qip04_shortest_paths

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Quantum query complexity
 of some graph problems


   C. Dürr        Univ. Paris-Sud
   M. Heiligman   National Security Agency
   P. Høyer       Univ. of Calgary
   M. Mhalla      Institut IMAG, Grenoble
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                General algorithm
Tree T={v0} covering vertices S={v0}
while |S|<n
     add cheapest border edge (u,v)∈E∩Sx(V\S) to A
     add v to S

Definition cost of edge (u,v)
     =shortest path weight(v0,u) + edge weight(u,v)


                                                    •
                                            •
      •     •                   •
                        •               •
•               •                                   •
          • •               •       •           •
  •
 v0                 •
                                                    •
                                                     1


 Quantum procedure for finding
    cheapest border edge
Consider the decomposition of |S| into powers of 2
Decompose S into P1∪…∪Pk s.t.
   ●|P |>…>|P |
      1       k
   ●and each |P | is a power of 2
                i




                                                •
                                        •
           •     •            •
                           •       •
   •            P1 •          P2   P3           •
               • •           • •            •
       •               •
                                                •
                                                            1


 Quantum procedure for finding
    cheapest border edge
Consider the decomposition of |S| into powers of 2
Decompose S into P1∪…∪Pk s.t.
   ●|P |>…>|P |
      1       k
   ●and each |P | is a power of 2
                i
   ●Suppose for every P we computed A : the |P | cheapest
                         i                i     i
   border edges of Pi with distinct targets
   (for edges with source∈Pi and target∉P1∪…∪Pi)


                                                 •
                                         •
           •     •            •
                           •       •
   •            P1 •          P2   P3            •
               • •           • •        A3   •
       •               •
                                                 •
                           A1      A2
                                                                 1




                            Observations
●Ai∩Sx(V\S) (restricted to targets∉S) is non empty for every i
●The cheapest border edge of S (u,v) has its source u∈P for
                                                        i
some i, and therefore v∈Ai
●Thus (A ∪…∪A )∩Sx(V\S)
        1      k
contains the cheapest border edge of S




                                                     •
                              u              •
            •     •            •         v
                            •       •
    •            P1 •          P2   P3               •
                • •           • •        A3      •
        •               •
                                                     •
                             A1     A2
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posted:10/27/2011
language:English
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