Graphbots: Mobility in Discrete Spaces. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 2 Mobility in Discrete Spaces Move beyond robots with simple.

Graphbots: Mobility Graphbots: Mobility in Discrete Spacesin Discrete Spaces

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Graphbots - 2סמינר 5 במדעי המחשב )236805(

Mobility in Discrete SpacesMobility in Discrete Spaces• Move beyond robots with simple Move beyond robots with simple

geometries )polygonal structure(.geometries )polygonal structure(.

• Move beyond simple spaces )planar Move beyond simple spaces )planar region containing polygonal obstacles(.region containing polygonal obstacles(.

• Teams of robots that operate in discrete Teams of robots that operate in discrete spaces like graphs.spaces like graphs.

• And that have discrete geometries And that have discrete geometries represented by subgraphs i.e they are represented by subgraphs i.e they are maintaining a formationmaintaining a formation!!

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Mobility DefinitionsMobility Definitions• A connected graph G -A connected graph G -

“the graph space” “the graph space”• A connected subgraph of G H –A connected subgraph of G H –

“a cooperating team of robots” “a cooperating team of robots”

• We represent the members of the team We represent the members of the team by single nodes.by single nodes.

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Mobility Definitions (Cont.)Mobility Definitions (Cont.)• A movement or motion of H from S to T A movement or motion of H from S to T

is defined by a sequence of subgraphs: is defined by a sequence of subgraphs: S=HS=H0, 0, HH11,…,H,…,Hkk=T =T all isomorphic to H.all isomorphic to H.

• The structure must be preserved when The structure must be preserved when the team moves.the team moves.

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The GoalsThe Goals• Find conditions for G to satisfy a free Find conditions for G to satisfy a free

movement of a given subgraph H in G.movement of a given subgraph H in G.

• Establish the complexity of finding a Establish the complexity of finding a motion with the fewest local motion with the fewest local displacements from S to T )if one exists(.displacements from S to T )if one exists(.

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Moving a TickMoving a Tick• A “Tick” is modeled by a two vertex A “Tick” is modeled by a two vertex

graph linked by a single edge.graph linked by a single edge.

• Theorem 2.1Theorem 2.1 - A tick can move freely in - A tick can move freely in any connected graph.any connected graph.

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Moving a ScorpionMoving a Scorpion• A “A “ScorpionScorpion” is modeled by a three ” is modeled by a three

vertex graph linked by two edges.vertex graph linked by two edges.• The degree 2 vertex is the “body”.The degree 2 vertex is the “body”.• The degree 1 vertices are the “feet”.The degree 1 vertices are the “feet”.• Theorem 2.2Theorem 2.2- A scorpion can move - A scorpion can move

freely in G iff G does not contain a freely in G iff G does not contain a vertex v with two neighbors of degree 1. vertex v with two neighbors of degree 1.

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Proff of Theorem 2.2 Proff of Theorem 2.2 • If G has such a vertex v, we can place If G has such a vertex v, we can place

the scorpion with b on v, and the f 's on the scorpion with b on v, and the f 's on the neighbors vthe neighbors vii of v. of v.

• Any movement of the feet requires that Any movement of the feet requires that both must move to v. If we move one both must move to v. If we move one foot to v, then b must leave v foot to v, then b must leave v the the other foot will not be adjacent to b's other foot will not be adjacent to b's new locationnew location..

b

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G ST

Proff of Theorem 2.2 Proff of Theorem 2.2 • There is no such vertex v.There is no such vertex v.

• S-{vS-{v11,v,v22,v,v33} , T-{u} , T-{u11,u,u22,u,u33}}

• If there is a path joining a foot of the If there is a path joining a foot of the scorpion in the initial S location to a foot scorpion in the initial S location to a foot at the final T location, without passing at the final T location, without passing through uthrough u22 and v and v22 , then the scorpion , then the scorpion

can “creep” along this path .can “creep” along this path .

f b f f b f

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Proff of Theorem 2.2 Proff of Theorem 2.2 (Cont.)(Cont.)• The only path from S to T goes through The only path from S to T goes through

either ueither u22 or v or v22 )or both(: )or both(:

G S

u2

u1 u3 Tv1 v3

v2

• The degree of uThe degree of u11 is not 1 and it has a is not 1 and it has a

neighbor uneighbor u00 )other than u )other than u22((

• f goes from uf goes from u11 to u to u00 , b goes to u , b goes to u11 and f and f

goes from ugoes from u33 to u to u22 . Now the scorpion . Now the scorpion

can creep along the path to T.can creep along the path to T.

u0

u3u1

u2

u0

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To corner a scorpion

you have to completely

immobilize it at its

starting location!

b

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Some DefinitionsSome Definitions• A single vertex in a connected graph A single vertex in a connected graph

whose deletion disconnects the graph is whose deletion disconnects the graph is called a called a cut vertexcut vertex..

• Biconnected Graphs Biconnected Graphs - A graph with no - A graph with no cut vertices is called cut vertices is called biconnectedbiconnected. .

- - Connected Graphs Connected Graphs - A graph is said - A graph is said to be to be Connected Connected if the deletion of any if the deletion of any subset of subset of -1 vertices leaves the graph -1 vertices leaves the graph connected.connected.

