Trapezoid graphs and their coloring
Discrete Applied Mathematics
An optimal shortest path parallel algorithm for permutation graphs
Journal of Parallel and Distributed Computing
On powers of m-trapezoid graphs
Discrete Applied Mathematics
Steiner set and connected domination in trapezoid graphs
Information Processing Letters
Parallel computation: models and methods
Parallel computation: models and methods
Measuring the vulnerability for classes of intersection graphs
Discrete Applied Mathematics
Efficient Dispersion Algorithms for Geometric Intersection Graphs
WG '00 Proceedings of the 26th International Workshop on Graph-Theoretic Concepts in Computer Science
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In this paper, we consider parallel algorithms for shortest paths and related problems on trapezoid graphs under the CREW PRAM model. Given a trapezoid graph with its corresponding trapezoid diagram, 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(d2n= 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(d2n= log d) processors.