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Lower Bound for Graph Exploration by Swarm of Robots

Exploring a graph using a swarm of oblivious robots poses challenges that are tackled in this work. The study focuses on determining the minimum number of robots required to effectively explore graphs of varying sizes. The concepts of distinguishable configurations and automorphisms are pivotal in understanding the lower bound for robot exploration. Theoretical frameworks and proofs are provided to establish the optimal algorithm for graph exploration tasks, specifically in the context of robotic systems with limited communication capabilities.

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Lower Bound for Graph Exploration by Swarm of Robots

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  1. A Lower Bound for Graph Exploration by a Swarm of Oblivious Robots Stéphane Devismes Joint work with: Anissa Lamani, Franck Petit, and Sébastien Tixeuil

  2. Roadmap • Models and Exploration problems • The lower bound • Application: lower bound for the ring case • Proof of the result • Optimal algorithm for the ring case GRASTA 2018, Berlin 2

  3. Roadmap • Models and Exploration problems • The lower bound • Application: lower bound for the ring case • Proof of the result • Optimal algorithm for the ring case GRASTA 2018, Berlin 3

  4. Robots • A swarm of K robots – Motion actuators – Visibility sensors – Uniform & anonymous – Oblivious – No communication mean GRASTA 2018, Berlin 4

  5. Discrete environment • A simple graph G = (V,E) – Connected – Unoriented – Anonymous Possible moves Locations GRASTA 2018, Berlin 5

  6. Discrete environment • A simple graph G = (V,E) – Connected – Unoriented – Anonymous Possible moves Locations GRASTA 2018, Berlin 6

  7. View • Instantaneous snapshot of the system – Full or Distance d – Multiplicity (information about each location) • Weak (0, 1, or Tower) • Strong (0, 1, or Tower of x robots) GRASTA 2018, Berlin 7

  8. View • Instantaneous snapshot of the system – Full or Distance d – Multiplicity (information about each location) • Weak (0, 1, or Tower) • Strong (0, 1, or Tower of x robots) GRASTA 2018, Berlin 8

  9. Symmetry ? GRASTA 2018, Berlin 9

  10. Symmetry • Worst case decision: chosen by an adversary ? GRASTA 2018, Berlin 10

  11. Cycle • Look: Instantaneous Snapshot • Compute: Based on this observation, decides (deterministically or probabilistically) to either stay idle or move to one of the neighboring nodes • Move: Move toward the destination GRASTA 2018, Berlin 11

  12. Models (from the stronger to the weaker) • FSYNC: each robot executes a full cycle at each step • SSYNC: at each step, a nonempty subset of robots executes a full cycle (+ fairness) • ASYNC: Look, Compute and Move are atomic however the time between Look, Compute, and Move is finite but unbounded Remark: in any snapshot, no robot on edges GRASTA 2018, Berlin 12

  13. (Terminating) Exploration • Deterministic version: starting from any towerless configuration (i.e., at most one robot per node) – Exploration: Each node must be visited by at least one robot – Termination: Within finite time, every robot stays idle • Probabilistic version (Las Vegas): termination with probability one GRASTA 2018, Berlin 13

  14. Challenge • What is the minimal number of robots K necessary to explore a graph of size n > K? (under a distributed scheduler) GRASTA 2018, Berlin 14

  15. Challenge • What is the minimal number of robots K necessary to explore a graph of size n > K? (under a distributed scheduler) Termination Detection Need to use configurations as an implicit memory Oblivious Robots GRASTA 2018, Berlin 15

  16. Roadmap • Models and Exploration problems • The lower bound • Application: lower bound for the ring case • Proof of the result • Optimal algorithm for the ring case GRASTA 2018, Berlin 16

  17. Our result Theorem: deterministic) exploration with – K robots – on a graph of n>K nodes is possible only if • K>2 and there exists a set S of at least n-K+1 configurations such that: – any two different distinguishable, and – in every configuration in S, there is a tower of less than K robots. Terminating (probabilistic or configurations in S are GRASTA 2018, Berlin 17

  18. Distinguishable Configurations • Two configurations C1and C2are indistinguisable if there exists an automorphism f on G = (V,E) such that ∀p∈V, Multiplicity(C1,p) = Multiplicity(C2,f(p)) • If C1and C2are not indistinguishable, then they are distinguisable • Automorphism: f: G → G such that ∀p,q∈V, {p,q}∈E ⟺ {f(p), f(q)}∈E GRASTA 2018, Berlin 18

