Optimal Pheromone Utilization

1. Optimal Pheromone Utilization Roman Kecher Joint work with: Yehuda Afekand Moshe Sulamy Tel-Aviv University

2. Ants Nearby Treasure Search N • Infinite grid • k ants • Initially at the origin • Food at distance D • Ants have to findthe food • Optimal run-time:Ω(D + D2/k)[Feinerman, Korman, Lotker, Sereni, 2012] D E W k (0,0) S

3. Pheromones N • Ants emit pheromones[Lenzen, Radeva, 2013] • Or not • And sense them • No other communication • Biological resource • Goal: minimizepheromone count E W S

4. Ground Rules • Every ant runs same algorithm (locally) • With same initial state • Only uniform algorithms,ants have no knowledge of: • k, total number of ants • D, distance to the food

5. Synchronous Model N • Rounds:all ants move onceper round E W S

6. Synchronous Model N • Rounds:all ants move onceper round • Assumption: antemission scheme[Emek, Langner, Uitto,Wattenhofer, 2013] • At most one ant isemitted in each round E W S

7. Asynchronous Model N • Adversary repeatedlyschedules one ant • Test&Set: • Sense and emit a pheromone is oneatomic step • Definition of Rounds: • Round ends when everyant took at least one step • Only for (time) complexity E W S

8. Ants Models • FSM: Finite State Machines Constant size memory • TM: Turing Machines Unlimited memory • Both deterministic

9. Results Previously known: O(D2) pheromones [Lenzen, Radeva, 2013]

10. Results Previously known: O(D2) pheromones [Lenzen, Radeva, 2013]

11. Results Previously known: O(D2) pheromones [Lenzen, Radeva, 2013] Results hold for Synchronous and Asynchronous models

12. Layers N • Definition: layer L • All grid cells at distance L from origin L E W S

13. FSM Need Ω(D) Pheromones N • Assume FSM withS states • Uses o(D) pheromones • S+1 consecutivepheromone-free layers exist • Path starts and ends in same state • Infinite loop E W S

14. FSM Algorithms N • Problem:FSM can’t count • Solution:Use pheromonesas turning points • Similar to the idea of guides[Emek, Langner,Uitto, Wattenhofer, 2013] E W S

15. Asynchronous FSM Algorithm N • Mark E, S, W, N • Explore from N • N never longer thanE, S or W • Test&Setpreventsmultiple ants fromexploring same layer E W S

16. Synchronous FSM Algorithm N • Emission schemebreaks initialsymmetry • But what happensif two ants collide? • Veteran ants behavedifferently thanNewbie ants Veteran Newbie E W S

17. Results Previously known: O(D2) pheromones [Lenzen, Radeva, 2013] Results hold for Synchronous and Asynchronous models

18. Results Previously known: O(D2) pheromones [Lenzen, Radeva, 2013] Results hold for Synchronous and Asynchronous models

19. Async TM Need Ω(k) Pheromones N • Assume oneant does not emitpheromones E W S

20. Async TM Need Ω(k) Pheromones N • Assume oneant does not emitpheromones • Consider samescheduling butwith extra ants • All new ants followthat one ant • Runtime remains thesame (but more ants) E W S

21. Sync TM Need Ω(k) Pheromones N • Emit one ant • Until all pheromonesare placed • Emit second ant • Until all pheromonesare placed • Continue • Delay is constant • If no new pheromonesare placed, all followingants behave the same E W S

22. Asynchronous TM Algorithm N • TM can count! • Use pheromonestoassign IDs to ants • Static partition • Explore layersL = ID (mod Total) • Occasionally updateestimated Total • Also works for thesynchronous model 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 ID = 2 2 2 2 1 2 1 2 2 2 ID = 1 1 1 2 2 1 2 1 1 2 2 2 2 2 1 1 1 1 E W 2 2 1 2 1 2 1 1 1 2 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 S

23. Thanks Questions?