A Survey of Computational Geometry

ly, the motion planning problem is that of computing a path between two points in the topological space FP. In the rst two of ve seminal papers on the \Piano Movers' Problem" ([220{222, 229, 223], collected in the book [131]), Schwartz and Sharir show that the boundary of FP is a semialgebraic set (assuming the original constraints of the problem are semialgebraic). This then allows the motion planning problem to be written as a decision question in the theory of real closed elds (see Tarski [241]), which can be solved by adding appropriate adjacency information to the cylindrical decomposition that is (symbolically) computed by the algorithm of Collins [67]. For any xed d and xed degree of the polynomials describing the constraints, the complexity of the resulting motion planning algorithm is polynomial in n, the combinatorial size of the problem description. Instead of computing a cell decomposition of FP, an alternative paradigm in motion planning is to compute a lower dimensional subspace, FP 0 FP, and to de ne a \retraction function" that maps FP onto FP 0 . O'D unlaing and Yap [194] and O'D unlaing, Sharir, and Yap [195, 192, 193] have computed such retractions on the basis of Voronoi diagrams, obtaining e cient solutions to several low-dimensional motion planning problems. Most recently, Canny [43] has described a method of reducing the motion planning problem to a (one-dimensional) graph search problem, by means of a \roadmap"; this is the currently best known method for general motion planning problems. The bottom line is that the motion planning problem can be solved in polynomial time (polynomial, that is, in the combinatorial complexity of the set of obstacles), for any xed number of degrees of freedom of the robot. Many lower bounds have also been established on motion planning problems. The rst such results were by Reif [213], who showed that the generalized movers' problem (with many independently movable objects) is PSPACE-hard. Hopcroft, Joseph, and Whitesides [130] give PSPACE-hardness and NP-hardness results for several planar motion planning problems. See also the recent lower bounds paper by Canny and Reif [46]. 5. Matching, Traveling Salesman, and Watchman Routes Matching and traveling salesman are among the best known problems in combinatorial optimization. In this section, we survey some results on these problems where the underlying graph is induced by a geometric input. Ch. 1. A Survey of Computational Geometry 35