Dynamic Shortest Paths

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Dynamic Shortest Paths

• Camil Demetrescu
• University of Rome La Sapienza
• Giuseppe F. Italiano
• University of Rome Tor Vergata

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Dynamic (All Pairs) Shortest Paths

• Given a weighted directed graph
  G(V,E,w),perform any intermixed sequence of the
  following operations

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Previous work on dynamic APSP
papers
65-69
70-74
75-79
80-84
85-89
90-94
95-99
00-
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Previous work on fully dynamic APSP
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A new approach
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A new fully dynamic algorithm
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?(n2) changes per update
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-1
1
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Main ingredients of previous results
(results for general graphs with integer weights
in 0,C)
(results for general graphs with S real weights
per edge)
(all fully dynamic algorithms on general graphs)
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Uniform paths
UNIFORM
NOT UNIFORM
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Properties of Uniform paths
Theorem I
Shortest paths ? Uniform paths
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Properties of Uniform paths
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Properties of Uniform paths
(Ties can be broken by adding a tiny fraction to
the weight of each edge)
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Properties of Uniform paths
Theorem III
There are at most n-1 uniform paths connecting x,y
This is a consequence of vertex-disjointess
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Dynamic graphs
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Uniform paths in dynamic graphs
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Uniform paths in dynamic graphs
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Uniform paths in dynamic graphs
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Uniform paths in dynamic graphs
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Uniform paths in dynamic graphs
Theorem V
For any pair (x,y), the amortized number of
uniform paths pxy appearing with a new weight
per update in a fully dynamic sequence is O(log
n)
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Sketch of proof of Theorem VI (1/3)
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Sketch of proof of Theorem VI (2/3)
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Sketch of proof of Theorem VI (3/3)
Map each edge (?1, ?2) of H into a point p(?1,
?2) in the square k ? k
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Shortest paths and edge weight updates
How does a shortest path change after an update?
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Shortest paths and edge weight updates
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A new approach to dynamic APSP
NO
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How to pay only once?
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Looking at the substructure
but if we removed the edge it would get a
shortest path again!
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Zombies
A path is a zombie if it used to be a shortest
path, and its edges have not been updated since
then
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Potentially uniform paths
Relaxed notion of uniformity Subpaths do not
need to be shortest at the same time
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Properties of potentially uniform paths
Potentiallyuniform paths
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Properties of potentially uniform paths
Theorem II
O(zn2) zombies at any time
O(zn2) new potentially uniform paths per update
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How many zombies can we have?
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Reducing of zombies Smoothing
At each update we pick an edge with the maximum
number of zombies passing through it, and we
remove and reinsert it
zombies
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A new approach to dynamic APSP (II)
There exists and update algorithm that spends
O(log n) time per potentially uniform path
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Handling the hard case
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The update algorithm
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Conclusions