scholarly journals Optimal Algorithms for Range Searching over Multi-Armed Bandits

2021 ◽  
Author(s):  
Siddharth Barman ◽  
Ramakrishnan Krishnamurthy ◽  
Saladi Rahul
Keyword(s):  
High Probability ◽  
Range Searching ◽  
Dimensional Vector ◽  
Basic Form ◽  
Maximum Weight ◽  
Underlying Distribution ◽  
Hitting Set ◽  
Probably Approximately Correct ◽  
Set Of Points ◽  
The Given

This paper studies a multi-armed bandit (MAB) version of the range-searching problem. In its basic form, range searching considers as input a set of points (on the real line) and a collection of (real) intervals. Here, with each specified point, we have an associated weight, and the problem objective is to find a maximum-weight point within every given interval. The current work addresses range searching with stochastic weights: each point corresponds to an arm (that admits sample access) and the point's weight is the (unknown) mean of the underlying distribution. In this MAB setup, we develop sample-efficient algorithms that find, with high probability, near-optimal arms within the given intervals, i.e., we obtain PAC (probably approximately correct) guarantees. We also provide an algorithm for a generalization wherein the weight of each point is a multi-dimensional vector. The sample complexities of our algorithms depend, in particular, on the size of the {optimal hitting set} of the given intervals. Finally, we establish lower bounds proving that the obtained sample complexities are essentially tight. Our results highlight the significance of geometric constructs (specifically, hitting sets) in our MAB setting.

scholarly journals Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

2021 ◽  
Author(s):  
Eswara Venkata Kumar Dhulipala
Keyword(s):  
Euclidean Distance ◽  
Travelling Salesman Problem ◽  
Dynamic Programming Algorithm ◽  
Constant Factor ◽  
Programming Algorithm ◽  
Worst Case ◽  
Angle Bisector ◽  
Set Of Points ◽  
The Given ◽  
Tour Length

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

Spanning Properties of Yao and 𝜃-Graphs in the Presence of Constraints

2019 ◽  
Vol 29 (02) ◽  
pp. 95-120 ◽  
Author(s):  
Prosenjit Bose ◽  
André van Renssen
Keyword(s):  
Line Segment ◽  
Large Family ◽  
Upper Bounds ◽  
The Other ◽  
Additional Property ◽  
Graph Partitions ◽  
Set Of Points ◽  
The Given

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

GENERALIZED MELZAK'S CONSTRUCTION IN THE STEINER TREE PROBLEM

2002 ◽  
Vol 12 (06) ◽  
pp. 481-488 ◽  
Author(s):  
JIA F. WENG
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Steiner Tree Problem ◽  
Minimal Tree ◽  
Steiner Minimal Tree ◽  
The Core ◽  
Steiner Minimal Trees ◽  
Minimum Networks ◽  
Set Of Points ◽  
The Given

For a given set of points in the Euclidean plane, a minimum network (a Steiner minimal tree) can be constructed using a geometric method, called Melzak's construction. The core of the Melzak construction is to replace a pair of terminals adjacent to the same Steiner point with a new terminal. In this paper we prove that the Melzak construction can be generalized to constructing Steiner minimal trees for circles so that either the given points (terminals) are constrained on the circles or the terminal edges are tangent to the circles. Then we show that the generalized Melzak construction can be used to find minimum networks separating and surrounding circular objects or to find minimum networks connecting convex and smoothly bounded objects and avoiding convex and smoothly bounded obstacles.

The density recovery profile: A method for the analysis of points in the plane applicable to retinal studies

1991 ◽  
Vol 6 (2) ◽  
pp. 95-111 ◽  
Author(s):  
R. W. Rodieck
Keyword(s):  
Amacrine Cells ◽  
Effective Radius ◽  
Spatial Density ◽  
Two Dimensional ◽  
Underlying Distribution ◽  
Packing Factor ◽  
Recovery Profile ◽  
Starburst Amacrine Cells ◽  
The Mean ◽  
Set Of Points

