Graphbots: Mobility in Discrete Spaces

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# Graphbots: Mobility in Discrete Spaces - PowerPoint PPT Presentation

Sharon Lahav. Graphbots: Mobility in Discrete Spaces. Mobility in Discrete Spaces. Move beyond robots with simple geometries (polygonal structure). Move beyond simple spaces (planar region containing polygonal obstacles). Teams of robots that operate in discrete spaces like graphs.

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Sharon Lahav

Graphbots: Mobility in Discrete Spaces

Mobility in Discrete Spaces
• Move beyond robots with simple geometries (polygonal structure).
• Move beyond simple spaces (planar region containing polygonal obstacles).
• Teams of robots that operate in discrete spaces like graphs.
• And that have discrete geometries represented by subgraphs i.e they are maintaining a formation!

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Mobility Definitions
• A connected graph G - “the graph space”
• A connected subgraph of G H – “a cooperating team of robots”
• We represent the members of the team by single nodes.

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Mobility Definitions (Cont.)
• A movement or motion of H from S to T is defined by a sequence of subgraphs: S=H0, H1,…,Hk=T all isomorphic to H.
• The structure must be preserved when the team moves.

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The Goals
• Find conditions for G to satisfy a free movement of a given subgraph H in G.
• Establish the complexity of finding a motion with the fewest local displacements from S to T (if one exists).

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Moving a Tick
• A “Tick” is modeled by a two vertex graph linked by a single edge.
• Theorem 2.1 - A tick can move freely in any connected graph.

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Moving a Scorpion
• A “Scorpion” is modeled by a three­vertex graph linked by two edges.
• The degree­2 vertex is the “body”.
• The degree­1 vertices are the “feet”.
• Theorem 2.2- A scorpion can move freely in G iff G does not contain a vertex v with two neighbors of degree 1.

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b

Proff of Theorem 2.2 
• If G has such a vertex v, we can place the scorpion with b on v, and the f \'s on the neighbors vi of v.
• Any movement of the feet requires that both must move to v. If we move one foot to v, then b must leave v  the other foot will not be adjacent to b\'s new location.

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G

S

T

Proff of Theorem 2.2 
• There is no such vertex v.
• S-{v1,v2,v3} , T-{u1,u2,u3}
• If there is a path joining a foot of the scorpion in the initial S location to a foot at the final T location, without passing through u2 and v2 , then the scorpion can “creep” along this path .

f b f

f b f

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u2

G

u2

v1

v3

S

T

u3

u1

u3

u1

v2

u0

u0

Proff of Theorem 2.2  (Cont.)
• The only path from S to T goes through either u2 or v2 (or both):
• The degree of u1 is not 1 and it has a neighbor u0 (other than u2)
• f goes from u1 to u0 , b goes to u1 and f goes from u3 to u2 . Now the scorpion can creep along the path to T.

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To corner a scorpion

you have to completely

immobilize it at its

starting location!

b

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Some Definitions
• A single vertex in a connected graph whose deletion disconnects the graph is called a cut vertex.
• Biconnected Graphs - A graph with no cut vertices is called biconnected.
• - Connected Graphs - A graph is said to be ­ Connected if the deletion of any subset of -1 vertices leaves the graph connected.

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Some Definitions (Cont.)
• A Chordal Graph is a graph in which each cycle of length at least 4 has a Chord.
• A Chord is an edge that connects two vertices that are not adjacent in the cycle.

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Some Definitions (Cont.)
• A perfect elimination ordering(peo) is a numbering of the vertices from {1,…,n} such that for each i, the higher numbered neighbors of vertex i form a clique.
• A peo is represented by a sequence  of vertices.
• Theorem 2.3 (Fulkerson and Gross) A graph G has a peo iff G is chordal.

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Moving a Trilobite
• A “Trilobite” is modeled by a three­vertex graph linked by three edges. (a clique of size three).
• Theorem 2.4- A Trilobite can move freely in a biconnected chordal graph, that has at least three vertices.

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chordality is sufficient,

but not necessary.

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Moving a Spider
• A spider is modeled by a (k+1)­vertex graph having a central vertex denoting its “body” linked by edges to vertices f1,…,fk representing its “feet”.
• K=1 is a “Tick”
• K=2 is a “Scorpion”
• Theorem 2.6 A k­legged Spider can move freely in a (K-1)­ Connected Chordal Graph.

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Moving a Four-Legged Spider
• We can move a three­legged spider in a biconnected chordal graph, this follows from the theorem 2.6 with k = 3.
• A stronger result yet!Theorem 2.10 A four­legged spider can also move freely in a biconnected chordal graph.

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A five­legged spider

cannot move freely

in a biconnected

chordal graph!

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Finding Shortest Motion
• If G has n vertices and H has l vertices. There are at most O(nl) possible locations of H in G.
• G’ = (V’,E’) in which each vertex corresponds to a possible valid location of H in G.
• There is an edge in E’ between u,vV’ if there is a local displacement between u and v of H.

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Finding Shortest Motion (Cont.)
• G’ can be constructed in polynomial time in the size of any fixed graph H.
• By finding the shortest path in G’ from S to T, we can determine the motion with the least number of local displacements in polynomial time.

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NP­ Completenesswhen H is part of the input !
• Clique(G,k) : is the problem of checking if the graph G contains a clique of size k. This problem is known to be NP­complete!
• We reduced the clique problem to checking to see if there exists a motion that moves H = K 2k from a start location to a target location.

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NP­ Completenesswhen H is part of the input (Cont.)
• By constructing a new graph G’ =(V’,E’) V’=V(x1,…,x2k)(y2k+1,…,y4k). E’=EExEyE’’ Ex=[(xi,xj)|1ij2k] Ey=[(yi,yj)|2k+1ij 4k] E’’=[(v,xi),(v,yj)| vV,1i2k ,2k+1j4k]

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סוף

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