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A Lower Bound for Graph Exploration by a Swarm of Oblivious Robots

Stéphane Devismes and team explore the minimal number of robots required to efficiently explore graphs using oblivious robots in a workshop on Mobile Robots and Verification. The research discusses deterministic and probabilistic exploration scenarios, distinguishable configurations, and the necessity of certain robot quantities for graph exploration. Results are presented regarding the minimum number of robots needed for successful exploration based on different graph configurations.

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A Lower Bound for Graph Exploration by a Swarm of Oblivious Robots

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

  2. Robots • A swarm of K robots – Motion actuators – Visibility sensors – Uniform & anonymous – Oblivious – No communication mean Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 2

  3. Discrete environment • A simple graph G = (V,E) – Connected – Unoriented – Anonymous Possible moves Locations Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 3

  4. 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) Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 4

  5. 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) Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 5

  6. Symmetry ? Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 6

  7. Symmetry • Worst case decision: chosen by an adversary ? Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 7

  8. Cycle • Look: Instantaneous Snapshot • Compute: Based on this observation, decides to either stay idle or move to one of the neighboring nodes • Move: Move toward the destination Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 8

  9. 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 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 9

  10. (Terminating) Exploration • Deterministic version: starting from any towerless – 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 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 10

  11. Challenge • What is the minimal number of robots K necessary to explore a graph of size n > K? Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 11

  12. Challenge • What is the minimal number of robots K necessary to explore a graph of size n > K? Termination Detection Need to use configurations as an implicit memory Oblivious Robots Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 12

  13. 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 configurations such that: • any two different configurations in S are distinguishable, and • in every configuration in S, there is a tower of less than K robots. Terminating (probabilistic or of at least n-K+1 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 13

  14. 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 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 14

  15. 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) Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 15

  16. Example of distinguishable configurations Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 16

  17. 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 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 17

  18. 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) Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 18

  19. 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 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 any torus of size l * L where 7≤I≤L [NETYS, 2015] Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 19

  20. 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 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 any torus of size l * L where 7≤I≤L [NETYS, 2015] Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 20

  21. Detailed Application • In a ring with K=3 robots (K=1 and K=2 are trivial insufficient using our theorem) Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 21

  22. Detailed Application • In a ring with K=3 robots Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 22

  23. Detailed Application • In a ring with K=3 robots Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 23

  24. Detailed Application • In a ring with K=3 robots Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 24

  25. Detailed Application • In a ring with K=3 robots Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 25

  26. Detailed Application • In a ring with K=3 robots Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 26

  27. Detailed Application • In a ring with K=3 robots • For n > 4, 4 robots are necessary Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 27

  28. 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 configurations such that: • any two different configurations in S are distinguishable, and • in every configuration in S, there is a tower of less than K robots. Terminating (probabilistic or of at least n-K+1 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 28

  29. Proof of the Result • Let P be any exploration protocol for K robots on a ring of n > K nodes Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 29

  30. 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. Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 30

  31. 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. Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 31

  32. Proof of the Result • Let E = C1C2… Cxbe a sequential terminating execution of P Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 32

  33. Proof of the Result • Let E = C1C2… Cxbe a sequential terminating execution of P • By Remark 1, Cxcontains a tower and K>1 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 33

  34. 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’ Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 34

  35. 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 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 35

  36. 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 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 36

  37. 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. Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 37

  38. 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 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 contradicting Lemma 1 less than K robots, Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 38

  39. Proof of the Result • Assume Ciand Cjare indistinguishable by the automorphism f then – C1… CiCj+1’ … Cx’ is a terminating execution of P where ∀y∈ [j+1..x], Cyand Cy’ are indistinguishable by f Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 39

  40. Algorithm for Rings • Special cases for size 4 to 8 • General case for n > 8 Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 40

  41. Algorithm for Rings • Special cases for size 4 to 8 • General case for n > 8 – Alignment (no tower creation) Workshop MoRoVer: Mobile Robots and Verification 41 15/11/2017

  42. Algorithm for Rings • Special cases for size 4 to 8 • General case for n > 8 – Alignment – Arrow Creation Workshop MoRoVer: Mobile Robots and Verification 42 15/11/2017

  43. Algorithm for Rings • Special cases for size 4 to 8 • General case for n > 8 – Alignment – Arrow Creation – Exploration Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 43

  44. Alignment: Overview • In most of the cases, deterministic moves, e.g. Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 44

  45. Alignment: Overview • In case of symmetry, probabilistic moves to break the symmetry, e.g. Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 45

  46. Thank you! Workshop MoRoVer: Mobile Robots and Verification 15/11/2017 46

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