Spanning paths in hypercubes

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Spanning pathsin hypercubes
Tomá Dvorák Petr Gregor Vaek Koubek

• Charles University Prague

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n-dimensional hypercube Qn

• V(Qn) binary vectors of length n
• E(Qn) two vertices are adjacent whenever the
  corresponding vectors differ in exactly one
  coordinate

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Hypercubic architectures for parallel computing

• simple recursive structure ? supports
  divide-and-conquer strategies
• low diameter n ? fast broadcasting
• n-connected graph ? high fault-tolerance
• can simulate other networks (array, binary tree,
  mesh of trees) with only a constant slowdown
  (array, binary tree, mesh of trees)
• high genus ?(Qn) (n - 4) ?2n-3 1, n ? 4

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Qn is Hamiltonian for any n?2
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Qn is Hamiltonian for any n?2
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Qn is Hamiltonian for any n?2
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Qn is Hamiltonian for any n?2
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Qn is Hamiltonian for any n?2

• (Louis Gros, 1872) a solution of the Chinese
  ring puzzle

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Qn is Hamilton-laceable for any n?1

• (I.Havel,1984) Qn contains a Hamiltonian path
  between vertices u and v iff u and v belong to
  different partite classes

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Hamiltonian cycles in hypercubes with faulty edges

• Given a set of faulty edges in Qn
• Does there exist a Hamiltonian cycle of Qn
  avoiding each of the faulty edges?

?
?
?
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Hamiltonian cycles in hypercubes with faulty edges

• (Chan,Lee,1991) Let FE ? E(Qn), FE 2n-5. Then
  Qn- FE remains Hamiltonian provided its minimum
  degree 2.

• (Tsai,2004) Similar result for Hamiltonian paths
  with prescribed endvertices.

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Hamiltonian cycles in hypercubes with faulty
vertices

• Known results of the type F ? V(Qn), F f
• ?
• Qn- F ? a cycle of length at least 2n 2f
• (Fu, 2003) f ? 2n-4

?
?
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Hamiltonian cycles in hypercubes with faulty
vertices

• Natural necessary condition Qn- F contains a
  Hamiltonian cycle ? vertex set F is balanced

• ( contains the same number of vertices from each
  of the partite classes)
• Problem When is the natural necessary condition
  also sufficient?

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Spanning paths in hypercubes

• More general problem

u1
u2
v1
u3
v3
v2
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Spanning paths in hypercubes

• More general problem

Such spanning paths Pi exist ? ui,vii1k
need to be balanced
u1
u2
v1
u3
v3
ui,vi both ui and vi belong to partite class
A ui,vi both ui and vi belong to
partite class B
v2
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Spanning paths in hypercubes upper bounds

• When is this natural necessary condition also
  sufficient, i.e.
• ui,vii1k balanced ? spanning paths Pi1k of
  Qn
  with endvertices ui,vii1k exist?

• ?
• number of pairs k lt dimension n for n 3

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Spanning paths for odd families of vertices

• Restrict to the case when
• k ? n - 1
• ui and vi belong to different partite classes,
  i.e. the distance d(ui,vi) is odd for each i
  1,..., k
• n 2 ?
• n 3 ?
• n 4

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Spanning paths for odd families of vertices
main result

• Theorem n ? 2, ui,vii1k pairwise distinct
  vertices of Qn such that d(ui,vi) is odd, k ?
  n-1.
• Then there are spanning paths Pii1k of Qn such
  that ui and vi are endvertices of Pi except the
  case

• Does not hold for k gt n-1

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Sketch of the proof the easy case
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Sketch of the proof neglect repair
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Sketch of the proof neglect repair
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Sketch of the proof neglect repair
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Sketch of the proof dividing into quarters
u1
v4
v2
u2
u3
v1
v3
u4
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Sketch of the proof dividing into quarters
u1
u3
v4
v2
u2
v1
u4
v3
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Sketch of the proof dividing into halves
neglect repair
u1
v1
u3
v2
u2
v3
u4
v4
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Sketch of the proof dividing into halves
neglect repair
u1
v1
u3
v2
u2
v3
u4
v4
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Sketch of the proof dividing into halves
neglect repair different parities
u1
v1
u3
v3
u2
v2
u4
v4
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Spanning paths general case

• Conjecture n ? 2, ui,vii1k vertices of Qn,
  ui,vi?uj,vj ? for i ? j, ui ? vi for at
  least one i, k ? n - 1.
• Then there are spanning paths Pii1k of Qn with
  endvertices ui and vi ? ui,vii1k is
  balancedexcept the cases

• verified for n ? 5 by a computer search
• corollary for u1 ? v1, ui vi, i 2,,k

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Spanning paths general case partial result

• (Caha, Koubek, 2003) n ? 2, ui,vii1k pairwise
  distinct vertices of Qn, k ? (n1)/3. Then

u1
? ui,vii1k form a balanced family
u2
v1
u3
v3
v2
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Complexity

• Hamiltonian cycle in hypercubes with faulty
  edges
• Input Qn with a set FE of faulty edges
• Question Is Qn FE Hamiltonian?

?NPC (Chan,Lee,1991)

• Hamiltonian cycle in hypercubes with faulty
  vertices
• Input Qn with a set FV of faulty vertices
• Question Is Qn FV Hamiltonian?

?NPC?
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Complexity

• Spanning paths in general graphs
• Input Graph G and vertex pairs ui,vii1k
• Question Does G contain spanning paths with
  endvertices ui,vii1k?

?NPC

• Spanning paths in hypercubes
• Input Qn and vertex pairs ui,vii1k
• Question Does Qn contain spanning paths with
  endvertices ui,vii1k?

?NPC?