  19. Example of indistinguishable configurations a b c f(i) f(h) f(g) d e f f(f) f(e) f(d) g h i f(c) f(b) f(a) GRASTA 2018, Berlin 19

  20. Example of distinguishable configurations GRASTA 2018, Berlin 20

  21. Property 1 • Let E = C1C2… Cxbe a factor of execution • If C1’ is indistinguishable from C1 by the automorphism f then – C1’ C2’ … Cx’ is a factor of execution such that ∀i∈ [1..x], Ciand Ci’ are indistinguishable by f i.e.∀p∈V, Multiplicity(Ci,p) = Multiplicity(Ci’,f(p)) GRASTA 2018, Berlin 21

  22. Illustration of the property a a b a b c b c c d d e d e f e f f g g h g h i h i i f(i) f(h) f(g) f(i) f(h) f(g) f(i) f(h) f(g) f(f) f(e) f(d) f(f) f(e) f(d) f(f) f(e) f(d) f(c) f(b) f(a) f(c) f(b) f(a) f(c) f(b) f(a) GRASTA 2018, Berlin 22

  23. Roadmap • Models and Exploration problems • The lower bound • Application: lower bound for the ring case • Proof of the result • Optimal algorithm for the ring case GRASTA 2018, Berlin 23

  24. Applications of the Result • Assuming full visibility and weak multiplicity – In SSYNC, 4 robots are necessary and sufficient to probabilistically explore any ring of size n > 4 [TCS, 2013] – In ASYNC, 3 robots are necessary and sufficient to deterministically explore almost all grids [SSS, 2012] • (2*2 requires 4 and 3*3 requires 5) – In SSYNC, 4 robots are necessary and sufficient to probabilistically explore a torus of size l * L where 7 ≤ l ≤ L [NETYS, 2015] GRASTA 2018, Berlin 24

  25. Applications of the Result • Assuming full visibility and weak multiplicity – In SSYNC, 4 robots are necessary and sufficient to probabilistically explore any ring of size n > 4 [TCS, 2013] – In ASYNC, 3 robots are necessary and sufficient to deterministically explore almost all grids [SSS, 2012] • (2*2 requires 4 and 3*3 requires 5) – In SSYNC, 4 robots are necessary and sufficient to probabilistically explore a torus of size l * L where 7 ≤ l ≤ L [NETYS, 2015] GRASTA 2018, Berlin 25

  26. Detailed Application • In a ring with K=3 robots (K=1 and K=2 are trivial insufficient using our theorem) GRASTA 2018, Berlin 26

  27. Detailed Application • In a ring with K=3 robots GRASTA 2018, Berlin 27

  28. Detailed Application • In a ring with K=3 robots GRASTA 2018, Berlin 28

  29. Detailed Application • In a ring with K=3 robots GRASTA 2018, Berlin 29

  30. Detailed Application • In a ring with K=3 robots GRASTA 2018, Berlin 30

  31. Detailed Application • In a ring with K=3 robots GRASTA 2018, Berlin 31

  32. Detailed Application • In a ring with K=3 robots GRASTA 2018, Berlin 32

  33. Detailed Application • In a ring with K=3 robots • For n > 4, 4 robots are necessary – N.b. this does not exclude the case K=3 and n=4 GRASTA 2018, Berlin 33

  34. Roadmap • Models and Exploration problems • The lower bound • Application: lower bound for the ring case • Proof of the result • Optimal algorithm for the ring case GRASTA 2018, Berlin 34

  35. Proof of the Result Theorem: deterministic) exploration with – K robots – on a graph of n>K nodes is possible only if • K>2 and there exists a set S of at least n-K+1 configurations such that: – any two different distinguishable, and – in every configuration in S, there is a tower of less than K robots. Terminating (probabilistic or configurations in S are GRASTA 2018, Berlin 35

  36. Proof of the Result • Let P be any exploration protocol for K robots on a ring of n > K nodes GRASTA 2018, Berlin 36

  37. Proof of the Result • Let P be any exploration protocol for K robots on a ring of n > K nodes • As n > K and robots are oblivious, any terminal configuration should be distinguishable from any initial (towerless) configuration, hence: Remark 1: Any terminal configuration contains a tower. GRASTA 2018, Berlin 37

