On Computing the Fréchet Distance Between Surfaces.

A contour interpolation algorithm based on Fréchet distance

A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the {O}(n^2) dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to O(n^2/log^{2}n) in several ways and using a variety of ingenious tricks. This line of research, also known as the art of shaving log factors, lacks a tool for proving negative results. Specifically, how can we show that it is unlikely that LCS can be solved in time O(n^2/log^3n)?\nPerhaps the only approach for such results was suggested in a recent paper of Abboud, Hansen, Vassilevska W. and Williams (STOC'16). The authors blame the hardness of shaving logs on the hardness of solving satisfiability on boolean formulas (Formula-SAT) faster than exhaustive search. They show that an O(n^2/log^{1000} n) algorithm for LCS would imply a major advance in circuit lower bounds. Whether this approach can lead to tighter barriers was unclear.\nIn this paper, we push this approach to its limit and, in particular, prove that a well-known barrier from complexity theory stands in the way for shaving five additional log factors for fundamental combinatorial problems. For LCS, regular expression pattern matching, as well as the Fréchet distance problem from Computational Geometry, we show that an O(n^2/log^{7+epsilon}{n}) runtime would imply new Formula-SAT algorithms.\nOur main result is a reduction from SAT on formulas of size s over n variables to LCS on sequences of length N=2^{n/2} * s^{1+o(1)}. Our reduction is essentially as efficient as possible, and it greatly improves the previously known reduction for LCS with N=2^{n/2} * s^c, for some c >= 100.

In the last decade, drones became frequently used to provide eye-in-the-sky overview in the outdoor environment. Their main advantage compared to the other types of robots is that they can fly above obstacles and rough terrains and they can quickly cover large areas. These properties also open a new application; drones could provide a multi-hop, line of sight communication for groups of ground users. The aim of this thesis is to develop a drone team that will establish wireless ad-hoc network between users on the ground and distributively adapt links and spatial arrangement to the requirements and motion of the ground users. For this application, we use fixed wing drones. Such platforms can be easily and quickly deployed. Fixed wing drones have higher forward speed and higher battery life than hovering platforms. On the other hand, fixed wing drones have unicycle dynamics with constrained forward speed which makes them unable to hover or perform sharp turns. The first challenge consists in bridging unicycle dynamics of the fixed wing drones. Some control strategies have been proposed and validated in simulations using the average distance between the target and the drone as a performance metric. However, besides the distance metric, energy expenditure of the flight also plays an important role in assessing the overall performance of the flight. We propose a new methodology that introduces a new metric (energy expenditure), we compare existing methods on a large set of target motion patterns and present a comparison between the simulation and field experiments on proposed target motion patterns. The second challenge consists in developing a formation control algorithm that will allow fixed wing robots to provide a wide area coverage and to relay data in a wireless ad-hoc network. In such applications fixed wing drones have to be able to regulate an inter-drone distance. Their reduced maneuverability presents the main challenge to design a formation algorithm that will regulate an inter-drone distance. To address this challenge, we present a distributed control strategy that relies only on local information. Each drone has its own virtual agent, it follows the virtual agent by performing previously evaluated and selected target tracking strategy, and flocking interaction rules are implemented between virtual agents. It is shown in simulation and in field experiments with a team of fixed wing drones that using this distributed formation algorithm, drones can cover an area by creating an equilateral triangular lattice and regulate communication link quality between neighboring drones. The third challenge consists in allowing connectivity between independently moving ground users using fixed wing drone team. We design two distributed control algorithms that change drones' spatial arrangement and interaction topology to maintain the connectivity. We propose a potential field based strategy which adapts distance between drones to shrink and expand the fixed wing drones' formation. In second approach, market-based adaptation, drones distributively delete interaction links to expand the formation graph to a tree graph. In simulations and field experiments we show that our proposed strategies successfully maintain independently moving ground users connected. Overall, this thesis presents synthesis of distributed algorithms for fixed wing drones to establish and maintain wireless ad-hoc communication networks.

