perfect codes in generalized sierpi nski graphs
play

Perfect codes in Generalized Sierpi nski Graphs Aline Parreau - PowerPoint PPT Presentation

Perfect codes in Generalized Sierpi nski Graphs Aline Parreau Institut Fourier - Universit e de Grenoble - France Join work with Sylvain Gravier and Matjaz Kov se CID 2011- September 20 th , 2011 ANR IDEA 1/21 Outline Sierpi nski


  1. Perfect codes in Generalized Sierpi´ nski Graphs Aline Parreau Institut Fourier - Universit´ e de Grenoble - France Join work with Sylvain Gravier and Matjaz Kovˇ se CID 2011- September 20 th , 2011 ANR IDEA 1/21

  2. Outline Sierpi´ nski Graphs: → Graph on { 1 , ..., k } n with good metric and coding properties. Idea : generalize those graphs to have new (and good?) metrics on { 1 , ..., k } n 2/21

  3. Recursive construction of Sierpi´ nski graph S ( n , k ) 1. Start with the complete graph: S (1 , k ) = K k . 3 1 2 3/21

  4. Recursive construction of Sierpi´ nski graph S ( n , k ) 1. Start with the complete graph: S (1 , k ) = K k . 2. Copy it k times. 3/21

  5. Recursive construction of Sierpi´ nski graph S ( n , k ) 1. Start with the complete graph: S (1 , k ) = K k . 2. Copy it k times. 3. Add one edge between each pair of copies to get S (2 , k ). 3/21

  6. Recursive construction of Sierpi´ nski graph S ( n , k ) 1. Start with the complete graph: S (1 , k ) = K k . 2. Copy it k times. 3. Add one edge between each pair of copies to get S (2 , k ). 4. ”New” vertex i is vertex i of copy i . 3 1 2 3/21

  7. Recursive construction of Sierpi´ nski graph S ( n , k ) 1. Start with the complete graph: S (1 , k ) = K k . 2. Copy it k times. 3. Add one edge between each pair of copies to get S (2 , k ). 4. ”New” vertex i is vertex i of copy i . 5. Repeat to obtain S (3 , k ), S (4 , k ),... 3 3 1 2 1 2 3/21

  8. Examples of Sierpi´ nski graphs S ( n , k ) : k vertices in the complete graph, n iterations �� �� ���� ���� ���� ���� ���� ���� �� �� �� �� �� �� � � � � � � � � � � � � �� �� �� �� �� �� � � � � � � � � � � � � �� �� �� �� �� �� � � � � � � �� �� � � �� �� �� �� ���� ���� ���� ���� � � � � � � � � � � � � � � � � ���� ���� ���� ���� � � � � � � � � �� �� � � �� �� �� �� ���� ���� ���� ���� � � � � � � � � � � � � � � � � ���� ���� ���� ���� � � � � � � � � � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� � � �� �� �� �� � � � � � � � � � � � � �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � � � �� �� �� �� � � � � � � � � � � � � � � � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� � � � � � � � � � � � � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� � � � � � � � � � � � � � � � � � � �� �� �� �� �� �� � � �� �� �� �� ���� ���� � � � � � � ���� ���� ���� ���� � � � � � � ���� ���� � � � � � � � � �� �� �� �� �� �� �� �� � � �� �� �� �� � � � � � � ���� ���� ���� ���� � � � � � � � � � � � � � � � � �� �� � � ���� ���� ���� ���� � � � � �� �� � � ���� ���� ���� ���� � � � � � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� � � �� �� � � � � ���� ���� ���� ���� � � � � � � � � � � ���� ���� ���� ���� �� �� �� �� � � �� �� � � � � ���� ���� ���� ���� � � � � � � � � � � �� �� �� �� �� �� �� �� � � � � � � � � � � � � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� � � � � � � � � � � � � � � � � � � � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� �� �� � � � � � � � � � � � � � � � � � � �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� �� �� � � � � � � � � � � � � � � � � � � �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� � � �� �� ���� ���� ���� ���� � � � � � � � � � � � � ���� ���� ���� ���� � � � � � � �� �� �� �� �� �� � � � � � � � � � � � � �� �� � � ���� ���� ���� ���� � � � � ���� ���� ���� ���� � � �� �� �� �� �� �� � � � � � � � � � � � � �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� � � � � � � � � � � � � � � � � � � �� �� �� �� �� �� �� �� ���� ���� � � � � � � ���� ���� � � � � � � ���� ���� � � � � � � �� �� �� �� �� �� �� �� �� �� � � � � � � � � � � � � � � � � � � �� �� �� �� S(4,3) S(5,2) 4/21

  9. Sierpi´ nski graphs: definition with words nski graphs: { 1 , . . . , k } n Vertex set of Sierpi´ Edge between u 1 u 2 ... u n and v 1 v 2 ... v n , if there is 1 ≤ j ≤ n s.t: • u i = v i if i < j , • u j � = v j , • u i = v j and v i = u j if i > j y . . . y u = w x y v = w x . . . x Extreme vertex x: vertex x ... x 5/21

  10. Sierpi´ nski graphs: definition with words 333 331 332 313 323 311 322 312 321 133 233 131 132 231 232 113 123 213 223 111 222 112 121 122 211 212 221 y . . . y u = w x y v = w x . . . x 6/21

  11. About Sierpi´ nski Graphs • Introduced in 1997 by Klavˇ zar and Milutinovi´ c • S ( n , 3) are Hano¨ ı graphs ( n = number of disks): 7/21

  12. About Sierpi´ nski Graphs • Introduced in 1997 by Klavˇ zar and Milutinovi´ c • S ( n , 3) are Hano¨ ı graphs ( n = number of disks): ↔ 7/21

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend