Walking an Unknown Street with Limited Sensing

2019 ◽  
Vol 33 (13) ◽  
pp. 1959042
Author(s):  
Qi Wei ◽  
Xuehou Tan ◽  
Yonggong Ren
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Mobile Robot ◽  
Shortest Path ◽  
Competitive Strategy ◽  
Visible Region ◽  
Simple Polygon ◽  
Limited Sensing

This paper studies a searching problem in an unknown street. A simple polygon [Formula: see text] with two distinguished vertices, [Formula: see text] and [Formula: see text], is called a street if the two boundary chains from [Formula: see text] to [Formula: see text] are mutually weakly visible. We use a mobile robot to locate [Formula: see text] starting from [Formula: see text]. Assume that the robot has a limited sensing capability that can only detect the constructed edges (also called gaps) on the boundary of its visible region, but cannot measure any angle or distance. The robot does not have knowledge of the street in advance. We present a new competitive strategy for this problem and prove that the length of the path generated by the robot is at most 9-times longer than the shortest path. We also propose a matching lower bound to show that our strategy is optimal. Compared with the previous strategy, we further relaxed the restriction that the robot should take a marking device and use the data structure S-GNT. The analysis of our strategy is tight.

LOGARITHMIC-TIME LINK PATH QUERIES IN A SIMPLE POLYGON

1995 ◽  
Vol 05 (04) ◽  
pp. 369-395 ◽  
Author(s):  
ESTHER M. ARKIN ◽  
JOSEPH S.B. MITCHELL ◽  
SUBHASH SURI
Keyword(s):  
Data Structure ◽  
Shortest Path ◽  
Simple Polygon ◽  
Additive Constant ◽  
Time And Space ◽  
Link Distance ◽  
Path Distance ◽  
Path Queries ◽  
Shortest Path Distance ◽  
Small Additive

We develop a data structure for answering link distance queries between two arbitrary points in a simple polygon. The data structure requires O(n3) time and space for its construction and answers link distance queries in O(log n) time, after which a minimum-link path can be reported in time proportional to the number of links. Here, n denotes the number of vertices of the polygon. Our result extends to link distance queries between pairs of segments or polygons. We also propose a simpler data structure for computing a link distance approximately, where the error is bounded by a small additive constant. Finally, we also present a scheme for approximating the link and the shortest path distance simultaneously.

scholarly journals Optimal Shortest Path and Minimum-Link Path Queries between Two Convex Polygons inside a Simple Polygonal Obstacle

1997 ◽  
Vol 07 (01n02) ◽  
pp. 85-121 ◽  
Author(s):  
Yi-Jen Chiang ◽  
Roberto Tamassia
Keyword(s):  
Data Structure ◽  
Shortest Path ◽  
Dynamic Environment ◽  
Simple Polygon ◽  
Efficient Algorithms ◽  
Optimal Time ◽  
Convex Polygons ◽  
Separation Problem ◽  
Dynamic Case ◽  
Path Queries

We present efficient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O( log h + log n) (plus O(k) time for reporting the k edges of the path) using a data structure with O(n) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O( log h + log n) (plus O(k) to report the k links), with O(n3) space and preprocessing time. We also extend our results to the dynamic case, and give a unified data structure that supports both queries for convex polygons in the same region of a connected planar subdivision [Formula: see text]. The update operations consist of insertions and deletions of edges and vertices. Let n be the current number of vertices in [Formula: see text]. The data structure uses O(n) space, supports updates in O( log 2 n) time, and performs shortest-path and minimum-link-path queries in times O( log h+ log 2n) (plus O(k) to report the k edges of the path) and O( log h + k log 2 n), respectively. Performing shortest-path queries is a variation of the well-studied separation problem, which has not been efficiently solved before in the presence of obstacles. Also, it was not previously known how to perform minimum-link-path queries in a dynamic environment, even for two-point queries.

FINDING AN OPTIMAL BRIDGE BETWEEN TWO POLYGONS

2002 ◽  
Vol 12 (03) ◽  
pp. 249-261 ◽  
Author(s):  
XUEHOU TAN
Keyword(s):  
Hierarchical Structure ◽  
Shortest Path ◽  
Line Segment ◽  
Euclidean Distance ◽  
Geodesic Distance ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Search Trees ◽  
Bridge Problem ◽  
Path Queries

Let π(a,b) denote the shortest path between two points a, b inside a simple polygon P, which totally lies in P. The geodesic distance between a and b in P is defined as the length of π(a,b), denoted by gd(a,b), in contrast with the Euclidean distance between a and b in the plane, denoted by d(a,b). Given two disjoint polygons P and Q in the plane, the bridge problem asks for a line segment (optimal bridge) that connects a point p on the boundary of P and a point q on the boundary of Q such that the sum of three distances gd(p′,p), d(p,q) and gd(q,q′), with any p′ ∈ P and any q′ ∈ Q, is minimized. We present an O(n log 3 n) time algorithm for finding an optimal bridge between two simple polygons. This significantly improves upon the previous O(n2) time bound. Our result is obtained by making substantial use of a hierarchical structure that consists of segment trees, range trees and persistent search trees, and a structure that supports dynamic ray shooting and shortest path queries as well.

