR$ is both reliable and secure and that it may be efficiently implemented based on its typical communication complexity.

cs.CR↗

arXiv2023

Visibility Extension via Reflection

This paper studies a variant of the Art Gallery problem in which the ``walls" can be replaced by \emph{reflecting edges}, which allows the guards to see further and thereby see a larger portion of the gallery. Given a simple polygon $\cal P$, first, we consider one guard as a point viewer, and we intend to use reflection to add a certain amount of area to the visibility polygon of the guard. We study visibility with specular and diffuse reflections where the specular type of reflection is the mirror-like reflection, and in the diffuse type of reflection, the angle between the incident and reflected ray may assume all possible values between $0$ and $π$. Lee and Aggarwal already proved that several versions of the general Art Gallery problem are $NP$-hard. We show that several cases of adding an area to the visible area of a given point guard are $NP$-hard, too. Second, we assume all edges are reflectors, and we intend to decrease the minimum number of guards required to cover the whole gallery. Chao Xu proved that even considering $r$ specular reflections, one may need $\lfloor \frac{n}{3} \rfloor$ guards to cover the polygon. Let $r$ be the maximum number of reflections of a guard's visibility ray. In this work, we prove that considering $r$ \emph{diffuse} reflections, the minimum number of \emph{vertex or boundary} guards required to cover a given simple polygon $\cal P$ decreases to { $\bf \lceil \fracα{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}, where $α$ indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the $\mathcal{O}(\log n)$-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection.

cs.CG↗

arXiv2023

Agent-Cells with DNA Programming: A Dynamic Decentralized System

This paper introduces a new concept. We intend to give life to a software agent. A software agent is a computer program that acts on a user's behalf. We put a DNA inside the agent. DNA is a simple text, a whole roadmap of a network of agents or a system with details. A Dynamic Numerical Abstract of a multiagent system. It is also a reproductive part for an \emph{agent} that makes the agent take actions and decide independently and reproduce coworkers. By defining different DNA structures, one can establish new agents and different nets for different usages. We initiate such thinking as \emph{DNA programming}. This strategy leads to a new field of programming. This type of programming can help us manage large systems with various elements with an incredibly organized customizable structure. An agent can reproduce another agent. We put one or a few agents around a given network, and the agents will reproduce themselves till they can reach others and pervade the whole network. An agent's position or other environmental or geographical characteristics make it possible for an agent to know its active set of \emph{genes} on its DNA. The active set of genes specifies its duties. There is a database that includes a list of functions s.t. each one is an implementation of what a \emph{gene} represents. To utilize a decentralized database, we may use a blockchain-based structure. This design can adapt to a system that manages many static and dynamic networks. This network could be a distributed system, a decentralized system, a telecommunication network such as a 5G monitoring system, an IoT management system, or even an energy management system. The final system is the combination of all the agents and the overlay net that connects the agents. We denote the final net as the \emph{body} of the system.

cs.MA↗

arXiv2023

Trajectory Range Visibility

Consider two entities with constant but not necessarily equal velocities, moving on two given piece-wise linear trajectories inside a simple polygon $P$. The Trajectory Range Visibility problem deals with determining the sub-trajectories on which two entities become visible to each other. A more straightforward decision version of this problem is called Trajectory Visibility, where the trajectories are line segments. The decision version specifies whether the entities can see one another. This version was studied by P. Eades et al. in 2020, where they supposed given constant velocities for the entities. However, the approach presented in this paper supports non-constant complexity trajectories. Furthermore, we report every pair of constant velocities with which the entities can see each other. In particular, for every constant velocity of a moving entity, we specify: $(1)$ All visible parts of the other entity's trajectory. $(2)$ All possible constant velocities of the other entity to become visible. Regarding line-segment trajectories, we present $\mathcal{O}(n \log n)$ running time algorithm which obtains all pairs of sub-trajectories on which the moving entities become visible to one another, where $n$ is the complexity of $P$. Regarding the general case, we provide an algorithm with $\mathcal{O}(n \log n + m(\log m + \log n))$ running time, where $m$ indicates the complexity of both trajectories. We offer $\mathcal{O}(\log n)$ query time for line segment trajectories and $\mathcal{O}(\log m + k)$ for the non-constant complexity ones s.t. $k$ is the number of velocity ranges reported in the output. Interestingly, our results require only $\mathcal{O}(n + m)$ space for non-constant complexity trajectories.

