How Many Ants Does It Take to Find the Food? Jara Uitto ETH Zurich - - PowerPoint PPT Presentation

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How Many Ants Does It Take to Find the Food? Jara Uitto ETH Zurich - - PowerPoint PPT Presentation

Nov 22, 2022 184 likes β€’1.11k views

How Many Ants Does It Take to Find the Food? Jara Uitto ETH Zurich Distributed Computing www.disco.ethz.ch Ants Nearby Treasure Search Introduced by Feinerman, Korman, Lotker and Sereni [PODC 2012]. mobile agents,

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ETH Zurich – Distributed Computing – www.disco.ethz.ch

Jara Uitto

How Many Ants Does It Take to Find the Food?

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  • Introduced by Feinerman, Korman, Lotker and

Sereni [PODC 2012].

  • π‘œ mobile agents, controlled by Turing

machines, search for a treasure.

  • Communication not allowed.

Ants Nearby Treasure Search

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Model

  • Infinite integer grid.
  • Each ant initially located in the origin.
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Model

  • Adversarially hidden treasure/food.
  • (Manhattan) distance to treasure is 𝐸.
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  • How many rounds until the

treasure is found?

  • We study the number of ants

needed to find the treasure at all.

Ants Nearby Treasure Search

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Model

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Model

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Model

  • One Turing Machine is enough. No

communication needed.

=

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Model

  • Ants are controlled by

(randomized) finite state machines.

  • Communicate by sensing the

states of nearby ants.

  • Run-time studied by Emek,

Langner, Uitto and Wattenhofer [ICALP2014].

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  • Synchrony vs. Asynchrony
  • A deterministic protocol?

Model

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Model

  • Individual algorithm for

each ant.

  • An algorithm works

correctly if the ants find the treasure in expected finite time.

3 2 1

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Deterministic + Asynchronous

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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Triangle Search

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  • Can we perform better if the ants have a

common sense of time?

Synchronization?

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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Rectangle Search

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  • How about random coin tosses?

Randomization

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Geometric Search

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NE

Geometric Search

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NE 1

Geometric Search

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NE 11

Geometric Search

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NE 111

Geometric Search

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NE 1110

Geometric Search

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NE 11101

Geometric Search

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NE 111011

Geometric Search

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NE 1110110

Geometric Search

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NE 1110110

Geometric Search

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NE 1110110

Geometric Search

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NE 1110110

Geometric Search

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NE 1110110

Geometric Search

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NE 1110110

Geometric Search

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Geometric Search

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Run-Time

  • For every search 𝑗, we have a probability of at

least Ai =

1 4 βˆ™ 2βˆ’(𝐸+1) to find the treasure.

  • Let 𝐢𝑗 be the event that the treasure is not

found during any search π‘˜ < 𝑗.

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  • Let π‘ˆ be the total time required.
  • 𝐹 π‘ˆ ≀

𝑄(𝐡𝑗+1 βˆ™ 𝐢𝑗)

∞ 𝑗=1

(𝑃 𝑗 + 𝑃(𝐸)).

  • 𝑄 𝐡𝑗+1 βˆ™ 𝐢𝑗 ≀ 2βˆ’ 𝐸+3 βˆ™ 1 βˆ’ 2βˆ’ 𝐸+3

𝑗.

  • 𝐹 π‘ˆ ≀ 2βˆ’ 𝐸+3

1 βˆ’ 2βˆ’(𝐸+3) 𝑗 𝑃 𝑗 + 𝑃(𝐸)

∞ 𝑗=1

= 𝑃 2𝐸 .

Run-Time

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  • Can we do better? In the deterministic and

synchronous case, the answer is no.

  • Let us start with showing that one ant is not

enough.

Lower Bounds?

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  • A finite state machine repeats its behavior.

One Ant

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q

One Ant

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q q

One Ant

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q q q

One Ant

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A band of constant width

𝑑

One Ant

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  • One ant can only discover a band of constant

width.

  • How about two ants?

One Ant

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  • Let 𝑒 be the time of the last meeting.

Two Ants

  • Both agents (alone) discover a band after 𝑒.
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  • Lemma: The ants meet infinitely often in some

pair of states (π‘Ÿ, π‘Ÿβ€²).

  • Observation: the time between two such

meetings is bounded by a constant.

Two Ants

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Two Ants

(π‘Ÿ, π‘Ÿβ€²)

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Two Ants

(π‘Ÿ, π‘Ÿβ€²) (π‘Ÿ, π‘Ÿβ€²)

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Two Ants

(π‘Ÿ, π‘Ÿβ€²) (π‘Ÿ, π‘Ÿβ€²) (π‘Ÿ, π‘Ÿβ€²)

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Two Ants

(π‘Ÿ, π‘Ÿβ€²) (π‘Ÿ, π‘Ÿβ€²) (π‘Ÿ, π‘Ÿβ€²)

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Two Ants

  • Two deterministic ants can only discover a band
  • f constant width.
  • Two deterministic ants cannot find the food.

𝑑′ (π‘Ÿ, π‘Ÿβ€²) (π‘Ÿ, π‘Ÿβ€²) (π‘Ÿ, π‘Ÿβ€²)

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  • Three asynchronous ants?
  • Two randomized ants?

Conclusion

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Conclusion

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Questions?

Thanks to my co-authors Yuval Emek, Tobias Langner, David Stolz and Roger Wattenhofer