A New Algorithm for Euclidean Shortest Path of Visiting Segments in a Polygon

2014 ◽  
Vol 644-650 ◽  
pp. 1891-1894
Author(s):  
Li Juan Wang ◽  
An Sheng Deng ◽  
Bo Jiang ◽  
Qi Wei
Keyword(s):  
Test Data ◽  
Shortest Path ◽  
Time Complexity ◽  
Simple Polygon ◽  
Rubber Band ◽  
Crossing Over ◽  
Euclidean Shortest Path ◽  
Adjacent Segments ◽  
Improved Algorithm

Let s and t be two points on the boundary of a simple polygon, how to compute the Euclidean shortest path between s and t which visits a sequence of segments given in the simple polygon is the problem to be discussed, especially, the situation of the adjacent segments intersect is the focus of our study. In this paper, we first analyze the degeneration applying rubber-band algorithm to solve the problem. Then based on rubber-band algorithm, we present an improved algorithm which can solve the degeneration by the method of crossing over two segments to deal with intersection and in our algorithm the adjacent segments order can be changed when they intersect. Particularly, we have implemented the algorithm and have applied a large of test data to test it. The experiments demonstrate that our algorithm is correct and efficient, and it has the same time complexity as the rubber-band algorithm.

scholarly journals m-Bonsai: A Practical Compact Dynamic Trie

2018 ◽  
Vol 29 (08) ◽  
pp. 1257-1278 ◽  
Author(s):  
Andreas Poyias ◽  
Simon J. Puglisi ◽  
Rajeev Raman
Keyword(s):  
Data Structure ◽  
Lower Bound ◽  
Upper Bound ◽  
Hash Functions ◽  
Information Theoretic ◽  
Expected Time ◽  
Speed Performance ◽  
Practical Performance

We consider the problem of implementing a space-efficient dynamic trie, with an emphasis on good practical performance. For a trie with [Formula: see text] nodes with an alphabet of size [Formula: see text], the information-theoretic space lower bound is [Formula: see text] bits. The Bonsai data structure is a compact trie proposed by Darragh et al. (Softw. Pract. Exper. 23(3) (1993) 277–291). Its disadvantages include the user having to specify an upper bound [Formula: see text] on the trie size in advance (which cannot be changed easily after initalization), a space usage of [Formula: see text] (which is asymptotically non-optimal for smaller [Formula: see text] or if [Formula: see text]) and a lack of support for deletions. It supports traversal and update operations in [Formula: see text] expected time (based on assumptions about the behaviour of hash functions), where [Formula: see text] and has excellent speed performance in practice. We propose an alternative, m-Bonsai, that addresses the above problems, obtaining a trie that uses [Formula: see text] bits in expectation, and supports traversal and update operations in [Formula: see text] expected time and [Formula: see text] amortized expected time, for any user-specified parameter [Formula: see text] (again based on assumptions about the behaviour of hash functions). We give an implementation of m-Bonsai which uses considerably less memory and is slightly faster than the original Bonsai.

Navigation of a mobile robot by locally optimal trajectories

Robotica ◽  
1999 ◽  
Vol 17 (5) ◽  
pp. 553-562 ◽  
Author(s):  
Kokou Djath ◽  
Ali Siadet ◽  
Michel Dufaut ◽  
Didier Wolf
Keyword(s):  
Mobile Robot ◽  
Shortest Path ◽  
Laser Scanner ◽  
Navigation Algorithm ◽  
Map Construction ◽  
Unstructured Environment ◽  
Environment Perception ◽  
Simple Equations ◽  
Holonomic Mobile Robot ◽  
Local Map

This paper proposes a navigation system for a non-holonomic mobile robot. The navigation is based on a “look and move” approach. The aim is to define intermediate points called sub-goals through which the robot must pass. This algorithm is particularly suitable for navigation in an unknown environment and obstacle avoidance. Between two successive sub-goals, a shortest path planning solution is adopted. We have adopted the “Dubins' car” because of the environment perception sensor, a 180° laser scanner. In order to minimize the calculation time, the theoretical results of shortest path are approximated by simple equations. The navigation algorithm proposed can be used either in a structured or unstructured environment. In this context the local map construction is based on the segmentation of a structured environment; so for an unstructured environment, a suitable algorithm must be used instead.

scholarly journals Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

2018 ◽  
Vol 50 (01) ◽  
pp. 35-56 ◽  
Author(s):  
Nicolas Chenavier ◽  
Olivier Devillers
Keyword(s):  
Lower Bound ◽  
Point Process ◽  
Delaunay Triangulation ◽  
Shortest Path ◽  
Upper Bound ◽  
Poisson Point Process ◽  
Large Intensity ◽  
Stretch Factor ◽  
Expected Length

Abstract Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

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