Martin Gardner in the early 1970’s described the game of RaceTrack [M. Gardner, Mathematical games—Sim, Chomp and Race Track: new games for the intellect (and not for Lady Luck), Scientific American, 228(1):108–115, Jan. 1973]. Here we study the complexity of deciding whether a RaceTrack player has a winning strategy. We first prove that the complexity of RaceTrack reachability, i.e., whether the finish line can be reached or not, crucially depends on whether the car can touch the edge of the carriageway (racetrack): the non-touching variant is NL-complete while the touching variant is equivalent to the undirected grid graph reachability problem, a problem in L but not known to be L-hard. Then we show that single-player RaceTrack is NL-complete, regardless of whether driving on the track boundary is allowed or not, and that deciding the existence of a winning strategy in Gardner’s original two-player game is P-complete. Hence RaceTrack is an example of a game that is interesting to play despite the fact that deciding the existence of a winning strategy is most likely not NP-hard.
- Turing Machine
- Winning Strategy
- Graph Reachability
- Reachability Problem
- Virtual Edge
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Tax calculation will be finalised at checkout
Purchases are for personal use onlyLearn about institutional subscriptions
Unable to display preview. Download preview PDF.
Allender, E., Mix Barrington, D.A., Chakraborty, T., Datta, S., Roy, S.: Planar and grid graph reachability problems. Theory of Computing Systems 45(4), 675–723 (2009)
Blum, M., Kozen, D.: On the power of the compass (or, why mazes are easier to search than graphs). In: Proceedings of the 19th Symposium on Foundations of Computer Science, Ann Arbor, Michigan, USA, pp. 132–142. IEEE Computer Society, Los Alamitos (1978)
Erickson, J.: How hard is optimal racing? (2009), http://3dpancakes.typepad.com/ernie/2009/06/how-hard-is-optimal-racing.html
Gardner, M.: Mathematical games—Sim, Chomp and Race Track: new games for the intellect (and not for Lady Luck). Scientific American 228(1), 108–115 (1973)
Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, Oxford (1995)
Jones, N.D., Laaser, W.T.: Complete problems for deterministic polynomial time. Theoretical Computer Science 3, 105–117 (1977)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Reingold, O.: Undirected connectivity in log-space. Journal of the ACM Article 17, 55(4), 24 (2008)
Schmid, J.: VectorRace (2005), http://schmid.dk/articles/vectorRace.pdf
Editors and Affiliations
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Holzer, M., McKenzie, P. (2010). The Computational Complexity of RaceTrack . In: Boldi, P., Gargano, L. (eds) Fun with Algorithms. FUN 2010. Lecture Notes in Computer Science, vol 6099. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13122-6_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13121-9
Online ISBN: 978-3-642-13122-6