AbstractThe density recovery profile is a plot of the spatial density of a set of points as a function of the distance of each of those points from all the others. It is based upon a two-dimensional point autocorrelogram. If the points are randomly distributed, then the profile is flat, with a value equal to the mean spatial density. Thus, any deviation from this value indicates that the presence of the object represented by the point alters the probability of encountering nearby objects of the same set. Increased value near an object indicates clustering, decreased value near an object indicates anticlustering. The method appears to be unique in its ability to provide quantitative measures of the anticlustered state. Two examples are presented. The first is based upon a sample of the distribution of the somata of starburst amacrine cells in the macaque retina; the second is based upon the distribution of the terminal enlargements on the dendrites of a single macaque ganglion cell that projects to the superior colliculus. In both cases, the density recovery profile is initially lower than the mean density, and increases up to the plateau at the value of the mean density. Two useful measures can be derived from this profile: an intensive parameter termed the effective radius, which quantifies the extent of the region of decreased probability and is insensitive to random undersampling of the underlying distribution, and an extensive parameter termed the packing factor, which quantifies the degree of packing possible for a given effective radius, and is insensitive to scaling. An extension of this method, applicable to correlations between two superimposed distributions, and based upon a two-dimensional point cross-correlogram, is also described.

scholarly journals Pre-crastination: time uncertainty increases walking effort

2020 ◽  
Author(s):  
E Hong Tiew ◽  
Nidhi Seethapathi ◽  
Manoj Srinivasan
Keyword(s):  
Energy Cost ◽  
High Probability ◽  
Time Estimation ◽  
Optimal Feedback Control ◽  
Time Constraints ◽  
Energy Costs ◽  
Time Duration ◽  
Time Uncertainty ◽  
Walking Speeds ◽  
The Given

AbstractIn many circumstances, humans walk in a manner that approximately minimizes energy cost. Here, we performed human subject experiments to examine how having a time constraint affects the speeds at which humans walk. First, we measured subjects’ preferred walking speeds to travel a given distance in the absence of any time constraints. Then, we asked subjects to travel the same distance under different time constraints. That is, they had to travel the given distance within the time duration provided – they can arrive early, but not late. Under these constraints, subjects systematically arrived earlier than necessary. Surprisingly, even when the time duration provided was large enough to walk at their unconstrained preferred speeds, subjects walked systematically faster than their unconstrained preferred speed. We propose that this faster-than-energy optimal speeds may be due to human uncertainty in time estimation. We show that a model assuming that humans perform stochastic optimal feedback control to arrive on time with high probability while minimizing expected energy costs predicts walking speeds higher than energy optimal, as observed in experiment.

scholarly journals CHARACTERIZATION OF THE EFFICIENCY OF THE FEATURES AGGREGATE IN FUZZY PATTERN RECOGNITION TASK

1997 ◽  
Vol 1 ◽  
pp. 78
Author(s):  
R. Grekov ◽  
A. Borisov
Keyword(s):  
Pattern Recognition ◽  
Decision Rule ◽  
Recognition Task ◽  
Dimensional Vector ◽  
Fuzzy Pattern Recognition ◽  
Fuzzy Pattern ◽  
The Given ◽  
Pattern Recognition Task

Let a set of objects exist each of which is described by N features X1? ..., XN, where each feature X} is a real number. So each object is set by N-dimensional vector (Xl5 ..., XN) and represents a point in the space of object descriptions, RN.There are also set objects for which degrees of membership in either class are unknown. A decision rule should be determined that could enable estimation of the membership of either object with unknown degrees of membership in the given classes (Ozols and Borisov, 1996). To determine the decision rule, such features should be found which give a possibility to distinguish objects belonging to different classes, i.e. features that are specific for each class. That is why a subtask of estimation of the efficiency of features should be solved. A function 5 should be determined which could enable estimation of the efficiency of both separate features and of features groups.Thus, the task is reduced to the determination of a number of features from set N that will best describe groups of objects and will enable possibly correct recognition of the object's membership in a class.

scholarly journals Numerical limitations of the attainment of the orientation of geological planes

2018 ◽  
Vol 10 (1) ◽  
pp. 395-402 ◽  
Author(s):  
Michał Michalak
Keyword(s):  
Statistical Criterion ◽  
Local Orientation ◽  
Combinatorial Algorithm ◽  
Floating Point Arithmetic ◽  
Reliable Source ◽  
Set Of Points ◽  
The Given ◽  
The Relationship ◽  
Orientation Parameters ◽  
Point Arithmetic

