# Computing minimum-link path in a homotopy class amidst semi-algebraic obstacles in the plane

## Abstract

Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct a polygonal path between these points which has the minimum number of segments (links) and the minimum ‘total turn’, that is the sum of absolute values of angles of turns of the consecutive polygon links. We describe an algorithm that solves the problem spending polynomial time to construct one segment of the minimum-link and minimum-turn polygon if to use a modification of real RAMs which permits to handle the solutions of algebraic equations. It is known that the number of segments in such a minimum-link polygon can be exponential as function of the length of the input data or even of the degree of polynomials representing the semi-algebraic set. In fact, we describe how to construct a minimum-link-turn path for a given class of homotopy(whose shortest path has no self-intersections), and provide a rigorous and rather a universal way of reasoning about homotopy classes in contexts related to algorithms. It was previously shown by Heintz-Krick-Slissenko-Solernó that a shortest path in the situation under consideration is semi-algebraic, and an extended real RAM that is able to compute integrals of algebraic functions can find it in polytime.

## Keywords

Short Path Homotopy Class Homotopy Type Angle Function Calculated Weight## Preview

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## References

- [AM88]M. E. Alonso and Raimondo M. The computation of the topology of a planar semialgebraic set.
*Rend. Sem. Mat. Univers. Politecn. Torino*, 46(3):327–342, 1988.Google Scholar - [BCR87]J. Bochnak, M. Coste, and M.-F. Roy.
*Géométrie algébrique réelle*. Springer-Verlag, 1987.Google Scholar - [BSS89]L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over real numbers: NP-completeness, recursive functions and universal machines.
*Bull. Amer. Math. Soc.*, 1:1–46, 1989.Google Scholar - [CGM+95]V. Chandru, S. K. Ghosh, A. Maheshwari, V. T. Rajan, and S. Saluja.
*NC*-algorithms for minimum link path and related problems.*J. of Algorithms*, 19:173–203, 1995.Google Scholar - [CR87]J. Canny and J. Reif. New lower bound technique for robot motion planning problemms. In
*Proc. 28th Annu. IEEE Symp. on Foundations of Comput. Sci.*, pages 49–60, 1987.Google Scholar - [Gri88]D. Yu. Grigoriev. Complexity of deciding Tarski algebra.
*J. Symb. Comput.*, 5:65–108, 1988.Google Scholar - [GV92]D. Yu. Grigoriev and N. N. Vorobjov. Counting connected components of a semi-algebraic set in subexponential time.
*Computational Complexity*, 2(2):133–184, 1992.Google Scholar - [HKSS93]J. Heintz, T. Krick, A. Slissenko, and P. Solernó. Une borne inférieure pour la construction de chemins polygonaux dans
*R*^{n}. In*Publications du département de mathématiques de l'Université de Limoges*, pages 94–100. Université de Limoges, France, 1993.Google Scholar - [HKSS94]J. Heintz, T. Krick, A. Slissenko, and P. Solernó. Search for shortest path around semialgebraic obstacles in the plane.
*J. Math. Sciences*,**70**(4):1944–1949, 1994. Translation into English of the paper published in*Zapiski Nauchn. Semin. LOMI*, vol. 192(1991), p. 163–173.Google Scholar - [HRS90]J. Heintz, M.-F. Roy, and P. Solernó. Sur la complexité du principe de Tarski-Seidenberg.
*Bull. Soc. Math. de France*, 118:101–126, 1990.Google Scholar - [HRS94]J. Heintz, M.-F. Roy, and P. Solernó. Single exponential path finding in semi-algebraic sets. part 2: the general case. In Ch. L. Bajaj, editor,
*Algebraic Geometry and Its Applications*, 1994.Google Scholar - [HS94]J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class.
*Computational Geometry*, 4:63–97, 1994.Google Scholar - [MPA92]J. S. B. Mitchell, C. Piatko, and E. M. Arkin. Computing a shortest
*k*-link path in a polygon. In*Proc. 33rd IEEE FOCS*, pages 573–582, 1992.Google Scholar - [MRW92]J. S. B. Mitchell, G. Rote, and G. Woeginger. Minimum-link paths among obstacles in the plane.
*Algorithmica*8:431–459, 1992.Google Scholar - [MS95]J. S. B. Mitchell and Suri S.
*A survey on computational geometry*, volume 7 of*Hanbooks in Operations Research and Management Sciences*, chapter 7, pages 425–479. Elsevier Science B. V., 1995.Google Scholar - [Pap94]C. H. Papadimitriou.
*Computational complexity*. Addison-Wesley, 1994.Google Scholar - [Ren92]J. Renegar. On the computational complexity and geometry of the first-order theory of the reals. parts 1–3.
*J. Symb. Comput.*, 13(3):255–352, 1992.Google Scholar - [ST80]H. Seifert and W. Threlfall.
*A Textbook of Topology*. Academic Press, 1980.Google Scholar