Cartesian graph factorization at logarithmic cost per edge
LetG be a connected graph withn vertices andm edges. We develop an algorithm that finds the (unique) prime factors ofG with respect to the Cartesian product inO(m logn) time andO(m) space. This shows that factoringG is at most as costly as sorting its edges. The algorithm gains its efficiency and practicality from using only basic properties of product graphs and simple data structures.
Key wordsCartesian product graphs Factorization Algorithms
Unable to display preview. Download preview PDF.
- F. Aurenhammer, and J. Hagauer,Computing equivalence classes among the edges of a graph with applications, Proc. Int. Conf. Algebraic Graph Theory, 1989; and Discrete Math. Vol. 110, 1992.Google Scholar
- T. Feder,Product Graph Representations, J. of Graph Theory, 1992 (to appear).Google Scholar
- J. Feigenbaum,Product Graphs: Some algorithmic and combinatorial properties, Ph. D. Thesis, Dept. Comput Sci., Stanford. Univ., CA, 1986.Google Scholar
- J. Feigenbaum, J. Hershberger, andA. Schäffer,A polynomial time algorithm for finding the prime factors of Cartesian-product graphs, Discrete Appl. Math. 12 1985, 123–138.Google Scholar
- A. Gibbons,Algorithmic Graph Theory, Cambridge University Press, 1985.Google Scholar
- R. L. Graham, andP. M. Winkler,On isometric embeddings of graphs, Trans. AMS 288 1985, 527–536.Google Scholar
- B. Hochstrasser,A note on Winkler's algorithm for factoring a connected graph, Proc. Int. Conf. Algebraic Graph. Theory, 1989; Discrete Math. Vol. 110, 1992.Google Scholar
- G. Sabidussi,Graph multiplication, Math Z. 72 1960, 446–457.Google Scholar
- R. Sedgewick Algorithms, Addison-Wesley, Reading MA, 1983.Google Scholar
- V. G. Vizing,The Cartesian product of graphs., Engl. transl.: Comp. El. Syst. 2 1966, 352–365.Google Scholar
- P. M. Winkler,Factoring a graph, in polynomial time, Europ. J. Combin. 8 1987, 209–212.Google Scholar