On the Competitive Complexity of Navigation Tasks

Icking, Christian; Kamphans, Thomas; Klein, Rolf; Langetepe, Elmar

doi:10.1007/3-540-45993-6_14

Cited by 25 publications

(10 citation statements)

References 34 publications

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“…Also note that the universal lower bound refers to the worst-case performance of any specific online algorithm solving P . When the competitive upper bound of a specific algorithm for a task P matches up to constants the universal lower bound for P , the bound itself becomes the competitive complexity of the task [17]. Using a theoretical area-coverage approach, the competitive complexity of several mobile robot tasks is discussed in [15].…”

Section: Basic Setup and Definition Of Competitivenessmentioning

confidence: 99%

CBUG: A Quadratically Competitive Mobile Robot Navigation Algorithm

Gabriely

Rimon

2008

IEEE Trans. Robot.

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This paper is concerned with online navigation of a size D mobile robot in an unknown planar environment. A formal means for assessing algorithms for online tasks is competitiveness. For the navigation task, competitiveness measures the algorithm's path length relative to the optimal offline path length. While competitiveness usually means constant relative performance, it is measured in this paper in terms of a quadratic relationship between online performance and optimal offline solution. An online navigation algorithm for a size D robot called CBUG is described. The competitiveness of CBUG is analyzed and shown to be quadratic in the length of the shortest offline path. Moreover, it is shown that, in general, quadratic competitiveness is the best achievable performance over all online navigation algorithms. Thus, up to constants, CBUG achieves optimal competitiveness. The algorithm is improved with some practical speedups, and the usefulness of its competitiveness in terms of path stability is illustrated in office-like environments.

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Section: Basic Setup and Definition Of Competitivenessmentioning

confidence: 99%

CBUG: A Quadratically Competitive Mobile Robot Navigation Algorithm

Gabriely

Rimon

2008

IEEE Trans. Robot.

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“…The robot imposes an on line discretization of the continuous area into a grid of D-size cells [9,16]. The grid consists only of free cells 4 and is surrounded by partially occupied cells.…”

Section: Mrsam Motion Algorithm To An Unknown Targetmentioning

confidence: 99%

Classifying the multi robot path finding problem into a quadratic competitive complexity class

Sarid

Shapiro

2008

Ann Math Artif Intell

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We explore two motion planning problems where a group of mobile robots has to reach a target located in an a priori unknown environment while on-line planning the next step. In the first problem the target position is unknown and should be found by the robots, while in the second problem the target position is known and only a path to it should be found. We focus on optimizing the cost of the task in terms of motion time, which, under the assumption of uniform velocity of all the robots, correlates to the path length passed by the robot which reaches the target. The performance of an on-line algorithm is usually expressed in terms of Competitiveness, the constant ratio between the on-line and the optimal off-line solutions. Specifically, the ratio between the lengths of the actual path made by the robot which reached the target to the shortest path to the target. We use generalized competitiveness, i.e., the ratio is not necessarily constant, but could be any function. Classification of a motion planning task in the sense of performance is done by finding an upper and a lower bounds on the competitiveness of all algorithms solving that task. If the two bounds belong to the same functional class this is the Competitive Complexity Class of the task. We find the two bounds for the aforementioned common on-line motion planning problems, and classify them into competitive classes. It is shown that in general any on-line motion planning algorithm that tries to solve these problems must have at least a quadratic competitive performance. This is a lower bound of 170 S. Sarid, A. Shapiro the problems. This paper describes two new on-line navigation algorithm which solve the problems under discussion. The first is called MRSAM, short for MultiRobot Search Area Multiplication, and the second is called MRBUG, short for MultiRobot BUG which extends Lumelsky famous BUG algorithm. Both algorithms have quadratic upper bounds, which prove that the problems they solve have quadratic upper bounds. Thus it is shown that navigation in an unknown environment by a group of robots belongs to a quadratic competitive class. MRSAM and MRBUG have a quadratic competitive performance and thus have optimal competitiveness. The algorithms' performance is simulated in office-like environments.

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“…More precisely, we call a search strategy competitive with a factor C, if |Π| ≤ C · |Π opt | holds for every location of the hider, where |Π| denotes the length of the searcher's path and |Π opt | the shortest path to the hider. For analysing the efficiency of a search startegy we use the competitive framework which was introduced by Sleator and Tarjan [28], and used in many settings since then, see for example the survey by Fiat and Woeginger [10] or, for the field of online robot motion planning, see the surveys [24,17].…”

Section: Introductionmentioning

confidence: 99%

On the Optimality of Spiral Search

Langetepe

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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Searching for a point in the plane is a well-known search game problem introduced in the early eighties. The best known search strategy is given by a spiral and achieves a competitive ratio of 17.289 . . . It was shown by Gal [14] that this strategy is the best strategy among all monotone and periodic strategies. Since then it was unknown whether the given strategy is optimal in general. This paper settles this old open fundamental search problem and shows that spiral search is indeed optimal. The given problem can be considered as the continuous version of the well-known m-ray search problem and also appears in several non-geometric applications and modifications. Therefore the optimality of spiral search is an important question considered by many researchers in the last decades. We answer the logarithmic spiral conjecture for the given problem. The lower bound construction might be helpful for similar settings, it also simplifies existing proofs on classical m-ray search.

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On the Competitive Complexity of Navigation Tasks

Cited by 25 publications

References 34 publications

CBUG: A Quadratically Competitive Mobile Robot Navigation Algorithm

CBUG: A Quadratically Competitive Mobile Robot Navigation Algorithm

Classifying the multi robot path finding problem into a quadratic competitive complexity class

On the Optimality of Spiral Search

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