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Maintaining Shortest Paths in Digraphs with Arbitrary Arc Weights

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Maintaining Shortest Paths in Digraphs with Arbitrary Arc Weights ... Nodes Processed by Dijkstra's Algo. The Range of Arc Weights Is Important. k ... – PowerPoint PPT presentation

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Title: Maintaining Shortest Paths in Digraphs with Arbitrary Arc Weights


1
Maintaining Shortest Paths in Digraphs with
Arbitrary Arc Weights
C. Demetrescu D. Frigioni A. Marchetti-Spaccamela
U. Nanni
University of Rome La Sapienza
2
Fully Dynamic Single-source Shortest Paths
3
A Simple-minded Method
After each Increase or Decrease
use best static algorithm Bellman-Ford to
recompute from scratch shortest paths in G
? O(mn) worst-case time ( nV, mE )
Can we do any better?
4
An Asymptotically Faster Method
After each Increase or Decrease
5
A Reweighting Technique
Edmonds, Karp
6
A Reweighting Technique Ramalingam and Reps
7
Weight decrease Ramalingam and Reps
Decreasing the weight of an edge might allow to
find better paths out of T(v)
Ramalingam and Reps apply Dijkstras alg. to the
graph G (with modified weights)
u

-e
v
T(v)
8
Weight decrease (cont.)
There exists a negat. cycle if and only if v is
labelled again
u
Ideas of ownership and k-bounded account. fct.
can be applied reducing w.c.running time
v
T(v)
9
Weight increase Output Bounded Analysis
Increase(u,v,e)
e
10
Algorithms Under Evaluation
Update time
Technique
Name
11
DF vs RR/DFMN
Increase
L ? nodes in T(v)
Heuristic
RR/DFMN
Remove from L nodes which dont change distance

Output- Bounded
DF
Compute G induced by nodes in L
Reweighting
Run Dijkstra on G
12
Goals of Experimentation
Look for hints about questions like
but what about constant factors?
but is it useful in practice?
13
DF vs RR/DFMN
Increase
L ? nodes in T(v)
Heuristic
RR/DFMN
Remove from L nodes which dont change distance

Output- Bounded
DF
Compute G induced by nodes in L
Reweighting
Run Dijkstra on G
14
Experimental Setup
15
Constant Factors Are Small
16
Nodes Processed by Dijkstras Algo
17
The Range of Arc Weights Is Important
18
Small Arc Weights Range
19
Large Arc Weights Range
20
Extreme Case All Zero-length Cycles...
21
Conclusions
Dynamic algorithms based on the reweighting
technique are very useful
In general, the simpler, the faster
Output bounded is useful for small ranges of arc
weights
What happens on real test sets?
What happens on larger test sets?
22
Algorithms in a Nutshell (Increase)
BF
Bellman-Ford on G
Heuristic
DF
L ? nodes in T(v)
Compute G induced by nodes in L
Run Dijkstra on G
RR DFMN
L ? nodes in T(v)
Heuristic
Remove from L nodes which dont change distance
Output Bounded
Compute G induced by nodes in L
Run Dijkstra on G
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