حفظ محیط زیست روستاها از جمله ضرورت‌های توسعه روستایی محسوب می‌شود. یکی از مسایلی که محیط زیست روستاها را تهدید می‌کند عدم توجه به نحوه جمع آوری و دفن زباله‌ها در محیط روستایی است. در این مقاله با توجه به پیچیدگی عوامل تأثیرگذار در انتخاب محل دفن بهداشتی زباله، ضرورت استفاده از فن‌آوری های اطلاعات جغرافیایی (GIS) و تلفیق آن با سایر امور مدیریتی و برنامه ریزی مطرح شده است. در این مقاله از سیستم‌های پشتیبانی تصمیم سازی فضایی (SDSS) برای مکان‌یابی محل دفن زباله‌های سکونتگاه‌های روستایی نمونه در شهرستان خواف استفاده شده است. پس از تجزیه و تحلیل لایه‌های اطلاعاتی مؤثر در مکان یابی محل دفن زباله و همپوشانی لایه‌ها، محدوده روستاها به صورت کاملا مناسب، مناسب، نسبتا مناسب و نامناسب برای مکان یابی محل دفن زباله پهنه‌بندی گردید. در مقایسه تطبیقی بین یافته‌های پژوهش و مکان فعلی دفن زباله در روستاهای نمونه، مشخص گردید که مکان دفن زباله در اکثریت روستاهای مورد مطالعه در پهنه نسبتا مناسب و نامناسب قرار دارند و فقط در سه روستای چمن آباد - ولی آباد، سیجاوند و چاه زول مکان دفن زباله در پهنه کاملا مناسب و مناسب قرار گرفته است. با توجه به یافته‌های تحقیق و جهت جلوگیری از آلودگی محیط زیست، برقراری یک سیستم مدیریتی امن جهت ساماندهی پسماندهای تولیدی منطقه مورد مطالعه امری لازم و ضروری می‌باشد؛ و از آن جایی که مکان یابی محل دفن زباله در نواحی روستایی مقوله صرفا خاص برای یک روستا نیست، پیشنهاد می‌شود یک مکان مناسب برای مجموعه‌ای از روستاهای همگن و نزدیک به هم مورد استفاده قرار گیرد.

We study the problem of computing correlation between two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in three dimensions---polyhedral terrains---can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in $O(n^{4/3}\\operatorname{polylog}n)$ expected time, where $n$ is the total number of vertices in the graphs of the two functions. We also present approximation algorithms for minimizing the mean distance between the graphs of univariate and bivariate functions. For univariate functions we present a $(1+\\varepsilon)$-approximation algorithm that runs in $O(n (1 + \\log^2 (1/\\varepsilon)))$ expected time for any fixed $\\varepsilon >0$. The $(1+\\varepsilon)$-approximation algorithm for bivariate functions runs in $O(n/\\varepsilon)$ time, for any fixed $\\varepsilon >0$, provided the two functions are defined over the same triangulation of their domain.

The 2D real time optimally adapting meshes (ROAM) algorithm has had wide success in the field of terrain visualization, because of its efficient error-controlling properties. In this paper, we propose a generalization of ROAM in 3D suitable for scalable volume visualization. Therefore, we perform a straightforward 2D/3D analogy, replacing the triangle of 2D ROAM by its 3D equivalent, the tetrahedron. Although work in the field of hierarchical tetrahedral meshes was widely undertaken, the produced meshes were not used for volumetric rendering purposes. We explain how to compute a bounded error inside the tetrahedron to build a hierarchical tetrahedral mesh and how to refine this mesh in real time to adapt it to the viewing conditions. We further show how to achieve cell sorting in linear time, thus yielding real time view-dependent display of the volumetric object. We present examples of large volume data sets and compare our approach with a similar one. Our results outline the high quality and computational efficiency of our approach.

We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Bresar-Klep theorem [Bresar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q over rationals. 1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q . Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem. 2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f. 3) The Bresar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d. Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].

A fundamental problem in computer and communication networks is the wavelength assignment (WA) problem: given a set of routing paths on a network, assign wavelengths (channels) to the paths such that the paths with the same wavelength are edge-disjoint. The optimization problem here is to minimize the number of wavelengths. A popular network topology is a tree of rings. It is known NP-hard to find the minimum number of wavelengths for the WA problem on a tree of rings. Let L be the maximum number of paths on any edge in the network. Then L is a lower bound on the number of wavelengths for the WA problem. We give a polynomial time algorithm which uses at most 3L wavelengths for the WA problem on a tree of rings with node degree at most eight. This improves the previous result of 4L. We also show that some instances of the WA problem require at least 3L wavelengths on a tree of rings, implying that the 3L upper bound is optimal for the worst case instances. In addition, we prove that our algorithm has approximation ratios 2 and 2.5 for a tree of rings with node degrees at most four and six, respectively.