cs.CG↗

arXiv2021

A Constant-Factor Approximation Algorithm for Point Guarding an Art Gallery

Given a simple polygon $\cal P$, in the Art Gallery problem the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the edges of $\cal P$. This paper studies a variant of the Art Gallery problem in which guards are restricted to lie on a dense grid inside $\cal P$. In the general problem, guards can be anywhere inside or on the boundary of $\cal P$. The general problem is called the \emph{point} guarding problem. It was proved that the point guarding problem is APX-complete, meaning that we cannot do better than a constant-factor approximation algorithm unless $P = NP$. A huge amount of research is committed to the studies of combinatorial and algorithmic aspects of this problem, and as of this time, we could not find a constant factor approximation for simple polygons. The last best-known approximation factor for point guarding a simple polygon was $\mathcal{O}(\log (|OPT|))$ introduced by E. Bonnet and T. Miltzow in 2020, where $|OPT|$ is the size of the optimal solution. Here, we propose an algorithm with a constant approximation factor for the point guarding problem where the location of guards is restricted to a grid. The running time of the proposed algorithm depends on the number of cells of the grid. The approximation factor is constant regardless of the grid we use, the running time could be super-polynomial if the grid size becomes exponential.

cs.CG↗

arXiv2021

Art Gallery Plus Single Specular-reflection

Given a simple polygon $\cal P$, in the Art Gallery problem, the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the edges of $\cal P$. This paper studies a variant of the Art Gallery problem in which the boundaries of $\cal P$ are replaced by single specular-reflection edges, allowing the view rays to reflect once per collision with an edge. This property allows the guards to see through the reflections, thereby viewing a larger portion of the polygon. For this problem, the position of the guards in $\cal P$ can be determined with our proposed $\mathcal{O}(\log n)$-approximation algorithm. Besides presenting an algorithm with the mentioned approximation factor, we will see that reflection can decrease the number of guards in practice. The proposed algorithm converts the generalized problem to the Set Cover problem.

cs.CG↗

arXiv2018

Visibility Extension via Reflective Edges to an Exact Quantity

We consider extending the visibility polygon of a given point $q$, inside a simple polygon $P$ by converting some edges of $P$ to mirrors. We will show that several variations of the problem of finding mirror-edges to add precisely $k$ units of area to $VP(q)$ are NP-complete. The optimal cases are NP-hard. We are unaware of any result on adding an exact number to a polygon, or covering an area with an exact surface. We deal with both single and multiple reflecting mirrors for both specular or diffuse types of reflections.

cs.CC↗

arXiv2017

Expanding Visibility Polygons by Mirrors upto at least K units

We consider extending visibility polygon $(VP)$ of a given point $q$ $(VP(q))$, inside a simple polygon $¶$ by converting some edges of $¶$ to mirrors. We will show that several variations of the problem of finding mirror-edges to add at least $k$ units of area to $VP(q)$ are NP-complete, or NP-hard. Which $k$ is a given value. We deal with both single and multiple reflecting mirrors, and also specular or diffuse types of reflections. In specular reflection, a single incoming direction is reflected into a single outgoing direction. In this paper diffuse reflection is regarded as reflecting lights at all possible angles from a given surface. The paper deals with finding mirror-edges to add \emph{at least} $k$ units of area to $VP(q)$. In the case of specular type of reflections we only consider single reflections, and the multiple case is still open. Specular case of the problem is more tricky. We construct a simple polygon for every given instance of a 3-SAT problem. There are some specific spikes which are visible only by some particular mirror-edges. Consequently, to have minimum number of mirror-edges it is required to choose only one of these mirrors to see a particular spike. There is a reduction polygon which contains a clause-gadget corresponding to every clause, and a variable-gadget corresponding to every variable. 3-SAT formula has $n$ variables and $m$ clauses, so the minimum number of mirrors required to add an area of at least $k$ to $V P(q)$ is $l = 3m+n+1$ if and only if the 3-SAT formula is satisfiable. This reduction works in these two cases: adding at least $k$ vertex of $¶$ to $VP(q)$, and expanding $VP(q)$ at least $k$ units of area.

cs.CG↗