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Some Definitions (Cont.)Some Definitions (Cont.)• A Chordal GraphA Chordal Graph is a graph in which is a graph in which

each cycle of length at least 4 has a each cycle of length at least 4 has a ChordChord..

• A ChordA Chord is an edge that connects two is an edge that connects two vertices that are not adjacent in the vertices that are not adjacent in the cycle.cycle.

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Some Definitions (Cont.)Some Definitions (Cont.)• A perfect elimination orderingA perfect elimination ordering (peo)(peo)

is a numbering of the vertices from {1,is a numbering of the vertices from {1,…,n} such that for each i, the higher …,n} such that for each i, the higher numbered neighbors of vertex i form a numbered neighbors of vertex i form a clique.clique.

• A A peopeo is represented by a sequence is represented by a sequence of vertices. of vertices.

• Theorem 2.3Theorem 2.3 )Fulkerson and Gross( )Fulkerson and Gross( A graph G has a A graph G has a peopeo iff G is iff G is chordalchordal. .

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Moving a TrilobiteMoving a Trilobite• A “Trilobite” is modeled by a three vertex A “Trilobite” is modeled by a three vertex

graph linked by three edges. )a clique of graph linked by three edges. )a clique of size three(.size three(.

• Theorem 2.4Theorem 2.4- A Trilobite can move - A Trilobite can move freely in a biconnected chordal graph, freely in a biconnected chordal graph, that has at least three vertices.that has at least three vertices.

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chordality is sufficient,

but not necessary.

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Moving a SpiderMoving a Spider• A spider is modeled by a )k+1( vertex A spider is modeled by a )k+1( vertex

graph having a central vertex denoting graph having a central vertex denoting its “body” linked by edges to vertices fits “body” linked by edges to vertices f11,,

…,f…,fkk representing its “feet”. representing its “feet”.

• K=1 is a K=1 is a “Tick”“Tick”

• K=2 is a K=2 is a “Scorpion”“Scorpion”

• Theorem 2.6Theorem 2.6 A k legged A k legged SpiderSpider can can move freely in a move freely in a (K-1) Connected (K-1) Connected Chordal Graph.Chordal Graph.

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Moving a Four-Legged SpiderMoving a Four-Legged Spider• We can move a We can move a three legged spiderthree legged spider in in

a a biconnected chordalbiconnected chordal graph, this graph, this follows from the theorem 2.6 with k = 3.follows from the theorem 2.6 with k = 3.

• A stronger result yet!A stronger result yet! Theorem 2.10 Theorem 2.10 A A four legged spiderfour legged spider can also move can also move freely in a freely in a biconnected chordalbiconnected chordal graph. graph.

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A five legged spider

cannot move freely

in a biconnected

chordal graph!

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Finding Shortest Motion Finding Shortest Motion • If G has n vertices and H has If G has n vertices and H has ll vertices. vertices.

There are at most O)nThere are at most O)nll( possible ( possible locations of H in G.locations of H in G.

• G’ = )V’,E’( in which each vertex G’ = )V’,E’( in which each vertex corresponds to a possible valid location corresponds to a possible valid location of H in G.of H in G.

• There is an edge in E’ between u,vThere is an edge in E’ between u,vV’ if V’ if there is a local displacement between u there is a local displacement between u and v of H.and v of H.

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Finding Shortest Motion (Cont.)Finding Shortest Motion (Cont.)• G’ can be constructed in polynomial G’ can be constructed in polynomial

time in the size of any time in the size of any fixedfixed graph H. graph H.

• By finding the shortest path in G’ By finding the shortest path in G’ from S to T, we can determine the from S to T, we can determine the motion with the least number of local motion with the least number of local displacements in polynomial time.displacements in polynomial time.

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NP CompletenessNP Completenesswhen H is part of the input !when H is part of the input !

• Clique)G,k( : is the problem of checking Clique)G,k( : is the problem of checking if the graph G contains a clique of size if the graph G contains a clique of size k. This problem is known to be NP k. This problem is known to be NP complete!complete!

• We reduced the clique problem to We reduced the clique problem to checking to see if there exists a motion checking to see if there exists a motion that moves H = K that moves H = K 2k2k from a start location from a start location

to a target location.to a target location.

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NP CompletenessNP Completenesswhen H is part of the input (Cont.)when H is part of the input (Cont.)

• By constructing a new graph G’ =)V’,E’( By constructing a new graph G’ =)V’,E’( V’=VV’=V(x(x11,…,x,…,x2k2k))(y(y2k+12k+1,…,y,…,y4k4k). ).

E’=EE’=EEExxEEyyE’’E’’

E Exx=[(x=[(xii,x,xjj)|1)|1iijj2k] 2k]

EEyy=[(y=[(yii,y,yjj)|2k+1)|2k+1iijj 4k] 4k] E’’=[(v,x E’’=[(v,xii),),

(v,y(v,yjj)| v)| vV,1V,1ii2k ,2k+12k ,2k+1jj4k]4k]

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Graphbots: Mobility in Discrete Spaces. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 2 Mobility in Discrete Spaces Move beyond robots with simple.

Documents

graph g

threevertex graph

chordal graph

graph spacea

vertex v

g iff g

subgraph of g h

cut vertex