FUN 2014: Fun with Algorithms pp 194-205

Excuse Me! or The Courteous Theatregoers’ Problem

(Extended Abstract)
• Konstantinos Georgiou
• Evangelos Kranakis
• Danny Krizanc
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8496)

Abstract

Consider a theatre consisting of m rows each containing n seats. Theatregoers enter the theatre along aisles and pick a row which they enter along one of its two entrances so as to occupy a seat. Assume they select their seats uniformly and independently at random among the empty ones. A row of seats is narrow and an occupant who is already occupying a seat is blocking passage to new incoming theatregoers. As a consequence, occupying a specific seat depends on the courtesy of theatregoers and their willingness to get up so as to create free space that will allow passage to others. Thus, courtesy facilitates and may well increase the overall seat occupancy of the theatre. We say a theatregoer is courteous if (s)he will get up to let others pass. Otherwise, the theatregoer is selfish. A set of theatregoers is courteous with probability p (or p-courteous, for short) if each theatregoer in the set is courteous with probability p, randomly and independently. It is assumed that the behaviour of a theatregoer does not change during the occupancy of the row. Thus, p = 1 represents the case where all theatregoers are courteous and p = 0 when they are all selfish.

In this paper, we are interested in the following question: what is the expected number of occupied seats as a function of the total number of seats in a theatre, n, and the probability that a theatregoer is courteous, p? We study and analyze interesting variants of this problem reflecting behaviour of the theatregoers as entirely selfish, and p-courteous for a row of seats with one or two entrances and as a consequence for a theatre with m rows of seats with multiple aisles. We also consider the case where seats in a row are chosen according to the geometric distribution and the Zipf distibrution (as opposed to the uniform distribution) and provide bounds on the occupancy of a row (and thus the theatre) in each case. Finally, we propose several open problems for other seating probability distributions and theatre seating arrangements.

Keywords

(p-)Courteous Theatregoers Theatre occupancy Seat Selfish Row Uniform distribution Geometric distribution Zipf distribution

References

1. 1.
Aronson, J., Dyer, M., Frieze, A., Suen, S.: Randomized greedy matching. II. Random Structures & Algorithms 6(1), 55–73 (1995)
2. 2.
Baxter, R.J.: Planar lattice gases with nearest-neighbour exclusion. Annals of Combin. 3, 191–203 (1999)
3. 3.
Bouttier, J., Di Francesco, P., Guitte, E.: Critical and tricritical hard objects on bicolorable random lattices: Exact solutions. J. Phys. A35, 3821–3854 (2012), Also available as arXiv:cond-mat/0201213Google Scholar
4. 4.
Bouttier, J., Di Francesco, P., Guitte, E.: Combinatorics of hard particles on planar graphs. J. Phys. A38, 4529–4559 (2005), Also available as arXiv:math/0501344v2Google Scholar
5. 5.
Calkin, N.J., Wilf, H.S.: The number of independent sets in a grid graph. SIAM J. Discret. Math. 11(1), 54–60 (1998)
6. 6.
Dyer, M., Frieze, A.: Randomized greedy matching. Random Structures & Algorithms 2(1), 29–45 (1991)
7. 7.
Finch, S.R.: Several Constants Arising in Statistical Mechanics. Annals of Combinatorics, 323–335 (1999)Google Scholar
8. 8.
Flory, P.J.: Intramolecular reaction between neighboring substituents of vinyl polymers. Journal of the American Chemical Society 61(6), 1518–1521 (1939)
9. 9.
Freedman, D., Shepp, L.: An unfriendly seating arrangement (problem 62-3). SIAM Review 4(2), 150 (1962)
10. 10.
Friedman, H.D., Rothman, D.: Solution to: An unfriendly seating arrangement (problem 62-3). SIAM Review 6(2), 180–182 (1964)
11. 11.
Georgiou, K., Kranakis, E., Krizanc, D.: Random maximal independent sets and the unfriendly theater seating arrangement problem. Discrete Mathematics 309(16), 5120–5129 (2009)
12. 12.
Georgiou, K., Kranakis, E., Krizanc, D.: Excuse Me! or The Courteous Theatregoers’ Problem, eprint arXiv, primary class cs.DM (2014), http://arxiv.org/abs/1403.1988
13. 13.
Kranakis, E., Krizanc, D.: Maintaining privacy on a line. Theory of Computing Systems 50(1), 147–157 (2012)
14. 14.
MacKenzie, J.K.: Sequential filling of a line by intervals placed at random and its application to linear adsorption. The Journal of Chemical Physics 37(4), 723–728 (1962)
15. 15.
Mitzenmacher, M., Upfal, E.: Probability and computing: Randomized algorithms and probabilistic analysis. Cambridge University Press (2005)Google Scholar
16. 16.
Olson, W.H.: A markov chain model for the kinetics of reactant isolation. Journal of Applied Probability, 835–841 (1978)Google Scholar
17. 17.
Strogatz, S.H.: The Joy of X: A Guided Tour of Math, from One to Infinity. Eamon Dolan/Houghton Mifflin Harcourt (2012)Google Scholar
18. 18.
Zipf, G.K.: Human behavior and the principle of least effort. Addison-Wesley (1949)Google Scholar

© Springer International Publishing Switzerland 2014

Authors and Affiliations

• Konstantinos Georgiou
• 1
• Evangelos Kranakis
• 2
• Danny Krizanc
• 3
1. 1.Department of Combinatorics & OptimizationUniversity of WaterlooCanada
2. 2.School of Computer ScienceCarleton UniversityCanada
3. 3.Department of Mathematics & Computer ScienceWesleyan UniversityUSA