Anonymous Graph Exploration with Binoculars Jérémie Chalopin Emmanuel Godard Antoine Naudin LIF , CNRS & Aix-Marseille Université GRASTA-MAC 2015 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 1/23
Graph Exploration ◮ An agent is moving along the edges of a graph ◮ Goal : visit all the nodes and stop GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 2/23
Graph Exploration ◮ An agent is moving along the edges of a graph ◮ Goal : visit all the nodes and stop GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 2/23
Graph Exploration ◮ An agent is moving along the edges of a graph ◮ Goal : visit all the nodes and stop GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 2/23
How to navigate in the graph ? 5 3 2 4 6 2 6 3 1 4 3 6 6 2 1 2 3 4 2 2 5 ◮ Anonymous graph ◮ Port-numbering ◮ The agent knows its incoming port number ◮ It has an infinite memory GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 3/23
Exploration without information Exploration of a graph G Visit every node of G and stop Question What graphs can we explore without information ? GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 4/23
Exploration without information Exploration of a graph G Visit every node of G and stop Question What graphs can we explore without information ? An algorithm A is an exploration algorithm ◮ for every graph G , if A stops, then the agent has visited all the nodes of G GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 4/23
Exploration without information Exploration of a graph G Visit every node of G and stop Question What graphs can we explore without information ? An algorithm A is an exploration algorithm for a family F ◮ for every graph G , if A stops, then the agent has visited all the nodes of G ◮ for every graph G ∈ F , A visits all nodes of G and stops GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 4/23
Known Results [Folklore] If nodes can be marked : ◮ every graph is explorable by a DFS in O ( m ) moves If nodes cannot be marked : ◮ Trees can be explored by a DFS in O ( n ) moves ◮ Non tree graphs : it is impossible to detect when all nodes have been visited GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 5/23
Graph Coverings Definition A graph covering is a locally bijective homomorphism ϕ : G → H 4 2 3 4 3 4 2 2 2 3 3 4 2 3 4 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 6/23
Lifting Lemma Lifting Lemma (from Angluin) If G is a graph cover of H , then an agent cannot decide if it starts on v ∈ V ( G ) or on ϕ ( v ) ∈ V ( H ) 4 2 3 4 3 4 2 2 2 3 3 4 2 3 4 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 7/23
Lifting Lemma Lifting Lemma (from Angluin) If G is a graph cover of H , then an agent cannot decide if it starts on v ∈ V ( G ) or on ϕ ( v ) ∈ V ( H ) 4 2 3 4 3 4 2 2 2 3 3 4 2 3 4 Corollary If an exploration algorithm A stops in r steps in H , r ≥ | V ( G ) | GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 7/23
Explorable graphs without global information G is explorable ⇐ ⇒ G has a unique graph cover (itself) ⇐ ⇒ G has no infinite graph cover ⇐ ⇒ G is a tree GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 8/23
Our model : Mobile Agent with Binoculars ◮ the agent sees the graph induced by its neighbors 5 3 6 2 4 2 6 4 1 3 3 6 2 1 6 2 3 2 2 4 5 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 9/23
Our model : Mobile Agent with Binoculars ◮ the agent sees the graph induced by its neighbors ◮ One can detect triangles ◮ Graph coverings are no longer the good notion 5 3 6 2 4 2 6 4 1 3 3 6 2 1 6 2 3 2 2 4 5 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 9/23
What can we do with binoculars ? ◮ Can we explore every graph ? ◮ NO Cycles of length ≥ 4 cannot be explored GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 10/23
What can we do with binoculars ? ◮ Can we explore every graph ? ◮ NO Cycles of length ≥ 4 cannot be explored ◮ Can we characterize explorable graphs ? ◮ YES ◮ using clique complexes and simplicial coverings ◮ a universal exploration algorithm GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 10/23
