# Parallel Algorithms for Shortest Paths and Related Problems on Trapezoid Graphs

## Abstract

In this paper, we consider parallel algorithms for shortest pa- ths and related problems on trapezoid graphs under the CREW PRAM model. Given a trapezoid graph with its corresponding trapezoid dia- gram, we present parallel algorithms solving the following problems: For the *single-source shortest path problem*, the algorithm runs in *O*(log *n*) time using *O*(*n*) processors and space. For the *all-pair shortest path query problem*, after spending *O*(log *n*) preprocessing time using *O*(*n* log *n*) space and *O*(*n*) processors, the algorithm can answer the query in *O*(log δ) time using one processor. Here δ denotes the distance between two queried vertices. For the *minimum cardinality Steiner set problem*, the algorithm runs in *O*(log *n*) time using *O*(*n*) processors and space.

We also extend our results to the generalized trapezoid graphs. The single-source shortest path problem and the minimum cardinality Steiner set problem on *d-trapezoid graphs* and *circular d-trapezoid graphs* can both be solved in *O*(log *n* log *d*) time using *O*(*nd*) space and *O*(*d* ^{2} *n*/log *d*) processors. The all-pair shortest path query problem on *d*-trapezoid graphs and circular *d*-trapezoid graphs can be answered in *O*(*d* log δ) time using one processor after spending *O*(log *n* log *d*) preprocessing time using *O*(*nd* log *n*) space and *O*(*d* ^{2} *n*/log *d*) processors.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.S.G. Akl, “Parallel computation: models and methods”, Prentice Hall, Upper Saddle River, New Jersey, 1997.Google Scholar
- 2.I. Dagan, M.C. Golumbic and R.Y. Pinter, “Trapezoid graphs and their coloring”,
*Discr. Applied Math*.,21:35–46,1988.zbMATHCrossRefMathSciNetGoogle Scholar - 3.D.Z. Chen, D. T. Lee, R. Sridhar and C. N. Sekharan, “Solving the all-pair shortest path query problem on interval and circular-arc graphs”,
*Networks*, pp. 249–257, 1998.Google Scholar - 4.Flotow, “On Powers of m-Trapezoid Graphs”,
*Discr. Applied Math*., 63:187–192, 1995.zbMATHCrossRefMathSciNetGoogle Scholar - 5.H.S. Chao, F.R. Hsu and R.C.T. Lee, “On the Shortest Length Queries for Permutation Graphs”,
*Proc. Of the 1998 International Computer Symposium, Workshop on Algorithms*, NCKU, Taiwan, pp. 132–138, 1998.Google Scholar - 6.O.H. Ibarra and Q. Zheng, “An optimal shortest path parallel algorithm for permutation graphs”,
*J. of Parallel and Distributed Computing*, 24:94–99, 1995.CrossRefGoogle Scholar - 7.Dieter Kratsch, Ton Kloks and Haiko Müller, “Measuring the vulnerability for classes of intersection graphs”,
*Discr. Applied Math*., 77:259–270, 1997.zbMATHCrossRefGoogle Scholar - 8.Y. D. Liang, “Steiner set and connected domination in trapezoid graphs”,
*Information Processing Letters*, 56(2):101–108, 1995.CrossRefMathSciNetGoogle Scholar