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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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Copyright information

© 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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