On motion planning in graphs

Kalpesh Kapoor, Associate Professor, Department of Mathematics, Indian Institute of Technology Guwahati for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for a degree. In addition, I would like to thank all the research friends from the Department of Mathematics, IIT Guwahati, for their presence during my research.

Introduction

Scope and Basic Terminology

We give the minimum number of moves required for the motion planning problem in Cartesian Product and Lexicographic Product of graphs. We also provide bounds for the minimum number of moves required for the motion planning problem in Strong Product.

Motivation

Vehicles may only stop at the stations and not in the middle of the track. The goal is to prepare a plan for the coordinated movement of the vehicles, which allows the special vehicle to move from the source station to the destination station.

Literature Survey

The goal of Robot Motion Planning in Graph (RMPG) is to focus on the combinatorial aspects and ignore the geometric constraints of the general robot motion planning problem. A straightforward generalization of the RMPG problem is the problem where we have k different robots with corresponding destinations, and is also known as the multi-robot motion planning problem (MRMPG).

Organization of the Thesis

We then provide bounds on the diameter of trees reachable by mRJ and the entire trees reachable by mRJ. Finally, we provide a characterization of complete trees reachable by 2RJ and complete trees reachable by 3RJ in Sections 2.3 and 2.4, respectively.

Background

It suffices to show that Cvu and Cyx are reachable from each other by mRJ. Since T is mRJ-reachable from the Cvu configuration, we can take the robot to point x starting from the Cvu configuration.

Diameter and mRJ -Reachability

Then xP(u, v) has an odd length and therefore xP(u, v) is 2RJ-reachable from the configuration Cux. Since T is fully 3RJ reachable, there is a sequence consisting of simple and 3RJ moves in T that takes the configuration Cwu to the configuration Cwu′′. Let S be a sequence with a minimum number of moves among all such sequences that brings the configuration Cwu to the configuration Cwu ′ ′.

This is the sequence of moves S′ leads the configuration Cwu to the configuration Cwu′′ inT′. Let S be a sequence of the minimum number of moves among all such sequences that leads the configuration Cwu to the configuration Cwu′′. The modified sequence also leads the Cwu configuration to the Cwu′′ configuration. vii) Suppose that S contains a sequence of moves of the form v1i ←−r.

The sequence S′ consists of moves involving only vertices of the tree T′ and it takes the configuration Cwu to the configuration Cwu′′. Let S be a sequence of minimum number of moves between all such sequences that take the configuration Cwu to the configuration Cwu′′ in T′. Since T′ is fully 3RJ-reachable, there exists a sequence consisting of simple moves and 3RJ moves in T′ that takes the configuration Cwu to the configuration Cwu′′.

Let S be a sequence with a minimum number of moves among all such sequences that brings the configurationCwu to the configuration Cwu′′ inT′.

Figure 2.4: A minimal complete 5RJ -reachable tree.

Preamble

A connected graph G is fully S-reachable if and only if every two configurations in G are S-reachable with respect to each other. Now given two configurations Cvu and Cvu′′, there exists a pair of vertices x, y ∈ V(G− w) such that Cvu, Cxw are S-reachable of each other and Cvu′′,Cyw are S-reachable of each other. We also can't move the hole to another vertex of the graph without disrupting the robot ATV.

Thus, starting with the configuration Cvu of G, we cannot take the robot to any other vertex of G. Every vertex u in the component of G−v that intersects block B is said to be on the B side of v. If G is complete S-reachable, then there exists a path P(u, v) of length m1+ 1 connecting v to a vertex u on side B of v.

Then, to move the robot to any other point G, we must first move it to a point on the B side of v. A block in a graph is called a hanging block if it has only one cut point.

Complete {m}-Reachability

Note that if u is an unintersected vertex in the graph G, then for any set of nonnegative integers S, the configurations Cxu and Cyu are S-reachable from each other (only through obstacle moves). Thus, the vertex set of a doubly connected graph can be partitioned in such a way that Cxu and Cyv S- are reachable from each other if and only if u, v belongs to the same set of the partition. But according to Corollary 3.2 we can claim that Cyx,Cyα{m}- are reachable of each other, and that Cyx′′, Cyβ′ {m}- are reachable of each other.