  38. Proof of the Result Let S = C1C2… Cxbe a sequence of configurations MRS(S) is the maximal subsequence of S where no two consecutive configurations are identical Lemma 1: Let E = C1C2… Cxbe a sequential terminating execution of P. K > 2 and MRS(E) has at least n-K+1 configurations containing a tower of less than K robots. GRASTA 2018, Berlin 38

  39. Proof of the Result • Let E = C1C2… Cxbe a sequential terminating execution of P GRASTA 2018, Berlin 39

  40. Proof of the Result • Let E = C1C2… Cxbe a sequential terminating execution of P • By Remark 1, Cxcontains a tower and K>1 GRASTA 2018, Berlin 40

  41. Proof of the Result • Let E = C1C2… Cxbe a sequential terminating execution of P • By Remark 1, Cxcontains a tower and K>1 • Let E’ = Ci… Cxbe the suffix of E such that – Ciis the only towerless configuration of E’ GRASTA 2018, Berlin 41

  42. Proof of the Result • Let E = C1C2… Cxbe a sequential terminating execution of P • By Remark 1, Cxcontains a tower and K>1 • Let E’ = Ci… Cxbe the suffix of E such that – Ciis the only towerless configuration of E’ • E’ is a sequential terminating execution of P GRASTA 2018, Berlin 42

  43. Proof of the Result • Let E = C1C2… Cxbe a sequential terminating execution of P • By Remark 1, Cxcontains a tower and K>1 • Let E’ = Ci… Cxbe the suffix of E such that – Ciis the only towerless configuration of E’ • E’ is a sequential terminating execution of P • MRS(E’) is a sequential terminating execution of P too GRASTA 2018, Berlin 43

  44. Proof of the Result • For each step C→C’ of MRS(E’), 3 cases: a) C is towerless (the first step). C’ contains a tower and no new node is visited in this step b) C contains a tower and C’ contains a tower of K robots. No new node is visited in this step c) C contains a tower and C’ contains a tower of less than K robots. At most 1 new node is visited in this step If K=2, case c) do not exist: except the K initially visited nodes, no other node can be visited If K>2: Initially, K nodes are visited. In case a), which appears exactly once, C’ contains a tower of less than K robots. Case c) should appear at least n-K times. Hence, MRS(E’), and so MRS(E), has at least n-K+1 configurations containing a tower of less than K robots. GRASTA 2018, Berlin 44

  45. Proof of the Result Lemma 2: Let E = C1C2… Cxbe a sequential terminating execution of P. K > 2 and MRS(E) has at least n-K+1 configurations containing a tower of less than K robots and any two of them are distinguishable. By the contradiction: – at least n-K+1 configurations containing a tower of less than K robots – But less than n-K+1 of them are pairwise distinguisable By inductively applying Property 1, we can construct a sequential terminating execution E’ from E that has less than n-K+1 configurations containing a tower of less than K robots, contradicting Lemma 1 GRASTA 2018, Berlin 45

  46. Proof of the Result • Assume Ciand Cj – contains a tower of less than K robots and – are indistinguishable by the automorphism f in C1… Ci… CjCj+1… Cx • C1… CiCj+1’ … Cx’ is a terminating execution of P where ∀y∈ [j+1..x], Cyand Cy’ are indistinguishable by f i.e.∀p∈V, Multiplicity(Cy,p) = Multiplicity(Cy’,f(p)) • We removed Ci GRASTA 2018, Berlin 46

  47. Back to the case K=3 and n=4 • Our result only deals with of sequential solutions – (a particular case of the distributed scheduler) • So, the result on the ring does not exclude sequential solution for K=3 and n=4 GRASTA 2018, Berlin 47

  48. Sequential solution for K=3 and n=4 GRASTA 2018, Berlin 48

  49. Sequential solution for K=3 and n=4 • However, no distributed solution – (Combinatorial study) GRASTA 2018, Berlin 49

  50. Roadmap • Models and Exploration problems • The lower bound • Application: lower bound for the ring case • Proof of the result • Optimal algorithm for the ring case GRASTA 2018, Berlin 50

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