Embedding Fibonacci Cubes into Hypercubes with Ω(2cn) Faulty Nodes

• Rostislav Caha
• Petr Gregor
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)

Abstract

Fibonacci Cubes are special subgraphs of hypercubes based on Fibonacci numbers. We present a construction of a direct embedding of a Fibonacci Cube of dimension n into a faulty hypercube of dimension n with less or equal $$2^{\left\lceil {\tfrac{n} {4}} \right\rceil - 1}$$ faults. In fact, there exists a direct embedding of a Fibonacci Cube of dimension n into a faulty hypercube of dimension n with at most $$\tfrac{{2^n }} {{4f_n }}$$ faults (f n is the n-th Fibonacci number). Thus the number ø(n) of tolerable faults grows exponentially with respect to dimension n,ø(n) = Ω(2cn), for c = 2−log2(1+$$\sqrt 5$$) ≐ 0:31. On the other hand, ø(n) = O(2dn), for d = (8−3 log2 3)=4 ≐ 0:82. As a corollary, there exists a nearly polynomial algorithm constructing a direct embedding of a Fibonacci Cube of dimension n into a faulty hypercube of dimension n (if it exists) provided that faults are given on input by enumeration. However, the problem is NP-complete, if faults are given on input with an asterisk convention.

Keywords

Tolerable Fault Binary String Polynomial Algorithm Fibonacci Number Subgraph Isomorphic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

1. 1.
J. Bergum, B. Cong, and S. Sharma: Simulation of Tree Structures on Fibonacci Cubes. Proc. First Int’l Conf. Computer Comm. and Networks (1992) 279–283Google Scholar
2. 2.
M. Y. Chan and S. J. Lee: Fault-Tolerant Embedding of Complete Binary Trees in Hypercubes. IEEE Trans. Parallel and Distributed Systems 4 (1993) 277–288
3. 3.
M. J. Chung and W.-J. Hsu: Generalized Fibonacci Cubes. Proc. 1993 Int’l Conf. Parallel Processing 1 (1993) 299–302Google Scholar
4. 4.
B. Cong and S. Q. Zheng: Near-Optimal Embeddings of Trees into Fibonacci Cubes. Proc. 28th IEEE Southeastern Symp. System Theory (1996) 421–426Google Scholar
5. 5.
B. Cong, S. Sharma, and S. Q. Zheng: On Simulations of Linear Arrays, Rings, and 2-D Meshes on Fibonacci Cube Networks. Proc. 7th Int’l Parallel Processing Symp. (1993) 748–751Google Scholar
6. 6.
R. L. Graham, D. E. Knuth, and O. Patashnik: “Special numbers,” in Concrete Mathematics. Reading, Addison-Wesley, Massachusetts (1989)Google Scholar
7. 7.
P. Gregor: Embeddings of Special Graph Classes into Hypercubes and their Generalizations. Charles University, Prague, master thesis (1999)Google Scholar
8. 8.
J. Hastad, F. T. Leighton, and M. Newman: Reconfiguring a hypercube in the presence of faults. Proc. 19th Annu. ACM Symp. Theory Comput. (1987) 274–284Google Scholar
9. 9.
S.-J. Horng, F.-S. Jiang, T.-W. Kao: Embedding of Generalized Fibonacci Cubes in Hypercubes with Faulty Nodes. IEEE Trans. Parallel and Distributed Systems 8 (1997) 727–737
10. 10.
W.-J. Hsu: Fibonacci Cubes-A New Interconnection Topology. IEEE Trans. Parallel and Distributed Systems 4 (1993) 3–12
11. 11.
W.-J. Hsu and J. Liu: Fibonacci Codes as Formal Languages. Technical Report CPS-91-05, Michigan State University (1991)Google Scholar
12. 12.
F. T. Leighton: Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes Morgan Kaufmann, San Mateo, California(1992)
13. 13.
C. S. Raghavendra and P.-J. Yang: Embedding and Reconfiguration of Binary Trees in Faulty Hypercubes. IEEE Trans. Parallel and Distributed Systems 7 (1996) 237–245
14. 14.
N. F. Tzeng: Structural Properties of Incomplete Hypercube Computers. Proc. 10th IEEE Int’l Conf. Distributed Computing Systems (1990) 262–269Google Scholar
15. 15.
J. Wu: Extended Fibonacci Cubes. IEEE Trans. Parallel and Distributed Systems 8 (1997) 1203–1210
16. 16.
M. R. Garey and D. J. Johnson: Computers and Intractability. Bell Laboratories, New Jersey (1979)

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

• Rostislav Caha
• 1
• Petr Gregor
• 1
1. 1.Dep. of Theoretical Computer ScienceCharles UniversityPrague 1Czech Republic

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