We consider the classic scheduling problem of minimizing the total weighted flow-time on a single machine (min-WPFT), when preemption is allowed. In this problem, we are given a set of $n$ jobs, each job having a release time $r_j$, a processing time $p_j$, and a weight $w_j$. The flow-time of a job is defined as the amount of time the job spends in the system before it completes; that is, $F_j = C_j - r_j$, where $C_j$ is the completion time of job. The objective is to minimize the total weighted flow-time of jobs. This NP-hard problem has been studied quite extensively for decades. In a recent breakthrough, Batra, Garg, and Kumar presented a {\em pseudo-polynomial} time algorithm that has an $O(1)$ approximation ratio. The design of a truly polynomial time algorithm, however, remained an open problem. In this paper, we show a transformation from pseudo-polynomial time algorithms to polynomial time algorithms in the context of min-WPFT. Our result combined with the result of Batra, Garg, and Kumar settles the long standing conjecture that there is a polynomial time algorithm with $O(1)$-approximation for min-WPFT.

Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al. 2006), this is essentially the only class of linear programs for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call “distance-bounded network design convex programs”. These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in O((D/e) log n) rounds in the LOCAL model and with high probability finds a (1+e)-approximation to the optimal LP solution for any 0 < e ≤ 1, where D is the largest distance constraint. While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic 3- and 4-Spanner, Directed k-Spanner, Lowest Degree k-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any “heavy” computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.

In this thesis, we study the theoretical approach on energy-efficient scheduling problems arising in demand response management in the modern electrical smart grid. Consumers send in power requests with flexible feasible timeslots during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. We study the smart grid scheduling problem in different models. For the offline model, we prove the problem is NP-hard for the general case. We propose a polynomial time algorithm for special input where jobs have unit power request and unit time duration. By adapting the polynomial time algorithm for unit-size jobs, we propose an approximation algorithm for more general input. On the other hand, we also present an exact algorithm to find the optimal schedule for the problem with general input. For the online model, we propose an online algorithm for jobs with jobs with arbitrary power request, arbitrary time duration, and arbitrary contiguous feasible intervals. We also show a lower bound of the competitive ratio for the smart grid scheduling problem with unit height and arbitrary width. For special cases, we design different online algorithms with better competitive ratios. Finally, we look at other optimization problems and show how to solve them by adapting our techniques. We prove that our online algorithm can solve the machine minimization problem with an asymptotically optimal competitive ratio. We also show that our exact algorithm can be adapted to solve other demand response management problems.

We propose a new measure to capture similarity between polygonal curves, called the minimum backward Frechet distance. It is a natural optimization on the weak Frechet distance, a variant of the well-known Frechet distance. More specifically, for a given threshold e, we are searching for a pair of walks for two entities on the two input curves, T1 and T2, such that the union of the portions of backward movements is minimized and the distance between the two entities, at any time during the walk, is less than or equal to e. Our algorithm detects if no such pair of walks exists. This natural optimization problem appears in many applications in Geographical Information Systems, mobile networks and robotics. We provide an exact algorithm with time complexity of O(n2 log n) and space complexity of O(n2), where n is the maximum number of segments in the input polygonal curves.

As for a distribute target,the change of radar look angle and topographical slopes will arouse the polarization orientation angles shifts.On the contrary,when the radar look angle is known,topographical slopes can be estimated from the variance of the polarization orientation angle.Thus,in this paper,we review and summarize all kinds of estimation techniques proposed for the polarization orientation angles induced by terrain slopes so far,and evaluate the merit and deficiency of those methods.At the same time,besides the selection of algorithms,others factors that affect the accuracy of orientation angle estimations are discussed,and the current research and applications fields of orientation angles will also be presented.

We show that the Frechet distance between two piecewise linear surfaces can be decided in finite time, hence, the problem is decidable. For the special case that one of the surfaces is a triangle, we show that the problem is in PSPACE. In both cases, our computational model is a Turing Machine, and our algorithms rely on Canny's result [STOC 1988] that the existential theory of the real numbers is decidable in PSPACE.

Planetary rovers are required to explore rough and steep areas such as craters and cliffs to achieve scientific goals. Most of the proposed rovers have some wheels to move on a planetary surface. However, it is difficult to traverse on a rough terrain flexibly. A legged rover can provide higher capability of moving in such a rough terrain than wheeled robots. Therefore the authors have studied a small and lightweight rover which can move across a rough terrain such as a steep slope inside craters. This paper presents the system design concept, architecture, and configuration of the walking rover for planetary exploration. This paper also proposes a new walking algorithm on a rough terrain