The simple grid polygon exploration problem

Abstract

This paper considers an on-line exploration problem. We use a mobile robot to explore an unknown simple grid polygon. The robot is assumed to have limited sensing capability that can only detect the four basic cells adjacent to it. The robot’s task is to explore each cell and to return to the start. To explore a cell, the robot has to enter it. We ask for a short exploration tour, that is, minimizing the number of multiple-visit cells. The competitive ratio is used to measure the performance of our strategy. It is the ratio between the length of the on-line exploration path and the length of the shortest path achieved with full knowledge of the polygon. As our main result, we present a new on-line exploration strategy and prove that the robot’s path is at most 7/6 times longer than the shortest path. This matches the previously known lower bound.

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References

  1. Almeida JPLSD, Nakashima RT, Neves-Jr F, Arruda LVRD (2019) Bio-inspired on-line path planner for cooperative exploration of unknown environment by a multi-robot system. Robot Auton Syst 112:32–48

    Article  Google Scholar 

  2. Chalopin J, Das S, Disser Y, Mihalák M, Widmayer P (2015) Mapping simple polygons: The power of telling convex from reflex. ACM Trans Algorithms 11(4):33:1–33:16

    MathSciNet  Article  Google Scholar 

  3. Deng X, Kameda T, Papadimitriou C (1991) How to learn an unknown environment. In: Proceedings 32nd annual symposium of foundations of computer science, pp. 298–303

  4. Dereniowski D, Disser Y, Kosowski A, Paja̧k D, Uznański P (2015) Fast collaborative graph exploration. Inf Comput 243:37–49

  5. Doi K, Yamauchi Y, Kijima S, Yamashita M (2018) Exploration of finite 2d square grid by a metamorphic robotic system. In: Izumi T, Kuznetsov P (eds.) Stabilization, Safety, and Security of Distributed Systems, Lecture Notes in Computer Science, vol. 11201, pp. 96–110. Springer, Berlin

  6. Foerster KT, Wattenhofer R (2016) Lower and upper competitive bounds for online directed graph exploration. Theoret Comput Sci 655:15–29

    MathSciNet  Article  Google Scholar 

  7. Gabriely Y, Rimon E (2003) Competitive on-line coverage of grid environments by a mobile robot. Comput Geom 24(3):197–224

    MathSciNet  Article  Google Scholar 

  8. Ghosh SK, Klein R (2010) Online algorithms for searching and exploration in the plane. Comput Sci Rev 4(4):189–201

    Article  Google Scholar 

  9. Hagius R, Icking C, Langetepe E (2004) Lower bounds for the polygon exploration problems. In: 20th European Workshop Computer Geometry, pp. 135–138

  10. Hoffmann F, Icking C, Klein R, Kriegel K (2002) The polygon exploration problem. SIAM J Comput 31(2):577–600

    MathSciNet  Article  Google Scholar 

  11. Icking C, Kamphans T, Klein R, Langetepe E (2002) On the competitive complexity of navigation tasks. In: Hager GD, Christensen HI, Bunke H, Klein R (eds) Sensor Based Intelligent Robots, vol 2238. Lecture Notes in Computer Science. pp 245–258. Springer, Berlin

  12. Icking C, Kamphans T, Klein R, Langetepe E (2005) Exploring simple grid polygons. In: Wang L (ed) Computing and Combinatorics, vol 3595. Lecture Notes in Computer Science. pp 524–533. Springer, Berlin

  13. Keshavarz-Kohjerdi F, Bagheri A (2016) Hamiltonian paths in l-shaped grid graphs. Theoret Comput Sci 621:37–56

    MathSciNet  Article  Google Scholar 

  14. Megow N, Mehlhorn K, Schweitzer P (2012) Online graph exploration: New results on old and new algorithms. Theoret Comput Sci 463:62–72

    MathSciNet  Article  Google Scholar 

  15. Ortolf C, Schindelhauer C (2012) Online multi-robot exploration of grid graphs with rectangular obstacles. In: Proceedings of the Twenty-fourth Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’12, pp 27–36. ACM, New York, USA

  16. Strom DP, Bogoslavskyi I, Stachniss C (2017) Robust exploration and homing for autonomous robots. Robot Auton Syst 90:125–135

    Article  Google Scholar 

  17. Tan X, Wei Q (2015) An improved on-line strategy for exploring unknown polygons. Combinatorial Optimization and Applications, vol 9486. Lecture Notes in Computer Science. pp 163–177. Springer, Berlin

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Correspondence to Qi Wei or Yonggong Ren.

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This work was supported in part by National Natural Science Foundation of China (No. 61702242, 61976109) and Doctoral Scientific Research Foundation of Liaoning Province (No. 2019-BS-153, 2019-BS-014).

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Wei, Q., Sun, J., Tan, X. et al. The simple grid polygon exploration problem. J Comb Optim (2021). https://doi.org/10.1007/s10878-021-00705-5

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Keywords

  • Path planning
  • On-line Exploration
  • Competitive analysis
  • Grid polygon
  • Mobile robot