Abstract The paper discusses limitations of analytical attainment of the attitude of a geological plane by using three non-collinear points. We present problems that arise during computing the orientation of a plane generated by almost collinear points. We referred these errors to floating-point arithmetic inaccuracies. To demonstrate the problem, we examined a surface of constant orientation. We used Delaunay triangulation to calculate its local orientation parameters. We introduced a new measure of collinearity applicable for collecting attitude of planar triangles. Using this measure we showed that certain planes generated by the triangulation cannot be treated as a reliable source of measurement. To examine the relationship between collinearity and orientation, we used a combinatorial algorithm to obtain all possible planes from the given set of points. A statistical criterion of rejecting almost collinear planes was suggested.

scholarly journals An example of an infinite Steiner tree connecting an uncountable set

2015 ◽  
Vol 8 (3) ◽  
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov ◽  
Yana Teplitskaya
Keyword(s):  
Unique Solution ◽  
Infinite Number ◽  
Steiner Tree ◽  
Steiner Trees ◽  
Steiner Problem ◽  
Finite Sets ◽  
Branching Points ◽  
Set Of Points ◽  
The Given

AbstractWe construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, which are unique connected solutions of the Steiner problem for some given finite sets of points, and with growing complexity (i.e. the number of branching points).

Images of Linear Conditions on a Manhattan Plane

2020 ◽  
Vol 8 (1) ◽  
pp. 3-14
Author(s):  
V. Yurkov
Keyword(s):  
Plane Partition ◽  
Point Sets ◽  
Geometric Problem ◽  
Line Segments ◽  
Finite Set ◽  
Isolated Points ◽  
Finite Sum ◽  
Lattice Construction ◽  
Set Of Points ◽  
The Given

In this paper are considered planar point sets generated by linear conditions, which are realized in rectangular or Manhattan metric. Linear conditions are those expressed by the finite sum of the products of distances by numerical coefficients. Finite sets of points and lines are considered as figures defining linear conditions. It has been shown that linear conditions can be defined relative to other planar figures: lines, polygons, etc. The design solutions of the following general geometric problem are considered: for a finite set of figures (points, line segments, polygons...) specified on a plane with a rectangular metric, which are in a common position, it is necessary to construct sets that satisfy any linear condition. The problems in which the given sets are point and segment ones have been considered in detail, and linear conditions are represented as a sum or as relations of distances. It is proved that solution result can be isolated points, broken lines, and areas on the plane. Sets of broken lines satisfying the given conditions form families of isolines for the given condition. An algorithm for building isoline families is presented. The algorithm is based on the Hanan lattice construction and the isolines behavior in each node and each sub-region of the lattice. For isoline families defined by conditions for relation of distances, some of their properties allowing accelerate their construction process are proved. As an example for application of the described theory, the problem of plane partition into regions corresponding to a given set of points, lines and other figures is considered. The problem is generalized problem of Voronoi diagram construction, and considered in general formulation. It means the next: 1) the problem is considered in rectangular metric; 2) all given points may be integrated in various figures – separate points, line segments, triangles, quadrangles etc.; 3) the Voronoi diagram’s property of proximity is changed for property of proportionality. Have been represented examples for plane partition into regions, determined by two-point sets.

Numerical and Analytical Investigations of Nonlinear Synthesis Problems of Antenna with the Flat Radiating Aperture

2016 ◽  
Vol 7 ◽  
pp. 18 ◽  
Author(s):  
Petro Olexandrovych Savenko
Keyword(s):  
Optimal Design ◽  
Review Article ◽  
Two Dimensional ◽  
Nonlinear Integral Equations ◽  
Energy Characteristics ◽  
Nonlinear Spectral Problem ◽  
Set Of Points ◽  
The Given ◽  
Branching Solutions ◽  
Specific Synthesis

The review article contains results of researches of the synthesis problems of radiating systems with incomplete input information arising in particular during optimal design of radio, acoustic or other types of radiating systems (RS) by the given requirements to energy characteristics of radiated field. In mathematical terms, these tasks are reduced to study and nume­rical solution of one class of two-dimensional nonlinear integral equations of Hammerstein type that depend on two real para­meters. It was established that a characteristic feature of this class of equations is nonuniqueness and the branching (or bifur­cation) of existing solutions. Methods of solving two-parametric nonlinear spectral problem, which is necessary to finding the set of points of branching are proposed. Algorithms and numerical methods for the finding of branching solutions are built and founded. Numerous examples of specific synthesis problems are presented.

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