1. Graphbots: Mobility in Discrete Spaces

2. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 2 Mobility in Discrete Spaces Move beyond robots with simple geometries (polygonal structure).Move beyond robots with simple geometries (polygonal structure). Move beyond simple spaces (planar region containing polygonal obstacles).Move beyond simple spaces (planar region containing polygonal obstacles). Teams of robots that operate in discrete spaces like graphs.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!And that have discrete geometries represented by subgraphs i.e they are maintaining a formation!

3. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 3

4. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 4 Mobility Definitions A connected graph G - “the graph space”A connected graph G - “the graph space” A connected subgraph of G H – “a cooperating team of robots”A connected subgraph of G H – “a cooperating team of robots” We represent the members of the team by single nodes.We represent the members of the team by single nodes.

5. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 5 Mobility Definitions (Cont.) A movement or motion of H from S to T is defined by a sequence of subgraphs: S=H 0, H 1,…,H k =T all isomorphic to H.A movement or motion of H from S to T is defined by a sequence of subgraphs: S=H 0, H 1,…,H k =T all isomorphic to H. The structure must be preserved when the team moves.The structure must be preserved when the team moves.

6. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 6 The Goals Find conditions for G to satisfy a free movement of a given subgraph H in G.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).Establish the complexity of finding a motion with the fewest local displacements from S to T (if one exists).

7. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 7 Moving a Tick A “Tick” is modeled by a two vertex graph linked by a single edge.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.Theorem 2.1 - A tick can move freely in any connected graph.

8. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 8 Moving a Scorpion A “Scorpion” is modeled by a three­ vertex graph linked by two edges.A “Scorpion” is modeled by a three­ vertex graph linked by two edges. The degree­2 vertex is the “body”.The degree­2 vertex is the “body”. The degree­1 vertices are the “feet”.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.Theorem 2.2- A scorpion can move freely in G iff G does not contain a vertex v with two neighbors of degree 1.

9. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 9 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 v i of v.If G has such a vertex v, we can place the scorpion with b on v, and the f 's on the neighbors v i 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.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. b

10. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 10 G S T Proff of Theorem 2.2  There is no such vertex v.There is no such vertex v. S-{v 1,v 2,v 3 }, T-{u 1,u 2,u 3 }S-{v 1,v 2,v 3 }, T-{u 1,u 2,u 3 } 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 u 2 and v 2, then the scorpion can “creep” along this path.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 u 2 and v 2, then the scorpion can “creep” along this path. f b f

11. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 11 Proff of Theorem 2.2  (Cont.) The only path from S to T goes through either u 2 or v 2 (or both):The only path from S to T goes through either u 2 or v 2 (or both): G S u2u2 u1u1 u3u3 T v1v1 v3v3 v2v2 The degree of u 1 is not 1 and it has a neighbor u 0 (other than u 2 )The degree of u 1 is not 1 and it has a neighbor u 0 (other than u 2 ) f goes from u 1 to u 0, b goes to u 1 and f goes from u 3 to u 2. Now the scorpion can creep along the path to T.f goes from u 1 to u 0, b goes to u 1 and f goes from u 3 to u 2. Now the scorpion can creep along the path to T. u0u0 u3u3 u1u1 u2u2 u0u0

12. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 12 To corner a scorpion you have to completely immobilize it at its starting location! b

13. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 13 Some Definitions A single vertex in a connected graph whose deletion disconnects the graph is called a cut vertex.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.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. - Connected Graphs - A graph is said to be ­ Connected if the deletion of any subset of -1 vertices leaves the graph connected.

14. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 14 Some Definitions (Cont.) A Chordal Graph is a graph in which each cycle of length at least 4 has a Chord.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.A Chord is an edge that connects two vertices that are not adjacent in the cycle.

15. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 15 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 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.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.Theorem 2.3 (Fulkerson and Gross) A graph G has a peo iff G is chordal.

16. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 16 Moving a Trilobite A “Trilobite” is modeled by a three­vertex graph linked by three edges. (a clique of size three).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.Theorem 2.4- A Trilobite can move freely in a biconnected chordal graph, that has at least three vertices.

17. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 17 chordality is sufficient, but not necessary.

18. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 18 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 f 1,…,f k representing its “feet”.A spider is modeled by a (k+1)­vertex graph having a central vertex denoting its “body” linked by edges to vertices f 1,…,f k representing its “feet”. K=1 is a “Tick”K=1 is a “Tick” K=2 is a “Scorpion”K=2 is a “Scorpion” Theorem 2.6 A k­legged Spider can move freely in a (K-1)­ Connected Chordal Graph.Theorem 2.6 A k­legged Spider can move freely in a (K-1)­ Connected Chordal Graph.

19. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 19 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.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.A stronger result yet! Theorem 2.10 A four­legged spider can also move freely in a biconnected chordal graph.

20. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 20 A five­legged spider cannot move freely in a biconnected chordal graph!

21. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 21 Finding Shortest Motion If G has n vertices and H has l vertices. There are at most O(n l ) possible locations of H in G.If G has n vertices and H has l vertices. There are at most O(n l ) possible locations of H in G. G’ = (V’,E’) in which each vertex corresponds to a possible valid location 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.There is an edge in E’ between u,v  V’ if there is a local displacement between u and v of H.

22. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 22 Finding Shortest Motion (Cont.) G’ can be constructed in polynomial time in the size of any fixed graph H.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.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.

23. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 23 NP­ Completeness when 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 NP­ complete!Clique(G,k) : is the problem of checking if the graph G contains a clique of size k. This problem is known to be NP­ complete! 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.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.

24. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 24 NP­ Completeness when H is part of the input (Cont.) By constructing a new graph G’ =(V’,E’) V’=V  (x 1,…,x 2k )  (y 2k+1,…,y 4k ). E’=E  E x  E y  E’’ E x =[(x i,x j )|1  i  j  2k] E y =[(y i,y j )|2k+1  i  j  4k] E’’=[(v,x i ),(v,y j )| v  V,1  i  2k,2k+1  j  4k]By constructing a new graph G’ =(V’,E’) V’=V  (x 1,…,x 2k )  (y 2k+1,…,y 4k ). E’=E  E x  E y  E’’ E x =[(x i,x j )|1  i  j  2k] E y =[(y i,y j )|2k+1  i  j  4k] E’’=[(v,x i ),(v,y j )| v  V,1  i  2k,2k+1  j  4k]

25. 2001, שרון להב סמינר 5 במדעי המחשב (236805)Graphbots - 25