# Searching without communicating: tradeoffs between performance and selection complexity

- 159 Downloads

## Abstract

We consider the ANTS problem (Feinerman et al.) in which a group of agents collaboratively search for a target in a two-dimensional plane. Because this problem is inspired by the behavior of biological species, we argue that in addition to studying the *time complexity* of solutions it is also important to study the *selection complexity*, a measure of how likely a given algorithmic strategy is to arise in nature due to selective pressures. Intuitively, the larger the \(\chi \) value, the more complicated the algorithm, and therefore the less likely it is to arise in nature. In more detail, we propose a new selection complexity metric \(\chi \), defined for algorithm \({\mathscr {A}}\) such that \(\chi ({\mathscr {A}}) = b + \log \ell \), where *b* is the number of memory bits used by each agent and \(\ell \) bounds the fineness of available probabilities (agents use probabilities of at least \(1/2^\ell \)). In this paper, we study the trade-off between the standard performance metric of speed-up, which measures how the expected time to find the target improves with *n*, and our new selection metric. Our goal is to determine the thresholds of algorithmic complexity needed to enable efficient search. In particular, consider *n* agents searching for a treasure located within some distance *D* from the origin (where *n* is sub-exponential in *D*). For this problem, we identify the threshold \(\log \log D\) to be crucial for our selection complexity metric. We first prove a new upper bound that achieves a near-optimal speed-up for \(\chi ({\mathscr {A}}) \approx \log \log D + \mathscr {O}(1)\). In particular, for \(\ell \in \mathscr {O}(1)\), the speed-up is asymptotically optimal. By comparison, the existing results for this problem (Feinerman et al.) that achieve similar speed-up require \(\chi ({\mathscr {A}}) = \varOmega (\log D)\). We then show that this threshold is tight by describing a lower bound showing that if \(\chi ({\mathscr {A}}) < \log \log D - \omega (1)\), then with high probability the target is not found in \(D^{2-o(1)}\) moves per agent. Hence, there is a sizable gap with respect to the straightforward \(\varOmega (D^2/n + D)\) lower bound in this setting.

## Keywords

Distributed algorithms Biology-inspired algorithms Search algorithms Markov chains## Notes

### Acknowledgments

We express our gratitude to Yoav Rodeh and the anonymous reviewers who provided us with many insights, ideas how to strengthen our results and helped us improve the presentation of the material.

## References

- 1.Afek, Y., Alon, N., Barad, O., Hornstein, E., Barkai, N., Bar-Joseph, Z.: A biological solution to a fundamental distributed computing problem. Science
**331**(6014), 183–185 (2011)MathSciNetCrossRefMATHGoogle Scholar - 2.Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput.
**29**(4), 1164–1188 (2000)MathSciNetCrossRefMATHGoogle Scholar - 3.Alon, N., Avin, C., Kouckỳ, M., Kozma, G., Lotker, Z., Tuttle, M.R.: Many random walks are faster than one. Comb., Prob. Comput.
**20**(04), 481–502 (2011)MathSciNetCrossRefMATHGoogle Scholar - 4.Ambuhl, C., Gasieniec, L., Pelc, A., Radzik, T., Zhang, X.: Tree exploration with logarithmic memory. ACM Trans. Algorithms
**7**(2), 17 (2011)MathSciNetCrossRefMATHGoogle Scholar - 5.Arbilly, M., Motro, U., Feldman, M.W., Lotem, A.: Co-evolution of learning complexity and social foraging strategies. J. Theor. Biol.
**267**(4), 573–581 (2010)MathSciNetCrossRefGoogle Scholar - 6.Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.: The power of a pebble: Exploring and mapping directed graphs. In: Proceedings of the ACM Symposium on Theory of Computing, pp. 269–278. ACM (1998)Google Scholar
- 7.Brauer, A.: On a problem of partitions. Am. J. Math.
**64**(1), 299–312 (1942)CrossRefMATHGoogle Scholar - 8.Deng, X., Papadimitriou, C.H.: Exploring an unknown graph. In: Proceedings of the Symposium on Foundations of Computer Science, pp. 355–361. IEEE (1990)Google Scholar
- 9.Diks, K., Fraigniaud, P., Kranakis, E., Pelc, A.: Tree exploration with little memory. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 588–597. Society for Industrial and Applied Mathematics (2002)Google Scholar
- 10.Emek, Y., Langner, T., Uitto, J., Wattenhofer, R.: Solving the ANTS problem with asynchronous finite state machines. In: Proceedings of the International Colloquium, pp. 471–482 (2014)Google Scholar
- 11.Emek, Y., Wattenhofer, R.: Stone age distributed computing. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, pp. 137–146. ACM (2013)Google Scholar
- 12.Feinerman, O., Korman, A.: Memory lower bounds for randomized collaborative search and implications for biology. In: Distributed Computing, pp. 61–75. Springer (2012)Google Scholar
- 13.Feinerman, O., Korman, A.: Theoretical distributed computing meets biology: A review. In: Distributed Computing and Internet Technology, pp. 1–18. Springer (2013)Google Scholar
- 14.Feinerman, O., Korman, A., Lotker, Z., Sereni, J.S.: Collaborative search on the plane without communication. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, pp. 77–86. ACM (2012)Google Scholar
- 15.Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, Hoboken (2008)MATHGoogle Scholar
- 16.Fraigniaud, P., Gasieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks
**48**(3), 166–177 (2006)MathSciNetCrossRefMATHGoogle Scholar - 17.Giraldeau, L.A., Caraco, T.: Social Foraging Theory. Princeton University Press, Princeton (2000)Google Scholar
- 18.Harkness, R., Maroudas, N.: Central place foraging by an ant (Cataglyphis bicolor Fab.): a model of searching. Anim. Behav.
**33**(3), 916–928 (1985)CrossRefGoogle Scholar - 19.Holder, K., Polis, G.: Optimal and central-place foraging theory applied to a desert harvester ant
*Pogonomyrmex californicus*. Oecologia**72**(3), 440–448 (1987)CrossRefGoogle Scholar - 20.Lenzen, C., Lynch, N., Newport, C., Radeva, T.: Trade-offs between selection complexity and performance when searching the plane without communication. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, pp. 252–261. ACM (2014)Google Scholar
- 21.McLeman, M., Pratt, S., Franks, N.: Navigation using visual landmarks by the ant leptothorax albipennis. Insectes Sociaux
**49**(3), 203–208 (2002)CrossRefGoogle Scholar - 22.O’Brien, C.: Solving ANTS with loneliness detection and constant memory. M.Eng Thesis, MIT EECS Department (2014)Google Scholar
- 23.Panaite, P., Pelc, A.: Exploring unknown undirected graphs. J. Algorithms
**33**(2), 281–295 (1999)MathSciNetCrossRefMATHGoogle Scholar - 24.Reingold, O.: Undirected connectivity in log-space. J. ACM (JACM)
**55**(4), 17 (2008)MathSciNetCrossRefMATHGoogle Scholar - 25.Robinson, E.J., Jackson, D.E., Holcombe, M., Ratnieks, F.L.: Insect communication: “no entry” signal in ant foraging. Nature
**438**(7067), 442–442 (2005)CrossRefGoogle Scholar - 26.Rosenthal, J.S.: Rates of convergence for data augmentation on finite sample spaces. Ann. Appl. Probab.
**3**(3), 819–839 (1993)Google Scholar