Lee Distance and Topological Properties of kary ncubes

1
Lee Distance and Topological Propertiesof k-ary
n-cubes

• Interconnection Networks
• ???e??d?? Te?d?s???
• ?.?. 99

2
??sa????

• ???tas? t?p???????? ?d??t?t?? t?? k-ary n-cube
  (Qnk) µe ß?s? t?? Lee distance.
• F?s??? µet???? ??a t?? ???sµ? ?a? t? µe??t? t???
  s?µf??a µe t??? s????afe??.
• ??et??eta? ? a?t?st????s? d?af???? ???f?? se Qnk.
• ???ß??µa t?p???????? t?p???t?s?? p????.

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Ge???? - ???sµ??

• ? s???e???µ??? d?µ? ???s?µ?p??e?ta? se concurrent
  systems
• ?e????? d?af?????? e??s?se??
• ?p?te?esµat???? ?p?????sµ?? ??. ???eß?a?.
• ?n CPUs µe µ??µ? 2n s??d?se??.
• ?p?stas? Lee
• A n-??f?? d????sµa ? ß?s??.
• Lee weight
• Lee distance A,B WL(A-B)
• Bitwise difference mod k.

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Ge???? ???sµ?? 2

• G?a ?2,3 Dh(A,B)DL(A,B).
• ?gt3, DL(A,B)gt Dh(A,B).
• ??a? Qnk e??a? 2n ta?t???? ???f?? µe ?n ??µß???.
• ?eta?? ?,? ??µß?? ?p???e? a?µ? a? DL(A,B)1.
• ?? s??t?µ?te?? µ???p?t? µeta?? d?? ??µß?? ??e?
  µ???? ?s? µe t? Lee ap?stas?.

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?d??t?te? Qnk

• ???µet??? Dnfloor(k/2).
• ??µ??
• Lnkn.
• ?p?f??e?a
• ?????

• ?a?tes?a?? ????µe?? ??????.
• ?????te??? ??ß?? a?ad??µ???.

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???a µ???p?t?a

• Fault-tolerance.
• Te???µa
• G?a ?,? ????f?? se ??a Qnk ?a? lDL(A,B),
  hDh(A,B), WiDL(Ai,Bi), 0ltiltn-1 µe ?gt2
  ?p?????? 2n ???a µeta?? t??? pa??????a µ???p?t?a
  e? t?? ?p???? h µ????? l, 2(n-h) µ????? l2 ?a?
  ??a ???e Wigt0 ?p???e? µ???p?t? µ????? lk-2Wi.

?a??de??µa ?a 6 ???a µeta?? t??? pa??????a
µ???p?t?a µeta?? t?? 013 ?a? t?? 034 st? Q35.
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??t?st????se?? - ?a?t?????

• ?????? µ????? ?, ??n e??a? ?????? Hamilton.
• ?p????µe ?a pa?????µe ????? Hamilton µ?s? Gray
  code.
• ?atas?e?ast??? d?ad??as?a
• ?p????µe ?a p????µe ?a? t?? f-1 a? ??e?aste?.

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??t?st????se?? ?a?t????? 2

• ???s?µ?p????ta? t? p??????µe?? ?e???µa µp????µe
  ?a pa?????µe ????? Hamilton ??a ?p???d?p?te Qnk
  ep?t???????ta? t?? a?t?st????s?.
• ?a??de??µa efa?µ???? t?? ?e???µat?? se ??a Q24.

????µe 4 R d??t? se ???e Qnk ????µe ??????
µeta?? t??? k-ary da?t?????? se ???e d??stas?.
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??t?st????se?? ?a?t????? 3

• ?p?s?? ?p???e? ?a? ? f ??a pa?a???? Gray code.
• ???a? a?a??ast???
• ??a
• ? ??t??? ????µe Hamiltonian ?????.
• ?????? Hamiltonian µ???p?t?.

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??t?st????se?? - ????µa

• ?fa?µ??eta? d?ad????? µet???µas?a t?? ????f?? µe
  s?????s? se ???e d??stas?.
• ? a?t?st????s? ep?t?????eta? ?a? p??? µ?s? Gray
  code.

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??t?st????se?? - ?pe???ß??

• ??t?st????s? t?? Qn se Qceiling(n/2)4.
• ????e?µ???? ?a ???e? ? a?t?st????s? a????????µe
  ta e???
• ?st? x ??µß?? (anan-1a1) t?? Qn . ?? t? n e??a?
  pe??tt? µet???µ????µe t? x se (0anan-1a1).
• ??te ? x st?? Qceiling(n/2)4 ??e? d?e????s?
  (bjbj-1b1) µe jceiling(n/2), bif(a2ia2i-1)
  1ltiltj µe t?? f ?p?? ????eta? st? ??µµa 1.

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??t?st????se?? ?pe???ß?? 2

• ?a??de??µa

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??p???t?s? p????

• ???ß??µa d?aµ???asµ?? pe?????sµ???? a???µ?? p????
  se ??a s?st?µa ?ste ?a ?p???e? ?s?t?µ? p??sßas?
  ap? ?????.
• j-adjacency ???e ??µß?? e?te ??e? t?? p???, e?te
  ?e?t??e?e? µe j ??µß??? p?? ????? t?? p??? ?a?
  de? ?p?????? d?? ?e?t?????? ??µß?? p?? ?a ?????
  t?? p??? ?a? ?? d??.
• t-embedding ???e ??µß?? p?? de? ??e? t?? p???
  ??e? resource distance 1 ?a? de? ?p?????? ??µß??
  p?? ?a µ?? ????? t?? p??? µe µeta?? t??? ap?stas?
  2t1.
• jt1 t? p??ß??µa e??a? ?d?? ?a? st?? d??
  pe??pt?se??.

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??p???t?s? p???? 2

• ???s????s? t?? p??ß??µat?? µe error correcting
  code se Qn.
• Hamming single error correction, perfect.
• ??? d?s???? ? a?t?st????s? µe ??d??e? se Qnk µe
  t?? ap?stas? Lee ?? ??d??a.
• ??s? µe Lee distance single error correcting
  codes st? 1-adjacency p??ß??µa.

Perfect embedding nonresource nodes in Qn have
resource distance 1 and no two nodes are adjacent.
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??p???t?s? p???? 3

• ????d?? ?st? ? pe??tt?? a???a??? ?a? n(kr-1)/2
  ??a a???a?? r. O Lee ??d??a? p????pte? ?? e???
• ?st? ? rxn p??a?a? (parity check matrix).
• ?? st??e? t?? ? ap?te????ta? ap? µ? µ?de????
  d?a??sµata rx1 Vivi,1vi,2vi,rT ?ste
  0ltvi,jltfloor(k/2) µe j t?? µ????te?? a???a??
  ?ste vi,j d??f??? t?? 0.
• ??te ??a n-??f?? d????sµa ? ß?s?? ? e??a?
  ??d??????? a? ?a? µ??? a? ???0.

?a??de??µa ?p?????? 510 ????????a d?a??sµata
p?? s??µat????? t?? ??d??a µa?