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The ANTS problem

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Abstract

We introduce the Ants Nearby Treasure Search problem, which models natural cooperative foraging behavior such as that performed by ants around their nest. In this problem, k probabilistic agents, initially placed at a central location, collectively search for a treasure on the two-dimensional grid. The treasure is placed at a target location by an adversary and the agents’ goal is to find it as fast as possible as a function of both k and D, where D is the (unknown) distance between the central location and the target. We concentrate on the case in which agents cannot communicate while searching. It is straightforward to see that the time until at least one agent finds the target is at least \({\varOmega }(D+D^2/k)\), even for very sophisticated agents, with unrestricted memory. Our algorithmic analysis aims at establishing connections between the time complexity and the initial knowledge held by agents (e.g., regarding their total number k), as they commence the search. We provide a range of both upper and lower bounds for the initial knowledge required for obtaining fast running time. For example, we prove that \(\log \log k+{\varTheta }(1)\) bits of initial information are both necessary and sufficient to obtain asymptotically optimal running time, i.e., \({\mathcal {O}}(D+D^2/k)\). We also we prove that for every \(0<\epsilon < 1\), running in time \({\mathcal {O}}(\log ^{1-\epsilon } k\cdot (D+D^2/k))\) requires that agents have the capacity for storing \({\varOmega }(\log ^\epsilon k)\) different states as they leave the nest to start the search. To the best of our knowledge, the lower bounds presented in this paper provide the first non-trivial lower bounds on the memory complexity of probabilistic agents in the context of search problems. We view this paper as a “proof of concept” for a new type of interdisciplinary methodology. To fully demonstrate this methodology, the theoretical tradeoff presented here (or a similar one) should be combined with measurements of the time performance of searching ants.

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Notes

  1. Note that a lower bound of \(x\ge 1\) bits on the advice size implies a lower bound of \(2^{x-1}\) on the number of states used by the agents already when commencing the search.

  2. To measure the time complexity, we adopt the terminology of competitive analysis, and say that an algorithm is c-competitive, if its time complexity is at most c times the straightforward lower bound, that is, at most \(c(D+ D^2/k)\). In particular, an \({\mathcal {O}}(1)\)-competitive algorithm is an algorithm that runs in \({\mathcal {O}}(D+ D^2/k)\) time.

  3. The spiral search is a particular deterministic search algorithm that starts at a node v and enables the agent to visit all nodes at distance \({\varOmega }(\sqrt{x})\) from v by traversing x edges, for every integer x (see, e.g., [9]). For our purposes, since we are interested with asymptotic results only, we can replace this atomic navigation protocol with any procedure that guarantees such a property. For simplicity, in what follows, we assume that for any integer x, the spiral search of length x starting at a node v visits all nodes at distance at most \(\sqrt{x}/2\) from v.

  4. It is not clear whether or not a probabilistic oracle is strictly more powerful than a deterministic one. Indeed, the oracle assigning the advice is unaware of D, and may thus potentially use the randomization to reduce the size of the advices by balancing between the efficiency of the search for small values of D and larger values.

  5. We note that even though we consider a very liberal setting, and allow a very powerful oracle, the oracles we use for our upper bounds constructions are very simple and rely on much weaker assumptions. Indeed, these oracles are not only deterministic but also assign the same advice to each of the k agents.

  6. For example, it can simulate to following very liberal setting. Assume that in the preprocessing stage, the k agents are organized in a clique topology, and that each agent can send a separate message to each other agent. Furthermore, even though the agents are identical, in this preprocessing stage, let us assume that agents can distinguish the messages received from different agents, and that each of the k agents may use a different probabilistic protocol for this preliminary communication. In addition, no restriction is made either on the memory and computation capabilities of agents or on the preprocessing time, that is, the preprocessing stage takes finite, yet unlimited, time.

  7. Although the treasure is placed far, in principle, it could have also been placed much closer to the source. In fact, had the treasure been placed at a distance of roughly the number of agents \(k<K\), the algorithm would have had enough time T to find it. This is the intuition behind of choice of the value of parameter T.

  8. For example, note that the functions of the form \(\alpha _0+\alpha _1\log ^{\beta _1} x+\alpha _2\log ^{\beta _2}\log x+ \alpha _3 2^{\log ^{\beta _3 }\log x} \log x+\alpha _4\log ^{\beta _4} x\log ^{\beta _5}\log x\), (for non-negative constants \(\alpha _i\) and \(\beta _i\), \(i=1,2,3,4,5\) such that \(\sum _{i=1}^4\alpha _i>0\)) are all relatively-slow.

  9. The scenario is called imaginary, because, instead of letting the oracle assign the advice for the agents, we impose a particular advice to each agent, and let the agents perform the search with our advices. Note that even though such a scenario cannot occur by the definition of the model, each individual agent with advice a cannot distinguish this case from the case that the number of agents was some \(k'\) and the oracle assigned it the advice a.

  10. Since we may have more bits than we need here, we can just pad zero’s as the most significant bits.

  11. This procedure is straightforward. Simply follow a binary search in \(S(\alpha )\) for the element \(k_{i^*}(\alpha )\), up to depth B(k).

  12. The name harmonic was chosen because of structure resemblances to the celebrated harmonic algorithm for the k-server problem, see, e.g., [10].

  13. The bait itself may be “imaginary” in the sense that it need not be placed physically on the terrain. Instead, a camera can just be placed filming the location, and detecting when searchers get nearby.

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Acknowledgments

The authors are thankful to Jean-Sébastien Sereni and Zvi Lotker for helpful discussions, that were particularly useful for facilitating the proofs of Theorems 8 and 10. In addition, the authors are thankful to the anonymous reviewers for helping to improve the presentation of the paper, and for providing us with an idea that was used to prove the upper bound in Sect. 5.2.2.

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  1. Weizmann Institute of Science, Rehovot, Israel

    Ofer Feinerman

  2. CNRS and University Paris Diderot, Paris, France

    Amos Korman

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  1. Ofer Feinerman
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Correspondence to Amos Korman.

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O.F., incumbent of the Shlomo and Michla Tomarin Career Development Chair, was partially supported by the Clore Foundation and the Israel Science Foundation (grant 833/15). A.K. was partially supported by the ANR projects DISPLEXITY and PROSE, and by the INRIA project GANG.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648032).

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Feinerman, O., Korman, A. The ANTS problem. Distrib. Comput. 30, 149–168 (2017). https://doi.org/10.1007/s00446-016-0285-8

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