What can we do with binoculars ? ◮ Can we explore every graph ? ◮ NO Cycles of length ≥ 4 cannot be explored ◮ Can we characterize explorable graphs ? ◮ YES ◮ using clique complexes and simplicial coverings ◮ a universal exploration algorithm ◮ Can we find an efficient universal algorithm for explorable graphs ? ◮ NO ◮ the exploration time cannot be bounded by a computable function GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 10/23
Clique complexes Definition The clique complex K ( G ) of G is a simplicial complex s.t. the simplices of K ( G ) are the cliques of G K ( G ) G GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 11/23
Simplicial coverings Definition A simplicial covering is a locally bijective simplicial map ψ : K → K ′ If K ( G ) is a simplicial cover of K ( H ) , we say that G is a simplicial cover of H GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 12/23
Simplicial Lifting Lemma Simplicial Lifting Lemma If K ( G ) is a simplicial cover of K ( H ) , then an agent cannot decide if it starts on v ∈ V ( G ) or on ϕ ( v ) ∈ V ( H ) GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 13/23
Simplicial Lifting Lemma Simplicial Lifting Lemma If K ( G ) is a simplicial cover of K ( H ) , then an agent cannot decide if it starts on v ∈ V ( G ) or on ϕ ( v ) ∈ V ( H ) Corollary If an exploration algorithm A stops in r steps in H , r ≥ | V ( G ) | GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 13/23
Exploration with binoculars : Characterization INF (All others) FC Not Explorable Explorable ⇐ ⇒ K ( G ) has only ⇐ ⇒ K ( G ) has an finite simplicial covers infinite simplicial cover G 10 G 6 G 3 G 11 ψ G 7 G 9 G 7 ψ G 9 G 10 G 6 G 13 G 8 G 5 G 5 G 8 G 12 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 14/23
Examples INF : K ( G ) has an infinite simplicial cover ϕ Fig. by Ag2gaeh via Wikimedia Commons GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 15/23
Examples SC : K ( G ) has a unique cover (itself) Fig. by Tilman Piesk via Wikimedia Commons 5 4 6 1 2 8 3 3 7 7 4 6 5 2 8 1 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 15/23
Examples FC : K ( G ) has only finite covers ◮ in this case, K ( G ) has a finite number of covers ◮ SC � FC 5 4 6 1 2 1 8 2 8 3 3 7 7 3 7 4 4 6 6 5 5 2 8 1 ϕ GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 15/23
How to distinguish the two classes ? ◮ INF : { G | K ( G ) has an infinite simplicial cover } ◮ FC : { G | K ( G ) has only finite simplicial covers } ◮ SC : { G | K ( G ) has a unique finite simplicial cover } Proposition (from Topology) G is in FC ⇐ ⇒ G has a finite simplicial cover in SC Theorem (from Topology) G is in SC ⇐ ⇒ K ( G ) is simply connected GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 16/23
Contractible Cycles ◮ a cycle is contractible if it is related with the empty cycle (a vertex) by a sequence s of elementary deformations : ◮ Pushing across a triangle ◮ Pushing across an isolated vertex ◮ c is k -contractible if | s | ≤ k GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 17/23
Contractible Cycles ◮ a cycle is contractible if it is related with the empty cycle (a vertex) by a sequence s of elementary deformations : ◮ Pushing across a triangle ◮ Pushing across an isolated vertex ◮ c is k -contractible if | s | ≤ k Simple Connectivity K ( G ) is simply connected ⇐ ⇒ all cycles of G are contractible ⇐ ⇒ K ( G ) has a unique simplicial cover GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 17/23
Our Exploration Algorithm Explore B ( v 0 , 2 k ) by computing the view T G ( v 0 , 2 k ) v 0 B ( v 0 , 2 k ) look for compute v 0 ˜ 2 k v 0 the view H ? G Look for a graph H such that ◮ | V ( H ) | ≤ k ◮ ∃ ˜ v 0 ∈ V ( H ) s.t. T H (˜ v 0 , 2 k ) ≃ T G ( v 0 , 2 k ) ◮ simple cycles of H are k − contractible If there is no such H , increment k and repeat the procedure GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 18/23
Views We compute the view T G ( v ) of v in G where every node u is labeled with B G ( u , 1 ) v T G ( v, k ) G 2 2 1 2 2 1 1 1 5 5 4 4 5 5 4 4 3 2 2 3 3 2 3 2 1 1 1 1 1 1 1 1 4 5 1 2 1 3 5 4 3 2 3 1 2 1 1 1 4 5 5 4 2 3 2 1 1 1 5 4 2 3 k 1 1 2 1 GRASTA-MAC 2015 Anonymous Graph Exploration with Binoculars 19/23
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