A Fast Algorithm to Calculate Powers of a Boolean Matrix for Diameter Computation of Random Graphs

In this paper, a fast algorithm is proposed to calculate kth power of an n × n Boolean matrix that requires O(kn3p) addition operations, where p is the probability that an entry of the matrix is 1. The algorithm generates a single set of inference rules at the beginning. It then selects entries (specified by the same inference rule) from any matrix Ak-1 and adds them up for calculating corresponding entries of Ak. No multiplication operation is required. A modification of the proposed algorithm can compute the diameter of any graph and for a massive random graph, it requires only O(n2(1-p)E[q]) operations, where q is the number of attempts required to find the first occurrence of 1 in a column in a linear search. The performance comparisons say that the proposed algorithms outperform the existing ones.

[1]  Robin Wilson,et al.  Graph theory and combinatorics , 1979 .

[2]  Giuseppe Cattaneo,et al.  Algorithm engineering , 1999, CSUR.

[3]  Alfred V. Aho,et al.  Data Structures and Algorithms , 1983 .

[4]  Sheldon M. Ross,et al.  Introduction to Probability Models (4th ed.). , 1990 .

[5]  Roy D. Yates,et al.  Probability and stochastic processes , 1998 .

[6]  V. Ramachandran,et al.  The diameter of sparse random graphs , 2007 .

[7]  Ning Li,et al.  A matrix-based fast calculation algorithm for estimating network capacity of MANETs , 2005, 2005 Systems Communications (ICW'05, ICHSN'05, ICMCS'05, SENET'05).

[8]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[9]  B. S. Robinson.  Number 9 , November 2005 Toward an Optimal Algorithm for Matrix Multiplication , 2005 .

[10]  Jean-Philippe Domenger,et al.  Efficient drawing of RNA secondary structure , 2006, J. Graph Algorithms Appl..

[11]  Donald F. Towsley,et al.  On the diameter of a class of random graphs , 1990, IEEE Trans. Inf. Theory.

[12]  Sheldon M. Ross.  Introduction to Probability Models. , 1995 .

[13]  Linyuan Lu,et al.  The diameter of random massive graphs , 2001, SODA '01.

[14]  Fan Chung Graham,et al.  Random evolution in massive graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[15]  Linyuan Lu,et al.  Random evolution in massive graphs , 2001 .

[16]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[17]  Alexandru Nicolau,et al.  Fractal Matrix Multiplication: A Case Study on Portability of Cache Performance , 2001, Algorithm Engineering.

[18]  Panos M. Pardalos,et al.  Handbook of Massive Data Sets , 2002, Massive Computing.

[19]  Béla Bollobás,et al.  The Diameter of Random Graphs , 1981 .

[20]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[21]  Mi Zhou,et al.  GeneInfoViz: Constructing and visualizing gene relation networks , 2004, Silico Biol..

[22]  E. Marinari,et al.  On the number of circuits in random graphs , 2006, cond-mat/0603657.