On the two-dimensional cow search problem

Jeż, Artur; Łopuszański, Jakub

doi:10.1016/j.ipl.2009.01.020

Cited by 29 publications

(11 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The paper also provides a matching upper bound, improving upon the previous upper bound of 12.5406 by Jez [15]. In [16], the optimality of spiral search is shown for a related problem of searching for a point in plane, in which a point is found if it lies on the line segment connecting the agent's current position with the O.…”

Section: Related Workmentioning

confidence: 92%

Improved Lower Bounds for Shoreline Search

Dobrev¹,

Královič²,

Pardubská³

2020

Structural Information and Communication Complexity

View full text Add to dashboard Cite

Shoreline search is a natural and well-studied generalisation of the classical cow-path problem: k initially co-located unit speed agents are searching for a line (called shoreline) in 2 dimensional Euclidean space. The shoreline is at (a possibly unknown) distance δ from the starting point O of the agents. The goal is to minimize the competitive ratio T δ δ , where T δ is the worst case (over all possible locations of the shoreline at distance δ) time until the shoreline is found.Upper bounds conjectured to be optimal have been established for all k ≥ 1[4], however lower bounds have been severely lacking. Recent paper [1] showed an improved lower bound for k = 2 and gave the first non-trivial lower bounds for k ≥ 3. While for k ≥ 4 the lower bounds match the best known upper bounds, that is not the case for k < 4.In this paper we improve the lower bound for k = 2 from 3 to (1 + √ 3 + π/6) ≈ 3.2556, and for k = 3 from √ 3 to 2. These lower bounds apply for known δ, matching the corresponding upper bounds. In fact, for k = 3 our lower bound matches the upper bound for unknown δ as well.We achieve these results by employing a novel simple virtual colouring technique, allowing us to transform the problem of covering the (uncountably many) points of the circle of radius δ (whose tangents represent all possible shorelines at distance δ) to a combinatorially much simpler problem of finding the shortest path from the centre to three specific tangents of this circle.

show abstract

Section: Related Workmentioning

confidence: 92%

Improved Lower Bounds for Shoreline Search

Dobrev¹,

Královič²,

Pardubská³

2020

Structural Information and Communication Complexity

View full text Add to dashboard Cite

show abstract

“…Below we give representative and selective examples, with an attempt to cite relatively recent results. Variations of search-type problems that share many similarities range from the type of search domain (for example, 1 or 2-dimensional [26,32], d-dimensional grid [17], cycle [37], polygons [22], graphs [6], grid [14], m-rays [12]), to the number of searchers (1 or more [36]), to the criterion for termination (for example, search, evacuation [13], priority evacuation [19], fetching [30]) to the communication model (for example, wireless or face-to-face [18]) to the type of the objective (for example, minimize worst case or average case [16]) to cost specs (for example, turning costs [25], cost for revisiting [10]), to the measure of efficiency (for example, time, energy [23]) to the knowledge of the input (none or partial [11]) and to other robots' specs (for example, speeds [21], faults [31], memory [38]), just to name a few. More recently, Fraigniaud et al considered in [27] a Bayesian search problem in a discrete space, where a set of searchers are trying to locate a treasure placed, according to some distribution, in one of the boxes indexed by positive integers.…”

Section: Related Workmentioning

confidence: 99%

Probabilistically Faulty Searching on a Half-Line

Bonato¹,

Georgiou²,

MacRury³

et al. 2020

Preprint

View full text Add to dashboard Cite

We study p-Faulty Search, a variant of the classic cow-path optimization problem, where a unit speed robot searches the half-line (or 1-ray) for a hidden item. The searcher is probabilistically faulty, and detection of the item with each visitation is an independent Bernoulli trial whose probability of success p is known. The objective is to minimize the worst case expected detection time, relative to the distance of the hidden item to the origin. A variation of the same problem was first proposed by Gal [28] in 1980. Alpern and Gal [3] proposed a so-called monotone solution for searching the line (2-rays); that is, a trajectory in which the newly searched space increases monotonically in each ray and in each iteration. Moreover, they conjectured that an optimal trajectory for the 2-rays problem must be monotone. We disprove this conjecture when the search domain is the half-line (1-ray). We provide a lower bound for all monotone algorithms, which we also match with an upper bound. Our main contribution is the design and analysis of a sequence of refined search strategies, outside the family of monotone algorithms, which we call t-sub-monotone algorithms. Such algorithms induce performance that is strictly decreasing with t, and for all p ∈ (0, 1). The value of t quantifies, in a certain sense, how much our algorithms deviate from being monotone, demonstrating that monotone algorithms are sub-optimal when searching the half-line.

show abstract

“…Typical tasks include exploring and mapping an unknown environment, finding a (mobile or immobile) target (e.g. cops and robbers games [8] and pursuit-evasion games [21]; the "lost at sea" problem [15]; the cow-path problem and plane-searching problem [2,3,4,9,16,17,20,22]), rendezvous or gathering of mobile agents [18,19], and evacuation [11,12,14]. (Note that we distinguish between the distributed version of evacuation problems involving a search for an unknown exit, and centralized versions, typically modeled as (dynamic) capacitated flow problems on graphs, where the exit is known.)…”

Section: Related Workmentioning

confidence: 99%

Fast Two-Robot Disk Evacuation with Wireless Communication

Lamprou¹,

Martin²,

Schewe³

2016

Lecture Notes in Computer Science

View full text Add to dashboard Cite

In the fast evacuation problem, we study the path planning problem for two robots who want to minimize the worst-case evacuation time on the unit disk. The robots are initially placed at the center of the disk. In order to evacuate, they need to reach an unknown point, the exit, on the boundary of the disk. Once one of the robots finds the exit, it will instantaneously notify the other agent, who will make a beeline to it.The problem has been studied for robots with the same speed [12]. We study a more general case where one robot has speed 1 and the other has speed s ≥ 1. We provide optimal evacuation strategies in the case that s ≥ c 2.75 ≈ 2.75 by showing matching upper and lower bounds on the worst-case evacuation time. For 1 ≤ s < c 2.75 , we show (non-matching) upper and lower bounds on the evacuation time with a ratio less than 1.22. Moreover, we demonstrate that a generalization of the two-robot search strategy from [12] is outperformed by our proposed strategies for any s ≥ c 1.71 ≈ 1.71.

show abstract

On the two-dimensional cow search problem

Cited by 29 publications

References 6 publications

Improved Lower Bounds for Shoreline Search

Improved Lower Bounds for Shoreline Search

Probabilistically Faulty Searching on a Half-Line

Fast Two-Robot Disk Evacuation with Wireless Communication

Contact Info

Product

Resources

About