Graphbots: Mobility in Discrete Spaces

1. Sharon Lahav Graphbots: Mobility in Discrete Spaces

2. 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! סמינר 5 במדעי המחשב (236805)

3. סמינר 5 במדעי המחשב (236805)

4. 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. סמינר 5 במדעי המחשב (236805)

5. 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. סמינר 5 במדעי המחשב (236805)

6. 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). סמינר 5 במדעי המחשב (236805)

7. 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. סמינר 5 במדעי המחשב (236805)

8. 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. סמינר 5 במדעי המחשב (236805)

9. 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. סמינר 5 במדעי המחשב (236805)

10. 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 סמינר 5 במדעי המחשב (236805)

11. 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. סמינר 5 במדעי המחשב (236805)

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

13. 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. סמינר 5 במדעי המחשב (236805)

14. 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. סמינר 5 במדעי המחשב (236805)

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 peo is represented by a sequence  of vertices. • Theorem 2.3 (Fulkerson and Gross) A graph G has a peo iff G is chordal. סמינר 5 במדעי המחשב (236805)

16. 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. סמינר 5 במדעי המחשב (236805)

17. chordality is sufficient, but not necessary. סמינר 5 במדעי המחשב (236805)

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 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. סמינר 5 במדעי המחשב (236805)

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. • A stronger result yet!Theorem 2.10 A four­legged spider can also move freely in a biconnected chordal graph. סמינר 5 במדעי המחשב (236805)

20. A five­legged spider cannot move freely in a biconnected chordal graph! סמינר 5 במדעי המחשב (236805)

21. 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. סמינר 5 במדעי המחשב (236805)

22. 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. סמינר 5 במדעי המחשב (236805)

23. 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. סמינר 5 במדעי המחשב (236805)

24. 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] סמינר 5 במדעי המחשב (236805)

25. סוף סמינר 5 במדעי המחשב (236805)