LATIN 2004: LATIN 2004: Theoretical Informatics pp 329-338

# Encoding Homotopy of Paths in the Plane

• Sergei Bespamyatnikh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

## Abstract

We study the problem of encoding homotopy of simple paths in the plane. We show that the homotopy of a simple path with k edges in the presence of n obstacles can be encoded using O(n log(n + k)) bits. The bound is tight if k=Ω(n 1 + ε). We present an efficient algorithm for encoding the homotopy of a path. The algorithm can be applied to find homotopic paths among a set of simple paths. We show that the homotopy of a general (not necessary simple) path can be encoded using O(k logn) bits. The bound is tight. The code is based on a homotopic minimum-link path and we present output-sensitive algorithms for computing a path and the code.

## References

1. 1.
Bar-Yehuda, R., Chazelle, B.: Triangulating disjoint Jordan chains. Internat. J. Comput. Geom. Appl. 4(4), 475–481 (1994)
2. 2.
Bespamyatnikh, S.: Computing homotopic shortest paths in the plane. J. Algorithms 49(2), 284–303 (2003)
3. 3.
Cabello, S., Liu, Y., Mantler, A., Snoeyink, J.: Testing homotopy for paths in the plane. In: Proc. 18th Annu. ACM Sympos. Comput. Geom., pp. 160–169 (2002)Google Scholar
4. 4.
Chazelle, B.: A theorem on polygon cutting with applications. In: Proc. 23rd Annu. IEEE Sympos. Found. Comput. Sci., pp. 339–349 (1982)Google Scholar
5. 5.
Chazelle, B.: An algorithm for segment-dragging and its implementation. Algorithmica 3, 205–221 (1988)
6. 6.
Efrat, A., Kobourov, S.G., Lubiw, A.: Computing homotopic shortest paths efficiently. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 411–423. Springer, Heidelberg (2002)
7. 7.
Ghosh, S.K.: Computing visibility polygon from a convex set and related problems. J. Algorithms 12, 75–95 (1991)
8. 8.
Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl. 4, 63–98 (1994)
9. 9.
Keeler, K., Westbrook, J.: Short encodings of planar graphs and maps. Discrete Applied Mathematics 58(3), 239–252 (1995)
10. 10.
Lee, D.T., Preparata, F.P.: Euclidean shortest paths in the presence of rectilinear barriers. Networks 14, 393–410 (1984)
11. 11.
Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993)
12. 12.
Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier Science Publishers B.V. North-Holland, Amsterdam (2000)
13. 13.
Ian Munro, J., Raman, V.: Succinct representation of balanced parentheses and static trees. SIAM J. Comput. 31(3), 762–776 (2001)