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# qip04_shortest_paths by gegeshandong

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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                 •
•
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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
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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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