Finally, since d(α, β) =m+ 1, the configuration Cyβ′ is {m}-reachable from Cyα using a sequence of moves. Finally, by Corollary 3.2 we can claim that Cyx and Cyx′′ are {m}-reachable from each other. Thus, from all these observations we can conclude that any two configurations in C1 are {1,2}-reachable from each other.

Note that for any vertex u, x, y ∈V(G), the configurations Cxu and Cyu are {1,2}-reachable by a sequence of moves consisting only of obstacle moves. If v is on the path connecting P(u, x), then by Lemma 3.5, the configuration Cvx is {1,2}-reachable from Cvu and this again implies that Cyx is {1,2}-reachable from Cvu .

Figure 3.4: A graph that is not complete {11}- {11}-reachable.

Preamble

The four standard product graphs are: the Cartesian product, the direct product, the lexicographic product and the strong product. For example, one of the important common properties of the four standard products is that if we take the product of two simple graphs, we will get a simple graph. Also, all the four products are associative and except for the lexicographic product, all the other three standard products are commutative.

It is known that, computing the minimum number of moves for the robot motion planning problem on a graph is NP-complete in general and also when restricted to planar graphs [24]. In this chapter, we focus on the four standard products and show that the graph motion planning problem can be solved efficiently when restricted to these product graphs. Given an initial configuration of a graph G with the robot at u, a sequence of moves that takes the robot from u to a vertex v in G is called minimal if it is a sequence with the minimum number of moves among all sequences of moves that the robot takes from was seeded in G.

It is easy to see that the minimum number of moves required to take the robot from u to v is 21 and the path traced by the robot in any minimum sequence of moves is not the shortest path from u tov . In the case of each of the product graphs discussed here, the path traced by the robot in a minimum sequence of moves that takes the robot from a source vertex to a target vertex is a shortest path.

Figure 4.1: The graph P 6 and the grid graph of order 6 2 .

Graph Cartesian Product

Therefore, to get the robot from ui to vj, we need at least dH(u, v) number of moves H. From Proposition 4.5, we need at least three moves to make the first move G of the robot. That is, S includes at least 5k+p+ 1 moves if the robot's first move is aH-move.

That is, S includes at least 5k+moves if the first robot move is a G move. Thus, the number of moves in S is at least 5k+2, if it includes H robot moves. Let S be a sequence of moves that takes the robot from ui to vj to GH.

For some j ∈ GH, let S be the minimal sequence of moves that takes the robot from ui to vj. For some j ∈ GH, let S be the minimal sequence of moves that takes the robot from ui to vj.

Graph Strong Product

For some v ∈ N(u), the minimum number of movements required to take the robot from uj tovk is then 3. For some j ∈ V(G), S is a minimum set of movements that take the robot from ui to uj. 4.1) There is also a set of movements satisfying 4.1, where each robot movement is a G-movement or an N-movement. So q+ 1 moves are sufficient to move the robot from ui to uj (q moves to take the gap from vi to uj without disturbing the robot at ui plus the robot move uj ←−r ui).

Note also that every move of the robot in the sequence that takes the robot from ui to vj along the path P is a move G or a move N. For some j ∈ V(G), let S be the minimal sequence of moves that takes the robot from ui to uj . Let S be a sequence of moves that takes the robot from ui to vi along path P.

Let S′ be a sequence with a minimum number of moves that takes the robot from ui to vj along the path P. Let S′ be a sequence with a minimum number of moves that takes the robot from ui to vj along the path P.

Graph Lexicographic Product

Therefore at least three moves are required to take the robot from vj to week. To move the robot from vi to wi, we must first move the hole to wi. Let S be a minimal sequence of moves that takes the robot from ui to vj for some j ∈ V(G).

Since H moves of the robot always keep the robot in the same copy of H, S involves at least one robot move that is not an H move. The first robot move in S is not an H move: Let yp ←−r xp be the first H move of the robot in S. So each of the robot moves in S that appears before the move yp ←−r xp is either a G-move or an N-move.

The first move of the robot in S is an H move: Let yp ←−r xi be the first non-H move of the robot in S. Let S be a minimal sequence of moves that takes the robot from ui to vj for some j ∈ V(G).