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## Details

Document Type: | Book |
---|---|

All Authors / Contributors: |
John E Hopcroft; Gordon Thomas Wilfong |

OCLC Number: | 30062013 |

Notes: | "July 1984." |

Description: | 28 pages ; 28 cm. |

Series Title: | Cornell University.; Department of Computer Science.; Technical report |

Responsibility: | John E. Hopcroft, Gordon Wilfong. |

### Abstract:

In this paper we study the motion planning problem for multiple objects where an object is a 2-dimensional body whose faces are line segments parallel to the axes of $R[superscript]{2}$ and translations are the only motions allowed. Towards this end we analyze the structure of configuration space, the space of points that correspond to positions of the objects. In particular, we consider CONNECTED, the set of all points in configuration space that correspond to configurations of the objects where the objects form one connected component. We show that CONNECTED consists of faces of various dimensions such that if there is a path in CONNECTED between two 0-dimensional faces (vertices) of CONNECTED then there is a path between them along 1-dimensional faces (edges) of CONNECTED. It is known that if there is a motion between the configurations. Thus by the result of this paper the existence of a motion between two vertices of CONNECTED implies a motion corresponding to a path along edges of CONNECTED. Hence we have reduced the motion planning problem from a search of a high dimensional space to a graph searching problem.

Searching the graph of vertices and edges of CONNECTED for a path has a prohibitive worse-case complexity because of the large number of vertices and edges. However, if the search generates edges and vertices only as they are needed, a practical and efficient algorithm may be possible using some effective heuristic.

From this result it is shown that motion planning for rectangles in a rectangular boundary is in PSPACE. Since it is known that the problem is PSPACE-hard we conclude it is a PSPACE-complete problem.

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